Finite Group Databasebeta

← #30648641

Subgroups — #30648641

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550383C211yesmaximal, characteristic
2#117×D127C211yesmaximal, characteristic, complemented
3#22550216C211yesmaximal, characteristic, complemented
4#22550215C211yesmaximal, characteristic, complemented
5#22550210C211yesmaximal, characteristic, complemented
6#22550205C211yesmaximal, characteristic, complemented
7#22550199C211yesmaximal, characteristic, complemented
8#22550198C211yesmaximal, characteristic, complemented
9#22550193C211yesmaximal, characteristic, complemented
10#22550188C211yesmaximal, characteristic, complemented
11#22549914C211yesmaximal, characteristic, complemented
12#22549912C211yesmaximal, characteristic, complemented
13#22549872C211yesmaximal, characteristic, complemented
14C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
15#416×C127C211yesmaximal, characteristic, complemented
16C22×C4×D127C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18#16861190—21no—
19#16861185C2211yescharacteristic
20#16861185C2211yescharacteristic
21#16861180C2211yescharacteristic
22#16861180C2211yescharacteristic
23#16861177—21no—
24C4×C22⋊D127—21no—
25C4×C22⋊D127—21no—
26#16861168C2211yescharacteristic
27#16861168C2211yescharacteristic
28#16861167C2211yescharacteristic
29#16861167C2211yescharacteristic
30#16861166—21no—
31#16861165—21no—
32C4⋊C4×D127C2211yescharacteristic
33C4⋊C4×D127C2211yescharacteristic
34C22⋊C4×D127—21no—
35C42×D127—21no—
36#16861143C2211yescharacteristic, complemented
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41#16861141C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43#16861140C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861130C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic, complemented
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54#117×C127C2211yescharacteristic
55#103×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59C23×D127D411yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C22×Dic127D411yescharacteristic
63C2×C4×D127—21no—
64C2×C4×D127—21no—
65C2×C4×D127—22no—
66C2×C4×D127—22no—
67C2×C4×D127D411yescharacteristic
68C2×C4×D127D411yescharacteristic
69C2×C4×D127C2311yescharacteristic
70C2×C4×D127C2311yescharacteristic
71C2×C127⋊C8—22no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90C4×Dic127—21no—
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127—21no—
94#41×C127—22no—
95#41×C127—22no—
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C22×C4×C127C2311yescharacteristic
101C2×C8×C127—22no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C22×D127—22no—
109C22×D127—22no—
110C22×D127—41no—
111C22×D127C2×D411yescharacteristic
112C22×D127C2×D411yescharacteristic
113C22⋊D127—42no—
114C22⋊D127—42no—
115C2×Dic127—22no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C4×D127—22no—
123C4×D127—22no—
124C4×D127—21no—
125C4×D127—21no—
126C4×D127—21no—
127C4×D127—21no—
128C4×D127—42no—
129C4×D127—42no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C23×C127—21no—
133C23×C127C2×D411yescharacteristic
134Q8×C127—21no—
135Q8×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—42no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—42no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127#4112yes—
145C2×C4×C127#4112yes—
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127#12312yes—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—44no—
162C2×D127#13512yes—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127#12312yes—
167C4×C127C2×#4111yescharacteristic
168C4×C127C2×#4111yescharacteristic
169Dic127—21no—
170Dic127—21no—
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174Dic127—41no—
175C2×C127—22no—
176C2×C127—81no—
177C2×C127C2×#12311yescharacteristic
178C2×C127C2×#13511yescharacteristic
179C2×C127#57511yescharacteristic
180D127—22no—
181D127—41no—
182#3091—1271nomaximal, Sylow
183C127#309111yescharacteristic, complemented, Sylow
184#525—1271no—
185C2×#117—1271no—
186#485—1271no—
187#485—1271no—
188#484—1271no—
189#483—1271no—
190#482—1271no—
191#482—1271no—
192#481—1271no—
