Finite Group Databasebeta

← #29657652

Subgroups — #29657652

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic
2#21456401C211yesmaximal, characteristic, complemented
3#21455624C211yesmaximal, characteristic, complemented
4#21445075C212yesmaximal, complemented
5#21445026C212yesmaximal, complemented
6#21444991C212yesmaximal, complemented
7#21444953C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3507C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840165C2211yescharacteristic
14#16840141C2211yescharacteristic
15#16840114C2211yescharacteristic
16#16840112C2211yescharacteristic
17#16840069—22no—
18#16840057C2212yes—
19#16840052C2212yes—
20#16840047C2212yes—
21#16840044—21no—
22#16840001—22no—
23#16839989—22no—
24#16839967C2212yes—
25#16839959—21no—
26C4×#11758202—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839771—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#559—21no—
39C61×#557—21no—
40C61×#556—22no—
41C61×#556C2212yes—
42C61×#554C2212yes—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54C22×#92418—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519710—22no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—22no—
73#14519701—22no—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61C2311yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61D411yescharacteristic
88#126×C61—24no—
89#126×C61—24no—
90#125×C61—24no—
91#125×C61—22no—
92#125×C61—21no—
93#124×C61—22no—
94#124×C61—22no—
95#124×C61—21no—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61D412yes—
103#121×C61—21no—
104#121×C61D412yes—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C4×Q8×C61—21no—
110C4×Q8×C61D412yes—
111C4×Q8×C61D412yes—
112C4×Q8×C61C2312yes—
113C4×D4×C61—24no—
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61—21no—
117C4×D4×C61C2312yes—
118#117×C61—24no—
119#117×C61—21no—
120#117×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#41×C61—42no—
127#41×C61—42no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61C2×D412yes—
134C2×Q8×C61C2×D412yes—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×Dic122—21no—
147C2×Dic122—21no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C42×C61—24no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2411yescharacteristic
177C22⋊Dic61—41no—
178#11758202—44no—
179#11758200—82no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61—41no—
187#11758194—42no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—42no—
193C2×#92418—81no—
194C23×C61—22no—
195C23×C61#12011yescharacteristic
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—42no—
199Q8×C61—41no—
200Q8×C61—41no—
201Q8×C61—42no—
202Q8×C61—42no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—44no—
206D4×C61—42no—
207D4×C61—81no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—81no—
220C2×C4×C61—84no—
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#13512yes—
225C2×C4×C61#13712yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×Dic61—41no—
230C2×Dic61—41no—
231Dic122—42no—
232Dic122—41no—
233Dic122—41no—
234#92418—22no—
235#92418—41no—
236#92418—42no—
237#92418—42no—
238#18669—611nomaximal, Sylow
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61#45212yes—
245C22×C61C2×#12011yescharacteristic
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61#45212yes—
255C4×C61C2×#13511yescharacteristic
256C4×C61C2×#13711yescharacteristic
257C4×C61#57711yescharacteristic
258C4×C61#57811yescharacteristic
259Dic61—81no—
260Dic61—81no—
261#3507D6111yescharacteristic
262#3246—611no—
263#3170—611no—
264#3078—611no—
265#1650—612no—
266#1643—612no—
267#1638—612no—
268#1632—612no—
269#1550—611no—
270#1492—611no—
271C2×Q8⋊C8—611no—
272C2×C61—22no—
273C2×C61—82no—
274C2×C61C2×#45211yescharacteristic
275C2×C61#301311yescharacteristic
276C2×C61#301811yescharacteristic
277#559—21no—
278#557—21no—
279#556—22no—
280#556C2×D6112yes—
281#554C2×D6112yes—
282#552—22no—
283#539—22no—
284#536—21no—
285#533—611no—
286#533C2×D6111yescharacteristic
287#532C2×D6111yescharacteristic
288#529—22no—
289#520—21no—
290C2×C4×Q8C2×D6111yescharacteristic
291#500—611no—
292#494—611no—
293#487—611no—
294#487—611no—
