Finite Group Databasebeta

← #30642055

Subgroups — #30642055

Order 16256.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
2C2×#16861191C212yesmaximal, complemented
3C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549792C212yesmaximal, complemented
5#22549791C212yesmaximal, complemented
6#22549752C212yesmaximal, complemented
7#22549751C211yesmaximal, characteristic, complemented
8#22549748C211yesmaximal, characteristic, complemented
9#22549732C211yesmaximal, characteristic, complemented
10#22549731C211yesmaximal, characteristic, complemented
11#379×C127C211yesmaximal, characteristic, complemented
12C24×D127C2211yescharacteristic
13C22×C22⋊D127—24no—
14C22×C22⋊D127—28no—
15C22×C22⋊D127C2212yes—
16C22×C22⋊D127C2212yes—
17C23×Dic127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic, complemented
19C22×C4×D127C2211yescharacteristic, complemented
20#16861191—28no—
21C2×C4⋊Dic127C2212yescomplemented
22C2×Dic127⋊C4C2212yescomplemented
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2212yescomplemented
26C2×C22⋊Dic127C2212yescomplemented
27C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D127)⋊C4C2212yescomplemented
30C2×(C2×D127)⋊C4C2212yescomplemented
31C22⋊C4×D127—24no—
32C127⋊#95C2212yescomplemented
33C127⋊#95C2212yescomplemented
34C25×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2211yescharacteristic
37C2×C22⋊C4×C127C2212yescomplemented
38C2×C22⋊C4×C127C2212yescomplemented
39C23×D127—24no—
40C23×D127—22no—
41C23×D127—22no—
42C23×D127—42no—
43C23×D127D414yes—
44C23×D127C2311yescharacteristic
45C23×D127C2311yescharacteristic
46C2×C22⋊D127—232no—
47C2×C22⋊D127—24no—
48C2×C22⋊D127—24no—
49C2×C22⋊D127—24no—
50C2×C22⋊D127—24no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127D418yes—
53C22×Dic127—24no—
54C22×Dic127—22no—
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2312yes—
58C22×Dic127C2312yes—
59C2×C4×D127—24no—
60C2×C4×D127—22no—
61C2×C4×D127—22no—
62C2×C4×D127—42no—
63C4⋊Dic127—24no—
64Dic127⋊C4—24no—
65C22⋊Dic127—24no—
66C22⋊Dic127—22no—
67C22⋊Dic127—22no—
68C22⋊Dic127—42no—
69(C2×D127)⋊C4—24no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—42no—
73C24×C127—24no—
74C24×C127—22no—
75C24×C127—22no—
76C24×C127—22no—
77C24×C127—42no—
78C24×C127C2312yes—
79C24×C127C2311yescharacteristic
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2312yes—
83C22⋊C4×C127—24no—
84C22⋊C4×C127—22no—
85C22⋊C4×C127—22no—
86C22⋊C4×C127—42no—
87C22×D127—21no—
88C22×D127—21no—
89C22×D127—22no—
90C22×D127—28no—
91C22×D127—22no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—22no—
95C22×D127—28no—
96C22×D127—44no—
97C22×D127—48no—
98C22×D127—48no—
99C22×D127C2×D412yes—
100C22×D127C2×D414yes—
101C22×D127C2×D412yes—
102C22⋊D127—216no—
103C22⋊D127—216no—
104C2×Dic127—28no—
105C2×Dic127—22no—
106C2×Dic127—22no—
107C2×Dic127—22no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—22no—
111C2×Dic127—22no—
112C2×Dic127—22no—
113C2×Dic127—22no—
114C2×Dic127C2×D414yes—
115C4×D127—44no—
116C4×D127—44no—
117C23×C127—22no—
118C23×C127—22no—
119C23×C127—28no—
120C23×C127—22no—
121C23×C127—22no—
122C23×C127—22no—
123C23×C127—22no—
124C23×C127—22no—
125C23×C127—22no—
126C23×C127—48no—
127C23×C127—48no—
128C23×C127—44no—
129C23×C127—44no—
130C23×C127C2×D414yes—
131C23×C127C2×D412yes—
132C23×C127#4112yes—
133C23×C127#4112yes—
134C23×C127C2411yescharacteristic
135C2×C4×C127—22no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—22no—
142C22×C127—22no—
143C22×C127—22no—
144C22×C127—24no—
145C22×C127—24no—
146C22×C127—24no—
147C22×C127—24no—
148C22×C127—24no—
149C22×C127—416no—
150C22×C127—42no—
151C22×C127—42no—
152C22×C127—42no—
153C22×C127—42no—
154C22×C127#12314yescomplemented
155C22×C127C22×D411yescharacteristic
156C22×C127C22×D412yes—
157C22×C127C2×#4111yescharacteristic
158C22×C127C2×#4111yescharacteristic
159C22×C127H(5,𝔽2)12yes—
160C2×D127—22no—
161C2×D127—22no—
162C2×D127—24no—
163C2×D127—24no—
164C2×D127—22no—
165C2×D127—22no—
166C2×D127—22no—
167C2×D127—416no—
168C2×D127—42no—
169C2×D127—42no—
170C2×D127#12014yes—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127—42no—
174Dic127—24no—
175Dic127—41no—
176Dic127—41no—
177Dic127—42no—
178C2×C127—24no—
179C2×C127—44no—
180C2×C127C2×#12011yescharacteristic
181C2×C127C2×#12311yescharacteristic
182C2×C127#53411yescharacteristic
183C2×C127#54612yes—
184C2×C127#54612yes—
185D127—24no—