193#480—1271no—
194#427—1271no—
195#427—1271no—
196#416—1271no—
197#416D12711yescharacteristic, complemented
198C2×C22⋊C8—1271no—
199C2×#41C2×D12711yescharacteristic
200C23×C4—1271no—
201C22×C8—1271no—
202C22×C8C2×D12711yescharacteristic
203#123—2541no—
204C22⋊Q8—1271no—
205C22⋊Q8—1271no—
206#121—1271no—
207#121—1271no—
208#120—2541no—
209C4×D4—2541no—
210C4×D4—2541no—
211#117—1271no—
212#117—1271no—
213#117—1271no—
214#117—1271no—
215#117—2541no—
216#117—2541no—
217#117C2×D12711yescharacteristic
218C2×C4⋊C4—1271no—
219C2×C4⋊C4—1271no—
220C2×C22⋊C4—2541no—
221C2×C42—2541no—
222#107—1271no—
223#107—1271no—
224#107—1271no—
225#107—1271no—
226#106—1271no—
227#106—1271no—
228#106—1271no—
229#106—1271no—
230#103—1271no—
231#103—1271no—
232#103C2×D12711yescharacteristic, complemented
233#103C2×D12711yescharacteristic, complemented
234#102—1271no—
235#102—1271no—
236#102C2×D12711yescharacteristic, complemented
237#102C2×D12711yescharacteristic, complemented
238C22⋊C8—1271no—
239C22⋊C8—1271no—
240C22⋊C8—1271no—
241C22⋊C8—1271no—
242C24—1271no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D12711yescharacteristic
246C2×D4—21no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C22×C4—21no—
251C22×C4—1271no—
252C22×C4—1271no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—1271no—
256C22×C4—2541no—
257C22×C4—2541no—
258C22×C4—2542no—
259C22×C4—2542no—
260C22×C4C22×D12711yescharacteristic
261C2×C8—22no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8—2542no—
265C2×C8C22×D12711yescharacteristic
266C2×C8C22×D12711yescharacteristic
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—21no—
287C42—2541no—
288C42—2541no—
289C42—2541no—
290C42—2541no—
291C23—21no—
292C23—1271no—
293C23—1271no—
294C23—5081no—
295C23—2542no—
296C23—2542no—
297C23D4×D12711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—5082no—
304D4—5082no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—5082no—
312C2×C4—5082no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2542no—
318C2×C4—2542no—
319C2×C4—2542no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4D4×D12711yescharacteristic
325C2×C4#1452324712yes—
326C2×C4#1452324812yes—
327C2×C4C23×D12711yescharacteristic
328C8—21no—
329C8—21no—
330C8—2541no—
331C8—2541no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—1272no—
338C22—5084no—
339C22—2541no—
340C22—2541no—
341C22—2541no—
342C22—2541no—
343C22—2541no—
344C22#1686119112yes—
345C22C2×D4×D12711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—5081no—
350C4—5081no—
351C4—5081no—
352C4—5081no—
353C4—2541no—
354C4—2541no—
355C4#1686119112yes—
356C4C2×#1452324711yescharacteristic
357C4C2×#1452324811yescharacteristic
358C2—22no—
359C2—81no—
360C2—5081no—
361C2—2542no—
362C2C2×#1686119111yescharacteristic
363C2#135×D12711yescharacteristic
364C2#2255067311yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
prime radical#22550383C2
hypercenter#416D127
hypocenterC127#3091
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D127C2semidirect
3. #22550216C2semidirect
4. #22550215C2semidirect
5. #22550210C2semidirect
6. #22550205C2semidirect
7. #22550199C2semidirect
8. #22550198C2semidirect
9. #22550193C2semidirect
10. #22550188C2semidirect
11. #22549914C2semidirect
12. #22549912C2semidirect
13. #22549872C2semidirect
14. C2×(C2×D127)⋊C8C2semidirect
15. #416×C127C2semidirect
36. #16861143C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861142C22semidirect
40. #16861141C22semidirect
41. #16861141C22semidirect
42. #16861140C22semidirect
43. #16861140C22semidirect
44. #16861133C22semidirect
45. #16861133C22semidirect
46. #16861130C22semidirect
47. #16861130C22semidirect
55. #103×C127C22semidirect
56. #103×C127C22semidirect
57. #102×C127C22semidirect
58. #102×C127C22semidirect
183. C127#3091semidirect
197. #416D127semidirect
232. #103C2×D127semidirect
233. #103C2×D127semidirect
236. #102C2×D127semidirect
237. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
#117×D1272
#225502162
#225502152
#225502102
#225502052
#225501992
#225501982
#225501932
#225501882
#225499142
#225499122
#225498722
C2×(C2×D127)⋊C82
#416×C1272
#3091127