Finite Group Databasebeta

← #30648978

Subgroups — #30648978

Order 16256.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550486C211yesmaximal, characteristic, complemented
2#22550350C211yesmaximal, characteristic
3#22550217C211yesmaximal, characteristic, complemented
4#22550179C211yesmaximal, characteristic, complemented
5#22550175C211yesmaximal, characteristic, complemented
6#22550080C211yesmaximal, characteristic, complemented
7#22550080C211yesmaximal, characteristic, complemented
8#22550029C211yesmaximal, characteristic
9#22550026C211yesmaximal, characteristic
10#22549979C211yesmaximal, characteristic, complemented
11#22549975C211yesmaximal, characteristic, complemented
12#22549963C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#485×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21#16861207C2211yescharacteristic, complemented
22#16861202—21no—
23#16861202—21no—
24#16861200—21no—
25#16861200C2211yescharacteristic, complemented
26#16861199—21no—
27#16861188C2211yescharacteristic
28Q8×Dic127C2211yescharacteristic, complemented
29#16861174C2211yescharacteristic, complemented
30C4×D508—21no—
31C4×D508—21no—
32#16861171C2211yescharacteristic
33#16861171C2211yescharacteristic
34#16861171C2211yescharacteristic
35#16861171C2211yescharacteristic
36#16861168C2211yescharacteristic
37C22⋊C4×D127—21no—
38C22⋊C4×D127—21no—
39#16861142C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22⋊Q8×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C23×D127—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508—22no—
62C2×D508—22no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70#14523237—22no—
71#14523237—22no—
72#14523235—22no—
73#14523235—22no—
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C4⋊Dic127—21no—
77C4⋊Dic127—21no—
78C4⋊Dic127D411yescharacteristic
79C4⋊Dic127D411yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127D411yescharacteristic
88C22⋊Dic127D411yescharacteristic
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96C4×Dic127C2311yescharacteristic
97C2×Q8×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—24no—
111C22×D127—81no—
112C22×D127—81no—
113Dic254—21no—
114Dic254—21no—
115D508—24no—
116D508—21no—
117D508—21no—
118D508#4112yes—
119D508#4112yes—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C4×D127—81no—
127C4×D127—81no—
128C4×D127—81no—
129C4×D127—81no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C23×C127C2×D411yescharacteristic
133Q8×C127—21no—
134Q8×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127C22×D411yescharacteristic
149C2×D127—22no—
150C2×D127—22no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C2×D127—84no—
156C4×C127—21no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127C2×#4111yescharacteristic
162C4×C127C2×#4111yescharacteristic
163Dic127—21no—
164Dic127—21no—
165Dic127—21no—
166Dic127—21no—
167Dic127—41no—
168Dic127—41no—
169C2×C127—41no—
170C2×C127#52511yescharacteristic
171C2×C127#57711yescharacteristic
172C2×C127#57711yescharacteristic
173D127—42no—
174D127—42no—
175#3123—1271nomaximal, Sylow
176C127#312311yescharacteristic, complemented, Sylow
177#541—1271no—
178#518—1271no—
179#485—1271no—
180#485D12711yescharacteristic, complemented
181#478—1271no—
182#478—1271no—
183#461—1271no—
184#461—1271no—
185#450—1271no—
186#450—1271no—
187#440—1271no—
188#440—1271no—
189C4×SD16—1271no—
190C4×SD16—1271no—
191#433—1271no—
192C22×D4—1271no—
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C4⋊Q8—1271no—
198#126—2541no—
199#126—2541no—
200#125—1271no—
201#125—2541no—
202#125—2541no—
203C22⋊Q8—1271no—
204C22⋊Q8C2×D12711yescharacteristic
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×D4—1271no—
208C4×D4—1271no—
209C4×D4—1271no—
210C4×D4—1271no—
211C4×D4—2541no—
212C4×D4—2541no—
213#117—1271no—
214#117C2×D12711yescharacteristic
215C2×C22⋊C4—2541no—
216C2×C22⋊C4—2541no—
217#107—1271no—
218#107C2×D12711yescharacteristic, complemented
219#106—1271no—
220#106C2×D12711yescharacteristic, complemented
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223#103—1271no—
224#103—1271no—
225#103C2×D12711yescharacteristic, complemented
226#103C2×D12711yescharacteristic, complemented
227#102—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231C22⋊C8—1271no—
232C22⋊C8C2×D12711yescharacteristic
233#97—1271no—
234C4×C8—1271no—
235C24—2541no—
236C2×Q8—1271no—
237C2×Q8C22×D12711yescharacteristic
238C2×D4—1271no—
239C2×D4—1271no—
240C2×D4—2542no—
241C2×D4—2542no—
242C2×D4—2541no—
243C22×C4—2541no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4C22×D12711yescharacteristic
248SD16—2542no—
249SD16—2542no—
250SD16—2542no—
251SD16—2542no—
252C2×C8—1271no—
253C2×C8—1271no—
254C2×C8C22×D12711yescharacteristic
255C2×C8C22×D12711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—2541no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4C22×D12711yescharacteristic
268C4⋊C4C22×D12711yescharacteristic
269C4⋊C4C22×D12711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—1271no—
273C22⋊C4—1271no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C42—21no—
280C42—1271no—
281C42—1271no—
282C42—2541no—
283C42—2541no—
284C23—10161no—
285C23—10161no—
286C23—2544no—
287C23—2541no—
288C23—2541no—
289C23D4×D12711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—2541no—
293Q8—2541no—
294D4—1272no—
295D4—1272no—
296D4—2544no—
297D4—2541no—
298D4—2541no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—1271no—
306C2×C4—1271no—
307C2×C4—1271no—
308C2×C4—1271no—
309C2×C4—10161no—
310C2×C4—10161no—
311C2×C4—10161no—
312C2×C4—10161no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4D4×D12711yescharacteristic
316C2×C4C23×D12711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2541no—
320C8—2541no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—10164no—
325C22—5081no—
326C22—5081no—
327C22—5081no—
328C22—5081no—
329C22—2542no—
330C22—2542no—
331C22C2×D4×D12711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—5081no—
338C4—5081no—
339C4—2541no—
340C4—2541no—
341C4—2541no—
342C4—2541no—
343C4#41×D12711yescharacteristic
344C4#41×D12711yescharacteristic
345C2—41no—
346C2—5082no—
347C2—5082no—
348C2#2255038411yescharacteristic
349C2#2255068311yescharacteristic
350C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#485×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#485D127
hypocenterC127#3123
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #22550486C2semidirect
3. #22550217C2semidirect
4. #22550179C2semidirect
5. #22550175C2semidirect
6. #22550080C2semidirect
7. #22550080C2semidirect
10. #22549979C2semidirect
11. #22549975C2semidirect
12. #22549963C2semidirect
13. #22549961C2semidirect
14. #22549934C2semidirect
15. #485×C127C2semidirect
21. #16861207C22semidirect
25. #16861200C22semidirect
28. Q8×Dic127C22semidirect
29. #16861174C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861132C22semidirect
44. #16861132C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
56. #103×C127C22semidirect
57. #103×C127C22semidirect
176. C127#3123semidirect
180. #485D127semidirect
218. #107C2×D127semidirect
220. #106C2×D127semidirect
225. #103C2×D127semidirect
226. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504862
#225503502
#225502172
#225501792
#225501752
#225500802
#225500292
#225500262
#225499792
#225499752
#225499632
#225499612
#225499342
#485×C1272
#3123127