Finite Group Databasebeta

← #30641942

Subgroups — #30641942

Order 16256.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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×#14523247C211yesmaximal, characteristic, complemented
2C2×D4×Dic127C212yesmaximal, complemented
3C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
4#22549797C211yesmaximal, characteristic, complemented
5#22549792C212yesmaximal, complemented
6#22549790C211yesmaximal, characteristic, complemented
7#22549788C211yesmaximal, characteristic, complemented
8#22549784C212yesmaximal, complemented
9C22⋊C4×Dic127C212yesmaximal, complemented
10#22549723C211yesmaximal, characteristic, complemented
11#392×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—28no—
13C2×#14523247—24no—
14C22×Dic254C2211yescharacteristic
15C22×C22⋊D127C2211yescharacteristic
16C22×C22⋊D127C2212yes—
17C23×Dic127C2211yescharacteristic
18C23×Dic127C2212yes—
19C22×C4×D127C2211yescharacteristic
20D4×Dic127—28no—
21(C22⋊D127)⋊C4—24no—
22C2×C4⋊Dic127C2212yes—
23C2×Dic127⋊C4C2211yescharacteristic
24C2×C22⋊Dic127C2212yes—
25C2×C22⋊Dic127C2212yes—
26C2×C22⋊Dic127C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C2×C4×Dic127C2212yes—
31C127⋊#95C2212yes—
32C127⋊#95C2211yescharacteristic
33C127⋊#95C2212yes—
34C22×D4×C127C2212yes—
35C22×D4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2211yescharacteristic
37C2×C22⋊C4×C127C2212yes—
38#95×C127C2211yescharacteristic
39C23×D127C2311yescharacteristic
40#14523247—232no—
41C2×Dic254—24no—
42C2×Dic254—22no—
43C2×C22⋊D127—24no—
44C2×C22⋊D127—22no—
45C2×C22⋊D127—24no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—24no—
48C2×C22⋊D127—22no—
49C22×Dic127—24no—
50C22×Dic127—24no—
51C22×Dic127—22no—
52C22×Dic127—24no—
53C22×Dic127D418yescomplemented
54C22×Dic127C2311yescharacteristic
55C22×Dic127C2312yes—
56C22×Dic127C2312yes—
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C2×C4×D127—24no—
60C2×C4×D127D414yescomplemented
61C4⋊Dic127—24no—
62Dic127⋊C4—22no—
63C22⋊Dic127—24no—
64C22⋊Dic127—24no—
65C22⋊Dic127—24no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—24no—
68C4×Dic127—22no—
69C4×Dic127—24no—
70C24×C127C2312yes—
71C24×C127C2311yescharacteristic
72C2×D4×C127—24no—
73C2×D4×C127—24no—
74C2×D4×C127—24no—
75C2×D4×C127—22no—
76C2×D4×C127—24no—
77C22×C4×C127C2311yescharacteristic
78C22×C4×C127C2311yescharacteristic
79C22×C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22⋊C4×C127—24no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—21no—
86C22×D127C2×D412yes—
87Dic254—28no—
88C22⋊D127—28no—
89C22⋊D127—216no—
90C2×Dic127—21no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—216no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127—22no—
100C2×Dic127—44no—
101C2×Dic127C2×D414yes—
102C2×Dic127C2×D414yes—
103C2×Dic127C2×D412yes—
104C2×Dic127#4112yescomplemented
105C4×D127—28no—
106C23×C127—21no—
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—22no—
110C23×C127—22no—
111C23×C127—22no—
112C23×C127—22no—
113C23×C127—22no—
114C23×C127—22no—
115C23×C127—21no—
116C23×C127C2×D414yes—
117C23×C127#4112yescomplemented
118C23×C127C2411yescharacteristic
119D4×C127—216no—
120D4×C127—28no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—21no—
125C2×C4×C127—22no—
126C2×C4×C127C2×D412yes—
127C2×C4×C127#4114yescomplemented
128C22×C127—24no—
129C22×C127—24no—
130C22×C127—24no—
131C22×C127—22no—
132C22×C127—24no—
133C22×C127—41no—
134C22×C127—42no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—41no—
139C22×C127—42no—
140C22×C127C22×D412yes—
141C22×C127C22×D411yescharacteristic
142C22×C127C2×#4111yescharacteristic
143C22×C127C2×#4111yescharacteristic
144C22×C127C2×#4112yes—
145C2×D127—22no—
146C2×D127—22no—
147C2×D127—22no—
148C2×D127—42no—
149C2×D127—42no—
150C4×C127—22no—
151C4×C127—24no—
152C4×C127—41no—
153Dic127—24no—
154Dic127—22no—
155Dic127—41no—
156Dic127—42no—
157Dic127#12014yes—
