Finite Group Databasebeta

← #30648362

Subgroups — #30648362

Order 16256.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523232C211yesmaximal, characteristic, complemented
3C2×#16861188C211yesmaximal, characteristic, complemented
4#22550132C212yesmaximal, complemented
5#22550120C212yesmaximal, complemented
6#22550115C212yesmaximal, complemented
7#22550103C212yesmaximal, complemented
8#22549919C211yesmaximal, characteristic, complemented
9#22549884C211yesmaximal, characteristic, complemented
10#22549877C211yesmaximal, characteristic, complemented
11#525×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13#16861239—24no—
14C2×#14523238C2212yescomplemented
15C2×#14523236C2212yescomplemented
16C2×#14523232—24no—
17C2×#14523232C2212yes—
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20C22×C127⋊C8C2212yes—
21#16861188—21no—
22#16861188—21no—
23#16861188C2212yes—
24#16861188C2212yes—
25C2×C4⋊Dic127C2211yescharacteristic
26C2×Dic127⋊C4—21no—
27C2×C22⋊Dic127—21no—
28#16861142C2212yescomplemented
29#16861140C2212yescomplemented
30#16861133C2212yescomplemented
31#16861132C2212yescomplemented
32#16861132C2212yescomplemented
33#16861130C2212yescomplemented
34C2×#41×C127C2211yescharacteristic
35C23×C4×C127C2211yescharacteristic
36#123×C127—21no—
37C22⋊Q8×C127C2211yescharacteristic
38C22⋊Q8×C127C2211yescharacteristic
39#121×C127C2211yescharacteristic
40#121×C127C2211yescharacteristic
41#120×C127—21no—
42C4×D4×C127—21no—
43C4×D4×C127—21no—
44#117×C127C2211yescharacteristic
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254C2312yes—
49C22×Dic127—21no—
50C22×Dic127—21no—
51#14523238—24no—
52#14523236—24no—
53#14523232—24no—
54#14523232—22no—
55#14523232D414yescomplemented
56C2×C127⋊C8—24no—
57C2×C127⋊C8C2312yes—
58C2×C127⋊C8C2312yes—
59C4⋊Dic127—21no—
60C4⋊Dic127C2312yes—
61Dic127⋊C4—22no—
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127—22no—
65C24×C127D411yescharacteristic
66#41×C127—22no—
67#41×C127—22no—
68C2×Q8×C127C2311yescharacteristic
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127C2311yescharacteristic
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—21no—
76C22×C4×C127D411yescharacteristic
77C22×C4×C127D411yescharacteristic
78C22×C4×C127D411yescharacteristic
79C22×C4×C127C2311yescharacteristic
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C4⋊C4×C127—21no—
83C4⋊C4×C127—21no—
84C4⋊C4×C127C2311yescharacteristic
85C4⋊C4×C127C2311yescharacteristic
86C22⋊C4×C127—21no—
87C22⋊C4×C127—21no—
88C22⋊C4×C127—21no—
89C22⋊C4×C127—21no—
90C22⋊C4×C127—21no—
91C42×C127—21no—
92Dic254—22no—
93Dic254—22no—
94Dic254—42no—
95C2×Dic127—22no—
96C2×Dic127—22no—
97C2×Dic127—42no—
98C2×Dic127—42no—
99C127⋊C8—22no—
100C127⋊C8C2×D414yes—
101C23×C127—22no—
102C23×C127—22no—
103C23×C127—21no—
104C23×C127—41no—
105C23×C127C2×D411yescharacteristic
106C23×C127C2×D411yescharacteristic
107C23×C127C2×D411yescharacteristic
108Q8×C127—21no—
109Q8×C127—21no—
110D4×C127—21no—
111D4×C127—21no—
112D4×C127—42no—
113D4×C127—42no—
114D4×C127—42no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—42no—
126C2×C4×C127C2×D412yes—
127C2×C4×C127C2×D411yescharacteristic
128C2×C4×C127C2×D411yescharacteristic
129C2×C4×C127C2×D411yescharacteristic
130C2×C4×C127#4112yes—
131C2×C4×C127C2411yescharacteristic
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—44no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127#12112yes—
144C22×C127#13712yescomplemented
145C22×C127C22×D411yescharacteristic
146C4×C127—21no—
147C4×C127—21no—
148C4×C127—41no—
149C4×C127—41no—
150C4×C127—41no—
151C4×C127#12112yes—
152C4×C127C22×D411yescharacteristic
153C4×C127C2×#4111yescharacteristic
154Dic127—42no—
155Dic127—42no—
156C2×C127—22no—
157C2×C127—22no—
158C2×C127—41no—
159C2×C127—81no—
160C2×C127C2×#12111yescharacteristic
161C2×C127C2×#13711yescharacteristic
