Finite Group Databasebeta

← #29835350

Subgroups — #29835350

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456305C211yesmaximal, characteristic
2#21448062C212yesmaximal, complemented
3#21448058C212yesmaximal, complemented
4#21448053C212yesmaximal, complemented
5#21448044C212yesmaximal, complemented
6#21447534C211yesmaximal, characteristic, complemented
7#21447529C211yesmaximal, characteristic, complemented
8#21447524C211yesmaximal, characteristic, complemented
9C2×#16839562C211yesmaximal, characteristic, complemented
10#21447416C211yesmaximal, characteristic, complemented
11C61×#3164C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519799C2211yescharacteristic
14C22×D488C2211yescharacteristic
15#16840331—21no—
16C2×#14519717C2211yescharacteristic
17C22×#11758185C2211yescharacteristic
18#16840151—22no—
19#16840146C2212yes—
20#16840139C2212yes—
21#16839565C2211yescharacteristic, complemented
22#16839565C2211yescharacteristic, complemented
23#16839565C2212yescomplemented
24#16839562C2212yes—
25#16839562C2212yes—
26#16839562C2212yes—
27#16839562C2212yes—
28#16839562C2211yescharacteristic, complemented
29#16839562C2211yescharacteristic, complemented
30#16839562C2212yescomplemented
31#16839538C2212yescomplemented
32C4×#11758185C2212yescomplemented
33C61×#544—21no—
34C61×#501—22no—
35C61×#500C2212yescomplemented
36C61×#499C2212yescomplemented
37C61×#428—21no—
38C61×#427C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#405C2211yescharacteristic
41C2×#11758222—21no—
42C22×D244—21no—
43#14519805—44no—
44#14519799—22no—
45#14519799—22no—
46C2×Dic244C2311yescharacteristic
47C2×#11758190—21no—
48C2×#11758190—22no—
49C2×D488—22no—
50C2×D488—22no—
51C2×D488—21no—
52C2×D488—22no—
53C2×D488C2312yes—
54C2×D488C2311yescharacteristic
55#14519774—22no—
56#14519765—22no—
57#14519752—41no—
58#14519752—41no—
59#14519747—42no—
60#14519717—22no—
61C2×#11758185—22no—
62C2×#11758185C2311yescharacteristic
63C2×#11758185C2312yes—
64C2×#11758185C2311yescharacteristic
65C2×M16×C61D411yescharacteristic
66C22×C8×C61C2311yescharacteristic
67C4⋊Q8×C61—22no—
68C4⋊Q8×C61—41no—
69#125×C61—22no—
70#125×C61—41no—
71C22⋊Q8×C61—42no—
72#117×C61—21no—
73#117×C61D411yescharacteristic
74C2×C4⋊C4×C61—21no—
75#107×C61—21no—
76#107×C61—22no—
77#107×C61C2312yes—
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#106×C61—21no—
81#106×C61—22no—
82#97×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—41no—
85C2×Q8×C61—42no—
86C22×C4×C61—41no—
87C22×C4×C61C2×D411yescharacteristic
88C2×C22⋊D61—41no—
89#11758222—42no—
90#11758222—42no—
91C2×D244—22no—
92C2×D244—21no—
93C2×D244—42no—
94C2×D244—42no—
95C2×D244—41no—
96C2×D244—42no—
97C2×D244—42no—
98C2×C4×D61—41no—
99C2×Dic122—21no—
100M16×C61SD1614yes—
101C2×C8×C61C2×D411yescharacteristic
102C2×C8×C61C2×D412yes—
103C2×C8×C61#4112yes—
104C2×C8×C61#4112yes—
105C2×C8×C61C2411yescharacteristic
106C4⋊C4×C61—21no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—42no—
111C4⋊C4×C61—42no—
112C4⋊C4×C61—42no—
113C4⋊C4×C61D811yescharacteristic
114C4⋊C4×C61D811yescharacteristic
115C22⋊C4×C61—41no—
116C22⋊C4×C61D812yes—
117C42×C61—41no—
