Finite Group Databasebeta

← #29693926

Subgroups — #29693926

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456310C211yesmaximal, characteristic
2#21454734C211yesmaximal, characteristic
3#21454683C211yesmaximal, characteristic, complemented
4#21446733C211yesmaximal, characteristic, complemented
5#21446714C211yesmaximal, characteristic
6#21446357C211yesmaximal, characteristic, complemented
7#21446355C211yesmaximal, characteristic, complemented
8#21446340C211yesmaximal, characteristic, complemented
9#21446316C211yesmaximal, characteristic, complemented
10#21446311C211yesmaximal, characteristic, complemented
11#21446292C211yesmaximal, characteristic, complemented
12#21446289C211yesmaximal, characteristic, complemented
13#21445941C211yesmaximal, characteristic, complemented
14#21445573C211yesmaximal, characteristic, complemented
15C61×#1909C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519807C2211yescharacteristic
18C2×#14519797C2211yescharacteristic
19C22×#11758202C2211yescharacteristic
20#16840528C2211yescharacteristic
21#16840338C2211yescharacteristic
22#16840240C2211yescharacteristic
23#16840239C2211yescharacteristic
24#16840157C412yescomplemented
25#16840154C412yescomplemented
26#16840144C412yescomplemented
27#16840141C412yescomplemented
28#16839882—22no—
29#16839877—22no—
30#16839866—22no—
31#16839854—22no—
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839780C2211yescharacteristic
35#16839779C2211yescharacteristic
36#16839779C2211yescharacteristic
37#16839778C2211yescharacteristic
38#16839772C2211yescharacteristic
39#16839770C2211yescharacteristic
40C2×#14519696C2211yescharacteristic
41C2×#14519695C2211yescharacteristic
42#16839735C2211yescharacteristic
43#16839720C2211yescharacteristic
44#16839718C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839534C2211yescharacteristic
47#16839532C2211yescharacteristic
48#16839520C2211yescharacteristic
49#16839518C2211yescharacteristic
50#16839518C2211yescharacteristic
51#16839518C2211yescharacteristic
52C61×#568C2211yescharacteristic
53C61×#536C2211yescharacteristic
54C61×#419C2211yescharacteristic
55C61×#419C2211yescharacteristic
56C61×#410C2211yescharacteristic
57C61×H(3,ℤ/4)C2211yescharacteristic
58C61×H(3,ℤ/4)C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60C22×Dic122C2311yescharacteristic
61#14519808—41no—
62#14519808—41no—
63#14519807—41no—
64#14519807—41no—
65#14519797—21no—
66#14519797—21no—
67#14519797—21no—
68#14519797—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202C2×C412yes—
74C2×#11758202C2×C412yes—
75C2×#11758202C2×C411yescharacteristic
76C2×#11758202C2×C412yes—
77C2×#11758202C2×C412yes—
78C2×#11758200C2×C411yescharacteristic
79C2×#11758200C2×C412yes—
80C2×#11758201C2×C411yescharacteristic
81C2×#11758201C2×C412yes—
82C2×#11758199C2×C411yescharacteristic
83#14519785—41no—
84#14519783—22no—
85C2×#11758194C2311yescharacteristic
86C2×#11758194C2311yescharacteristic
87C2×#11758194C2311yescharacteristic
88M16×D61—41no—
89C22×#92418C2311yescharacteristic
90#14519777C2×C412yes—
91#14519765C2×C412yes—
92#14519761—41no—
93#14519757—41no—
94Q8×Dic61—22no—
95Q8×Dic61—41no—
96C4×Dic122—41no—
97#14519742—22no—
98#14519736—41no—
99#14519736C2311yescharacteristic
100#14519736C2311yescharacteristic
101#14519733—41no—
102#14519712—21no—
103#14519712—21no—
104#14519710—21no—
105#14519708—21no—
