Finite Group Databasebeta

← #30641944

Subgroups — #30641944

Order 16256.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16861171C211yesmaximal, characteristic, complemented
3C2×(C22⋊D127)⋊C4C212yesmaximal, complemented
4#22549796C211yesmaximal, characteristic, complemented
5#22549791C212yesmaximal, complemented
6#22549789C211yesmaximal, characteristic, complemented
7#22549785C212yesmaximal, complemented
8#22549784C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic127C212yesmaximal, complemented
10#22549723C211yesmaximal, characteristic, complemented
11#394×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—28no—
13C2×#14523247—24no—
14C22×Dic254C2211yescharacteristic
15C22×D508C2211yescharacteristic
16C22×C22⋊D127C2212yes—
17C22×C22⋊D127C2212yes—
18C23×Dic127C2212yes—
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21#16861171—24no—
22(C22⋊D127)⋊C4—28no—
23C2×Dic127⋊C4C2212yes—
24C2×C22⋊Dic127C2212yes—
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×C4×Dic127C2212yes—
30C2×C4×Dic127C2211yescharacteristic
31C127⋊#95C2211yescharacteristic
32C127⋊#95C2212yes—
33C127⋊#95C2212yes—
34C22×D4×C127C2211yescharacteristic
35C2×C4⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2212yes—
37C2×C22⋊C4×C127C2212yes—
38#95×C127C2211yescharacteristic
39C23×D127C2311yescharacteristic
40C23×D127C2311yescharacteristic
41#14523247—232no—
42C2×Dic254—24no—
43C2×Dic254—22no—
44C2×D508—22no—
45C2×D508—22no—
46C2×D508—22no—
47C2×C22⋊D127—24no—
48C2×C22⋊D127—24no—
49C2×C22⋊D127—24no—
50C2×C22⋊D127—24no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127—24no—
53C22×Dic127—24no—
54C22×Dic127—24no—
55C22×Dic127D418yescomplemented
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2312yes—
58C22×Dic127C2312yes—
59C22×Dic127C2311yescharacteristic
60C2×C4×D127—24no—
61C2×C4×D127—24no—
62C2×C4×D127—22no—
63C2×C4×D127D414yescomplemented
64Dic127⋊C4—24no—
65C22⋊Dic127—24no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—24no—
68(C2×D127)⋊C4—24no—
69(C2×D127)⋊C4—22no—
70C4×Dic127—24no—
71C4×Dic127—22no—
72C24×C127C2312yes—
73C2×D4×C127—24no—
74C2×D4×C127—22no—
75C22×C4×C127C2311yescharacteristic
76C22×C4×C127C2312yes—
77C22×C4×C127C2311yescharacteristic
78C22×C4×C127C2311yescharacteristic
79C4⋊C4×C127—22no—
80C22⋊C4×C127—24no—
81C22⋊C4×C127—24no—
82C22×D127—22no—
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—22no—
86C22×D127—22no—
87C22×D127—21no—
88C22×D127—21no—
89C22×D127—21no—
90C22×D127C2×D412yes—
91C22×D127#4112yescomplemented
92Dic254—28no—
93D508—28no—
94C22⋊D127—216no—
95C22⋊D127—216no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127—22no—
100C2×Dic127—216no—
101C2×Dic127—21no—
102C2×Dic127—22no—
103C2×Dic127C2×D414yes—
104C2×Dic127C2×D414yes—
105C2×Dic127C2×D412yes—
106C2×Dic127#4114yescomplemented
107C4×D127—28no—
108C4×D127—44no—
109C23×C127—22no—
110C23×C127—22no—
111C23×C127—22no—
112C23×C127—22no—
113C23×C127—22no—
114C23×C127—22no—
115C23×C127C2×D414yes—
116C23×C127C2411yescharacteristic
117D4×C127—28no—
118C2×C4×C127—21no—
119C2×C4×C127—22no—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—21no—
124C2×C4×C127—22no—
125C2×C4×C127—22no—
126C2×C4×C127C2×D412yes—
127C2×C4×C127#4112yescomplemented
128C22×C127—24no—
129C22×C127—24no—
130C22×C127—24no—
131C22×C127—42no—
132C22×C127—42no—
133C22×C127—42no—
134C22×C127—42no—
135C22×C127C22×D412yes—
136C22×C127C22×D411yescharacteristic
137C22×C127C2×#4111yescharacteristic
138C22×C127C2×#4112yes—
139C22×C127C2×#4111yescharacteristic
140C2×D127—22no—
141C2×D127—24no—
142C2×D127—22no—
143C2×D127—22no—
144C2×D127—22no—
145C2×D127—41no—
146C2×D127—42no—
147C2×D127—42no—
148C2×D127—42no—
149C2×D127—41no—
150C4×C127—22no—
151C4×C127—22no—
152C4×C127—41no—
153C4×C127—42no—
154Dic127—24no—
155Dic127—24no—
156Dic127—41no—
157Dic127#12014yes—
158C2×C127—44no—
159C2×C127C2×#12011yescharacteristic
160C2×C127#52512yes—
