Finite Group Databasebeta

← #30642050

Subgroups — #30642050

Order 16256.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 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, complemented
2C2×#16861179C212yesmaximal, complemented
3C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549787C211yesmaximal, characteristic, complemented
5#22549786C211yesmaximal, characteristic, complemented
6#22549785C211yesmaximal, characteristic, complemented
7#22549754C212yesmaximal, complemented
8#22549754C212yesmaximal, complemented
9#22549752C212yesmaximal, complemented
10#22549738C211yesmaximal, characteristic, complemented
11#393×C127C211yesmaximal, characteristic, complemented
12C24×D127C2211yescharacteristic
13C2×#14523249—28no—
14C2×#14523249—24no—
15C22×D508C2212yes—
16C22×D508C2212yes—
17C22×D508C2211yescharacteristic, complemented
18C22×C22⋊D127C2212yes—
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21C22×C4×D127C2212yescomplemented
22#16861179—28no—
23C2×C22⋊Dic127C2211yescharacteristic
24C2×(C2×D127)⋊C4C2211yescharacteristic
25C2×(C2×D127)⋊C4C2212yes—
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2212yescomplemented
29C2×(C2×D127)⋊C4C2212yescomplemented
30C22⋊C4×D127—24no—
31C127⋊#95C2211yescharacteristic, complemented
32C127⋊#95C2212yescomplemented
33C127⋊#95C2212yescomplemented
34C22×Q8×C127C2211yescharacteristic, complemented
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2212yes—
37#95×C127C2211yescharacteristic, complemented
38#95×C127C2212yescomplemented
39C23×D127—22no—
40C23×D127—22no—
41C23×D127—24no—
42C23×D127—22no—
43C23×D127—42no—
44C23×D127C2312yes—
45C23×D127C2311yescharacteristic
46C23×D127C2311yescharacteristic
47#14523249—232no—
48C2×D508—24no—
49C2×D508—24no—
50C2×D508—24no—
51C2×D508—24no—
52C2×D508—24no—
53C2×D508—22no—
54C2×D508—42no—
55C2×D508D418yescomplemented
56C2×C22⋊D127—24no—
57C2×C22⋊D127—24no—
58C2×C22⋊D127—42no—
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2312yes—
61C22×Dic127C2311yescharacteristic
62C2×C4×D127—24no—
63C2×C4×D127—24no—
64C2×C4×D127—24no—
65C2×C4×D127—22no—
66C2×C4×D127—24no—
67C2×C4×D127—42no—
68C2×C4×D127D414yescomplemented
69C22⋊Dic127—22no—
70(C2×D127)⋊C4—24no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—22no—
73(C2×D127)⋊C4—24no—
74(C2×D127)⋊C4—42no—
75(C2×D127)⋊C4—42no—
76C24×C127C2311yescharacteristic
77C2×Q8×C127—24no—
78C2×Q8×C127—22no—
79C22×C4×C127C2311yescharacteristic
80C22×C4×C127C2312yes—
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2312yes—
83C22⋊C4×C127—22no—
84C22⋊C4×C127—24no—
85C22×D127—21no—
86C22×D127—21no—
87C22×D127—22no—
88C22×D127—28no—
89C22×D127—21no—
90C22×D127—22no—
91C22×D127—22no—
92C22×D127—22no—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127—21no—
96C22×D127—22no—
97C22×D127—22no—
98C22×D127—22no—
99C22×D127—48no—
100C22×D127—48no—
101C22×D127—44no—
102C22×D127—44no—
103C22×D127C2×D414yes—
104C22×D127C2×D414yes—
105C22×D127C2×D412yes—
106C22×D127#4112yescomplemented
107D508—28no—