106#14519705—41no—
107#14519704—22no—
108#14519704—41no—
109#14519703—22no—
110#14519702—22no—
111#14519700—41no—
112#14519696—41no—
113#14519695—41no—
114#14519691—41no—
115#14519688—21no—
116#14519688C2×C412yes—
117#14519687D412yes—
118C4×#92418—21no—
119C4×#92418C2×C412yes—
120C8×Dic61D412yes—
121C2×#41×C61C2311yescharacteristic
122C22×Q8×C61C2311yescharacteristic
123#131×C61—22no—
124#131×C61—41no—
125C2×M16×C61—41no—
126C2×M16×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C4⋊Q8×C61C2×C412yes—
129#124×C61C2×C412yes—
130#123×C61—41no—
131C22⋊Q8×C61—41no—
132#117×C61C2311yescharacteristic
133#117×C61C2311yescharacteristic
134#117×C61C2311yescharacteristic
135#103×C61—22no—
136#103×C61—41no—
137#99×C61—41no—
138#41×C61—21no—
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C412yes—
148C2×Q8×C61C22×C411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C22×C411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—21no—
155C22×Dic61—21no—
156#11758222—21no—
157#11758222—21no—
158#11758222—21no—
159#11758222—21no—
160C2×D244C22×C411yescharacteristic
161C2×C4×D61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122C22×C412yes—
168C2×Dic122C22×C411yescharacteristic
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—21no—
172M16×C61—21no—
173M16×C61—42no—
174C2×C8×C61—21no—
175C2×C8×C61—42no—
176C2×C8×C61C2×D412yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—42no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—42no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C22×C412yes—
188C22⋊Dic61—21no—
189C22⋊Dic61—21no—
190#11758202—22no—
191#11758202—22no—
192#11758202—22no—
193#11758202—22no—
194#11758202—21no—
195#11758202—21no—
196#11758202—42no—
197#11758202—42no—
198#11758201—21no—
199#11758201—21no—
200#11758201—42no—
201#11758200—21no—
202#11758200—21no—
203#11758200—42no—
204#11758199—21no—
205#11758199—21no—
206#11758198—42no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—42no—
210C4⋊Dic61—42no—
211#11758196—42no—
212C4×Dic61—21no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61C2×D412yes—
216#11758194—21no—
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418C22×C412yes—
225C2×#92418C22×C411yescharacteristic
226C2×#92418C22×C411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229C2×#92418#4111yescharacteristic
230C2×#92418#4111yescharacteristic
231#11758189—42no—
232C8×D61—42no—
233C23×C61—21no—
234C23×C61H(5,𝔽2)11yescharacteristic
235C22×D61—21no—
236Q8×C61—21no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—21no—
240Q8×C61—41no—
241Q8×C61—41no—
242D4×C61—21no—
243D4×C61—21no—
244D4×C61—41no—
245D4×C61—41no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
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×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C22⋊D61—41no—
262C22⋊D61—41no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269D244—21no—
270D244—21no—
271C4×D61—41no—
272C4×D61—41no—
273Dic122—21no—
274Dic122—22no—
275Dic122—22no—
276Dic122—21no—
277Dic122—41no—
278Dic122—41no—
279C8×C61—22no—
280C8×C61—41no—
281#92418—22no—
282#92418—41no—
283#92418—41no—
284#92418C4×D411yescharacteristic
285#92418C4×D411yescharacteristic
286#92418C4×D411yescharacteristic
287#92418C4×D411yescharacteristic
288#22285—611nomaximal, Sylow
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—81no—
292C22×C61#51811yescharacteristic
