Finite Group Databasebeta

← #30648838

Subgroups — #30648838

Order 16256.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550405C211yesmaximal, characteristic, complemented
2C4×#14523249C211yesmaximal, characteristic, complemented
3#22550216C211yesmaximal, characteristic, complemented
4#22550199C211yesmaximal, characteristic, complemented
5#22550183C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550037C211yesmaximal, characteristic, complemented
9#22550020C211yesmaximal, characteristic
10#22549985C211yesmaximal, characteristic, complemented
11#22549983C211yesmaximal, characteristic, complemented
12#22549979C211yesmaximal, characteristic, complemented
13#22549977C211yesmaximal, characteristic, complemented
14#22549932C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19#16861202—21no—
20#16861201C2211yescharacteristic
21#16861191—21no—
22#16861185C2211yescharacteristic
23#16861180C2211yescharacteristic
24Q8×Dic127C2211yescharacteristic
25Q8×Dic127C2211yescharacteristic
26C4×D508—21no—
27C4×D508C2211yescharacteristic
28D4×Dic127C2211yescharacteristic
29#16861171—21no—
30#16861171C2211yescharacteristic
31(C22⋊D127)⋊C4—21no—
32#16861167—21no—
33#16861167C2211yescharacteristic
34#16861165—21no—
35C42×D127—21no—
36C42×D127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41C4⋊(C127⋊C8)C2211yescharacteristic
42#16861133C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50#16861119C2211yescharacteristic
51#16861119C2211yescharacteristic
52C2×Q16×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54#124×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—22no—
61C2×D508—21no—
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C2×C4×D127—21no—
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127D411yescharacteristic
69C2×C4×D127D411yescharacteristic
70C2×C4×D127C2311yescharacteristic
71#14523235—22no—
72#14523233—22no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C4⋊Dic127—21no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89C4×Dic127—21no—
90C4×Dic127—21no—
91C4×Dic127—21no—
92C4×Dic127D411yescharacteristic
93C4×Dic127D411yescharacteristic
94C4×Dic127C2311yescharacteristic
95C2×Q8×C127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98Q16×C127—22no—
99SD16×C127—22no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C42×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127C2×D411yescharacteristic
110D508—24no—
111D508—21no—
112D508#4112yes—
113C22⋊D127—42no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—42no—
118C2×Dic127C2×D411yescharacteristic
119C2×Dic127C2×D411yescharacteristic
120C2×Dic127C2×D411yescharacteristic
121C4×D127—24no—
122C4×D127—21no—
123C4×D127—21no—
124C4×D127—41no—
125C4×D127—41no—
126C4×D127—42no—
127C4×D127—42no—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C23×C127—21no—
131Q8×C127—21no—
132Q8×C127—21no—
133Q8×C127—21no—
134Q8×C127#4112yes—
135D4×C127—21no—
136D4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—41no—
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—
155C4×C127—22no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127C2×#4111yescharacteristic
161C4×C127C2×#4111yescharacteristic
162Dic127—21no—
163Dic127—21no—
164Dic127—21no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—81no—
169C2×C127#52511yescharacteristic
170C2×C127#57511yescharacteristic
171C2×C127#57511yescharacteristic
172D127—41no—
173D127—42no—
174#3111—1271nomaximal, Sylow
175C127#311111yescharacteristic, complemented, Sylow
176#529—1271no—
177C4×#41—1271no—
178#485—1271no—
179#482—1271no—
180#479—1271no—
181#478—1271no—
182#464D12711yescharacteristic, complemented
183#463—1271no—
184#452—1271no—
185#449—1271no—
186#442—1271no—
187#441—1271no—
188#440—1271no—
189#440—1271no—
190#432—1271no—
191C2×#41—1271no—
192C2×Q16C2×D12711yescharacteristic
193C2×SD16—1271no—
194C2×SD16C2×D12711yescharacteristic
195C2×D8—1271no—
196#126—2541no—
197#125—1271no—
198#124C2×D12711yescharacteristic
199#123—2541no—
200C22⋊Q8—1271no—
201#121—1271no—
202C4×Q8—1271no—
203C4×Q8—1271no—
204C4×Q8C2×D12711yescharacteristic
205C4×D4—1271no—
206C4×D4—1271no—
207C4×D4—1271no—
208C4×D4—2541no—
209C4×D4—2541no—
210C4×D4—2541no—
211#117—1271no—
212#117—2541no—
213#117—2541no—
214C2×C42—1271no—
215C2×C42—2541no—
216#107—1271no—
217#107—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8C2×D12711yescharacteristic
222#103—1271no—
223#103—1271no—
224#103—1271no—
225#103C2×D12711yescharacteristic
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102C2×D12711yescharacteristic
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232#97—1271no—
233#97—1271no—
234#41—2542no—
235#41—2542no—
236C2×Q8C22×D12711yescharacteristic
237C2×Q8C22×D12711yescharacteristic
238C2×D4—1271no—
239C2×D4—2541no—
240C2×D4—2541no—
241C2×D4C22×D12711yescharacteristic
242C22×C4—1271no—
243C22×C4—1271no—
244C22×C4—1271no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4—2541no—
248C22×C4—2541no—
249Q16—22no—
250SD16—22no—
251SD16—2542no—
252D8—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4C22×D12711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—2541no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C42—21no—
276C42—1271no—
277C42—1271no—
278C42—1271no—
279C42—2541no—
280C42—2541no—
281C42—2541no—
282C42C22×D12711yescharacteristic
283C23—21no—
284C23—1271no—
285C23—2541no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8#1452324712yes—
290D4—21no—
291D4—21no—
292D4—1272no—
293D4—5082no—
294D4—2544no—
295D4—2541no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—1271no—
300C2×C4—1271no—
301C2×C4—1271no—
302C2×C4—5082no—
303C2×C4—5081no—
304C2×C4—5081no—
305C2×C4—5082no—
306C2×C4—5082no—
307C2×C4—2544no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4D4×D12711yescharacteristic
314C2×C4D4×D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2541no—
319C8—2541no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—5081no—
324C22—5081no—
325C22—2542no—
326C22—2541no—
327C22—2541no—
328C22—2541no—
329C22C2×D4×D12711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—5081no—
336C4—5081no—
337C4—5081no—
338C4—2541no—
339C4—2541no—
340C4—2541no—
341C4#41×D12711yescharacteristic
342C4C2×#1452324711yescharacteristic
343C2—81no—
344C2—5082no—
345C2—5081no—
346C2#2255038611yescharacteristic
347C2#2255067311yescharacteristic
348C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
prime radical#22550020C2
hypercenter#464D127
hypocenterC127#3111
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #22550405C2semidirect
2. C4×#14523249C2semidirect
3. #22550216C2semidirect
4. #22550199C2semidirect
5. #22550183C2semidirect
6. #22550177C2semidirect
7. #22550088C2semidirect
8. #22550037C2semidirect
10. #22549985C2semidirect
11. #22549983C2semidirect
12. #22549979C2semidirect
13. #22549977C2semidirect
14. #22549932C2semidirect
15. #464×C127C2semidirect
175. C127#3111semidirect
182. #464D127semidirect

Maximal subgroups

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

subgroupindex
#225504052
C4×#145232492
#225502162
#225501992
#225501832
#225501772
#225500882
#225500372
#225500202
#225499852
#225499832
#225499792
#225499772
#225499322
#464×C1272
#3111127