Finite Group Databasebeta

← #29698422

Subgroups — #29698422

Order 15616.

1062 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758199C211yesmaximal, characteristic
2C2×#16839842C211yesmaximal, characteristic
3C2×#16839842C211yesmaximal, characteristic
4#21448728C211yesmaximal, characteristic, complemented
5#21448724C211yesmaximal, characteristic, complemented
6#21446551C211yesmaximal, characteristic, complemented
7#21446549C211yesmaximal, characteristic, complemented
8#21446083C211yesmaximal, characteristic, complemented
9#21446082C211yesmaximal, characteristic, complemented
10#21446077C211yesmaximal, characteristic, complemented
11#21445980C211yesmaximal, characteristic, complemented
12#21445828C211yesmaximal, characteristic, complemented
13#21445827C211yesmaximal, characteristic, complemented
14#21445680C211yesmaximal, characteristic, complemented
15C61×#1864C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C4×#11758199—24no—
23#16839842—24no—
24#16839842C418yescomplemented
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839636—21no—
41#16839635—21no—
42#16839634—21no—
43#16839633—21no—
44#16839627C2211yescharacteristic
45#16839627C2211yescharacteristic
46#16839620—21no—
47#16839619—21no—
48C4×#11758196C2211yescharacteristic
49C2×#14519684—21no—
50#95×D61—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×D244C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C2×#11758199C2×C414yes—
69C2×#11758199D414yes—
70C2×#11758199Q814yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C4×D244—22no—
74C4×D244—41no—
75D4×Dic61—22no—
76D4×Dic61—22no—
77D4×Dic61—41no—
78D4×Dic61C2×C414yescomplemented
79#14519739—22no—
80#14519739—22no—
81#14519739—41no—
82#14519739C2×C414yescomplemented
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196D411yescharacteristic
90C2×#11758196D411yescharacteristic
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519710—22no—
99#14519710—22no—
100#14519710C2×C414yescomplemented
101#14519709—41no—
102#14519706—22no—
103#14519699—22no—
104#14519699C2×C414yescomplemented
105#14519698—22no—
106#14519697—22no—
107C4×#92418—22no—
108C8×Dic61—22no—
109C8×Dic61C2×C414yescomplemented
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
120C23×C4×C61—21no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61—41no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C22⋊C4×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—41no—
132#102×C61—22no—
133#102×C61C2×C414yescomplemented
134#95×C61—21no—
135C24×C61—21no—
136C23×D61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61C22×C412yes—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×Q812yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—24no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—81no—
159C22×Dic61—81no—
160C22×Dic61—81no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C22×C412yes—
169C2×D244C2×D412yes—
170C2×D244C2×Q812yes—
171C2×C4×D61—24no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—81no—
177C2×C4×D61—81no—
178C2×C4×D61—81no—
179C2×C4×D61—81no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—81no—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—42no—
194C42×C61—21no—
195C42×C61—21no—
196C22⋊Dic61—24no—
197C22⋊Dic61—42no—
198#11758199C4⋊C4116yescomplemented
199#11758198—24no—
200#11758198—42no—
201#11758198—42no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—81no—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211#11758196—81no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61C22×C412yes—
216C4×Dic61#4112yes—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C2×#92418C2×D412yes—
222C2×#92418C2×Q812yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61C2×C4⋊C414yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61#13512yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×Q811yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—84no—
283C2×Dic61—84no—
284C2×Dic61C4×D412yes—
285C2×Dic61C4×D412yes—
286C2×Dic61#13512yes—
287D244—22no—
288D244—22no—
289D244—22no—
290D244C2×C4⋊C414yes—
291C4×D61—84no—
292C4×D61—84no—
293C4×D61—84no—
294C8×C61—22no—
295C8×C61—41no—
296#92418—41no—
297#92418C2×C4⋊C414yes—
298#22682—611nomaximal, Sylow
299C22×C61—24no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61C2×C4×D411yescharacteristic
307C22×C61D4211yescharacteristic
308C22×C61D4×Q811yescharacteristic
309C22×C61C2×#13511yescharacteristic
310C22×C61C2×#13511yescharacteristic
311C22×C61#57611yescharacteristic
312C22×C61#57611yescharacteristic
313C2×D61—24no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—41no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61C22×C4⋊C411yescharacteristic
326C4×C61#52011yescharacteristic
327C4×C61#54711yescharacteristic
328C4×C61#55411yescharacteristic
329Dic61—22no—
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—41no—
334Dic61—81no—
335Dic61—81no—
336C2×C4×D8—611no—
337C2×C4×D8—611no—
338C2×C4×D8—611no—
339#2349—611no—
340#2349—611no—
341#1942—611no—
342#1942—611no—
343#1864—611no—
344#1864—611no—
345#1864—611no—
346#1864D6111yescharacteristic, complemented
347#1846—611no—
348#1804—611no—
349#1804—611no—
350C8×C4⋊C4—611no—
351C2×C61—44no—
352C2×C61D4×C4⋊C411yescharacteristic
353C2×C61C4×#13511yescharacteristic
354C2×C61#294811yescharacteristic
355C2×C61#328011yescharacteristic
356C2×C61#328011yescharacteristic
357C2×C61#338011yescharacteristic
358C2×C61#338011yescharacteristic
359D61—44no—
360C22×D8—611no—