293C22×C61#53511yescharacteristic
294C22×C61#54011yescharacteristic
295C2×D61—41no—
296C2×D61—81no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61C2×C4×D411yescharacteristic
303C4×C61#51811yescharacteristic
304C4×C61#53511yescharacteristic
305C4×C61#54011yescharacteristic
306Dic61—42no—
307Dic61—42no—
308Dic61—41no—
309Dic61—81no—
310#3156—611no—
311#2957—611no—
312#2950—611no—
313#1979—611no—
314#1976—611no—
315#1912—611no—
316#1912—611no—
317#1909—611no—
318#1909D6111yescharacteristic, complemented
319#1905—611no—
320#1905—611no—
321#1902—611no—
322#1902—611no—
323#1833—611no—
324#1747—611no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—81no—
329C2×C61#232811yescharacteristic
330D61—81no—
331C2×#137—611no—
332C2×#137—611no—
333C2×#135—611no—
334C22×Q16—611no—
335#568—611no—
336#568C2×D6111yescharacteristic
337#536—611no—
338#536C2×D6111yescharacteristic
339#519—611no—
340#519—611no—
341#497—612no—
342#497—612no—
343#495—612no—
344#494—612no—
345#444—1222no—
346#443—1222no—
347#441—1222no—
348C4×Q16—1222no—
349C2×#108—611no—
350#427—611no—
351#421—611no—
352#421—611no—
353#421—611no—
354#421—611no—
355#419—611no—
356#419—611no—
357#419C2×D6111yescharacteristic
358#419C2×D6111yescharacteristic
359C2×#101—611no—
360C2×#101—611no—
361#410—611no—
362#410C2×D6111yescharacteristic
363#405—611no—
364#405—611no—
365#405—611no—
366#344—611no—
367#344—611no—
368#338—611no—
369H(3,ℤ/4)—611no—
370H(3,ℤ/4)—611no—
371H(3,ℤ/4)—611no—
372H(3,ℤ/4)C2×D6111yescharacteristic
373H(3,ℤ/4)C2×D6111yescharacteristic
374C61#2228511yescharacteristic, complemented, Sylow
375C2×#41—611no—
376C2×#41C22×D6111yescharacteristic
377C22×Q8—611no—
378C22×Q8C22×D6111yescharacteristic
379#137—2441no—
380#137—2441no—
381#137—2441no—
382#137—2441no—
383#135—1221no—
384#135—1221no—
385#135—1221no—
386#135—1221no—
387C2×Q16—612no—
388C2×Q16—612no—
389C2×Q16—611no—
390C2×Q16—612no—
391C2×Q16—612no—
392C2×Q16—1221no—
393C2×Q16—1221no—
394C2×Q16—1221no—
395C2×Q16—1221no—
396C2×SD16—611no—
397C2×SD16—611no—
398C2×SD16—612no—
399C2×SD16—612no—
400C2×D8—611no—
401#131—22no—
402#131—41no—
403#131—1222no—
404#131—2441no—
405C2×M16—41no—
406C2×M16—611no—
407C2×M16—611no—
408C2×M16—611no—
409C2×M16—2441no—
410C2×M16C22×D6111yescharacteristic
411C2×M16C22×D6111yescharacteristic
412C22×C8—611no—
413C4⋊Q8—612no—
414C4⋊Q8C4×D6112yes—
415#124—612no—
416#124C4×D6112yes—
417#123—41no—
418#123—2441no—
419C22⋊Q8—41no—
420C22⋊Q8—2441no—
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×Q8—2441no—
424C4×Q8—2441no—
425#117—611no—
426#117—611no—
427#117—2441no—
428#117—2441no—
429#117C22×D6111yescharacteristic
430#117C22×D6111yescharacteristic
431#117C22×D6111yescharacteristic
432#108—1221no—
433#108—1221no—
434#107—1221no—
435#106—1221no—
436#104—1222no—
437#104—1222no—
438#104—2441no—
439#104—2441no—
440#103—22no—
441#103—41no—
442#103—1222no—
443#103—2441no—
444#101—2441no—
445#101—2441no—
446#99—41no—
447#99—2441no—
448#97—612no—
449#97—612no—
450#97—1221no—
451C4×C8—612no—
452C4×C8—612no—
453C4×C8—1221no—
454#41—21no—
455#41—21no—
456#41—21no—
457#41—21no—
458#41—1221no—
459#41—1221no—
460#41—1221no—
461#41—1221no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—612no—
468C2×Q8—611no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8C2×C4×D6112yes—
475C2×Q8C2×C4×D6111yescharacteristic
476C2×D4—21no—
477C2×D4—611no—
