Finite Group Databasebeta

← #30648677

Subgroups — #30648677

Order 16256.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D127C211yesmaximal, characteristic
2#117×D127C211yesmaximal, characteristic
3#22550218C211yesmaximal, characteristic, complemented
4#22550217C211yesmaximal, characteristic, complemented
5#22550214C211yesmaximal, characteristic, complemented
6#22550213C211yesmaximal, characteristic, complemented
7#22550210C211yesmaximal, characteristic, complemented
8#22550208C211yesmaximal, characteristic, complemented
9#22550205C211yesmaximal, characteristic, complemented
10#22549917C211yesmaximal, characteristic, complemented
11#22549915C211yesmaximal, characteristic, complemented
12#22549883C211yesmaximal, characteristic, complemented
13#22549883C211yesmaximal, characteristic, complemented
14#22549838C211yesmaximal, characteristic
15#485×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20#16861189C2211yescharacteristic
21#16861188C2211yescharacteristic
22#16861187—21no—
23#16861186—21no—
24#16861185C2211yescharacteristic
25#16861168C2211yescharacteristic
26#16861167C2211yescharacteristic
27#16861167C2211yescharacteristic
28#16861166—21no—
29#16861165—21no—
30C4⋊C4×D127—21no—
31C4⋊C4×D127C2211yescharacteristic
32C4⋊C4×D127C2211yescharacteristic
33C4⋊C4×D127C2211yescharacteristic
34C22⋊C4×D127—21no—
35C22⋊C4×D127—21no—
36C42×D127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861133C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
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×D127D411yescharacteristic
60C2×Dic254—21no—
61C2×Dic254C2311yescharacteristic
62Q8×D127—22no—
63Q8×D127—22no—
64C22×Dic127D411yescharacteristic
65C2×C4×D127—22no—
66C2×C4×D127—22no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127D411yescharacteristic
72C2×C4×D127D411yescharacteristic
73C2×C4×D127C2311yescharacteristic
74C2×C4×D127C2311yescharacteristic
75#14523231—22no—
76#14523231—22no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
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—22no—
109C22×D127—22no—
110C22×D127—41no—
111C22×D127C2×D411yescharacteristic
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116C2×Dic127—22no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—22no—
124C4×D127—21no—
125C4×D127—21no—
126C4×D127—22no—
127C4×D127—42no—
128C4×D127—42no—
129C4×D127—42no—
130C4×D127—42no—
131C4×D127#4112yes—
132C4×D127#4112yes—
133C127⋊C8—21no—
134C127⋊C8—21no—
135C23×C127C2×D411yescharacteristic
136Q8×C127—21no—
137Q8×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127C22×D411yescharacteristic
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—44no—
157C2×D127#12312yes—
158C2×D127#13712yes—
159C4×C127—21no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127C2×#4111yescharacteristic
165C4×C127C2×#4111yescharacteristic
166Dic127—21no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171Dic127#12312yes—
172C2×C127—41no—
173C2×C127C2×#12311yescharacteristic
174C2×C127C2×#13711yescharacteristic
175C2×C127#57511yescharacteristic
176D127—22no—
177D127—22no—
178#3095—1271nomaximal, Sylow
179C127#309511yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—1271no—
181C2×#117—1271no—
182#485—1271no—
183#485—1271no—
184#485—1271no—
185#485D12711yescharacteristic, complemented
186#484—1271no—
187#484—1271no—
188#483—1271no—
189#483—1271no—
190#428—1271no—
191#428—1271no—
192#419—1271no—
193#419—1271no—
194#407—1271no—
195C22×Q8—1271no—
196C23×C4—1271no—
197C2×M16—1271no—
198C2×M16—1271no—
199C22⋊Q8—1271no—
200C22⋊Q8—1271no—
201C22⋊Q8—1271no—
202C22⋊Q8—2541no—
203C22⋊Q8—2541no—
204C22⋊Q8C2×D12711yescharacteristic
205#117—1271no—
206#117—1271no—
207#117—1271no—
208#117—2541no—
209#117—2541no—
