Finite Group Databasebeta

← #29665427

Subgroups — #29665427

Order 15616.

758 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#21456739C211yesmaximal, characteristic, complemented
2#21455299C211yesmaximal, characteristic, complemented
3#21455274C211yesmaximal, characteristic, complemented
4#21445263C212yesmaximal, complemented
5#21445256C212yesmaximal, complemented
6#21445179C212yesmaximal, complemented
7#21445174C212yesmaximal, complemented
8#21444594C211yesmaximal, characteristic, complemented
9#21444494C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#2988C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519797—21no—
14#16840312C2211yescharacteristic
15C2×#14519774C2211yescharacteristic
16#16840139C2212yes—
17#16840109C2212yes—
18#16839984—21no—
19#16839977—21no—
20#16839972C2211yescharacteristic, complemented
21#16839967C2211yescharacteristic, complemented
22#16839962—21no—
23#16839962C2212yes—
24#16839962C2211yescharacteristic, complemented
25#16839958—21no—
26#16839958—21no—
27#16839957—21no—
28#16839957C2212yes—
29#16839957C2211yescharacteristic, complemented
30#16839898—22no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839766—21no—
34#16839756—21no—
35#16839515C2212yescomplemented
36#16839515C2212yescomplemented
37#16839507C2212yescomplemented
38#16839505C2212yes—
39#16839505C2212yes—
40#16839505C2212yescomplemented
41C4×#41×C61C2211yescharacteristic
42C8×Q8×C61C2212yescomplemented
43C8×D4×C61—22no—
44C8×D4×C61C2212yescomplemented
45C61×#431—21no—
46C61×#431C2211yescharacteristic
47C2×C4⋊C8×C61—21no—
48C2×C4⋊C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53#14519802—44no—
54#14519797—44no—
55C2×#11758202—21no—
56C2×#11758200—21no—
57C2×#11758201—22no—
58C2×#11758201—21no—
59C2×D488—22no—
60C2×D488—22no—
61C2×#11758199—22no—
62C2×#11758199—21no—
63C2×#11758194—21no—
64C22×#92418—21no—
65#14519777C2311yescharacteristic
66#14519774—21no—
67#14519774—21no—
68#14519774—21no—
69#14519774C2312yes—
70#14519774C2311yescharacteristic
71#14519765—21no—
72#14519751—21no—
73#14519751—21no—
74C4×D244—21no—
75C4×D244—21no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2311yescharacteristic
81#14519700—21no—
82#14519700—21no—
83#14519699—22no—
84#14519699—22no—
85#14519697—22no—
86#14519697—22no—
87#14519697—21no—
88#14519697—21no—
89C2×#41×C61D411yescharacteristic
90C22×C8×C61—21no—
91C22×C8×C61—22no—
92C4×Q8×C61D411yescharacteristic
93C4×Q8×C61C2311yescharacteristic
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D411yescharacteristic
97C4×D4×C61C2311yescharacteristic
98#117×C61—21no—
99#117×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C4⋊C8×C61—21no—
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D411yescharacteristic
106C4⋊C8×C61C2312yes—
107C22⋊C8×C61—22no—
108C22⋊C8×C61D412yes—
109C4×C8×C61—22no—
110C4×C8×C61C2312yes—
111C23×D61—41no—
112#41×C61—22no—
113#41×C61D814yescomplemented
114C2×Q8×C61C2×D411yescharacteristic
115C2×D4×C61—21no—
116C2×D4×C61C2×D411yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C2×C22⋊D61—41no—
124#11758222—42no—
125#11758222—42no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—22no—
131C2×D244—21no—
132C2×D244—42no—
133C2×D244—42no—
134C2×D244—41no—
135C2×D244—41no—
136C2×D244—41no—
137C2×D244—81no—
138C2×D244—81no—
139C2×D244—82no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142C2×C8×C61—22no—
143C2×C8×C61—82no—
144C2×C8×C61—81no—
145C2×C8×C61C2×D412yes—
146C2×C8×C61C2×D412yes—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61C2×D411yescharacteristic
150C4⋊C4×C61C2×D411yescharacteristic
151C22⋊C4×C61—21no—
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61#4111yescharacteristic
160C42×C61C2411yescharacteristic
161#11758202—42no—
162#11758201—42no—
163#11758201—44no—