108D508—216no—
109D508—48no—
110C22⋊D127—48no—
111C2×Dic127—22no—
112C2×Dic127—22no—
113C2×Dic127—21no—
114C2×Dic127—22no—
115C2×Dic127—22no—
116C2×Dic127—22no—
117C2×Dic127C2×D412yes—
118C4×D127—216no—
119C4×D127—28no—
120C4×D127—44no—
121C23×C127—21no—
122C23×C127—22no—
123C23×C127—21no—
124C23×C127—22no—
125C23×C127#4112yescomplemented
126C23×C127C2411yescharacteristic
127Q8×C127—28no—
128C2×C4×C127—22no—
129C2×C4×C127—22no—
130C2×C4×C127—22no—
131C2×C4×C127—21no—
132C2×C4×C127—22no—
133C2×C4×C127—22no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127C2×D414yes—
137C2×C4×C127C2×D412yes—
138C22×C127—22no—
139C22×C127—24no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—42no—
143C22×C127C22×D411yescharacteristic
144C22×C127C22×D412yes—
145C22×C127C2×#4111yescharacteristic
146C22×C127C2×#4111yescharacteristic
147C22×C127H(5,𝔽2)12yes—
148C2×D127—22no—
149C2×D127—22no—
150C2×D127—24no—
151C2×D127—24no—
152C2×D127—24no—
153C2×D127—22no—
154C2×D127—24no—
155C2×D127—24no—
156C2×D127—24no—
157C2×D127—416no—
158C2×D127—41no—
159C2×D127—42no—
160C2×D127—42no—
161C2×D127—41no—
162C2×D127—42no—
163C2×D127—42no—
164C2×D127—42no—
165C2×D127#12014yes—
166C4×C127—24no—
167C4×C127—22no—
168C4×C127—41no—
169C4×C127—42no—
170Dic127—22no—
171Dic127—41no—
172Dic127—42no—
173C2×C127—42no—
174C2×C127C2×#12011yescharacteristic
175C2×C127#52511yescharacteristic
176C2×C127#53511yescharacteristic
177C2×C127#54612yes—
178C2×C127#54612yes—
179D127—24no—
180D127—44no—
181D127—42no—
182#2404—1271nomaximal, Sylow
183C127#240411yescharacteristic, complemented, Sylow
184C22×#41—1271no—
185C2×#120—1272no—
186C22×C22⋊C4—1271no—
187#393—1271no—
188#393D12711yescharacteristic, complemented
189#392—1271no—
190#392—1271no—
191#386—1272no—
192#386—1272no—
193#386—1272no—
194#381—1271no—
195C25—1271no—
196C2×#41—2544no—
197C2×#41—2548no—
198C22×Q8C2×D12711yescharacteristic, complemented
199C22×D4—1272no—
200C22×D4—1272no—
201C22×D4—1272no—
202C22×D4—1271no—
203C23×C4—1271no—
204C23×C4—1271no—
205C23×C4—1272no—
206#120—2548no—
207C2×C22⋊C4—1272no—
208C2×C22⋊C4—1272no—
209C2×C22⋊C4—1271no—
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C22⋊C4—1272no—
213C2×C22⋊C4—1272no—
214C2×C22⋊C4—2544no—
215C2×C22⋊C4C2×D12711yescharacteristic
216C2×C22⋊C4C2×D12712yes—
217#95—1272no—
218#95—1271no—
219#95—1272no—
220#95C2×D12711yescharacteristic, complemented
221#95C2×D12712yescomplemented
222C24—1271no—
223C24—1271no—
224C24—1272no—
225C24—5082no—
226C24—2544no—
227C24—2542no—
228C24—2542no—
229C24—2542no—
230C24C22×D12711yescharacteristic
231#41—25432no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—1278no—
235C2×D4—5082no—
236C2×D4—5082no—
237C2×D4—2544no—
238C2×D4—2544no—
239C2×D4—2544no—
240C2×D4—2544no—
241C2×D4—2544no—
242C2×D4—2544no—
243C2×D4—2544no—
244C2×D4—2542no—
245C22×C4—1271no—
246C22×C4—1274no—
247C22×C4—1272no—
248C22×C4—1271no—
249C22×C4—5082no—
250C22×C4—2544no—
251C22×C4—2542no—
252C22×C4—2544no—
253C22×C4—2544no—