210#117C2×D12711yescharacteristic
211C2×C4⋊C4—1271no—
212C2×C4⋊C4—1271no—
213C2×C4⋊C4—1271no—
214C2×C4⋊C4—2541no—
215C2×C22⋊C4—2541no—
216C2×C22⋊C4—2541no—
217C2×C42—2541no—
218#107—1271no—
219#107—1271no—
220#107—1271no—
221#107C2×D12711yescharacteristic, complemented
222#106—1271no—
223#106—1271no—
224#106—1271no—
225#106C2×D12711yescharacteristic, complemented
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103—1271no—
230#103—1271no—
231#103—1271no—
232#103C2×D12711yescharacteristic, complemented
233#103C2×D12711yescharacteristic, complemented
234C22⋊C8—1271no—
235C22⋊C8—1271no—
236C22⋊C8—1271no—
237C22⋊C8C2×D12711yescharacteristic
238C24—1271no—
239C2×Q8—1271no—
240C2×Q8—2541no—
241C2×Q8—2542no—
242C2×Q8—2542no—
243C2×Q8C22×D12711yescharacteristic
244C22×C4—1271no—
245C22×C4—1271no—
246C22×C4—1271no—
247C22×C4—1271no—
248C22×C4—1271no—
249C22×C4—2542no—
250C22×C4—2542no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4C22×D12711yescharacteristic
256M16—2542no—
257M16—2542no—
258C2×C8—1271no—
259C2×C8—1271no—
260C2×C8C22×D12711yescharacteristic
261C2×C8C22×D12711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—1271no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4C22×D12711yescharacteristic
273C4⋊C4C22×D12711yescharacteristic
274C4⋊C4C22×D12711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C42—21no—
284C42—2541no—
285C42—2541no—
286C42—2541no—
287C23—1271no—
288C23—1271no—
289C23—5081no—
290C23—2542no—
291C23—2542no—
292C23D4×D12711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—5082no—
296Q8—2541no—
297Q8—2541no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—1271no—
305C2×C4—1272no—
306C2×C4—1272no—
307C2×C4—1271no—
308C2×C4—5082no—
309C2×C4—5082no—
310C2×C4—5082no—
311C2×C4—5082no—
312C2×C4—2542no—
313C2×C4—2542no—
314C2×C4—2542no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4D4×D12711yescharacteristic
322C2×C4C23×D12711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2541no—
326C8—2541no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—1272no—
331C22—1272no—
332C22—5084no—
333C22—2541no—
334C22—2541no—
335C22—2541no—
336C22—2541no—
337C22C2×D4×D12711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—1272no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4—2541no—
349C4#41×D12711yescharacteristic
350C4#41×D12711yescharacteristic
351C2—41no—
352C2—2542no—
353C2—2542no—
354C2#123×D12711yescharacteristic
355C2#137×D12711yescharacteristic
356C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#485×C127C2
socleC22×C127C22×D4
prime radicalC23×D127D4
hypercenter#485D127
hypocenterC127#3095
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
3. #22550218C2semidirect
4. #22550217C2semidirect
5. #22550214C2semidirect
6. #22550213C2semidirect
7. #22550210C2semidirect
8. #22550208C2semidirect
9. #22550205C2semidirect
10. #22549917C2semidirect
11. #22549915C2semidirect
12. #22549883C2semidirect
13. #22549883C2semidirect
15. #485×C127C2semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861142C22semidirect
40. #16861141C22semidirect
41. #16861140C22semidirect
42. #16861140C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
45. #16861133C22semidirect
46. #16861133C22semidirect
47. #16861132C22semidirect
48. #16861132C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
56. #103×C127C22semidirect
57. #103×C127C22semidirect
179. C127#3095semidirect
185. #485D127semidirect
221. #107C2×D127semidirect
225. #106C2×D127semidirect
232. #103C2×D127semidirect
233. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1272
#117×D1272
#225502182
#225502172
#225502142
#225502132
#225502102
#225502082
#225502052
#225499172
#225499152
#225498832
#225498382
#485×C1272
#3095127