478C2×D4—1221no—
479C2×D4C2×C4×D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4C23×D6111yescharacteristic
485Q16—1222no—
486Q16—1222no—
487Q16—1222no—
488Q16—1222no—
489Q16—1221no—
490Q16—1221no—
491Q16—2442no—
492Q16—2442no—
493SD16—1221no—
494SD16—1221no—
495SD16—1221no—
496SD16—1221no—
497SD16—2442no—
498SD16—2442no—
499D8—1221no—
500D8—1221no—
501M16—21no—
502M16—21no—
503M16—21no—
504M16—21no—
505M16—42no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509M16—1221no—
510M16—1221no—
511M16—1221no—
512M16—2442no—
513C2×C8—21no—
514C2×C8—42no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—611no—
518C2×C8—611no—
519C2×C8—611no—
520C2×C8—611no—
521C2×C8—612no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—2442no—
525C2×C8D4×D6112yes—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—42no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—2442no—
542C42—21no—
543C42—21no—
544C42—612no—
545C42—1221no—
546C42—2442no—
547C42—2442no—
548C42C2×C4×D6112yes—
549C23—21no—
550C23—1221no—
551C23#1451982211yescharacteristic
552Q8—21no—
553Q8—22no—
554Q8—22no—
555Q8—21no—
556Q8—41no—
557Q8—41no—
558Q8—1221no—
559Q8—1222no—
560Q8—1222no—
561Q8—1221no—
562Q8—2441no—
563Q8—2441no—
564D4—21no—
565D4—21no—
566D4—41no—
567D4—41no—
568D4—1221no—
569D4—1221no—
570D4—2441no—
571D4—2441no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—1222no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4C22×C4×D6111yescharacteristic
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4#41×D6111yescharacteristic
592C2×C4#1451982211yescharacteristic
593C2×C4#1451982211yescharacteristic
594C2×C4#1451982311yescharacteristic
595C8—22no—
596C8—41no—
597C8—611no—
598C8—611no—
599C8—611no—
600C8—611no—
601C8—1222no—
602C8—2441no—
603C8—2441no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—4881no—
608C22—2441no—
609C22#1684023511yescharacteristic
610C22#1684032811yescharacteristic
611C22#1684036211yescharacteristic
612C4—42no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—4881no—
618C4—2442no—
619C4—2441no—
620C4—2442no—
621C4C4×D4×D6111yescharacteristic
622C4#1684023411yescharacteristic
623C4#1684032711yescharacteristic
624C4#1684036411yescharacteristic
625C2—21no—
626C2—21no—
627C2—21no—
628C2—81no—
629C2—4881no—
630C2#2144854011yescharacteristic
subgroupG/H
centerC2#21448540
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1909C2
socleC2×C61#2328
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1909D61
hypocenterC61#22285
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1909

Complements

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

NH ≅ G/Ntype
3. #21454683C2semidirect
4. #21446733C2semidirect
6. #21446357C2semidirect
7. #21446355C2semidirect
8. #21446340C2semidirect
9. #21446316C2semidirect
10. #21446311C2semidirect
11. #21446292C2semidirect
12. #21446289C2semidirect
13. #21445941C2semidirect
14. #21445573C2semidirect
15. C61×#1909C2semidirect
24. #16840157C4semidirect
25. #16840154C4semidirect
26. #16840144C4semidirect
27. #16840141C4semidirect
318. #1909D61semidirect
374. C61#22285semidirect

Maximal subgroups

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

subgroupindex
#214563102
#214547342
#214546832
#214467332
#214467142
#214463572
#214463552
#214463402
#214463162
#214463112
#214462922
#214462892
#214459412
#214455732
C61×#19092
#2228561