Finite Group Databasebeta

← #29833461

Subgroups — #29833461

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic
2C2×#16839885C211yesmaximal, characteristic, complemented
3#21447715C214yesmaximal, complemented
4#21447708C214yesmaximal, complemented
5#21447591C211yesmaximal, characteristic, complemented
6#21447568C211yesmaximal, characteristic, complemented
7#21447554C211yesmaximal, characteristic, complemented
8C2×#16839540C211yesmaximal, characteristic, complemented
9C61×#2943C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11C2×#14519783—21no—
12#16840203C2212yes—
13#16840202C412yescomplemented
14#16840198C2212yes—
15#16840197C412yescomplemented
16#16840192C414yescomplemented
17C2×#14519717C2211yescharacteristic
18C2×#14519717C2211yescharacteristic
19#16839885—22no—
20#16839885C2214yes—
21C2×#14519713C2211yescharacteristic
22C2×#14519705—21no—
23#16839584C2214yescomplemented
24#16839578C2212yescomplemented
25#16839575C2212yescomplemented
26#16839571C2214yes—
27#16839540C2214yescomplemented
28C2×#135×C61C2211yescharacteristic
29C4×#41×C61—21no—
30C61×#442C2212yescomplemented
31C61×#441C2212yescomplemented
32C61×#440—22no—
33C61×#427C2211yescharacteristic
34C61×#416—21no—
35C2×#97×C61C2211yescharacteristic
36C2×#11758222—21no—
37#14519799—21no—
38#14519799—21no—
39#14519799C2×C412yescomplemented
40#14519799C2312yes—
41C2×Dic244C2×C411yescharacteristic, complemented
42C2×#11758190—21no—
43C2×D488C2×C411yescharacteristic, complemented
44#14519783—24no—
45#14519783—42no—
46C2×#11758194—21no—
47C2×#11758189—41no—
48C2×C8×D61—41no—
49#14519723C2×C414yescomplemented
50#14519721C2×C414yescomplemented
51#14519720C2×C414yescomplemented
52#14519718C2×C414yescomplemented
53#14519717—21no—
54#14519717C2×C412yescomplemented
55#14519717C2312yes—
56#14519717C2312yes—
57C2×#11758185—21no—
58C2×#11758185C2×C412yescomplemented
59#14519713—21no—
60#14519713C2312yes—
61#14519705—24no—
62C2×#41×C61—21no—
63#135×C61—21no—
64#135×C61—21no—
65#135×C61C2×C412yes—
66#135×C61C2×C412yes—
67C2×Q16×C61C2311yescharacteristic
68C2×SD16×C61—21no—
69C2×D8×C61C2311yescharacteristic
70C22×C8×C61D411yescharacteristic
71C22×C8×C61C2311yescharacteristic
72C4×Q8×C61—22no—
73C4×D4×C61—22no—
74C4×D4×C61—42no—
75#117×C61—21no—
76#117×C61—41no—
77C2×C42×C61—41no—
78C2×C42×C61D411yescharacteristic
79#107×C61—21no—
80#106×C61C2312yes—
81#103×C61—22no—
82#102×C61—22no—
83#97×C61D412yes—
84#97×C61C2312yes—
85#41×C61—22no—
86#41×C61—22no—
87#41×C61—41no—
88#41×C61—41no—
89C2×Q8×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—41no—
97#11758222—22no—
98#11758222—22no—
99#11758222—41no—
100#11758222—41no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—21no—
104Q16×C61—21no—
105Q16×C61C22×C412yes—
106SD16×C61—22no—
107SD16×C61—22no—
108D8×C61—21no—
109D8×C61C22×C412yes—
110C2×C8×C61—21no—
111C2×C8×C61—21no—
112C2×C8×C61C22×C411yescharacteristic
113C2×C8×C61C22×C411yescharacteristic
114C2×C8×C61C2×D412yes—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C2×C8×C61#4111yescharacteristic
118C2×C8×C61C2411yescharacteristic
119C4⋊C4×C61—22no—
120C4⋊C4×C61—42no—
121C22⋊C4×C61—42no—
122C22⋊C4×C61—41no—
123C42×C61—21no—
124C42×C61—42no—
125C42×C61—41no—
126C42×C61C2×D412yes—
127#11758194—22no—
128#11758194—41no—