158C2×C127—44no—
159C2×C127—42no—
160C2×C127C2×#12011yescharacteristic
161C2×C127#52512yes—
162C2×C127#52511yescharacteristic
163C2×C127#53212yes—
164C2×C127#53211yescharacteristic
165D127—42no—
166#2388—1271nomaximal, Sylow
167C127#238811yescharacteristic, complemented, Sylow
168C22×#41—1271no—
169C2×C4×D4—1272no—
170C2×C4×D4—1271no—
171#395—1271no—
172#394—1271no—
173#394—1272no—
174#393—1271no—
175#392—1272no—
176#392D12711yescharacteristic, complemented
177C4×C22⋊C4—1272no—
178C4×C22⋊C4—1271no—
179C2×#41—2548no—
180C2×#41—2544no—
181C22×Q8—1271no—
182C22×D4—1271no—
183C22×D4—1272no—
184C22×D4C2×D12712yes—
185C22×D4C2×D12711yescharacteristic
186C23×C4—1272no—
187C23×C4—1271no—
188C23×C4—1271no—
189C4×D4—2544no—
190C4×D4—2548no—
191C2×C4⋊C4—1271no—
192C2×C4⋊C4—1272no—
193C2×C22⋊C4—1272no—
194C2×C22⋊C4—1272no—
195C2×C22⋊C4—1272no—
196C2×C22⋊C4—1271no—
197C2×C22⋊C4—1272no—
198C2×C22⋊C4C2×D12711yescharacteristic
199C2×C22⋊C4C2×D12712yes—
200C2×C42—1271no—
201C2×C42—1272no—
202#95—1272no—
203#95—1272no—
204#95—1271no—
205#95C2×D12711yescharacteristic
206C24—1271no—
207C24C22×D12712yes—
208C24C22×D12711yescharacteristic
209#41—25432no—
210C2×Q8—2544no—
211C2×Q8—2542no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—2542no—
218C2×D4—2544no—
219C2×D4—2542no—
220C2×D4—2544no—
221C2×D4—2544no—
222C2×D4—2542no—
223C22×C4—1271no—
224C22×C4—1278no—
225C22×C4—1271no—
226C22×C4—1272no—
227C22×C4—1274no—
228C22×C4—1272no—
229C22×C4—1271no—
230C22×C4—2544no—
231C22×C4—2542no—
232C22×C4—2544no—
233C22×C4—2544no—
234C22×C4—2544no—
235C22×C4C22×D12711yescharacteristic
236C22×C4C22×D12711yescharacteristic
237C22×C4C22×D12712yes—
238C4⋊C4—2544no—
239C4⋊C4—2542no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—2542no—
243C22⋊C4—2544no—
244C22⋊C4—2544no—
245C22⋊C4—2544no—
246C22⋊C4—2544no—
247C42—2544no—
248C42—2542no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—1272no—
260C23—2542no—
261C23—2541no—
262C23—2542no—
263C23—2541no—
264C23D4×D12714yes—
265C23#1452324712yescomplemented
266C23C23×D12711yescharacteristic
267Q8—2548no—
268D4—28no—
269D4—216no—
270D4—25416no—
271D4—2548no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—1274no—
278C2×C4—1272no—
279C2×C4—1274no—
280C2×C4—1272no—
281C2×C4—5084no—
282C2×C4—2548no—
283C2×C4—25416no—
284C2×C4—2542no—
285C2×C4—2542no—
286C2×C4—2541no—
287C2×C4—2542no—
288C2×C4—2542no—
289C2×C4—2541no—
290C2×C4—2542no—
291C2×C4—2542no—
292C2×C4—2541no—
293C2×C4D4×D12712yes—
294C2×C4#1452324714yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—5082no—
308C22—5082no—
309C22—2542no—
310C22—2542no—
311C22—2542no—
312C22C2×D4×D12712yes—
313C22C2×D4×D12711yescharacteristic
314C22#41×D12711yescharacteristic
315C22C2×#1452324711yescharacteristic
316C22C2×#1452324712yes—
317C4—22no—
318C4—24no—
319C4—41no—
320C4—1274no—
321C4—5081no—
322C4—5082no—
323C4—2544no—
324C4—2542no—
325C2—44no—
326C2—42no—
327C2—5082no—
328C2#120×D12711yescharacteristic
329C2#2255038212yes—
330C2#2255038611yescharacteristic
331C2#2255042112yes—
332C2#2255042411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#392×C127C2
hypercenter#392D127
hypocenterC127#2388
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C2×D4×Dic127C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. #22549797C2semidirect
5. #22549792C2semidirect
6. #22549790C2semidirect
7. #22549788C2semidirect
8. #22549784C2semidirect
9. C22⋊C4×Dic127C2semidirect
10. #22549723C2semidirect
11. #392×C127C2semidirect
53. C22×Dic127D4semidirect
60. C2×C4×D127D4semidirect
104. C2×Dic127#41semidirect
117. C23×C127#41semidirect
127. C2×C4×C127#41semidirect
167. C127#2388semidirect
176. #392D127semidirect
265. C23#14523247semidirect
294. C2×C4#14523247semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×D4×Dic1272
C2×(C22⋊D127)⋊C42
#225497972
#225497922
#225497902
#225497882
#225497842
C22⋊C4×Dic1272
#225497232
#392×C1272
#2388127