Finite Group Databasebeta

← #30642040

Subgroups — #30642040

Order 16256.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C23×C22⋊D127C211yesmaximal, characteristic, complemented
2C2×#16861178C212yesmaximal, complemented
3C2×#16861177C212yesmaximal, complemented
4#22549784C214yesmaximal, complemented
5#22549756C212yesmaximal, complemented
6#22549755C212yesmaximal, complemented
7#22549732C211yesmaximal, characteristic, complemented
8C23×D4×C127C211yesmaximal, characteristic, complemented
9C24×D127C2211yescharacteristic, complemented
10C22×C22⋊D127—28no—
11C22×C22⋊D127—24no—
12C22×C22⋊D127C2212yes—
13C22×C22⋊D127C2212yescomplemented
14C22×C22⋊D127C2214yescomplemented
15C23×Dic127C2211yescharacteristic, complemented
16#16861178—28no—
17#16861177—28no—
18C2×C22⋊Dic127C2214yescomplemented
19C2×C22⋊Dic127C2212yescomplemented
20C2×C22⋊Dic127C2212yescomplemented
21C2×C22⋊Dic127C2212yescomplemented
22C2×(C2×D127)⋊C4C2214yescomplemented
23C127⋊#95C2214yescomplemented
24C25×C127C2211yescharacteristic
25C25×C127C2211yescharacteristic, complemented
26C22×D4×C127—24no—
27C22×D4×C127—28no—
28C22×D4×C127C2212yes—
29C22×D4×C127C2212yescomplemented
30C23×C4×C127C2211yescharacteristic, complemented
31C23×D127—24no—
32C23×D127—22no—
33C23×D127—24no—
34C23×D127—42no—
35C23×D127C2312yes—
36C2×C22⋊D127—232no—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—28no—
39C2×C22⋊D127—24no—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—24no—
42C2×C22⋊D127—24no—
43C2×C22⋊D127—28no—
44C2×C22⋊D127—44no—
45C2×C22⋊D127D418yescomplemented
46C22×Dic127—24no—
47C22×Dic127—22no—
48C22×Dic127C2314yes—
49C22×Dic127C2312yes—
50C22⋊Dic127—28no—
51C22⋊Dic127—24no—
52C22⋊Dic127—24no—
53C22⋊Dic127—42no—
54(C2×D127)⋊C4—28no—
55C24×C127—24no—
56C24×C127—24no—
57C24×C127—22no—
58C24×C127—24no—
59C24×C127—24no—
60C24×C127—42no—
61C24×C127—42no—
62C24×C127D414yescomplemented
63C24×C127C2311yescharacteristic
64C24×C127C2312yes—
65C24×C127C2311yescharacteristic
66C24×C127C2311yescharacteristic
67C2×D4×C127—232no—
68C2×D4×C127—24no—
69C2×D4×C127—24no—
70C2×D4×C127—24no—
71C2×D4×C127—24no—
72C2×D4×C127—24no—
73C2×D4×C127D418yescomplemented
74C22×C4×C127—22no—
75C22×C4×C127—24no—
76C22×C4×C127C2312yes—
77C22×D127—28no—
78C22×D127—24no—
79C22×D127—22no—
80C22×D127—22no—
81C22×D127—24no—
82C22×D127—416no—
83C22×D127—44no—
84C22×D127—44no—
85C22×D127C2×D414yescomplemented
86C22⋊D127—216no—
87C22⋊D127—216no—
88C22⋊D127—416no—
89C2×Dic127—28no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—24no—
93C2×Dic127—24no—
94C2×Dic127—24no—
95C2×Dic127C2×D414yescomplemented
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—28no—
101C23×C127—28no—
102C23×C127—28no—
103C23×C127—22no—
104C23×C127—22no—
105C23×C127—22no—
106C23×C127—24no—
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—24no—
110C23×C127—44no—
111C23×C127—416no—
112C23×C127—416no—
113C23×C127—44no—
114C23×C127—44no—
115C23×C127C2×D414yes—
116C23×C127C2×D412yes—
117C23×C127C2×D412yes—
118C23×C127C2×D412yes—
119C23×C127C2×D414yes—
120C23×C127C2×D414yescomplemented
121C23×C127C2411yescharacteristic
122D4×C127—216no—
123D4×C127—216no—
124C2×C4×C127—28no—
125C2×C4×C127—22no—
126C2×C4×C127—22no—
127C2×C4×C127C2×D414yescomplemented
128C22×C127—22no—
129C22×C127—22no—
130C22×C127—22no—
131C22×C127—22no—
132C22×C127—28no—
133C22×C127—28no—
134C22×C127—22no—
135C22×C127—24no—
136C22×C127—22no—
137C22×C127—22no—
138C22×C127—28no—
139C22×C127—416no—
140C22×C127—416no—
141C22×C127—42no—
142C22×C127—42no—
143C22×C127—44no—
144C22×C127—44no—
145C22×C127#12014yescomplemented
146C22×C127#12014yescomplemented
147C22×C127C22×D412yes—
148C22×C127C22×D411yescharacteristic
149C22×C127C22×D412yes—
150C22×C127H(5,𝔽2)12yes—
151C2×D127—24no—
152C2×D127—28no—
153C2×D127—416no—
154C2×D127—42no—
155C2×D127—42no—
156C2×D127—44no—
157C4×C127—24no—
158Dic127—24no—
159Dic127—44no—
160C2×C127—24no—
161C2×C127—24no—
162C2×C127—42no—
163C2×C127—44no—
164C2×C127C2×#12011yescharacteristic
165C2×C127C2×#12011yescharacteristic
166C2×C127#53411yescharacteristic
167C2×C127D4214yes—
168D127—44no—
169#2403—1271nomaximal, Sylow
170C127#240311yescharacteristic, complemented, Sylow
171C23×D4—1271no—
172C23×D4D12711yescharacteristic, complemented