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×C4×D4—611no—
365C2×C4×D4—611no—
366C2×C4×D4C2×D6111yescharacteristic
367C4×D8—618no—
368C4×D8—1224no—
369C4×D8—1224no—
370C2×#107—611no—
371C2×#107—611no—
372C2×#107—611no—
373C2×#106—611no—
374C2×#106C2×D6111yescharacteristic
375C2×C4⋊C8—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102C2×D6111yescharacteristic
381C2×#102C2×D6111yescharacteristic
382C2×C4×C8—611no—
383C2×C4×C8—611no—
384C2×C4×C8—611no—
385#389—611no—
386#389—611no—
387#389—611no—
388#389C2×D6111yescharacteristic
389#385—1221no—
390#385—1221no—
391#385—1221no—
392#385—1221no—
393#382—611no—
394#382—611no—
395#380—1221no—
396#380—1221no—
397C4×C4⋊C4—611no—
398C2×#95—1221no—
399C2×#95—1221no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340C2×D6111yescharacteristic
404#336—611no—
405#336C2×D6111yescharacteristic
406C61#2268211yescharacteristic, complemented, Sylow
407C22×D4—611no—
408C22×D4C22×D6111yescharacteristic
409C23×C4—21no—
410C23×C4—1221no—
411C23×C4—1221no—
412C23×C4—1221no—
413C23×C4—1221no—
414C23×C4—1221no—
415C2×D8—614no—
416C2×D8—614no—
417C2×D8—614no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422C4×D4—22no—
423C4×D4—41no—
424C4×D4—614no—
425C4×D4—614no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—2441no—
432C4×D4—2441no—
433C4×D4—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42C22×D6111yescharacteristic
455#107—614no—
456#107—1222no—
457#107—1222no—
458#106—41no—
459#106—2441no—
460C4⋊C8—1222no—
461#102—22no—
462#102—614no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#102C4×D6114yescomplemented
467C4×C8—614no—
468C4×C8—1222no—
469C4×C8—1222no—
470#95—21no—
471#95—611no—
472#95—611no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480C24—21no—
481C24—1221no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—612no—
486C2×D4—612no—
487C2×D4—612no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4C2×C4×D6112yes—
492C2×D4D4×D6112yes—
493C2×D4Q8×D6112yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1224no—
505C22×C4—1224no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—4881no—
510C22×C4—4881no—
511C22×C4—4881no—
512C22×C4—4881no—
513C22×C4—4881no—
514C22×C4—4881no—
515C22×C4—4881no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525D8—6116no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—612no—
532C2×C8—612no—
533C2×C8—612no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8C2×C4×D6112yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—81no—
544C4⋊C4—612no—
545C4⋊C4—1222no—
546C4⋊C4—1221no—
547C4⋊C4—1222no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—4881no—
553C4⋊C4—4881no—
554C4⋊C4—4881no—
555C4⋊C4C2×C4×D6112yes—
556C22⋊C4—42no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—612no—
565C42—612no—
566C42—1221no—
567C42—1221no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—1222no—
576C23—1222no—
577C23—1222no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23C2×D4×D6111yescharacteristic
583D4—22no—
584D4—22no—
585D4—22no—
586D4—614no—
587D4—1222no—
588D4—1222no—
589D4—1222no—
590D4C4⋊C4×D6114yes—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—4884no—
620C2×C4—4884no—
621C2×C4—4884no—
622C2×C4—4884no—
623C2×C4—4884no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4#1451979812yes—
633C2×C4C22×C4×D6111yescharacteristic
634C2×C4C2×D4×D6111yescharacteristic
635C2×C4C2×Q8×D6111yescharacteristic
636C2×C4#41×D6111yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982411yescharacteristic
639C8—22no—
640C8—41no—
641C8—614no—
642C8—2441no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—1224no—
651C22—2442no—
652C22—2442no—
653C22—2442no—
654C22—2442no—
655C22—2442no—
656C22—2442no—
657C22C4×D4×D6111yescharacteristic
658C22#1684039411yescharacteristic
659C22#1684042511yescharacteristic
660C22#135×D6111yescharacteristic
661C22C2×#1451979811yescharacteristic
662C22#1684056611yescharacteristic
663C22#1684056611yescharacteristic
664C4—22no—
665C4—42no—
666C4—41no—
667C4—81no—
668C4—81no—
669C4—1222no—
670C4—1222no—
671C4—1222no—
672C4—4881no—
673C4—4881no—
674C4—2442no—
675C4—2441no—
676C4C2×C4⋊C4×D6111yescharacteristic
677C4#1684024111yescharacteristic
678C4#1684041011yescharacteristic
679C4#1684046611yescharacteristic
680C2—44no—
681C2—2444no—
682C2#2144871711yescharacteristic
683C2C4×#1451979811yescharacteristic
684C2#2145466811yescharacteristic
685C2#2145749711yescharacteristic
686C2#2145750211yescharacteristic
687C2#2145868111yescharacteristic
688C2#2145868511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenter#1864D61
hypocenterC61#22682
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
4. #21448728C2semidirect
5. #21448724C2semidirect
6. #21446551C2semidirect
7. #21446549C2semidirect
8. #21446083C2semidirect
9. #21446082C2semidirect
10. #21446077C2semidirect
11. #21445980C2semidirect
12. #21445828C2semidirect
13. #21445827C2semidirect
14. #21445680C2semidirect
15. C61×#1864C2semidirect
24. #16839842C4semidirect
78. D4×Dic61C2×C4semidirect
82. #14519739C2×C4semidirect
100. #14519710C2×C4semidirect
104. #14519699C2×C4semidirect
109. C8×Dic61C2×C4semidirect
133. #102×C61C2×C4semidirect
198. #11758199C4⋊C4semidirect
346. #1864D61semidirect
406. C61#22682semidirect
466. #102C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117581992
C2×#168398422
#214487282
#214487242
#214465512
#214465492
#214460832
#214460822
#214460772
#214459802
#214458282
#214458272
#214456802
C61×#18642
#2268261