162C2×C127#57511yescharacteristic
163#3056—1271nomaximal, Sylow
164C127#305611yescharacteristic, complemented, Sylow
165C2×#137—1271no—
166C22×M16—1271no—
167#525D12711yescharacteristic
168C2×C22⋊Q8—1271no—
169#470—1272no—
170#468—1272no—
171#467—1272no—
172#465—1272no—
173#428—1271no—
174#419—1271no—
175#417—1271no—
176C2×#41C2×D12711yescharacteristic
177C22×Q8—1271no—
178C23×C4C2×D12711yescharacteristic
179#137—2544no—
180C2×Q16—1272no—
181C2×SD16—1272no—
182C2×M16—1271no—
183C2×M16—1271no—
184C2×M16—1272no—
185C2×M16—2544no—
186C22×C8—1272no—
187#123—21no—
188C22⋊Q8—1272no—
189C22⋊Q8—1272no—
190C22⋊Q8—2541no—
191C22⋊Q8—2541no—
192C22⋊Q8C2×D12711yescharacteristic
193C22⋊Q8C2×D12711yescharacteristic
194#121C2×D12711yescharacteristic
195#121C2×D12711yescharacteristic
196#120—21no—
197C4×D4—21no—
198C4×D4—21no—
199#117C2×D12711yescharacteristic
200C2×C4⋊C4—1271no—
201C2×C4⋊C4—2541no—
202C2×C22⋊C4—2541no—
203#107—1272no—
204#106—1272no—
205#103—1272no—
206#103—1272no—
207#103—1272no—
208#102—1272no—
209C24C22⋊D12711yescharacteristic
210#41—22no—
211#41—22no—
212C2×Q8—1272no—
213C2×Q8—2542no—
214C2×Q8—2542no—
215C2×Q8—2541no—
216C2×Q8C22×D12711yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4C22×D12711yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—2541no—
225C22×C4—2541no—
226C22×C4C22⋊D12711yescharacteristic
227C22×C4C22⋊D12711yescharacteristic
228C22×C4C22⋊D12711yescharacteristic
229C22×C4C22×D12711yescharacteristic
230C22×C4C22×D12711yescharacteristic
231C22×C4C22×D12711yescharacteristic
232Q16—2544no—
233SD16—2544no—
234M16—1274no—
235M16—2544no—
236M16—2542no—
237C2×C8—1272no—
238C2×C8—1272no—
239C2×C8—2544no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—1272no—
243C4⋊C4—2542no—
244C4⋊C4—2542no—
245C4⋊C4—2541no—
246C4⋊C4C22×D12711yescharacteristic
247C4⋊C4C22×D12711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—2542no—
254C22⋊C4—2542no—
255C42—21no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—41no—
260C23C2×C22⋊D12711yescharacteristic
261C23C2×C22⋊D12711yescharacteristic
262C23C2×C22⋊D12711yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—5082no—
266Q8—2542no—
267Q8—2542no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—42no—
272D4—42no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—5082no—
285C2×C4—5082no—
286C2×C4—2542no—
287C2×C4—2542no—
288C2×C4D4×D12712yes—
289C2×C4C2×C22⋊D12711yescharacteristic
290C2×C4C2×C22⋊D12711yescharacteristic
291C2×C4C2×C22⋊D12711yescharacteristic
292C2×C4#1452324712yes—
293C2×C4C23×D12711yescharacteristic
294C8—1274no—
295C8—2542no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22#1686118112yes—
308C22#1686123912yescomplemented
309C22C22×C22⋊D12711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—5082no—
316C4—5082no—
317C4#1686118112yes—
318C4C2×D4×D12711yescharacteristic
319C4C2×#1452324711yescharacteristic
320C2—22no—
321C2—22no—
322C2—41no—
323C2—81no—
324C2C2×#1686118111yescharacteristic
325C2C2×#1686123911yescharacteristic
326C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3056
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523232C2semidirect
3. C2×#16861188C2semidirect
4. #22550132C2semidirect
5. #22550120C2semidirect
6. #22550115C2semidirect
7. #22550103C2semidirect
8. #22549919C2semidirect
9. #22549884C2semidirect
10. #22549877C2semidirect
14. C2×#14523238C22semidirect
15. C2×#14523236C22semidirect
28. #16861142C22semidirect
29. #16861140C22semidirect
30. #16861133C22semidirect
31. #16861132C22semidirect
32. #16861132C22semidirect
33. #16861130C22semidirect
55. #14523232D4semidirect
144. C22×C127#137semidirect
164. C127#3056semidirect
308. C22#16861239semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232322
C2×#168611882
#225501322
#225501202
#225501152
#225501032
#225499192
#225498842
#225498772
#525×C1272
#3056127