173C2×#120—1272no—
174C2×#120—1272no—
175#392—1274no—
176#386—1272no—
177#386—1272no—
178#379—1271no—
179C25—1271no—
180C25C2×D12711yescharacteristic
181C25C2×D12711yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—1272no—
185C22×D4—1274no—
186C22×D4—1272no—
187C22×D4—2548no—
188C22×D4—2544no—
189C22×D4C2×D12712yes—
190C22×D4C2×D12712yescomplemented
191C23×C4—1271no—
192C23×C4C2×D12711yescharacteristic, complemented
193#120—2548no—
194#120—2548no—
195C2×C22⋊C4—1272no—
196C2×C22⋊C4—1272no—
197C2×C22⋊C4—1272no—
198C2×C22⋊C4—1274no—
199C2×C22⋊C4—1274no—
200#95—1274no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—1272no—
209C24—5082no—
210C24—2544no—
211C24—2544no—
212C24—2542no—
213C24C22⋊D12714yescomplemented
214C24C22×D12711yescharacteristic
215C24C22×D12712yes—
216C24C22×D12711yescharacteristic
217C24C22×D12711yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—1278no—
225C2×D4—5084no—
226C2×D4—2544no—
227C2×D4—2548no—
228C2×D4—2548no—
229C2×D4—2544no—
230C2×D4—25432no—
231C2×D4—2544no—
232C2×D4—2544no—
233C2×D4—2544no—
234C2×D4C22⋊D12718yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—1272no—
238C22×C4—1274no—
239C22×C4—2544no—
240C22×C4—2542no—
241C22×C4C22×D12712yes—
242C22⋊C4—5082no—
243C22⋊C4—2548no—
244C22⋊C4—2544no—
245C22⋊C4—2544no—
246C22⋊C4—2548no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—28no—
260C23—28no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—1274no—
267C23—5084no—
268C23—50816no—
269C23—5084no—
270C23—2548no—
271C23—2542no—
272C23—2544no—
273C23—2542no—
274C23—2544no—
275C23D4×D12714yes—
276C23C2×C22⋊D12712yes—
277C23C2×C22⋊D12712yes—
278C23C2×C22⋊D12714yes—
279C23C2×C22⋊D12712yes—
280C23C2×C22⋊D12714yescomplemented
281C23C23×D12711yescharacteristic
282D4—216no—
283D4—216no—
284D4—50816no—
285D4—25416no—
286D4—25416no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—1274no—
291C2×C4—2544no—
292C2×C4—2544no—
293C2×C4—2544no—
294C2×C4—2548no—
295C2×C4—2542no—
296C2×C4—2542no—
297C2×C4C2×C22⋊D12714yescomplemented
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—28no—
303C22—28no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—22no—
308C22—28no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—42no—
313C22—44no—
314C22—44no—
315C22—50816no—
316C22—5082no—
317C22—5082no—
318C22—5084no—
319C22—2544no—
320C22—2548no—
321C22#1686117714yescomplemented
322C22#1686117814yescomplemented
323C22C2×D4×D12712yes—
324C22C22×C22⋊D12711yescharacteristic
325C22C22×C22⋊D12712yes—
326C22#1686125412yes—
327C4—24no—
328C4—5084no—
329C4—2544no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—5084no—
335C2C2×#1686117711yescharacteristic
336C2C2×#1686117811yescharacteristic
337C2#2255043511yescharacteristic
338C2D4×C22⋊D12714yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC23×D4×C127C2
hypercenterC23×D4D127
hypocenterC127#2403
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D127C2semidirect
2. C2×#16861178C2semidirect
3. C2×#16861177C2semidirect
4. #22549784C2semidirect
5. #22549756C2semidirect
6. #22549755C2semidirect
7. #22549732C2semidirect
8. C23×D4×C127C2semidirect
9. C24×D127C22semidirect
13. C22×C22⋊D127C22semidirect
14. C22×C22⋊D127C22semidirect
15. C23×Dic127C22semidirect
18. C2×C22⋊Dic127C22semidirect
19. C2×C22⋊Dic127C22semidirect
20. C2×C22⋊Dic127C22semidirect
21. C2×C22⋊Dic127C22semidirect
22. C2×(C2×D127)⋊C4C22semidirect
23. C127⋊#95C22semidirect
25. C25×C127C22semidirect
29. C22×D4×C127C22semidirect
30. C23×C4×C127C22semidirect
45. C2×C22⋊D127D4semidirect
62. C24×C127D4semidirect
73. C2×D4×C127D4semidirect
85. C22×D127C2×D4semidirect
95. C2×Dic127C2×D4semidirect
120. C23×C127C2×D4semidirect
127. C2×C4×C127C2×D4semidirect
145. C22×C127#120semidirect
146. C22×C127#120semidirect
170. C127#2403semidirect
172. C23×D4D127semidirect
181. C25C2×D127semidirect
190. C22×D4C2×D127semidirect
192. C23×C4C2×D127semidirect
213. C24C22⋊D127semidirect
234. C2×D4C22⋊D127semidirect
280. C23C2×C22⋊D127semidirect
297. C2×C4C2×C22⋊D127semidirect
321. C22#16861177semidirect
322. C22#16861178semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
C2×#168611782
C2×#168611772
#225497842
#225497562
#225497552
#225497322
C23×D4×C1272
#2403127