129C2×#92418—41no—
130Dic244—21no—
131Dic244C22×C412yes—
132D488—21no—
133D488C22×C412yes—
134#11758190—22no—
135#11758190—22no—
136#11758189—44no—
137C8×D61—44no—
138#11758185—22no—
139#11758185C22×C414yes—
140C23×C61—41no—
141C23×C61AGL(1,ℤ/8)11yescharacteristic
142C22×D61—41no—
143Q8×C61—22no—
144Q8×C61—41no—
145D4×C61—22no—
146D4×C61—42no—
147D4×C61—42no—
148D4×C61—41no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—81no—
155C2×C4×C61—84no—
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C22⋊D61—42no—
163C22⋊D61—42no—
164C2×Dic61—41no—
165D244—22no—
166D244—41no—
167C4×D61—42no—
168C4×D61—42no—
169Dic122—22no—
170Dic122—41no—
171C8×C61—22no—
172C8×C61C4×D411yescharacteristic
173C8×C61C4×D411yescharacteristic
174C8×C61C23×C411yescharacteristic
175C8×C61C2×#4111yescharacteristic
176#92418—42no—
177#33468—611nomaximal, Sylow
178C22×C61—41no—
179C22×C61—42no—
180C22×C61—82no—
181C22×C61#44211yescharacteristic
182C22×C61#44211yescharacteristic
183C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
184C2×D61—42no—
185C2×D61—82no—
186C4×C61—42no—
187C4×C61—42no—
188C4×C61—82no—
189C4×C61#44211yescharacteristic
190C4×C61#44211yescharacteristic
191C4×C61C2×C4×D411yescharacteristic
192C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
193Dic61—42no—
194#3414—611no—
195C2×#444—611no—
196#2943D6111yescharacteristic, complemented
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199#2155—611no—
200#2148—611no—
201#2145—611no—
202C2×#347—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—82no—
207C2×C61C2×#44211yescharacteristic
208D61—82no—
209C2×#135—611no—
210C2×#135C2×D6111yescharacteristic
211C2×#131—1221no—
212C4×#41—21no—
213#510—612no—
214#510—612no—
215#509—612no—
216#509—612no—
217#508—614no—
218C2×M32—611no—
219C2×M32—611no—
220#444—614no—
221#444—1222no—
222#442C2×D6112yescomplemented
223#441C2×D6112yescomplemented
224#440—22no—
225C2×#108—611no—
226#427C2×D6111yescharacteristic
227C2×#104—1221no—
228#416—21no—
229C2×#97C2×D6111yescharacteristic
230#365—614no—
231#362—612no—
232#361—612no—
233#360—614no—
234#347—614no—
235C61#3346811yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—1221no—
238#135—21no—
239#135—21no—
240#135—612no—
241#135—612no—
242#135—1221no—
243#135—1221no—
244#135C4×D6112yes—
245#135C4×D6112yes—
246C2×Q16—611no—
247C2×Q16C22×D6111yescharacteristic
248C2×SD16—21no—
249C2×SD16—1221no—
250C2×D8—611no—
251C2×D8C22×D6111yescharacteristic
252#131—1224no—
253#131—2442no—
254C2×M16—1221no—
255C2×M16—2441no—
256C22×C8—2441no—
257C22×C8C22⋊D6111yescharacteristic
258C22×C8C22×D6111yescharacteristic
259C4×Q8—22no—
260C4×D4—22no—
261C4×D4—42no—
262#117—21no—
263#117—41no—
264C2×C42—41no—
265C2×C42C22⋊D6111yescharacteristic
266Q32—614no—
267SD32—614no—
268SD32—614no—
269D16—614no—
270M32—612no—
271M32—612no—
272M32—612no—
273M32—1221no—
274C2×C16—612no—
275C2×C16—1221no—
276#108—612no—
277#108—1221no—
278#107—21no—
279#106C22×D6112yes—
280#104—1224no—
281#103—22no—
282#102—22no—
283#97C22⋊D6112yes—
284#97C22×D6112yes—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289#41—1222no—
290#41—1222no—
291#41—2441no—
292#41—2441no—
293C2×Q8—21no—
294C2×Q8—1221no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1221no—