164#11758200—42no—
165#11758199—42no—
166#11758199—44no—
167#11758198—41no—
168#11758198—41no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171#11758194—42no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—42no—
176D488—82no—
177D488—82no—
178C23×C61—21no—
179C23×C61#12111yescharacteristic
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—21no—
186Q8×C61C2×D812yes—
187D4×C61—24no—
188D4×C61—21no—
189D4×C61C2×D812yes—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—24no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—81no—
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61C2×D812yes—
204C2×C4×C61#13512yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C22⋊D61—82no—
209C2×Dic61—41no—
210D244—42no—
211D244—41no—
212D244—42no—
213D244—41no—
214D244—41no—
215D244—42no—
216D244—41no—
217D244—42no—
218D244—81no—
219D244—82no—
220D244—82no—
221D244—82no—
222D244—82no—
223C4×D61—82no—
224Dic122—41no—
225Dic122—41no—
226C8×C61—22no—
227C8×C61—22no—
228C8×C61—42no—
229#92418—41no—
230#92418—41no—
231#92418—42no—
232#19410—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61#46612yes—
239C22×C61C2×#12111yescharacteristic
240C2×D61—82no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—81no—
246C2×D61—164no—
247C4×C61—21no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61#46612yes—
254C4×C61C22×D811yescharacteristic
255C4×C61C2×#13511yescharacteristic
256C4×C61#57611yescharacteristic
257C4×C61#57711yescharacteristic
258Dic61—81no—
259#3205—611no—
260#3038—611no—
261#3035—611no—
262#2988D6111yescharacteristic, complemented
263#1682—612no—
264#1681—612no—
265#1668—612no—
266#1667—612no—
267#1571—611no—
268#1547—611no—
269C2×#331—611no—
270C2×C61—22no—
271C2×C61—42no—
272C2×C61#304511yescharacteristic
273C2×C61C2×#46611yescharacteristic
274C2×C61#306811yescharacteristic
275D61—161no—
276D61—162no—
277C2×AGL(1,ℤ/8)—1221no—
278C2×#135—1221no—
279#532—611no—
280C2×#127—611no—
281C4×#41C2×D6111yescharacteristic
282#493—612no—
283#486—612no—
284#464—1221no—
285#463—1221no—
286#462—611no—
287#461—611no—
288#460—612no—
289#460—611no—
290#460—1221no—
291#459—612no—
292#459—611no—
293#459—1221no—
294#459—1221no—
295#459—1221no—
296#447—1222no—
297C8×Q8C2×D6112yescomplemented
298C8×D4—22no—
299C8×D4C2×D6112yescomplemented
300#431—21no—
301#431C2×D6111yescharacteristic
302#423—611no—
303C2×C4⋊C8—21no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8C2×D6111yescharacteristic
306#418—1221no—
307#416—1221no—
308#335—612no—
309#335—612no—
310#332—612no—
311#331—612no—
312#331—612no—
313#331—612no—
314C61#1941011yescharacteristic, complemented, Sylow
315C2×#41—1221no—
316C2×#41D24411yescharacteristic
317C22×D4—1221no—
318C22×D4—1221no—
319C22×D4—1221no—
320AGL(1,ℤ/8)—2444no—
321#135—2444no—
322C2×Q16—1221no—
323C2×SD16—1221no—
324C2×SD16—1222no—
325C2×SD16—1221no—
326C2×D8—1221no—
327C2×D8—1222no—
328C2×D8—1222no—
329C2×D8—1222no—
330C2×M16—1221no—
331C22×C8—21no—
332C22×C8—22no—
333C22×C8—1221no—
334C4⋊Q8—611no—
335#127—612no—
336#127—611no—
337#127—1221no—
338#127—1221no—
339#127—1221no—
340#124—1221no—
341#121—1221no—
342#121—1221no—
343C4×Q8D24411yescharacteristic
344C4×Q8C22×D6111yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—1221no—
348C4×D4—1221no—
349C4×D4D24411yescharacteristic
350C4×D4C22×D6111yescharacteristic
351#117—21no—
352#117D24411yescharacteristic
353C2×C42—21no—
354C2×C42C22×D6111yescharacteristic
355C4⋊C8—22no—
356C4⋊C8—21no—
357C4⋊C8—611no—
358C4⋊C8—611no—
359C4⋊C8—612no—
360C4⋊C8—1221no—
361C4⋊C8—1221no—
362C4⋊C8C22⋊D6111yescharacteristic
363C4⋊C8C22⋊D6111yescharacteristic
364C4⋊C8C22×D6112yes—
365#103—1221no—
366#103—1221no—
367#102—1222no—
368#102—1222no—
369#102—1221no—
370#102—1221no—
371#102—1222no—
372#102—1222no—
373C22⋊C8—22no—
