Finite Group Databasebeta

← #29734554

Subgroups — #29734554

Order 15616.

950 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456372C211yesmaximal, characteristic, complemented
2#21456219C211yesmaximal, characteristic
3#21451006C211yesmaximal, characteristic, complemented
4#21447033C212yesmaximal, complemented
5#21447033C212yesmaximal, complemented
6#21446764C211yesmaximal, characteristic, complemented
7#21446739C211yesmaximal, characteristic
8#21446052C212yesmaximal, complemented
9#21446045C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C61×#3507C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840168—21no—
14#16840168—21no—
15#16840161—22no—
16#16840130—21no—
17#16840126—21no—
18#16840116—22no—
19#16839805C2212yes—
20#16839805C2212yes—
21C2×#14519704C2212yes—
22C2×#14519704C2212yes—
23#16839763C2212yes—
24#16839763C2212yes—
25C2×#14519696C2212yes—
26C4×#11758194C2211yescharacteristic
27C4×#11758194C2211yescharacteristic
28#16839695—22no—
29#16839669C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839646—22no—
32#16839631C2211yescharacteristic
33#16839627C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35#16839531C2212yes—
36#16839521C2212yes—
37#16839521C2212yes—
38C61×#559—21no—
39C61×#557—21no—
40C61×#556—21no—
41C61×#556—22no—
42C61×#554—21no—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536C2212yes—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520C2212yes—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53#14519769—21no—
54#14519769—21no—
55#14519769—21no—
56#14519769—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—22no—
62C2×#11758196—21no—
63C2×C4×Dic61C2311yescharacteristic
64C2×C4×Dic61C2311yescharacteristic
65#14519710—22no—
66#14519710—22no—
67#14519708—22no—
68#14519708—22no—
69#14519704—24no—
70#14519704—24no—
71#14519701—22no—
72#14519701—22no—
73#14519698—22no—
74#14519698—22no—
75#14519696—24no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418—21no—
80#14519684—22no—
81#14519684—22no—
82#14519684—22no—
83#14519684—21no—
84C2×#41×C61C2312yes—
85C22×Q8×C61C2311yescharacteristic
86C4⋊Q8×C61—22no—
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61D411yescharacteristic
89C4⋊Q8×C61D411yescharacteristic
90#127×C61D411yescharacteristic
91#126×C61—24no—
92#126×C61—24no—
93#125×C61—24no—
94#125×C61—22no—
95#125×C61D412yes—
96#124×C61—22no—
97#124×C61—22no—
98#124×C61D412yes—
99#123×C61—24no—
100#123×C61—24no—
101C22⋊Q8×C61—24no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106#121×C61—21no—
107#121×C61—21no—
108C4×Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×Q8×C61—21no—
111C4×Q8×C61—21no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C4×D4×C61—24no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121#117×C61—24no—
122#117×C61—22no—
123#117×C61C2312yes—
124C2×C4⋊C4×C61—22no—
125C2×C4⋊C4×C61D412yes—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61D412yes—
128C2×C42×C61C2311yescharacteristic
129#41×C61—22no—
130#41×C61—22no—
131#41×C61—22no—
132#41×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C2×D412yes—
144C22×C4×C61—24no—
145C22×C4×C61—22no—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—24no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189#11758196—44no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196#11758194—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194#4114yes—
200C2×#92418—22no—
201C2×#92418—22no—
202C23×C61—22no—
203C23×C61C22×D411yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—42no—
207Q8×C61—42no—
208Q8×C61—42no—
209Q8×C61—81no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—81no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—84no—
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C612−1+412yes—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247#92418—24no—
248#92418—42no—
249#25698—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61C2×#12011yescharacteristic
256C22×C61#53211yescharacteristic
257C22×C61#53611yescharacteristic
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61#54712yes—
268C4×C61#54812yes—
269Dic61—42no—
270Dic61—81no—
271Dic61—81no—
272Dic61—81no—
273#3507D6111yescharacteristic
274#3166—611no—
275#3142—611no—
276#2565—611no—
277#2038—612no—
278#2038—612no—
279#1986—611no—
280#1980—611no—
