Finite Group Databasebeta

← #29686536

Subgroups — #29686536

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839989C211yesmaximal, characteristic
2C2×#16839853C211yesmaximal, characteristic, complemented
3C2×#16839720C211yesmaximal, characteristic, complemented
4#21448514C211yesmaximal, characteristic
5#21448497C211yesmaximal, characteristic, complemented
6#21446674C211yesmaximal, characteristic, complemented
7#21446546C211yesmaximal, characteristic, complemented
8#21446513C211yesmaximal, characteristic, complemented
9#21446510C211yesmaximal, characteristic, complemented
10#21446252C211yesmaximal, characteristic, complemented
11#21445842C211yesmaximal, characteristic, complemented
12#21445707C211yesmaximal, characteristic, complemented
13#21445648C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic, complemented
15C61×#1895C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839989C418yescomplemented
23#16839853—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31#16839720—24no—
32C2×#14519687C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839665—21no—
38#16839665C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839656—21no—
43#16839655—21no—
44#16839650—21no—
45#16839649—21no—
46#16839642—21no—
47#16839637—21no—
48#16839622C2211yescharacteristic
49C4⋊C4×Dic61C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C22×M16×C61C2211yescharacteristic
56C2×#121×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200C2×C414yes—
67C22×#92418—22no—
68C22×#92418—22no—
69C22×#92418C2×C414yescomplemented
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519756—22no—
73#14519756C2×C414yescomplemented
74#14519742—22no—
75#14519742—22no—
76D4×Dic61—22no—
77D4×Dic61—22no—
78#14519736—22no—
79#14519736—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61D411yescharacteristic
95C2×C22⋊Dic61D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519708—22no—
102#14519708—22no—
103#14519708C2×C414yescomplemented
104#14519702—22no—
105#14519702—22no—
106#14519700C2×C414yescomplemented
107#14519698C2×C414yes—
108#14519687—22no—
109#14519687—22no—
110C8×Dic61—22no—
111C8×Dic61D414yescomplemented
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×D4×C61—21no—
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61C2311yescharacteristic
122C2×M16×C61—22no—
123C2×M16×C61—22no—
124C2×M16×C61—22no—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127#121×C61—22no—
128#121×C61C2×C414yes—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134#102×C61—22no—
135#102×C61—22no—
136#95×C61—21no—
137C24×C61—21no—
138C24×C61C2×D411yescharacteristic
139C2×D4×C61—24no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—81no—
146C2×D4×C61—81no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C22×C412yes—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4112yes—
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—42no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176M16×C61—28no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D412yes—
182C2×C8×C61#4112yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—24no—
190C22⋊C4×C61—81no—
191C22⋊Dic61—24no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—42no—
195C22⋊Dic61—81no—
196C22⋊Dic61—81no—
197C22⋊Dic61—81no—
198#11758200—28no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—81no—
207C4⋊Dic61C22×C412yes—
208#11758196—24no—
209#11758196—42no—
210#11758196—81no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—42no—
214C4×Dic61—81no—
215C4×Dic61—81no—
216C4×Dic61C2×D412yes—
217C4×Dic61#4112yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—28no—
221C2×#92418C22×C412yes—
222C2×#92418C22×C412yes—
223C2×#92418C2×D412yes—
224C2×#92418#4112yes—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—22no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61C4×D412yes—
239C23×C61C22×D411yescharacteristic
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—84no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C4×D412yes—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—84no—
290C2×Dic61#12112yes—
291C2×Dic61#12112yes—
292Dic122—22no—
293Dic122—22no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—22no—
297C8×C61—22no—
298#92418—22no—
299#92418C4×D414yes—
300#21523—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61C2×C4×D411yescharacteristic
316C22×C61C2×#12111yescharacteristic
317C22×C61#53211yescharacteristic
318C22×C61#57711yescharacteristic
319C22×C61#57711yescharacteristic
320C22×C61#57711yescharacteristic
321C22×C61#57711yescharacteristic
322C4×C61—22no—
323C4×C61—42no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61C2×C4×D411yescharacteristic
327C4×C61C4×#4111yescharacteristic
328C4×C61#52511yescharacteristic
329C4×C61#52911yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—41no—
334Dic61—41no—
335Dic61—42no—
336Dic61—81no—
337Dic61—81no—
338C2×#465—611no—
339C2×C4×SD16—611no—
340C2×#405—611no—
341#2324—611no—
342#2323—611no—
343#1968—611no—
344#1941—611no—
345#1936—611no—
346#1935—611no—
347#1895D6111yescharacteristic
348#1894—611no—
349#1807—611no—
350C4×#106—611no—
351#1763—611no—
352#1762—611no—
353C2×C61—42no—
354C2×C61—82no—
355C2×C61C4×#12111yescharacteristic
356C2×C61#294911yescharacteristic
357C2×C61#295211yescharacteristic
358C2×C61#305711yescharacteristic
359C2×C61#306011yescharacteristic
360C2×C61#317311yescharacteristic