374C22⋊C8C22⋊D6112yes—
375C4×C8—22no—
376C4×C8C22×D6112yes—
377C24—2441no—
378#41—22no—
379#41—2442no—
380#41—2442no—
381#41D48814yescomplemented
382C2×Q8—1221no—
383C2×Q8C2×D24411yescharacteristic
384C2×D4—21no—
385C2×D4—1222no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—1221no—
390C2×D4—1222no—
391C2×D4—4881no—
392C2×D4—4881no—
393C2×D4—4882no—
394C2×D4—2441no—
395C2×D4—2441no—
396C2×D4—2441no—
397C2×D4—2441no—
398C2×D4—2442no—
399C2×D4—2442no—
400C2×D4C2×D24411yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—2441no—
405C22×C4C2×D24411yescharacteristic
406C22×C4#1175822211yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—2442no—
409SD16—2442no—
410SD16—2444no—
411SD16—2442no—
412D8—4882no—
413D8—4882no—
414D8—2442no—
415D8—2444no—
416M16—2442no—
417C2×C8—22no—
418C2×C8—82no—
419C2×C8—81no—
420C2×C8—1222no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—2442no—
424C2×C8C2×C22⋊D6112yes—
425C2×C8C2×C22⋊D6112yes—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4C2×D24411yescharacteristic
431C4⋊C4C2×D24411yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42C2×D24411yescharacteristic
441C42#1175822211yescharacteristic
442C42C2×C22⋊D6111yescharacteristic
443C42C23×D6111yescharacteristic
444C23—21no—
445C23—4882no—
446C23—4882no—
447C23—4882no—
448C23—2441no—
449C23—2442no—
450C23#1451975111yescharacteristic
451Q8—21no—
452Q8—2441no—
453Q8—2441no—
454Q8C2×D48812yes—
455D4—24no—
456D4—21no—
457D4—4881no—
458D4—4882no—
459D4—4882no—
460D4—4882no—
461D4—4882no—
462D4—4882no—
463D4—2442no—
464D4—2441no—
465D4—2442no—
466D4—2441no—
467D4—2441no—
468D4—2442no—
469D4—2441no—
470D4—2442no—
471D4C2×D48812yes—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—4882no—
483C2×C4—2441no—
484C2×C4#1451975111yescharacteristic
485C2×C4#1451975111yescharacteristic
486C2×C4#1451975111yescharacteristic
487C2×C4C2×D48812yes—
488C2×C4#1451979912yes—
489C2×C4C22×C22⋊D6111yescharacteristic
490C2×C4C22×D24411yescharacteristic
491C2×C4C2×#1175822211yescharacteristic
492C8—22no—
493C8—22no—
494C8—42no—
495C8—2441no—
496C8—2441no—
497C8—2442no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—9764no—
504C22—4881no—
505C22—4882no—
506C22—4882no—
507C22—4881no—
508C22—4882no—
509C22—4881no—
510C22#1683999812yes—
511C22C2×#1451975111yescharacteristic
512C4—22no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—4881no—
519C4#1683999812yes—
520C4C22×D48811yescharacteristic
521C4C2×#1451979911yescharacteristic
522C4#1684056511yescharacteristic
523C4#1684057011yescharacteristic
524C2—22no—
525C2—42no—
526C2—9762no—
527C2—9761no—
528C2#2145535811yescharacteristic
529C2C2×#1683999811yescharacteristic
530C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2988C2
socleC22×C61C2×#121
hypercenter#2988D61
hypocenterC61#19410
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #21456739C2semidirect
2. #21455299C2semidirect
3. #21455274C2semidirect
4. #21445263C2semidirect
5. #21445256C2semidirect
6. #21445179C2semidirect
7. #21445174C2semidirect
8. #21444594C2semidirect
9. #21444494C2semidirect
10. C2×#16839505C2semidirect
11. C61×#2988C2semidirect
20. #16839972C22semidirect
21. #16839967C22semidirect
24. #16839962C22semidirect
29. #16839957C22semidirect
35. #16839515C22semidirect
36. #16839515C22semidirect
37. #16839507C22semidirect
40. #16839505C22semidirect
42. C8×Q8×C61C22semidirect
44. C8×D4×C61C22semidirect
113. #41×C61D8semidirect
262. #2988D61semidirect
297. C8×Q8C2×D61semidirect
299. C8×D4C2×D61semidirect
314. C61#19410semidirect
381. #41D488semidirect

Maximal subgroups

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

subgroupindex
#214567392
#214552992
#214552742
#214452632
#214452562
#214451792
#214451742
#214445942
#214444942
C2×#168395052
C61×#29882
#1941061