Finite Group Databasebeta

← #30648980

Subgroups — #30648980

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#22550487C211yesmaximal, characteristic, complemented
2#22550348C211yesmaximal, characteristic
3#22550216C211yesmaximal, characteristic, complemented
4#22550215C211yesmaximal, characteristic, complemented
5#22550177C211yesmaximal, characteristic, complemented
6#22550176C211yesmaximal, characteristic, complemented
7#22550078C211yesmaximal, characteristic, complemented
8#22550025C211yesmaximal, characteristic
9#22550025C211yesmaximal, characteristic
10#22549979C211yesmaximal, characteristic, complemented
11#22549977C211yesmaximal, characteristic, complemented
12SD16×Dic127C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14#22549936C211yesmaximal, characteristic, complemented
15#461×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19#16861204—21no—
20#16861204—21no—
21#16861201C2211yescharacteristic, complemented
22#16861199—21no—
23#16861199—21no—
24#16861185C2211yescharacteristic
25#16861185C2211yescharacteristic
26Q8×Dic127C2211yescharacteristic, complemented
27Q8×Dic127C2211yescharacteristic, complemented
28C4×D508C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30#16861171C2211yescharacteristic
31C4×C22⋊D127—21no—
32(C22⋊D127)⋊C4—21no—
33#16861167C2211yescharacteristic
34#16861166C2211yescharacteristic
35C2×(C2×D127)⋊C4—21no—
36C22⋊C4×D127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41C4⋊(C127⋊C8)C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861133C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C2×SD16×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C4⋊Q8×C127C2211yescharacteristic, complemented
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#103×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×D508C2311yescharacteristic
61C2×C22⋊D127—21no—
62D4×D127—22no—
63D4×D127—22no—
64C22×Dic127—21no—
65C2×C4×D127—21no—
66C2×C4×D127—21no—
67C2×C4×D127C2311yescharacteristic
68#14523235—22no—
69#14523235—22no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C4⋊Dic127—21no—
73C4⋊Dic127—21no—
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82Dic127⋊C4D411yescharacteristic
83C22⋊Dic127—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4D411yescharacteristic
90(C2×D127)⋊C4D411yescharacteristic
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127C2311yescharacteristic
95C2×Q8×C127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99SD16×C127—22no—
100SD16×C127—22no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127—21no—
104C4⋊C4×C127—21no—
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—24no—
110C22×D127—81no—
111C22×D127—81no—
112C22×D127C2×D411yescharacteristic
113D508—21no—
114D508#4112yes—
115C22⋊D127—24no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—21no—
124C4×D127—21no—
125C4×D127—81no—
126C4×D127—81no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C23×C127—21no—
130Q8×C127—21no—
131Q8×C127—21no—
132Q8×C127—21no—
133Q8×C127—21no—
134D4×C127—21no—
135D4×C127#4112yes—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—81no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—22no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127C22×D411yescharacteristic
149C2×D127—21no—
150C2×D127—22no—
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—41no—
154C2×D127—41no—
155C2×D127—84no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127C2×#4111yescharacteristic
162C4×C127C2×#4111yescharacteristic
163Dic127—21no—
164Dic127—21no—
165Dic127—21no—
166Dic127—41no—
167Dic127—41no—
168Dic127—41no—
169C2×C127—42no—
170C2×C127#52511yescharacteristic
171C2×C127#57711yescharacteristic
172C2×C127#57711yescharacteristic
173D127—41no—
174D127—42no—
175#3123—1271nomaximal, Sylow
176C127#312311yescharacteristic, complemented, Sylow
177#541—1271no—
178#518—1271no—