281#1860—612no—
282#1859—612no—
283C4×#101—611no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—82no—
287C2×C61#243211yescharacteristic
288C2×C61#301811yescharacteristic
289C2×C61#302011yescharacteristic
290#559—21no—
291#557—21no—
292#556—21no—
293#556—22no—
294#554—21no—
295#552—22no—
296#539—22no—
297#536C2×D6112yes—
298#533C2×D6111yescharacteristic
299#532C2×D6111yescharacteristic
300#529—22no—
301C2×#125—611no—
302#520C2×D6112yes—
303C2×C4×Q8C2×D6111yescharacteristic
304#501—1221no—
305#501—1221no—
306#499—1222no—
307#491—1221no—
308#490—1221no—
309#488—1222no—
310#428—612no—
311#428—612no—
312C2×#104—612no—
313C2×#104—612no—
314#417—612no—
315#417—612no—
316C2×#101—612no—
317C4×M16—611no—
318C4×M16—611no—
319#398—1222no—
320#391—611no—
321#391—611no—
322#387—1222no—
323#384—611no—
324#382—611no—
325C4×C4⋊C4—611no—
326#343—612no—
327#339—612no—
328#339—612no—
329C61#2569811yescharacteristic, complemented, Sylow
330C2×#41C22×D6112yes—
331C22×Q8C22×D6111yescharacteristic
332C2×M16—612no—
333C2×M16—612no—
334C4⋊Q8—22no—
335C4⋊Q8C22⋊D6111yescharacteristic
336C4⋊Q8C22⋊D6111yescharacteristic
337C4⋊Q8C22⋊D6111yescharacteristic
338#127C22⋊D6111yescharacteristic
339#126—24no—
340#126—24no—
341#125—22no—
342#125—24no—
343#125—1221no—
344#125—1221no—
345#125—1221no—
346#125—1221no—
347#125C22⋊D6112yes—
348#124—22no—
349#124—22no—
350#124C22⋊D6112yes—
351#123—24no—
352#123—24no—
353C22⋊Q8—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358#121—21no—
359#121—21no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—24no—
371C4×D4—21no—
372C4×D4—21no—
373#117—24no—
374#117—22no—
375#117C22×D6112yes—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4C22⋊D6112yes—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C42—611no—
386C2×C42—611no—
387C2×C42C22⋊D6112yes—
388C2×C42C22×D6111yescharacteristic
389#107—1222no—
390#107—1222no—
391#106—1222no—
392#106—1222no—
393#104—1224no—
394#104—1224no—
395#103—1222no—
396#103—1222no—
397#102—1222no—
398#102—1222no—
399#101—1224no—
400#97—1221no—
401#97—1221no—
402C4×C8—1221no—
403C4×C8—1221no—
404#95—1221no—
405#95—1222no—
406#95—1222no—
407#95—1222no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C22⋊D6112yes—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C22⋊D6112yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C23×D6111yescharacteristic
434M16—614no—
435M16—1222no—
436M16—1222no—
437M16—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—2444no—
458C4⋊C4—2442no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4C2×C22⋊D6112yes—
464C4⋊C4C2×C22⋊D6112yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—1222no—
476C42—1222no—
477C42C2×C22⋊D6111yescharacteristic
478C42C2×C22⋊D6111yescharacteristic
479C42C2×C22⋊D6111yescharacteristic
480C42C2×C22⋊D6111yescharacteristic
481C23—22no—
482C23C22×C22⋊D6111yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—81no—
489D4—22no—
490D4—22no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—81no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—1222no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451974611yescharacteristic
519C2×C4#1451974611yescharacteristic
520C2×C4#1451974611yescharacteristic
521C2×C4#1451974611yescharacteristic
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#41×D6112yes—
525C2×C4#1451982412yes—
526C8—1224no—
527C8—2442no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22C2×#1451974611yescharacteristic
534C22#1684030811yescharacteristic
535C22#1684033311yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—4881no—
546C4—4881no—
547C4—4881no—
548C4—2442no—
549C4#1684041112yes—
550C4#1684041912yes—
551C2—21no—
552C2—21no—
553C2—82no—
554C2#2144944311yescharacteristic
555C2#2145516811yescharacteristic
556C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#25698
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. #21456372C2semidirect
3. #21451006C2semidirect
4. #21447033C2semidirect
5. #21447033C2semidirect
6. #21446764C2semidirect
8. #21446052C2semidirect
9. #21446045C2semidirect
10. C4×#14519696C2semidirect
329. C61#25698semidirect

Maximal subgroups

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

subgroupindex
#214563722
#214562192
#214510062
#214470332
#214467642
#214467392
#214460522
#214460452
C4×#145196962
C61×#35072
#2569861