186D127—42no—
187#2405—1271nomaximal, Sylow
188C127#240511yescharacteristic, complemented, Sylow
189C23×D4—1271no—
190C2×#123—1272no—
191C22×C22⋊C4—1271no—
192#394—1272no—
193#394—1272no—
194#386—1272no—
195#385—1271no—
196#385—1271no—
197#379—1271no—
198#379—1271no—
199#379D12711yescharacteristic, complemented
200C25—1271no—
201C25C2×D12711yescharacteristic
202C22×D4—1272no—
203C22×D4—1272no—
204C22×D4—2548no—
205C22×D4—2544no—
206C23×C4—1271no—
207C23×C4—1271no—
208C23×C4—1271no—
209#123—2548no—
210C2×C4⋊C4—1272no—
211C2×C4⋊C4—1272no—
212C2×C22⋊C4—1272no—
213C2×C22⋊C4—1272no—
214C2×C22⋊C4—1271no—
215C2×C22⋊C4—1272no—
216C2×C22⋊C4—1271no—
217C2×C22⋊C4—1271no—
218C2×C22⋊C4—1271no—
219C2×C22⋊C4—1272no—
220C2×C22⋊C4—2544no—
221C2×C22⋊C4C2×D12711yescharacteristic
222C2×C22⋊C4C2×D12711yescharacteristic
223C2×C22⋊C4C2×D12712yescomplemented
224C2×C22⋊C4C2×D12712yescomplemented
225#95—1272no—
226#95—1272no—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—1271no—
233C24—1274no—
234C24—1271no—
235C24—5082no—
236C24—2544no—
237C24—2542no—
238C24—2542no—
239C24C22×D12712yes—
240C24C22×D12711yescharacteristic
241C2×D4—1278no—
242C2×D4—2544no—
243C2×D4—2544no—
244C2×D4—2544no—
245C2×D4—2544no—
246C2×D4—25432no—
247C2×D4—2544no—
248C22×C4—1272no—
249C22×C4—1271no—
250C22×C4—1272no—
251C22×C4—1271no—
252C22×C4—5082no—
253C22×C4—2542no—
254C22×C4—2542no—
255C22×C4—2544no—
256C22×C4—2544no—
257C22×C4—2542no—
258C22×C4C22×D12711yescharacteristic
259C22×C4C22×D12712yes—
260C22×C4C22×D12711yescharacteristic
261C4⋊C4—2544no—
262C4⋊C4—2544no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—5082no—
268C22⋊C4—5082no—
269C22⋊C4—2542no—
270C22⋊C4—2542no—
271C22⋊C4—2544no—
272C22⋊C4—2542no—
273C22⋊C4—2544no—
274C22⋊C4—2542no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—1274no—
289C23—1272no—
290C23—1272no—
291C23—5088no—
292C23—5088no—
293C23—5084no—
294C23—2548no—
295C23—2548no—
296C23—2542no—
297C23—2541no—
298C23—2541no—
299C23—2542no—
300C23—2541no—
301C23—2541no—
302C23—2542no—
303C23D4×D12714yes—
304C23D4×D12712yes—
305C23#1452324712yes—
306C23#1452324812yes—
307C23C23×D12711yescharacteristic
308D4—25416no—
309D4—25416no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—1274no—
318C2×C4—5084no—
319C2×C4—5084no—
320C2×C4—2542no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2542no—
324C2×C4—2542no—
325C2×C4—2542no—
326C2×C4—2542no—
327C2×C4—2548no—
328C2×C4—2542no—
329C2×C4—2542no—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—1274no—
343C22—50816no—
344C22—5082no—
345C22—5082no—
346C22—2542no—
347C22—2542no—
348C22—2542no—
349C22—2542no—
350C22—2542no—
351C22—2544no—
352C22—2544no—
353C22#1686119114yescomplemented
354C22C2×D4×D12711yescharacteristic
355C22C2×D4×D12712yes—
356C22C2×#1452324711yescharacteristic
357C22C2×#1452324811yescharacteristic
358C22#1686125412yes—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—5081no—
363C4—5081no—
364C4—5082no—
365C4—2544no—
366C2—24no—
367C2—44no—
368C2—5082no—
369C2—2544no—
370C2#120×D12711yescharacteristic
371C2C2×#1686119111yescharacteristic
372C2#2255043611yescharacteristic
373C2#2255051312yes—
374C2#2255051612yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#379×C127C2
prime radicalC23×C22⋊D127C2
hypercenter#379D127
hypocenterC127#2405
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#16861191C2semidirect
3. C2×C22⋊C4×D127C2semidirect
4. #22549792C2semidirect
5. #22549791C2semidirect
6. #22549752C2semidirect
7. #22549751C2semidirect
8. #22549748C2semidirect
9. #22549732C2semidirect
10. #22549731C2semidirect
11. #379×C127C2semidirect
18. C22×C4×D127C22semidirect
19. C22×C4×D127C22semidirect
21. C2×C4⋊Dic127C22semidirect
22. C2×Dic127⋊C4C22semidirect
25. C2×C22⋊Dic127C22semidirect
26. C2×C22⋊Dic127C22semidirect
27. C2×(C2×D127)⋊C4C22semidirect
28. C2×(C2×D127)⋊C4C22semidirect
29. C2×(C2×D127)⋊C4C22semidirect
30. C2×(C2×D127)⋊C4C22semidirect
32. C127⋊#95C22semidirect
33. C127⋊#95C22semidirect
37. C2×C22⋊C4×C127C22semidirect
38. C2×C22⋊C4×C127C22semidirect
154. C22×C127#123semidirect
188. C127#2405semidirect
199. #379D127semidirect
223. C2×C22⋊C4C2×D127semidirect
224. C2×C22⋊C4C2×D127semidirect
353. C22#16861191semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
C2×#168611912
C2×C22⋊C4×D1272
#225497922
#225497912
#225497522
#225497512
#225497482
#225497322
#225497312
#379×C1272
#2405127