Finite Group Databasebeta

← #29702364

Subgroups — #29702364

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839975C211yesmaximal, characteristic
2C2×#16839874C211yesmaximal, characteristic
3C2×#16839862C211yesmaximal, characteristic, complemented
4#21448889C211yesmaximal, characteristic, complemented
5#21448724C211yesmaximal, characteristic, complemented
6#21446629C211yesmaximal, characteristic, complemented
7#21446586C211yesmaximal, characteristic, complemented
8#21446519C211yesmaximal, characteristic, complemented
9#21446507C211yesmaximal, characteristic, complemented
10#21446151C211yesmaximal, characteristic, complemented
11#21446090C211yesmaximal, characteristic, complemented
12#21446064C211yesmaximal, characteristic, complemented
13#21445685C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#1879C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16839975C418yescomplemented
23#16839874—24no—
24#16839862—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839656—21no—
41#16839650—21no—
42#16839635—21no—
43#16839634—21no—
44#16839634—21no—
45#16839627C2211yescharacteristic
46#16839620—21no—
47C4⋊C4×Dic61C2211yescharacteristic
48C4×#11758196C2211yescharacteristic
49C22⋊C4×Dic61—21no—
50C2×#14519684—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C22×D8×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C23×Dic61—21no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200C2×C414yescomplemented
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519764—22no—
72#14519764C2×C414yescomplemented
73C4×Dic122—22no—
74C4×Dic122—41no—
75D4×Dic61—22no—
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—41no—
79D4×Dic61—41no—
80D4×Dic61C2×C414yescomplemented
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×#11758196D411yescharacteristic
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519710—22no—
99#14519709—22no—
100#14519706C2×C414yescomplemented
101#14519702—41no—
102#14519702C2×C414yescomplemented
103#14519700—22no—
104#14519698—22no—
105#14519698C2×C414yescomplemented
106#14519698D414yes—
107#14519688—22no—
108#14519687—22no—
109C8×Dic61—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×D4×C61C2311yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C2×D8×C61—22no—
123C2×D8×C61—22no—
124C2×D8×C61C2×C414yes—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4×D4×C61—22no—
128C4×D4×C61—41no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#102×C61—22no—
136#102×C61—41no—
137C24×C61—21no—
138C24×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C2×D4×C61C22×C412yes—
149C2×D4×C61C2×D412yes—
150C2×D4×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61—81no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—24no—
163C22×Dic61—21no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61—81no—
169C22×Dic61—81no—
170C22×Dic61—81no—
171C22×Dic61—81no—
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122C22×C412yes—
182D8×C61—28no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61C22×C412yes—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊C4×C61—42no—
194C22⋊C4×C61—42no—
195C22⋊C4×C61—81no—
196C42×C61—21no—
197C42×C61—21no—
198C22⋊Dic61—24no—
199C22⋊Dic61—24no—
200C22⋊Dic61—24no—
201C22⋊Dic61—42no—
202C22⋊Dic61—42no—
203#11758200—28no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C2×D412yes—
210#11758196—41no—
211#11758196—41no—
212#11758196—42no—
213#11758196—81no—
214#11758196—81no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—42no—
218C4×Dic61—81no—
219C4×Dic61C22×C412yes—
220C4×Dic61#4112yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C22×C412yes—
226C2×#92418C2×D412yes—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61C22×D411yescharacteristic