179#485—1271no—
180#485—1271no—
181#478—1271no—
182#478—1271no—
183#461—1271no—
184#461D12711yescharacteristic, complemented
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×SD16C2×D12711yescharacteristic
196C2×SD16C2×D12711yescharacteristic
197C4⋊Q8C2×D12711yescharacteristic, complemented
198#126—2541no—
199#126—2541no—
200#125—1271no—
201#125—2541no—
202#125—2541no—
203C22⋊Q8—1271no—
204C22⋊Q8—1271no—
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×D4—1271no—
208C4×D4—1271no—
209C4×D4—1271no—
210C4×D4—2541no—
211C4×D4—2541no—
212C4×D4C2×D12711yescharacteristic
213#117—1271no—
214#117—1271no—
215C2×C22⋊C4—2541no—
216C2×C22⋊C4—2541no—
217#107—1271no—
218#107—1271no—
219#106—1271no—
220#106—1271no—
221C4⋊C8—1271no—
222C4⋊C8C2×D12711yescharacteristic, complemented
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⋊C8—1271no—
233#97—1271no—
234C4×C8—1271no—
235C24—2541no—
236C2×Q8C22×D12711yescharacteristic
237C2×Q8C22×D12711yescharacteristic
238C2×D4—1271no—
239C2×D4—2541no—
240C2×D4—2542no—
241C2×D4—2542no—
242C2×D4C22×D12711yescharacteristic
243C22×C4—21no—
244C22×C4—1271no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4—2541no—
248SD16—22no—
249SD16—22no—
250SD16—2542no—
251SD16—2542no—
252C2×C8—1271no—
253C2×C8—1271no—
254C2×C8C22×D12711yescharacteristic
255C2×C8C22×D12711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—1271no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4C22×D12711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1271no—
272C22⋊C4—1271no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C42—1271no—
280C42—2541no—
281C42—2541no—
282C42—2541no—
283C42C22×D12711yescharacteristic
284C23—21no—
285C23—1271no—
286C23—10161no—
287C23—10161no—
288C23—2544no—
289C23—2541no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294D4—21no—
295D4—1272no—
296D4—2544no—
297D4—2541no—
298D4#1452324712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—1271no—
304C2×C4—1271no—
305C2×C4—1271no—
306C2×C4—10161no—
307C2×C4—10161no—
308C2×C4—10161no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4D4×D12711yescharacteristic
315C2×C4D4×D12711yescharacteristic
316C2×C4C23×D12711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2541no—
320C8—2541no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—10164no—
325C22—5081no—
326C22—5081no—
327C22—2541no—
328C22—2542no—
329C22—2541no—
330C22—2541no—
331C22C2×D4×D12711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—5081no—
338C4—5081no—
339C4—5081no—
340C4—2541no—
341C4—2541no—
342C4—2541no—
343C4#41×D12711yescharacteristic
344C4C2×#1452324711yescharacteristic
345C2—42no—
346C2—5081no—
347C2—5082no—
348C2#2255038611yescharacteristic
349C2#2255068211yescharacteristic
350C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#461D127
hypocenterC127#3123
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22550487C2semidirect
3. #22550216C2semidirect
4. #22550215C2semidirect
5. #22550177C2semidirect
6. #22550176C2semidirect
7. #22550078C2semidirect
10. #22549979C2semidirect
11. #22549977C2semidirect
12. SD16×Dic127C2semidirect
13. #22549961C2semidirect
14. #22549936C2semidirect
15. #461×C127C2semidirect
21. #16861201C22semidirect
26. Q8×Dic127C22semidirect
27. Q8×Dic127C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. C4⋊(C127⋊C8)C22semidirect
42. #16861133C22semidirect
43. #16861133C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
54. C4⋊Q8×C127C22semidirect
56. C4⋊C8×C127C22semidirect
57. #103×C127C22semidirect
58. #103×C127C22semidirect
176. C127#3123semidirect
184. #461D127semidirect
197. C4⋊Q8C2×D127semidirect
222. C4⋊C8C2×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
#225504872
#225503482
#225502162
#225502152
#225501772
#225501762
#225500782
#225500252
#225499792
#225499772
SD16×Dic1272
#225499612
#225499362
#461×C1272
#3123127