Finite Group Databasebeta

← #29669337

Subgroups — #29669337

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456538C211yesmaximal, characteristic
2#21456538C211yesmaximal, characteristic
3#21456366C211yesmaximal, characteristic, complemented
4#21456213C211yesmaximal, characteristic
5#21456201C211yesmaximal, characteristic, complemented
6#21445494C211yesmaximal, characteristic, complemented
7#21445482C211yesmaximal, characteristic, complemented
8#21445473C211yesmaximal, characteristic, complemented
9#21445463C211yesmaximal, characteristic, complemented
10#21445462C211yesmaximal, characteristic, complemented
11#21445444C211yesmaximal, characteristic, complemented
12#21445427C211yesmaximal, characteristic, complemented
13#21445422C211yesmaximal, characteristic, complemented
14#21444122C211yesmaximal, characteristic, complemented
15C61×#3185C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18#16840172—22no—
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840165C2211yescharacteristic
23#16840158C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840136C2211yescharacteristic
26#16840136C2211yescharacteristic
27#16840132—22no—
28#16840122—22no—
29#16840114C2211yescharacteristic
30#16840114C2211yescharacteristic
31#16840114C2211yescharacteristic
32#16840114C2211yescharacteristic
33#16840113C2211yescharacteristic
34#16840113C2211yescharacteristic
35#16840112C2211yescharacteristic
36#16840112C2211yescharacteristic
37#16840110C2211yescharacteristic
38#16840110C2211yescharacteristic
39#16840109C2211yescharacteristic
40#16840109C2211yescharacteristic
41#16840010—22no—
42#16840010—22no—
43#16839860—22no—
44#16839860—22no—
45#16839805—22no—
46#16839770—22no—
47#16839764—22no—
48C4×#11758194C2211yescharacteristic
49C4×#11758194C2211yescharacteristic
50#16839484C2211yescharacteristic
51#16839484C2211yescharacteristic
52#16839483C2211yescharacteristic
53C8×#92418C2211yescharacteristic
54C61×#533C2211yescharacteristic
55C61×#533C2211yescharacteristic
56C61×#498C2211yescharacteristic
57C61×#494C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#487C2211yescharacteristic
60C61×#470—22no—
61C61×#441—22no—
62C4×M16×C61C2211yescharacteristic
63C2×#11758222—21no—
64C22×Dic122—21no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758194D412yes—
68C2×#11758194D412yes—
69#14519777—21no—
70#14519777—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519774C2311yescharacteristic
76#14519765—21no—
77#14519751—21no—
78#14519751—21no—
79C4×D244—21no—
80C4×D244—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83#14519710—22no—
84#14519710—22no—
85#14519710—22no—
86#14519710—22no—
87#14519708—22no—
88#14519708—22no—
89#14519702—22no—
90#14519702—22no—
91#14519701—22no—
92#14519701—22no—
93#14519700—22no—
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519699—22no—
98#14519699—22no—
99#14519697—22no—
100#14519697—22no—
101#14519688D412yes—
102#14519688D412yes—
103C4×#92418C2311yescharacteristic
104C4×#92418C2311yescharacteristic
105C4×#92418C2311yescharacteristic
106C4×#92418C2311yescharacteristic
107C2×Q16×C61—22no—
108C2×M16×C61D412yes—
109C4⋊Q8×C61C2311yescharacteristic
110C4⋊Q8×C61C2311yescharacteristic
111C4⋊Q8×C61C2311yescharacteristic
112C4⋊Q8×C61C2311yescharacteristic
113#125×C61—21no—
114#125×C61—21no—
115C22⋊Q8×C61—21no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—21no—
118C22⋊Q8×C61—21no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123#117×C61—21no—
124#117×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#106×C61—22no—
127#103×C61—22no—
128#103×C61—22no—
129#97×C61D412yes—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141#11758222—44no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×C4×D61—41no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—44no—
150C2×Dic122—41no—
151Q16×C61—44no—
152M16×C61—42no—
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D412yes—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—41no—
164C4⋊C4×C61—41no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C42×C61—41no—
170C42×C61—41no—
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2411yescharacteristic
175#11758201—44no—
176#11758201—44no—
177#11758198—41no—
178#11758198—41no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—41no—
186C4⋊Dic61—41no—
187#11758194—24no—
188#11758194—42no—
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193C23×C61#12711yescharacteristic
194C22×D61—41no—
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198Q8×C61—42no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61#12711yescharacteristic
207C2×C4×C61#12711yescharacteristic
208C2×C4×C61#12711yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C22⋊D61—82no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—82no—
217D244—42no—
218D244—42no—
219D244—42no—
220C4×D61—82no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—82no—
225C8×C61—22no—
226C8×C61—22no—
227#92418—22no—
228#92418—22no—
229#92418—22no—
230#92418#13714yes—
231#19752—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61C2×#12711yescharacteristic