254C22×C4—2544no—
255C22×C4C22×D12712yes—
256C22×C4C22×D12711yescharacteristic
257C22×C4C22×D12711yescharacteristic
258C22×C4C22×D12712yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—5082no—
262C22⋊C4—5082no—
263C22⋊C4—2542no—
264C22⋊C4—2544no—
265C22⋊C4—2542no—
266C22⋊C4—2542no—
267C22⋊C4—2544no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—1274no—
273C23—1272no—
274C23—1274no—
275C23—1272no—
276C23—5088no—
277C23—5084no—
278C23—5084no—
279C23—5088no—
280C23—2548no—
281C23—2541no—
282C23—2542no—
283C23—2542no—
284C23—2542no—
285C23—2542no—
286C23—2542no—
287C23—2541no—
288C23—2542no—
289C23—2542no—
290C23—2542no—
291C23—2541no—
292C23—2541no—
293C23—2542no—
294C23#1452324912yescomplemented
295C23C23×D12711yescharacteristic
296Q8—28no—
297D4—5088no—
298D4—5088no—
299D4—25416no—
300D4—2548no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—1272no—
310C2×C4—5084no—
311C2×C4—2548no—
312C2×C4—25416no—
313C2×C4—2541no—
314C2×C4—2542no—
315C2×C4—2542no—
316C2×C4—2542no—
317C2×C4—2542no—
318C2×C4—2542no—
319C2×C4D4×D12712yes—
320C2×C4D4×D12714yes—
321C22—22no—
322C22—24no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—1274no—
327C22—50816no—
328C22—5081no—
329C22—5082no—
330C22—5081no—
331C22—5082no—
332C22—5082no—
333C22—5082no—
334C22—5082no—
335C22—2544no—
336C22—2542no—
337C22—2544no—
338C22—2544no—
339C22—2544no—
340C22—2542no—
341C22—2542no—
342C22—2544no—
343C22—2544no—
344C22C2×D4×D12711yescharacteristic
345C22C2×D4×D12712yes—
346C22#41×D12711yescharacteristic
347C22C2×#1452324911yescharacteristic
348C22#1686125512yes—
349C4—22no—
350C4—24no—
351C4—41no—
352C4—42no—
353C4—5081no—
354C4—5082no—
355C4—2542no—
356C2—42no—
357C2—5082no—
358C2—5084no—
359C2—2544no—
360C2#120×D12711yescharacteristic
361C2#2255038911yescharacteristic
362C2#2255044411yescharacteristic
363C2#2255051712yes—
364C2#2255051912yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#393×C127C2
hypercenter#393D127
hypocenterC127#2404
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C2×#16861179C2semidirect
3. C2×C22⋊C4×D127C2semidirect
4. #22549787C2semidirect
5. #22549786C2semidirect
6. #22549785C2semidirect
7. #22549754C2semidirect
8. #22549754C2semidirect
9. #22549752C2semidirect
10. #22549738C2semidirect
11. #393×C127C2semidirect
17. C22×D508C22semidirect
21. C22×C4×D127C22semidirect
28. C2×(C2×D127)⋊C4C22semidirect
29. C2×(C2×D127)⋊C4C22semidirect
31. C127⋊#95C22semidirect
32. C127⋊#95C22semidirect
33. C127⋊#95C22semidirect
34. C22×Q8×C127C22semidirect
37. #95×C127C22semidirect
38. #95×C127C22semidirect
55. C2×D508D4semidirect
68. C2×C4×D127D4semidirect
106. C22×D127#41semidirect
125. C23×C127#41semidirect
183. C127#2404semidirect
188. #393D127semidirect
198. C22×Q8C2×D127semidirect
220. #95C2×D127semidirect
221. #95C2×D127semidirect
294. C23#14523249semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168611792
C2×C22⋊C4×D1272
#225497872
#225497862
#225497852
#225497542
#225497522
#225497382
#393×C1272
#2404127