Finite Group Databasebeta

← #30642066

Subgroups — #30642066

Order 16256.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523249C211yesmaximal, characteristic
2C2×#16861194C212yesmaximal, complemented
3C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549795C211yesmaximal, characteristic, complemented
5#22549794C211yesmaximal, characteristic, complemented
6#22549793C211yesmaximal, characteristic, complemented
7#22549754C212yesmaximal, complemented
8#22549750C212yesmaximal, complemented
9#22549747C212yesmaximal, complemented
10#22549738C211yesmaximal, characteristic, complemented
11#395×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—28no—
13C2×#14523249—24no—
14C22×D508C2211yescharacteristic
15C22×D508C2212yes—
16C22×C4×D127C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C22×C4×D127C2212yes—
19C22×C4×D127C2211yescharacteristic
20#16861194—28no—
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2212yes—
24C2×Dic127⋊C4C2211yescharacteristic
25C4⋊C4×D127—24no—
26C2×(C2×D127)⋊C4C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2212yes—
29C2×(C2×D127)⋊C4C2212yes—
30C2×(C2×D127)⋊C4C2212yes—
31C127⋊#95C2211yescharacteristic
32C127⋊#95C2212yes—
33C127⋊#95C2212yes—
34C22×Q8×C127C2211yescharacteristic
35C2×C4⋊C4×C127C2211yescharacteristic
36C2×C4⋊C4×C127C2212yes—
37#95×C127C2211yescharacteristic
38#95×C127C2212yes—
39C23×D127C2312yes—
40C23×D127C2311yescharacteristic
41#14523249—232no—
42C2×D508—24no—
43C2×D508—24no—
44C2×D508—24no—
45C2×D508—22no—
46C2×D508D418yes—
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2312yes—
49C22×Dic127C2311yescharacteristic
50C22×Dic127C2311yescharacteristic
51C2×C4×D127—24no—
52C2×C4×D127—24no—
53C2×C4×D127—22no—
54C2×C4×D127—24no—
55C2×C4×D127—22no—
56C2×C4×D127—24no—
57C2×C4×D127—22no—
58C2×C4×D127—24no—
59C2×C4×D127—22no—
60C2×C4×D127D414yes—
61C4⋊Dic127—22no—
62Dic127⋊C4—22no—
63Dic127⋊C4—24no—
64Dic127⋊C4—22no—
65(C2×D127)⋊C4—24no—
66(C2×D127)⋊C4—24no—
67(C2×D127)⋊C4—24no—
68(C2×D127)⋊C4—24no—
69(C2×D127)⋊C4—24no—
70C2×Q8×C127—24no—
71C2×Q8×C127—22no—
72C22×C4×C127C2311yescharacteristic
73C22×C4×C127C2312yes—
74C22×C4×C127C2312yes—
75C22×C4×C127C2311yescharacteristic
76C22×C4×C127C2311yescharacteristic
77C4⋊C4×C127—24no—
78C4⋊C4×C127—22no—
79C22×D127—21no—
80C22×D127—21no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127—22no—
86C22×D127—22no—
87C22×D127—22no—
88C22×D127C2×D414yes—
89C22×D127C2×D414yes—
90C22×D127C2×D412yes—
91D508—28no—
92D508—216no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—21no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127—21no—
99C2×Dic127—22no—
100C2×Dic127—22no—
101C2×Dic127C2×D412yes—
102C2×Dic127#4112yescomplemented
103C4×D127—28no—
104C4×D127—216no—
105C4×D127—28no—
106C4×D127—44no—
107C23×C127C2411yescharacteristic
108Q8×C127—28no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127C2×D414yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127#4112yescomplemented
122C22×C127C22×D411yescharacteristic
123C22×C127C22×D412yes—
124C22×C127C2×#4111yescharacteristic
125C22×C127C2×#4111yescharacteristic
126C22×C1272−1+412yes—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—24no—
130C2×D127—24no—
131C2×D127—24no—
132C2×D127—24no—
133C2×D127—24no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—42no—
137C2×D127—42no—
138C2×D127#12014yes—
139C4×C127—24no—
140C4×C127—22no—
141C4×C127—22no—
142C4×C127—41no—
143C4×C127—42no—
144Dic127—22no—
145Dic127—22no—
146Dic127—41no—
147Dic127—42no—
148C2×C127C2×#12011yescharacteristic
149C2×C127#52511yescharacteristic
150C2×C127#53611yescharacteristic
151C2×C127#54712yes—
152C2×C127#54712yes—
153D127—24no—
