Finite Group Databasebeta

← #30648356

Subgroups — #30648356

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×#16861240C211yesmaximal, characteristic, complemented
2C22×#14523232C211yesmaximal, characteristic, complemented
3#22550387C211yesmaximal, characteristic
4#22550132C212yesmaximal, complemented
5#22550124C212yesmaximal, complemented
6#22550115C212yesmaximal, complemented
7#22550107C212yesmaximal, complemented
8#22549916C211yesmaximal, characteristic, complemented
9#22549885C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861240—24no—
14C2×#14523238C2212yescomplemented
15C2×#14523235C2212yescomplemented
16C2×#14523232—24no—
17C2×#14523232C2212yes—
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20C22×C127⋊C8C2212yes—
21#16861192—21no—
22#16861188C2211yescharacteristic
23#16861188C2211yescharacteristic
24#16861183C2211yescharacteristic
25#16861183C2211yescharacteristic
26#16861178—21no—
27C4×C22⋊D127—21no—
28C4×C22⋊D127—21no—
29#16861168C2211yescharacteristic
30#16861142C2212yescomplemented
31#16861140C2212yescomplemented
32#16861133C2212yescomplemented
33#16861133C2212yescomplemented
34#16861132C2212yescomplemented
35#16861129C2212yescomplemented
36C22×Q8×C127C2211yescharacteristic
37C23×C4×C127C2211yescharacteristic
38C22⋊Q8×C127—21no—
39C22⋊Q8×C127—21no—
40C22⋊Q8×C127C2212yes—
41C22⋊Q8×C127C2212yes—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127C2211yescharacteristic
44C2×C22⋊C4×C127—21no—
45#14523248—22no—
46#14523248—22no—
47C2×Dic254C2311yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C2×C22⋊D127—21no—
51C2×C22⋊D127—21no—
52C2×C4×D127—21no—
53#14523238—24no—
54#14523235—24no—
55#14523232—24no—
56#14523232—22no—
57#14523232D414yescomplemented
58C2×C127⋊C8—24no—
59C2×C127⋊C8C2312yes—
60C2×C127⋊C8C2312yes—
61C4⋊Dic127C2311yescharacteristic
62C4⋊Dic127C2311yescharacteristic
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66C22⋊Dic127—21no—
67C22⋊Dic127—21no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70C4×Dic127—21no—
71C24×C127D411yescharacteristic
72C2×Q8×C127—21no—
73C2×Q8×C127—22no—
74C2×Q8×C127—22no—
75C2×Q8×C127C2312yes—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127D411yescharacteristic
81C22×C4×C127D411yescharacteristic
82C22×C4×C127D411yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C4⋊C4×C127—22no—
87C4⋊C4×C127—22no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127C2312yes—
90C22⋊C4×C127—22no—
91C22⋊C4×C127—22no—
92C22×D127—21no—
93Dic254—21no—
94Dic254—21no—
95D508—21no—
96D508—21no—
97C22⋊D127—42no—
98C22⋊D127—42no—
99C22⋊D127—42no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C4×D127—42no—
104C127⋊C8—22no—
105C127⋊C8C2×D414yes—
106C23×C127—22no—
107C23×C127—22no—
108C23×C127—41no—
109C23×C127C2×D411yescharacteristic
110C23×C127C2×D411yescharacteristic
111C23×C127C2×D411yescharacteristic
112Q8×C127—22no—
113Q8×C127—22no—
114Q8×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—22no—
123C2×C4×C127—22no—
124C2×C4×C127—42no—
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#12112yes—
141C22×C127#13712yescomplemented
142C22×C127C22×D411yescharacteristic
143C2×D127—41no—
144C2×D127—41no—
145C2×D127—41no—
146C4×C127—21no—
147C4×C127—21no—
148C4×C127—42no—
149C4×C127—42no—
150C4×C127#12112yes—
151C4×C127C22×D411yescharacteristic
152C4×C127C2×#4111yescharacteristic
153Dic127—41no—
154Dic127—41no—
155Dic127—41no—
156C2×C127—22no—
157C2×C127—22no—
158C2×C127—41no—