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—22no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61C4×D414yes—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—84no—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—84no—
292C2×Dic61—84no—
293C2×Dic61—84no—
294C2×Dic61C4×D412yes—
295C2×Dic61C4×D412yes—
296Dic122—22no—
297Dic122—22no—
298Dic122—41no—
299Dic122—41no—
300C8×C61—22no—
301C8×C61—41no—
302#92418—22no—
303#92418—22no—
304#23005—611nomaximal, Sylow
305C22×C61—24no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—42no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61C2×C4×D411yescharacteristic
320C22×C61D4211yescharacteristic
321C22×C61#54811yescharacteristic
322C22×C61#57511yescharacteristic
323C22×C61#57511yescharacteristic
324C22×C61#57611yescharacteristic
325C22×C61#57711yescharacteristic
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61C2×C4×D411yescharacteristic
331C4×C61#52011yescharacteristic
332C4×C61#54711yescharacteristic
333C4×C61#55311yescharacteristic
334Dic61—22no—
335Dic61—22no—
336Dic61—42no—
337Dic61—42no—
338Dic61—41no—
339Dic61—42no—
340Dic61—41no—
341Dic61—81no—
342C2×#463—611no—
343C2×#442—611no—
344C2×#440—611no—
345#2367—611no—
346#2349—611no—
347#1958—611no—
348#1949—611no—
349#1936—611no—
350#1935—611no—
351#1879D6111yescharacteristic
352#1877—611no—
353#1865—611no—
354#1862—611no—
355#1770—611no—
356C4×#102—611no—
357C2×C61—44no—
358C2×C61—82no—
359C2×C61#236111yescharacteristic
360C2×C61#294311yescharacteristic
361C2×C61#294611yescharacteristic
362C2×C61#327311yescharacteristic
363C2×C61#327511yescharacteristic
364C2×C61#337011yescharacteristic
365C2×C61#337611yescharacteristic
366C22×SD16—611no—
367C22×D8C2×D6111yescharacteristic
368C2×#124—611no—
369C2×C4×Q8—611no—
370C2×C4×D4—611no—
371C2×C4×D4—611no—
372C2×C4×D4—611no—
373C2×C4×D4C2×D6111yescharacteristic
374#463—618no—
375#442—1224no—
376#440—1224no—
377C2×#107—611no—
378C2×#106—611no—
379C2×C4⋊C8—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#102—611no—
384C2×#102—611no—
385C2×#102—611no—
386C2×#102C2×D6111yescharacteristic
387C2×#102C2×D6111yescharacteristic
388C2×#97—611no—
389C2×#97—611no—
390C2×C4×C8—611no—
391#391—611no—
392#390C2×D6111yescharacteristic
393#389—611no—
394#389—611no—
395#388—1221no—
396#388—1221no—
397#385—1221no—
398#385—1221no—
399#385—1221no—
400#382—611no—
401#380—1221no—
402C4×C4⋊C4—611no—
403C4×C4⋊C4—611no—
404C4×C22⋊C4—1221no—
405C2×#95—1221no—
406#340—611no—
407#340—611no—
408#340—611no—
409#340C2×D6111yescharacteristic
410#336—611no—
411#336C2×D6111yescharacteristic
412C61#2300511yescharacteristic, complemented, Sylow
413C22×Q8—611no—
414C22×D4C22×D6111yescharacteristic
415C22×D4C22×D6111yescharacteristic
416C23×C4—21no—
417C23×C4—1221no—
418C23×C4—1221no—
419C23×C4—1221no—
420C2×SD16—614no—
421C2×SD16—1222no—
422C2×SD16—1222no—
423C2×D8—22no—
424C2×D8—22no—
425C2×D8C4×D6114yes—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8C22×D6111yescharacteristic
429C22×C8C22×D6111yescharacteristic
430#124—614no—
431#124—1222no—
432C4×Q8—1222no—
433C4×Q8—2441no—
434C4×D4—22no—
435C4×D4—41no—
436C4×D4—614no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1222no—
440C4×D4—2441no—
441C4×D4—2441no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—1221no—
463C2×C42—1221no—
464C2×C42C22×D6111yescharacteristic
465#107—1222no—
466#106—1222no—
467C4⋊C8—614no—
468#103—614no—
469#103—1222no—
470#103—2441no—
471#102—22no—
472#102—41no—
473#102—614no—
474#102—614no—
475#102—1222no—
476#97—1222no—
477#97—1222no—
478C4×C8—1222no—
479#95—611no—
480#95—611no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488C24—21no—
489C24—21no—