154D127—44no—
155#2406—1271nomaximal, Sylow
156C127#240611yescharacteristic, complemented, Sylow
157C22×#41—1271no—
158C2×#123—1272no—
159C22×C4⋊C4—1271no—
160#395—1271no—
161#395D12711yescharacteristic, complemented
162#394—1271no—
163#394—1271no—
164#386—1272no—
165#385—1272no—
166#385—1272no—
167#381—1271no—
168C2×#41—2544no—
169C2×#41—2548no—
170C22×Q8C2×D12711yescharacteristic
171C22×D4—1272no—
172C22×D4—1271no—
173C23×C4—1271no—
174C23×C4—1271no—
175C23×C4—1271no—
176C23×C4—1272no—
177#123—2548no—
178C2×C4⋊C4—1271no—
179C2×C4⋊C4—1271no—
180C2×C4⋊C4—1271no—
181C2×C4⋊C4—1272no—
182C2×C4⋊C4—2544no—
183C2×C4⋊C4C2×D12711yescharacteristic
184C2×C4⋊C4C2×D12712yes—
185C2×C22⋊C4—1272no—
186C2×C22⋊C4—1272no—
187C2×C22⋊C4—1272no—
188C2×C22⋊C4—1272no—
189C2×C22⋊C4—1272no—
190#95—1272no—
191#95—1271no—
192#95—1272no—
193#95C2×D12711yescharacteristic
194#95C2×D12712yes—
195C24—1271no—
196C24—1272no—
197#41—25432no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—1278no—
201C2×D4—2544no—
202C2×D4—2544no—
203C2×D4—2544no—
204C2×D4—2542no—
205C22×C4—1271no—
206C22×C4—1271no—
207C22×C4—1274no—
208C22×C4—1272no—
209C22×C4—1271no—
210C22×C4—2542no—
211C22×C4—2544no—
212C22×C4—2542no—
213C22×C4—2544no—
214C22×C4—2542no—
215C22×C4—2544no—
216C22×C4—2544no—
217C22×C4—2544no—
218C22×C4—2542no—
219C22×C4C22×D12712yes—
220C22×C4C22×D12711yescharacteristic
221C22×C4C22×D12711yescharacteristic
222C22×C4C22×D12712yes—
223C22×C4C22×D12711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—2542no—
227C4⋊C4—2542no—
228C4⋊C4—2542no—
229C4⋊C4—2544no—
230C22⋊C4—2544no—
231C22⋊C4—2544no—
232C22⋊C4—2544no—
233C22⋊C4—2544no—
234C22⋊C4—2544no—
235C23—1274no—
236C23—1272no—
237C23—1274no—
238C23—2542no—
239C23—2542no—
240C23—2542no—
241C23—2542no—
242C23—2542no—
243C23—2542no—
244C23—2541no—
245C23—2541no—
246C23—2542no—
247C23C23×D12711yescharacteristic
248Q8—28no—
249D4—25416no—
250D4—2548no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—1272no—
262C2×C4—1272no—
263C2×C4—5084no—
264C2×C4—2548no—
265C2×C4—25416no—
266C2×C4—2548no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2541no—
270C2×C4—2542no—
271C2×C4—2542no—
272C2×C4—2541no—
273C2×C4—2542no—
274C2×C4—2541no—
275C2×C4D4×D12712yes—
276C2×C4D4×D12714yes—
277C2×C4#1452324912yescomplemented
278C22—1274no—
279C22—5082no—
280C22—5082no—
281C22—5082no—
282C22—5082no—
283C22—2544no—
284C22—2544no—
285C22—2544no—
286C22—2544no—
287C22—2544no—
288C22—2542no—
289C22—2542no—
290C22C2×D4×D12712yes—
291C22C2×D4×D12711yescharacteristic
292C22#41×D12711yescharacteristic
293C22C2×#1452324911yescharacteristic
294C22#1686125712yes—
295C4—22no—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—5082no—
301C4—5081no—
302C4—2542no—
303C4—2542no—
304C2—5084no—
305C2—2544no—
306C2#120×D12711yescharacteristic
307C2#2255038911yescharacteristic
308C2#2255045111yescharacteristic
309C2#2255052812yes—
310C2#2255052912yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#395×C127C2
prime radicalC22×#14523249C2
hypercenter#395D127
hypocenterC127#2406
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#16861194C2semidirect
3. C2×C4⋊C4×D127C2semidirect
4. #22549795C2semidirect
5. #22549794C2semidirect
6. #22549793C2semidirect
7. #22549754C2semidirect
8. #22549750C2semidirect
9. #22549747C2semidirect
10. #22549738C2semidirect
11. #395×C127C2semidirect
102. C2×Dic127#41semidirect
121. C2×C4×C127#41semidirect
156. C127#2406semidirect
161. #395D127semidirect
277. C2×C4#14523249semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168611942
C2×C4⋊C4×D1272
#225497952
#225497942
#225497932
#225497542
#225497502
#225497472
#225497382
#395×C1272
#2406127