161C2×C127#52511yescharacteristic
162C2×C127#53212yes—
163C2×C127#53211yescharacteristic
164D127—42no—
165D127—42no—
166#2388—1271nomaximal, Sylow
167C127#238811yescharacteristic, complemented, Sylow
168C22×#41—1271no—
169C2×C4×D4—1271no—
170C2×C4×D4—1272no—
171#395—1271no—
172#394—1272no—
173#394D12711yescharacteristic, complemented
174#393—1271no—
175#392—1271no—
176#392—1272no—
177C4×C22⋊C4—1272no—
178C4×C22⋊C4—1271no—
179C2×#41—2548no—
180C2×#41—2544no—
181C22×Q8—1271no—
182C22×D4—1272no—
183C22×D4—1272no—
184C22×D4—1271no—
185C22×D4C2×D12711yescharacteristic
186C23×C4—1272no—
187C23×C4—1271no—
188C23×C4—1271no—
189C4×D4—2548no—
190C4×D4—2544no—
191C2×C4⋊C4—1272no—
192C2×C4⋊C4C2×D12711yescharacteristic
193C2×C22⋊C4—1272no—
194C2×C22⋊C4—1272no—
195C2×C22⋊C4—1271no—
196C2×C22⋊C4—1271no—
197C2×C22⋊C4—1272no—
198C2×C22⋊C4C2×D12712yes—
199C2×C22⋊C4C2×D12712yes—
200C2×C42—1272no—
201C2×C42—1271no—
202#95—1271no—
203#95—1272no—
204#95—1272no—
205#95C2×D12711yescharacteristic
206C24—1271no—
207C24—1271no—
208C24C22×D12712yes—
209#41—25432no—
210C2×Q8—2544no—
211C2×Q8—2542no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—2542no—
215C2×D4—2542no—
216C2×D4—2544no—
217C2×D4—2544no—
218C2×D4—2544no—
219C2×D4—2544no—
220C2×D4—2544no—
221C2×D4—2544no—
222C2×D4—2542no—
223C22×C4—1271no—
224C22×C4—1278no—
225C22×C4—1272no—
226C22×C4—1272no—
227C22×C4—1274no—
228C22×C4—1271no—
229C22×C4—2544no—
230C22×C4—2542no—
231C22×C4—2544no—
232C22×C4—2544no—
233C22×C4—2544no—
234C22×C4C22×D12712yes—
235C22×C4C22×D12711yescharacteristic
236C22×C4C22×D12711yescharacteristic
237C22×C4C22×D12711yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—2544no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—2544no—
243C22⋊C4—2544no—
244C22⋊C4—2542no—
245C22⋊C4—2544no—
246C22⋊C4—2542no—
247C42—2542no—
248C42—2544no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—1272no—
256C23—1272no—
257C23—2542no—
258C23—2541no—
259C23—2542no—
260C23—2542no—
261C23—2541no—
262C23—2542no—
263C23—2541no—
264C23—2541no—
265C23D4×D12714yes—
266C23C23×D12711yescharacteristic
267Q8—2548no—
268D4—28no—
269D4—2548no—
270D4—25416no—
271D4—25416no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—1274no—
281C2×C4—1272no—
282C2×C4—1274no—
283C2×C4—1274no—
284C2×C4—5084no—
285C2×C4—2548no—
286C2×C4—25416no—
287C2×C4—2541no—
288C2×C4—2542no—
289C2×C4—2542no—
290C2×C4—2542no—
291C2×C4—2542no—
292C2×C4—2542no—
293C2×C4D4×D12712yes—
294C2×C4#1452324712yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—5081no—
303C22—5082no—
304C22—5082no—
305C22—5082no—
306C22—5081no—
307C22—2542no—
308C22—2544no—
309C22—2542no—
310C22—2542no—
311C22—2542no—
312C22C2×D4×D12712yes—
313C22C2×D4×D12711yescharacteristic
314C22#41×D12711yescharacteristic
315C22#41×D12712yes—
316C22C2×#1452324711yescharacteristic
317C4—22no—
318C4—22no—
319C4—41no—
320C4—42no—
321C4—1274no—
322C4—5081no—
323C4—2544no—
324C4—2544no—
325C2—44no—
326C2—5082no—
327C2—5082no—
328C2#120×D12711yescharacteristic
329C2#2255038412yes—
330C2#2255038611yescharacteristic
331C2#2255042112yes—
332C2#2255042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#394×C127C2
hypercenter#394D127
hypocenterC127#2388
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C2×#16861171C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. #22549796C2semidirect
5. #22549791C2semidirect
6. #22549789C2semidirect
7. #22549785C2semidirect
8. #22549784C2semidirect
9. C22⋊C4×Dic127C2semidirect
10. #22549723C2semidirect
11. #394×C127C2semidirect
55. C22×Dic127D4semidirect
63. C2×C4×D127D4semidirect
91. C22×D127#41semidirect
106. C2×Dic127#41semidirect
127. C2×C4×C127#41semidirect
167. C127#2388semidirect
173. #394D127semidirect
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×#168611712
C2×(C22⋊D127)⋊C42
#225497962
#225497912
#225497892
#225497852
#225497842
C22⋊C4×Dic1272
#225497232
#394×C1272
#2388127