490C2×Q8—612no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4C2×C4×D6112yes—
505C2×D4C2×C4×D6112yes—
506C2×D4D4×D6112yes—
507C2×D4#1175822412yes—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—81no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1224no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—4881no—
523C22×C4—4881no—
524C22×C4—4881no—
525C22×C4—4881no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4D4×D6111yescharacteristic
531C22×C4D4×D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533SD16—1228no—
534D8—28no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—612no—
541C2×C8—612no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8C2×C4×D6112yes—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—612no—
552C4⋊C4—612no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—4881no—
558C4⋊C4—4881no—
559C4⋊C4—2441no—
560C4⋊C4—2441no—
561C4⋊C4—2442no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—81no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C22⋊C4—2442no—
569C22⋊C4—2442no—
570C42—21no—
571C42—21no—
572C42—612no—
573C42—612no—
574C42—1221no—
575C42—1221no—
576C42—4881no—
577C42—2442no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23C2×D4×D6111yescharacteristic
593Q8—1222no—
594Q8—1222no—
595Q8—2441no—
596Q8—2441no—
597D4—22no—
598D4—22no—
599D4—22no—
600D4—22no—
601D4—22no—
602D4—41no—
603D4—41no—
604D4#1451973714yes—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—84no—
618C2×C4—612no—
619C2×C4—612no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1222no—
633C2×C4—4884no—
634C2×C4—4884no—
635C2×C4—4884no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4C22×C4×D6111yescharacteristic
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4C2×#1175822411yescharacteristic
649C2×C4#1451982211yescharacteristic
650C2×C4#1451982411yescharacteristic
651C8—22no—
652C8—41no—
653C8—1222no—
654C8—1222no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C4×D4×D6111yescharacteristic
670C22#1684039411yescharacteristic
671C22#1684041811yescharacteristic
672C22#1684055811yescharacteristic
673C22#1684055811yescharacteristic
674C22#1684056311yescharacteristic
675C22#1684056911yescharacteristic
676C4—41no—
677C4—41no—
678C4—81no—
679C4—81no—
680C4—1222no—
681C4—1222no—
682C4—4881no—
683C4—2441no—
684C4—2442no—
685C4—2441no—
686C4—2442no—
687C4—2442no—
688C4C2×#1451973711yescharacteristic
689C4#1684024111yescharacteristic
690C4#1684041011yescharacteristic
691C4#1684045511yescharacteristic
692C2—44no—
693C2—82no—
694C2#2144883011yescharacteristic
695C2#2145461611yescharacteristic
696C2#2145464411yescharacteristic
697C2#2145743011yescharacteristic
698C2#2145744811yescharacteristic
699C2#2145856511yescharacteristic
700C2#2145864311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1879C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
cosocleC22×C4×C61C24
hypercenter#1879D61
hypocenterC61#23005
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#16839862C2semidirect
4. #21448889C2semidirect
5. #21448724C2semidirect
6. #21446629C2semidirect
7. #21446586C2semidirect
8. #21446519C2semidirect
9. #21446507C2semidirect
10. #21446151C2semidirect
11. #21446090C2semidirect
12. #21446064C2semidirect
13. #21445685C2semidirect
14. #102×Dic61C2semidirect
22. #16839975C4semidirect
68. C2×#11758200C2×C4semidirect
72. #14519764C2×C4semidirect
80. D4×Dic61C2×C4semidirect
100. #14519706C2×C4semidirect
102. #14519702C2×C4semidirect
105. #14519698C2×C4semidirect
412. C61#23005semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399752
C2×#168398742
C2×#168398622
#214488892
#214487242
#214466292
#214465862
#214465192
#214465072
#214461512
#214460902
#214460642
#214456852
#102×Dic612
C61×#18792
#2300561