361C2×C61#317411yescharacteristic
362C22×SD16—611no—
363C22×M16C2×D6111yescharacteristic
364C23×C8—611no—
365C2×C22⋊Q8—611no—
366C2×#121C2×D6111yescharacteristic
367C2×C4×Q8—611no—
368C2×C4×D4—611no—
369C2×#117—611no—
370#465—618no—
371C4×SD16—1224no—
372C2×#106—611no—
373C2×#106—611no—
374C2×#106—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#102—611no—
379C2×#102C2×D6111yescharacteristic
380C2×#102C2×D6111yescharacteristic
381#405—1224no—
382C2×#97—611no—
383C2×#97—611no—
384C2×C4×C8—611no—
385C2×C4×C8—611no—
386#391—611no—
387#390—611no—
388#390—1221no—
389#389—611no—
390#389—611no—
391#389—1221no—
392#388—1221no—
393#388—1221no—
394#388—1221no—
395#387—1221no—
396#386—1221no—
397#385—1221no—
398#380—611no—
399#380C2×D6111yescharacteristic
400C4×C4⋊C4—611no—
401C4×C4⋊C4—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340—611no—
406#340C2×D6111yescharacteristic
407#340C2×D6111yescharacteristic
408C61#2152311yescharacteristic, complemented, Sylow
409C22×Q8—611no—
410C22×D4—21no—
411C22×D4C22×D6111yescharacteristic
412C23×C4—1221no—
413C23×C4C22×D6111yescharacteristic
414C2×SD16—614no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C2×M16—22no—
418C2×M16—22no—
419C2×M16—22no—
420C22×C8—611no—
421C22×C8—614no—
422C22×C8—611no—
423C22×C8—1222no—
424C22×C8—1222no—
425C22×C8C22×D6111yescharacteristic
426C22×C8C22×D6111yescharacteristic
427C22⋊Q8—614no—
428C22⋊Q8—1222no—
429#121—22no—
430#121C4×D6114yes—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435#117—1222no—
436#117—1222no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—1221no—
461C2×C42—1221no—
462C2×C42—1221no—
463#106—614no—
464#106—1222no—
465#106—1222no—
466#103—614no—
467#103—1222no—
468#103—1222no—
469#102—22no—
470#102—22no—
471#102—614no—
472#97—1222no—
473#97—1222no—
474C4×C8—614no—
475C4×C8—1222no—
476#95—21no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484C24—21no—
485C24D4×D6111yescharacteristic
486C2×Q8—612no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—81no—
499C2×D4—81no—
500C2×D4C2×C4×D6112yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—4881no—
517C22×C4—4881no—
518C22×C4—2442no—
519C22×C4C2×C4×D6112yes—
520C22×C4D4×D6111yescharacteristic
521C22×C4#1175822412yes—
522C22×C4C23×D6111yescharacteristic
523SD16—1228no—
524M16—28no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—612no—
530C2×C8—612no—
531C2×C8—612no—
532C2×C8—612no—
533C2×C8—1228no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8D4×D6112yes—
537C2×C8#1175822412yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—612no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—1221no—
547C4⋊C4—1224no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—4881no—
553C4⋊C4—4881no—
554C4⋊C4—2442no—
555C4⋊C4C2×C4×D6112yes—
556C22⋊C4—24no—
557C22⋊C4—81no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1224no—
561C22⋊C4—4881no—
562C22⋊C4—4881no—
563C22⋊C4—4881no—
564C22⋊C4—2442no—
565C42—612no—
566C42—612no—
567C42—1221no—
568C42—1221no—
569C42—4881no—
570C42—4881no—
571C42—2442no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23#1451973712yes—
586C23C2×D4×D6111yescharacteristic
587Q8—1222no—
588Q8—1222no—
589Q8—2441no—
590Q8—2441no—
591D4—22no—
592D4—22no—
593D4—41no—
594D4—41no—
595D4—84no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—612no—
612C2×C4—612no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—4884no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4#1451973712yes—
637C2×C4C22×C4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4C2×#1175822411yescharacteristic
642C2×C4C2×#1175822411yescharacteristic
643C8—22no—
644C8—22no—
645C8—614no—
646C8—1222no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×#1451973711yescharacteristic
662C22#121×D6111yescharacteristic
663C22#1684030711yescharacteristic
664C22#1684057111yescharacteristic
665C22#1684057211yescharacteristic
666C22#1684057211yescharacteristic
667C22#1684057211yescharacteristic
668C4—22no—
669C4—42no—
670C4—81no—
671C4—81no—
672C4—1222no—
673C4—1222no—
674C4—4881no—
675C4—4881no—
676C4—2441no—
677C4—2442no—
678C4—2441no—
679C4—2442no—
680C4C4×D4×D6111yescharacteristic
681C4C4×#1175822411yescharacteristic
682C4#1684026811yescharacteristic
683C4#1684028911yescharacteristic
684C2—42no—
685C2—82no—
686C2#2144832911yescharacteristic
687C2#2145468111yescharacteristic
688C2#2145469811yescharacteristic
689C2#2145545611yescharacteristic
690C2#2145547911yescharacteristic
691C2#2145643111yescharacteristic
692C2#2145644211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#21523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×#16839853C2semidirect
3. C2×#16839720C2semidirect
5. #21448497C2semidirect
6. #21446674C2semidirect
7. #21446546C2semidirect
8. #21446513C2semidirect
9. #21446510C2semidirect
10. #21446252C2semidirect
11. #21445842C2semidirect
12. #21445707C2semidirect
13. #21445648C2semidirect
14. #21445646C2semidirect
22. #16839989C4semidirect
69. C22×#92418C2×C4semidirect
73. #14519756C2×C4semidirect
103. #14519708C2×C4semidirect
106. #14519700C2×C4semidirect
111. C8×Dic61D4semidirect
408. C61#21523semidirect

Maximal subgroups

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

subgroupindex
C2×#168399892
C2×#168398532
C2×#168397202
#214485142
#214484972
#214466742
#214465462
#214465132
#214465102
#214462522
#214458422
#214457072
#214456482
#214456462
C61×#18952
#2152361