159C2×C127C2×#12111yescharacteristic
160C2×C127C2×#13711yescharacteristic
161C2×C127#57511yescharacteristic
162D127—81no—
163#3056—1271nomaximal, Sylow
164C127#305611yescharacteristic, complemented, Sylow
165C2×#137—1271no—
166C22×M16—1271no—
167#525—1271no—
168C2×C22⋊Q8D12711yescharacteristic, complemented
169#470—1272no—
170#468—1272no—
171#467—1272no—
172#465—1272no—
173#428—1271no—
174#419—1271no—
175#417—1271no—
176C2×#41—1271no—
177C22×Q8C2×D12711yescharacteristic
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—2541no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—1271no—
191C22⋊Q8—1271no—
192C22⋊Q8C2×D12712yes—
193C22⋊Q8C2×D12712yes—
194#121—1271no—
195#121—1271no—
196#120—2541no—
197C4×D4—2541no—
198C4×D4—2541no—
199#117—1271no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×D12711yescharacteristic
202C2×C22⋊C4—21no—
203#107—1272no—
204#106—1272no—
205#103—1272no—
206#103—1272no—
207#103—1272no—
208#102—1272no—
209C24C22⋊D12711yescharacteristic
210#41—2542no—
211#41—2542no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—1271no—
216C2×Q8C22×D12712yes—
217C2×D4—1271no—
218C2×D4—2541no—
219C2×D4—2541no—
220C2×D4—2541no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
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—22no—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—1271no—
244C4⋊C4—1271no—
245C4⋊C4—2541no—
246C4⋊C4—2541no—
247C4⋊C4C22×D12712yes—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—2541no—
251C22⋊C4—2541no—
252C22⋊C4—2541no—
253C22⋊C4—2541no—
254C22⋊C4—2541no—
255C42—2541no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—2541no—
260C23C2×C22⋊D12711yescharacteristic
261C23C2×C22⋊D12711yescharacteristic
262C23C2×C22⋊D12711yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—42no—
266Q8—2541no—
267Q8—2541no—
268D4—5082no—
269D4—5082no—
270D4—5082no—
271D4—2541no—
272D4—2541no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—5082no—
285C2×C4—2541no—
286C2×C4—2541no—
287C2×C4—2541no—
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—5081no—
305C22—5081no—
306C22—5081no—
307C22#1686118112yes—
308C22#1686124012yescomplemented
309C22C22×C22⋊D12711yescharacteristic
310C4—21no—
311C4—21no—
312C4—42no—
313C4—42no—
314C4—5081no—
315C4—5081no—
316C4—5081no—
317C4#1686118112yes—
318C4C2×D4×D12711yescharacteristic
319C4C2×#1452324711yescharacteristic
320C2—22no—
321C2—22no—
322C2—41no—
323C2—10161no—
324C2C2×#1686118111yescharacteristic
325C2C2×#1686124011yescharacteristic
326C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC22×C127C22×D4
prime radical#22550387C2
hypercenterC2×C22⋊Q8D127
hypocenterC127#3056
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C22×#14523232C2semidirect
4. #22550132C2semidirect
5. #22550124C2semidirect
6. #22550115C2semidirect
7. #22550107C2semidirect
8. #22549916C2semidirect
9. #22549885C2semidirect
10. #22549875C2semidirect
11. C2×C22⋊Q8×C127C2semidirect
14. C2×#14523238C22semidirect
15. C2×#14523235C22semidirect
30. #16861142C22semidirect
31. #16861140C22semidirect
32. #16861133C22semidirect
33. #16861133C22semidirect
34. #16861132C22semidirect
35. #16861129C22semidirect
57. #14523232D4semidirect
141. C22×C127#137semidirect
164. C127#3056semidirect
168. C2×C22⋊Q8D127semidirect
308. C22#16861240semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C22×#145232322
#225503872
#225501322
#225501242
#225501152
#225501072
#225499162
#225498852
#225498752
C2×C22⋊Q8×C1272
#3056127