295#479—1222no—
296#477—612no—
297#476—612no—
298#475—612no—
299#475—1221no—
300#467—1222no—
301#465—1222no—
302#461—612no—
303#460—1221no—
304C4×Q16—1222no—
305#428—1221no—
306#423—611no—
307C2×C4⋊C8—611no—
308#419—1221no—
309C2×C4×C8—611no—
310#333—612no—
311#332—612no—
312Q8⋊C8—612no—
313Q8⋊C8—612no—
314D4⋊C8—612no—
315D4⋊C8—612no—
316C61#1866911yescharacteristic, complemented, Sylow
317C2×#41—21no—
318C22×Q8C22⋊D6111yescharacteristic
319C2×Q16—1222no—
320C2×SD16—1222no—
321C2×M16—1221no—
322C22×C8—1221no—
323C22×C8—1221no—
324C4⋊Q8—22no—
325C4⋊Q8—611no—
326C4⋊Q8—611no—
327C4⋊Q8C22⋊D6111yescharacteristic
328C4⋊Q8C22×D6111yescharacteristic
329C4⋊Q8C22×D6111yescharacteristic
330#127C22⋊D6111yescharacteristic
331#126—24no—
332#126—24no—
333#125—24no—
334#125—22no—
335#125—21no—
336#125—1221no—
337#124—22no—
338#124—22no—
339#124—21no—
340#123—24no—
341#123—24no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—24no—
345C22⋊Q8—21no—
346C22⋊Q8—1221no—
347C22⋊Q8—1221no—
348C22⋊Q8C22⋊D6112yes—
349#121—21no—
350#121C22⋊D6112yes—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—1221no—
357C4×Q8—1221no—
358C4×Q8C22⋊D6112yes—
359C4×Q8C22⋊D6112yes—
360C4×Q8C22×D6112yes—
361C4×D4—24no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—21no—
365C4×D4C22×D6112yes—
366#117—24no—
367#117—21no—
368#117—22no—
369#117—1221no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C42—21no—
374C2×C42C22×D6111yescharacteristic
375#107—1222no—
376#107—1222no—
377#106—1222no—
378#106—1222no—
379C4⋊C8—612no—
380C4⋊C8—612no—
381C4⋊C8—1221no—
382C4⋊C8—1221no—
383#103—1222no—
384#103—1222no—
385#103—1222no—
386#103—1222no—
387#103—1222no—
388#102—1222no—
389C4×C8—611no—
390C4×C8—611no—
391C4×C8—1221no—
392#41—42no—
393#41—42no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8C2×C22⋊D6112yes—
402C2×Q8C2×C22⋊D6112yes—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C2×C22⋊D6112yes—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414Q16—2444no—
415SD16—4882no—
416M16—2442no—
417C2×C8—1221no—
418C2×C8—1221no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—4881no—
422C2×C8—2442no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C4⋊C4C2×C22⋊D6112yes—
442C4⋊C4C2×C22⋊D6112yes—
443C4⋊C4C2×C22⋊D6112yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—2441no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—2441no—
456C42C2×C22⋊D6111yescharacteristic
457C42C2×C22⋊D6111yescharacteristic
458C42C2×C22⋊D6111yescharacteristic
459C42C23×D6111yescharacteristic
460C23—22no—
461C23#1451974611yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—42no—
465Q8—41no—
466Q8—41no—
467Q8—42no—
468Q8—42no—
469Q8—2442no—
470Q8—2441no—
471Q8—2441no—
472D4—22no—
473D4—22no—
474D4—44no—
475D4—42no—
476D4—81no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—81no—
489C2×C4—84no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4#1451974611yescharacteristic
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974611yescharacteristic
495C2×C4#1451979712yes—
496C2×C4#1451980712yes—
497C2×C4C22×C22⋊D6111yescharacteristic
498C2×C4C22×C22⋊D6111yescharacteristic
499C2×C4C22×C22⋊D6111yescharacteristic
500C8—1222no—
501C8—2441no—
502C8—2442no—
503C8—2442no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22#1683992412yes—
510C22C2×#1451974611yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—4881no—
520C4—4881no—
521C4#1683992412yes—
522C4C2×#1451979711yescharacteristic
523C4C2×#1451980711yescharacteristic
524C4#1684057011yescharacteristic
525C4#1684057711yescharacteristic
526C2—22no—
527C2—82no—
528C2C2×#1683992411yescharacteristic
529C2#2145513411yescharacteristic
530C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#18669
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #21456401C2semidirect
3. #21455624C2semidirect
4. #21445075C2semidirect
5. #21445026C2semidirect
6. #21444991C2semidirect
7. #21444953C2semidirect
8. #21444509C2semidirect
10. C2×Q8⋊#92418C2semidirect
316. C61#18669semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214564012
#214556242
#214450752
#214450262
#214449912
#214449532
#214445092
#214442702
C2×Q8⋊#924182
C61×#35072
#1866961