118C42×C61C2×D412yes—
119#11758198—42no—
120#11758198—42no—
121Dic244—21no—
122Dic244—21no—
123D488—22no—
124D488—22no—
125D488—21no—
126D488—21no—
127D488—42no—
128D488—44no—
129#11758190—42no—
130#11758190—44no—
131#11758185—21no—
132#11758185—21no—
133#11758185—22no—
134C23×C61C2×D811yescharacteristic
135C22×D61—42no—
136C22×D61—41no—
137C22×D61—82no—
138C22×D61—161no—
139C22×D61—161no—
140Q8×C61—162no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—161no—
145C2×C4×C61#12412yes—
146C2×C4×C61C2×D812yes—
147C2×C4×C61C2×D812yes—
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C2×SD1612yes—
150C2×C4×C61C22×D411yescharacteristic
151C22⋊D61—82no—
152C2×Dic61—41no—
153D244—41no—
154D244—42no—
155D244—42no—
156D244—41no—
157D244—82no—
158D244—162no—
159D244—162no—
160C4×D61—82no—
161Dic122—41no—
162Dic122—41no—
163C8×C61#12412yes—
164C8×C61C2×SD1614yes—
165C8×C61C2×#4111yescharacteristic
166C8×C61C2×#4111yescharacteristic
167#33702—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61#48612yes—
171C22×C61C22×D811yescharacteristic
172C2×D61—81no—
173C2×D61—82no—
174C2×D61—81no—
175C2×D61—82no—
176C2×D61—81no—
177C2×D61—82no—
178C2×D61—164no—
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—82no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61#48612yes—
185C4×C61C2×#12411yescharacteristic
186C4×C61C22×SD1611yescharacteristic
187Dic61—81no—
188#3164D6111yescharacteristic, complemented
189#3155—611no—
190#2244—612no—
191#2243—612no—
192#2242—612no—
193#2240—612no—
194#2140—611no—
195#2139—611no—
196#2138—611no—
197C2×#357—611no—
198#2108—611no—
199C2×C61—22no—
200C2×C61C2×#48611yescharacteristic
201C2×C61#341511yescharacteristic
202C2×C61#341611yescharacteristic
203D61—161no—
204D61—162no—
205C2×AGL(1,ℤ/8)—1221no—
206C2×#135—611no—
207C22×D8—611no—
208#544—21no—
209#535—1221no—
210C2×M32—611no—
211C22×C16—611no—
212#501—22no—
213#500C2×D6112yescomplemented
214#499C2×D6112yescomplemented
215#496—1222no—
216#495—612no—
217#493—612no—
218#428—21no—
219#427C2×D6111yescharacteristic
220C2×#107C2×D6111yescharacteristic
221#405C2×D6111yescharacteristic
222#358—611no—
223#358—611no—
224#358—612no—
225#357—612no—
226#357—612no—
227#357—611no—
228#357—611no—
229#357—612no—
230#357—612no—
231#357—612no—
232#346—612no—
233C4×C16—612no—
234C61#3370211yescharacteristic, complemented, Sylow
235C2×#41—1221no—
236C22×D4—1221no—
237AGL(1,ℤ/8)—2444no—
238#135—1222no—
239#135—1222no—
240C2×Q16—611no—
241C2×SD16—1221no—
242C2×SD16—1222no—
243C2×D8—612no—
244C2×D8—611no—
245C2×D8—1222no—
246C2×D8—1222no—
247C2×D8—1221no—
248C2×D8—1222no—
249C2×M16C22⋊D6111yescharacteristic
250C22×C8C22×D6111yescharacteristic
251C4⋊Q8—22no—
252C4⋊Q8—41no—
253#127—1222no—
254#125—22no—
255#125—41no—
256#124—1222no—
257C22⋊Q8—42no—
258#121—2441no—
259#121—2441no—
260#120—2442no—
261#117—21no—
262#117C22⋊D6111yescharacteristic
263C2×C4⋊C4—21no—
264M32—1222no—
265C2×C16—611no—
266C2×C16—612no—
267C2×C16—611no—
268C2×C16—1222no—
269#107—21no—
270#107—22no—
271#107C22×D6112yes—
272#107C22×D6111yescharacteristic
273#107C22×D6111yescharacteristic
274#106—21no—
275#106—22no—