298C2×D4—2441no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—2441no—
303C22×C4C2×C22⋊D6111yescharacteristic
304Q16—21no—
305Q16—612no—
306Q16—1221no—
307Q16C2×C4×D6112yes—
308SD16—22no—
309SD16—22no—
310SD16—1222no—
311SD16—1222no—
312D8—21no—
313D8—612no—
314D8—1221no—
315D8C2×C4×D6112yes—
316M16—1222no—
317M16—2444no—
318M16—2441no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—2444no—
322C2×C8—2441no—
323C2×C8C2×C4×D6111yescharacteristic
324C2×C8C2×C4×D6111yescharacteristic
325C2×C8#1175822211yescharacteristic
326C2×C8#1175822211yescharacteristic
327C2×C8C2×C22⋊D6112yes—
328C2×C8C2×C22⋊D6111yescharacteristic
329C2×C8C23×D6111yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—41no—
334C42—21no—
335C42—42no—
336C42—41no—
337C42C2×C22⋊D6112yes—
338C16—614no—
339C16—1222no—
340C23—41no—
341C23—2441no—
342C23#1451980311yescharacteristic
343Q8—22no—
344Q8—41no—
345Q8—1222no—
346Q8—2441no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—1222no—
352D4—2442no—
353D4—2442no—
354D4—2441no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4—2441no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4C4×C22⋊D6111yescharacteristic
366C2×C4C4×C22⋊D6111yescharacteristic
367C2×C4#1451980211yescharacteristic
368C2×C4#1451980211yescharacteristic
369C2×C4#1451980311yescharacteristic
370C2×C4C22×C22⋊D6111yescharacteristic
371C8—22no—
372C8—2442no—
373C8C4×C22⋊D6111yescharacteristic
374C8C4×C22⋊D6111yescharacteristic
375C8C22×C4×D6111yescharacteristic
376C8C2×#1175822211yescharacteristic
377C22—41no—
378C22—42no—
379C22—82no—
380C22—4882no—
381C22—2442no—
382C22#1683987211yescharacteristic
383C22#1683987211yescharacteristic
384C22C2×#1451980311yescharacteristic
385C4—42no—
386C4—42no—
387C4—82no—
388C4—2442no—
389C4#1683987211yescharacteristic
390C4#1683987211yescharacteristic
391C4C2×C4×C22⋊D6111yescharacteristic
392C4C2×#1451980211yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—4882no—
398C2C2×#1683987211yescharacteristic
subgroupG/H
centerC4#16839872
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2943C2
socleC2×C61C2×#442
prime radical#21458953C2
cosocleC2×C8×C61C24
hypercenter#2943D61
hypocenterC61#33468
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ #2943

Complements

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

NH ≅ G/Ntype
2. C2×#16839885C2semidirect
3. #21447715C2semidirect
4. #21447708C2semidirect
5. #21447591C2semidirect
6. #21447568C2semidirect
7. #21447554C2semidirect
8. C2×#16839540C2semidirect
9. C61×#2943C2semidirect
13. #16840202C4semidirect
15. #16840197C4semidirect
16. #16840192C4semidirect
23. #16839584C22semidirect
24. #16839578C22semidirect
25. #16839575C22semidirect
27. #16839540C22semidirect
30. C61×#442C22semidirect
31. C61×#441C22semidirect
39. #14519799C2×C4semidirect
41. C2×Dic244C2×C4semidirect
43. C2×D488C2×C4semidirect
49. #14519723C2×C4semidirect
50. #14519721C2×C4semidirect
51. #14519720C2×C4semidirect
52. #14519718C2×C4semidirect
54. #14519717C2×C4semidirect
58. C2×#11758185C2×C4semidirect
196. #2943D61semidirect
222. #442C2×D61semidirect
223. #441C2×D61semidirect
235. C61#33468semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168398852
#214477152
#214477082
#214475912
#214475682
#214475542
C2×#168395402
C61×#29432
#3346861