Finite Group Databasebeta

← #30650623

Subgroups — #30650623

Order 16256.

386 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
1#22550568C211yesmaximal, characteristic, complemented
2#22550532C211yesmaximal, characteristic
3#22550229C211yesmaximal, characteristic, complemented
4#22550220C211yesmaximal, characteristic, complemented
5#22550187C211yesmaximal, characteristic, complemented
6#22550186C211yesmaximal, characteristic, complemented
7#22550116C211yesmaximal, characteristic, complemented
8#22550096C211yesmaximal, characteristic, complemented
9#22550035C211yesmaximal, characteristic, complemented
10#22550031C211yesmaximal, characteristic, complemented
11#22549968C211yesmaximal, characteristic, complemented
12C8×C22⋊D127C211yesmaximal, characteristic, complemented
13#22549869C211yesmaximal, characteristic, complemented
14C2×#16861134C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×#14523238C2211yescharacteristic
19C2×#14523238C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861204—21no—
22#16861200—21no—
23#16861200C2211yescharacteristic
24#16861196C2211yescharacteristic
25#16861195—21no—
26#16861193—21no—
27#16861189C2211yescharacteristic
28#16861186—21no—
29#16861185C2211yescharacteristic
30#16861181—21no—
31Q8×Dic127C2211yescharacteristic
32#16861174C2211yescharacteristic
33D4×Dic127—21no—
34C4×C22⋊D127C2211yescharacteristic
35#16861167C2211yescharacteristic
36#16861165—21no—
37C2×C4⋊Dic127C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861142C2211yescharacteristic
40Dic127⋊C8C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—41no—
61C2×Dic254—22no—
62C2×Dic254—21no—
63C2×Dic254—41no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127D411yescharacteristic
68C22×Dic127—21no—
69C22×Dic127—21no—
70C22×Dic127—21no—
71C2×C4×D127C2311yescharacteristic
72#14523238—22no—
73#14523238—41no—
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C8×D127—41no—
77C4⋊Dic127—21no—
78C4⋊Dic127—21no—
79C4⋊Dic127—21no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127D411yescharacteristic
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127—21no—
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102SD16×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C22⋊C4×C127—21no—
111C22×D127C2×D411yescharacteristic
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116Dic254#4112yes—
117C22⋊D127—22no—
118C22⋊D127—42no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—42no—
124C2×Dic127—42no—
125C2×Dic127—42no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127—21no—
134C23×C127C2×D411yescharacteristic
135Q8×C127—21no—
136Q8×C127—21no—
137D4×C127—21no—
138D4×C127—21no—
139D4×C127—42no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127C2×D411yescharacteristic
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#13512yes—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—21no—
159C4×C127—21no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127C2×#4111yescharacteristic
164C4×C127H(5,𝔽2)11yescharacteristic
165Dic127—21no—
166Dic127—22no—
167Dic127—21no—
168Dic127—21no—
169Dic127—41no—
170Dic127—41no—
171Dic127—41no—
172C2×C127—22no—
173C2×C127—81no—
174C2×C127#54611yescharacteristic
175C2×C127C2×#13511yescharacteristic
176C2×C127#57811yescharacteristic
177D127—41no—
178#3300—1271nomaximal, Sylow
179C127#330011yescharacteristic, complemented, Sylow
180#552—1271no—
181#548—1271no—
182#488—1271no—
183#485—1271no—
184#480—1271no—
185#479—1271no—
186#467—1271no—
187#465D12711yescharacteristic, complemented
188#464—1271no—
189#452—1271no—
190#451—1271no—
191C4×Q16—1271no—
192C8×D4—1271no—
193#416—1271no—
194C2×#103—1271no—
195C2×#41—1271no—
196C22×Q8—1271no—
197C2×Q16—1271no—
198C2×Q16—1271no—
199C2×SD16C2×D12711yescharacteristic
200C22×C8—1271no—
201C22×C8C2×D12711yescharacteristic
202#126—2541no—
203#125—1271no—
204#125—2541no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—1271no—
209C22⋊Q8—1271no—
210C22⋊Q8—2541no—
211C22⋊Q8C2×D12711yescharacteristic
212#121—2541no—
213#121C2×D12711yescharacteristic
214C4×Q8—1271no—
215C4×Q8—1271no—
216C4×D4—1271no—
217C4×D4—2541no—
218#117—1271no—
219#117—2541no—
220C2×C4⋊C4—1271no—
221#107—1271no—
222#107—1271no—
223#106C2×D12711yescharacteristic
224C4⋊C8—1271no—
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103—1271no—
230#103—1271no—
231#103C2×D12711yescharacteristic
232#102—1271no—
233#102—1271no—
234#102C2×D12711yescharacteristic
235C22⋊C8—1271no—
236C22⋊C8—1271no—
237C4×C8—1271no—
238#41—5081no—
239#41—2542no—
240C2×Q8—1271no—
241C2×Q8—1271no—
242C2×Q8—5081no—
243C2×Q8—2542no—
244C2×Q8—2541no—
245C2×Q8C22×D12711yescharacteristic
246C2×D4—21no—
247C2×D4—1271no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C22×C4—1271no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4C22×D12711yescharacteristic
255Q16—5081no—
256Q16—2542no—
257SD16—41no—
258C2×C8—41no—
259C2×C8—1271no—
260C2×C8—1271no—
261C2×C8—5081no—
262C2×C8C22×D12711yescharacteristic
263C2×C8C22×D12711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1271no—
280C22⋊C4—1271no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—1271no—
287C42—2541no—
288C42—2541no—
289C42—2541no—
290C23—21no—
291C23—1271no—
292C23D4×D12711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1272no—
296Q8—5082no—
297Q8—2541no—
298Q8—2541no—
299Q8—2541no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—5082no—
304D4—2542no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—5082no—
314C2×C4—5082no—
315C2×C4—5082no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4D4×D12711yescharacteristic
323C2×C4C23×D12711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2541no—
327C8—2541no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—2541no—
334C22—2541no—
335C22—2541no—
336C22#1686123012yes—
337C22C2×D4×D12711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—5081no—
343C4—5081no—
344C4—5081no—
345C4—2541no—
346C4—2541no—
347C4—2541no—
348C4—2542no—
349C4#41×D12711yescharacteristic
350C4#1686125411yescharacteristic
351C2—22no—
352C2—81no—
353C2—5081no—
354C2#2255051511yescharacteristic
355C2C2×#1686123011yescharacteristic
356C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
prime radical#22550532C2
hypercenter#465D127
hypocenterC127#3300
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22550568C2semidirect
3. #22550229C2semidirect
4. #22550220C2semidirect
5. #22550187C2semidirect
6. #22550186C2semidirect
7. #22550116C2semidirect
8. #22550096C2semidirect
9. #22550035C2semidirect
10. #22550031C2semidirect
11. #22549968C2semidirect
12. C8×C22⋊D127C2semidirect
13. #22549869C2semidirect
14. C2×#16861134C2semidirect
15. #465×C127C2semidirect
179. C127#3300semidirect
187. #465D127semidirect

Maximal subgroups

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

subgroupindex
#225505682
#225505322
#225502292
#225502202
#225501872
#225501862
#225501162
#225500962
#225500352
#225500312
#225499682
C8×C22⋊D1272
#225498692
C2×#168611342
#465×C1272
#3300127