276#97C22⋊D6112yes—
277C4×C8C22×D6112yes—
278C24—2441no—
279#41—2442no—
280#41—2442no—
281C2×Q8—42no—
282C2×Q8—1221no—
283C2×D4—1222no—
284C2×D4—1221no—
285C2×D4—2441no—
286C2×D4—2442no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2441no—
290C2×D4—2442no—
291C22×C4—41no—
292C22×C4—2441no—
293C22×C4C2×C22⋊D6111yescharacteristic
294Q16—1221no—
295Q16—1221no—
296SD16—2442no—
297SD16—2444no—
298D8—1222no—
299D8—1222no—
300D8—1221no—
301D8—1221no—
302D8—2442no—
303D8—2444no—
304M16#1175820114yes—
305C2×C8#1175822612yes—
306C2×C8#1175822612yes—
307C2×C8C2×C22⋊D6111yescharacteristic
308C2×C8C2×C22⋊D6112yes—
309C2×C8C23×D6111yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4#1175819911yescharacteristic
318C4⋊C4#1175819911yescharacteristic
319C22⋊C4—41no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C22⋊C4#1175819912yes—
323C42—41no—
324C42C2×C22⋊D6112yes—
325C16—1221no—
326C16—1221no—
327C16—1222no—
328C23—9761no—
329C23—9761no—
330C23—4882no—
331C23—2442no—
332C23—2441no—
333C23C2×#1175819911yescharacteristic
334Q8—162no—
335Q8—2441no—
336Q8—2441no—
337D4—9762no—
338D4—9762no—
339D4—4882no—
340D4—4882no—
341D4—2441no—
342D4—2442no—
343D4—2442no—
344D4—2441no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—161no—
349C2×C4—4882no—
350C2×C4—2441no—
351C2×C4#1451976612yes—
352C2×C4C2×#1175819912yes—
353C2×C4C2×#1175819912yes—
354C2×C4C2×#1175819911yescharacteristic
355C2×C4C2×#1175820112yes—
356C2×C4C22×C22⋊D6111yescharacteristic
357C8#1451976612yes—
358C8C2×#1175820114yes—
359C8C2×#1175822611yescharacteristic
360C8C2×#1175822611yescharacteristic
361C22—21no—
362C22—21no—
363C22—9764no—
364C22—4881no—
365C22—4882no—
366C22—4881no—
367C22—4882no—
368C22—4881no—
369C22—4882no—
370C22#1684011012yes—
371C22C22×#1175819911yescharacteristic
372C4—22no—
373C4—22no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—4881no—
378C4#1684011012yes—
379C4C2×#1451976611yescharacteristic
380C4C22×#1175820111yescharacteristic
381C2—22no—
382C2—9762no—
383C2—9761no—
384C2C2×#1684011011yescharacteristic
385C2#2145895811yescharacteristic
386C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3164C2
socleC22×C61C22×D8
prime radical#21456305C2
hypercenter#3164D61
hypocenterC61#33702
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #21448062C2semidirect
3. #21448058C2semidirect
4. #21448053C2semidirect
5. #21448044C2semidirect
6. #21447534C2semidirect
7. #21447529C2semidirect
8. #21447524C2semidirect
9. C2×#16839562C2semidirect
10. #21447416C2semidirect
11. C61×#3164C2semidirect
21. #16839565C22semidirect
22. #16839565C22semidirect
23. #16839565C22semidirect
28. #16839562C22semidirect
29. #16839562C22semidirect
30. #16839562C22semidirect
31. #16839538C22semidirect
32. C4×#11758185C22semidirect
35. C61×#500C22semidirect
36. C61×#499C22semidirect
188. #3164D61semidirect
213. #500C2×D61semidirect
214. #499C2×D61semidirect
234. C61#33702semidirect

Maximal subgroups

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

subgroupindex
#214563052
#214480622
#214480582
#214480532
#214480442
#214475342
#214475292
#214475242
C2×#168395622
#214474162
C61×#31642
#3370261