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61C2×#13712yes—
247C4×C61#57512yes—
248C4×C61#57512yes—
249Dic61—81no—
250Dic61—81no—
251Dic61—81no—
252#3185D6111yescharacteristic, complemented
253#3182—611no—
254#3182—611no—
255#3165—611no—
256#3141—611no—
257#3139—611no—
258#1723—611no—
259#1720—611no—
260#1719—611no—
261#1718—611no—
262#1718—611no—
263#1714—611no—
264#1710—611no—
265#1708—611no—
266#1455—611no—
267C2×C61—41no—
268C2×C61#315211yescharacteristic
269C2×C61#315311yescharacteristic
270C2×C61#315311yescharacteristic
271D61—161no—
272#533C2×D6111yescharacteristic
273#533C2×D6111yescharacteristic
274#532—611no—
275C2×C4⋊Q8—611no—
276#501—1222no—
277#500—611no—
278#500—611no—
279#500—611no—
280#500—611no—
281#498—611no—
282#498—611no—
283#498C2×D6111yescharacteristic
284#494C2×D6111yescharacteristic
285#492—611no—
286#492—611no—
287#491—1222no—
288#489—1222no—
289#487—611no—
290#487—611no—
291#487—611no—
292#487—611no—
293#487—611no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487C2×D6111yescharacteristic
298#487C2×D6111yescharacteristic
299#486—611no—
300#486—611no—
301#486—611no—
302#486—611no—
303#470—22no—
304#468—1222no—
305#468—1222no—
306#441—22no—
307#440—1222no—
308#440—1222no—
309#428—1222no—
310#419—1222no—
311#417—1222no—
312C4×M16—611no—
313C4×M16—611no—
314C4×M16C2×D6111yescharacteristic
315#322—611no—
316#322—611no—
317#322—611no—
318C82—611no—
319C61#1975211yescharacteristic, complemented, Sylow
320C2×#41—1221no—
321C22×Q8—1221no—
322C2×Q16—22no—
323C2×SD16—1222no—
324C2×SD16—1222no—
325C2×M16—612no—
326C2×M16—612no—
327C2×M16C22⋊D6112yes—
328C4⋊Q8—611no—
329C4⋊Q8—611no—
330C4⋊Q8—611no—
331C4⋊Q8—1221no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8C22×D6111yescharacteristic
335C4⋊Q8C22×D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338#127—611no—
339#125—21no—
340#125—21no—
341#124—1221no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346#121—1221no—
347#121—1221no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×D4—1221no—
353C4×D4—1221no—
354#117—21no—
355#117—21no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C42C22×D6111yescharacteristic
359#107—1222no—
360#107—1222no—
361#107—1222no—
362#107—1222no—
363#106—22no—
364#106—1222no—
365#106—1222no—
366#103—22no—
367#103—22no—
368#103—1222no—
369#103—1222no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376#102—1222no—
377#102—1222no—
378#102—1222no—
379#102—1222no—
380#97—612no—
381#97—612no—
382#97C22⋊D6112yes—
383C4×C8—611no—
384C4×C8—611no—
385C4×C8—611no—
386C4×C8—611no—
387C4×C8C22×D6111yescharacteristic
388C4×C8C22×D6111yescharacteristic
389#41—2444no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×Q8—2444no—
398C2×Q8—2441no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—1221no—
402C2×D4—2441no—
403C22×C4—2441no—
404C22×C4—2441no—
405C22×C4D4×D6111yescharacteristic
406C22×C4D4×D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—44no—
409SD16—2444no—
410SD16—2444no—
411M16—42no—
412M16—1224no—
413M16—2442no—
414C2×C8—612no—
415C2×C8—612no—
416C2×C8—612no—
417C2×C8—612no—
418C2×C8C2×C22⋊D6112yes—
419C2×C8C2×C22⋊D6112yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C42—41no—
445C42—41no—
446C42D4×D6111yescharacteristic
447C42D4×D6111yescharacteristic
448C42C2×C22⋊D6111yescharacteristic
449C42C23×D6111yescharacteristic
450C23—2441no—
451C23#1451977311yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—4882no—
457Q8—2442no—
458Q8—2442no—
459Q8—2442no—
460D4—4882no—
461D4—2442no—
462D4—2442no—
463D4—2442no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—4882no—
472C2×C4—4882no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451977311yescharacteristic
477C2×C4#1451977311yescharacteristic
478C2×C4#1451977311yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4C22×C22⋊D6111yescharacteristic
482C8—22no—
483C8—22no—
484C8—614no—
485C8—1222no—
486C8—1222no—
487C8—1222no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—4881no—
492C22—4881no—
493C22—4881no—
494C22C2×#1451977311yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—4881no—
503C4—4881no—
504C4—4881no—
505C4#137×D6112yes—
506C4#1684056012yes—
507C4#1684056112yes—
508C2—41no—
509C2—9761no—
510C2#2145627411yescharacteristic
511C2#2145628611yescharacteristic
512C2#2145628711yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3185C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenter#3185D61
hypocenterC61#19752
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
3. #21456366C2semidirect
5. #21456201C2semidirect
6. #21445494C2semidirect
7. #21445482C2semidirect
8. #21445473C2semidirect
9. #21445463C2semidirect
10. #21445462C2semidirect
11. #21445444C2semidirect
12. #21445427C2semidirect
13. #21445422C2semidirect
14. #21444122C2semidirect
15. C61×#3185C2semidirect
252. #3185D61semidirect
319. C61#19752semidirect

Maximal subgroups

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

subgroupindex
#214565382
#214563662
#214562132
#214562012
#214454942
#214454822
#214454732
#214454632
#214454622
#214454442
#214454272
#214454222
#214441222
C61×#31852
#1975261