Finite Group Databasebeta

← #30647953

Subgroups — #30647953

Order 16256.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861232C212yesmaximal, complemented
3#22550389C211yesmaximal, characteristic, complemented
4#22550037C212yesmaximal, complemented
5#22550026C211yesmaximal, characteristic, complemented
6#22550024C212yesmaximal, complemented
7#22550015C211yesmaximal, characteristic, complemented
8#22550012C211yesmaximal, characteristic, complemented
9#22549871C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic, complemented
11#451×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—28no—
13C2×#14523249—21no—
14C2×#14523249—21no—
15C2×#14523249C2212yes—
16C2×#14523249C2212yes—
17C2×#14523249C2211yescharacteristic, complemented
18C22×D508—21no—
19C22×D508C2211yescharacteristic
20C22×C4×D127—21no—
21C22×C4×D127C2211yescharacteristic
22#16861232—28no—
23C2×#14523235C2212yescomplemented
24C2×#14523235C2212yescomplemented
25C2×D1016C2212yescomplemented
26C2×C8×D127C2212yescomplemented
27#16861194—21no—
28#16861185C2211yescharacteristic, complemented
29#16861184C2211yescharacteristic, complemented
30#16861183C2211yescharacteristic
31#16861179—21no—
32#16861171—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic, complemented
35#16861133C2212yescomplemented
36#16861131C2212yescomplemented
37#16861129C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic
39(C2×D127)⋊C8C2212yescomplemented
40C22×Q8×C127C2211yescharacteristic
41C2×Q16×C127C2212yescomplemented
42C22⋊Q8×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic
45C23×D127—21no—
46C23×D127D411yescharacteristic
47#14523249—28no—
48#14523249—28no—
49#14523249—24no—
50#14523249—24no—
51#14523249—24no—
52#14523249—24no—
53#14523249D418yescomplemented
54C2×D508—28no—
55C2×D508—22no—
56C2×D508—22no—
57C2×D508—24no—
58C2×D508—21no—
59C2×D508—21no—
60C2×D508—21no—
61C2×D508C2311yescharacteristic
62C2×D508C2312yes—
63C2×C22⋊D127—21no—
64C22×Dic127D411yescharacteristic
65C2×C4×D127—28no—
66C2×C4×D127—22no—
67C2×C4×D127—22no—
68C2×C4×D127—24no—
69C2×C4×D127—21no—
70C2×C4×D127D411yescharacteristic
71C2×C4×D127D411yescharacteristic
72C2×C4×D127C2311yescharacteristic
73C2×C4×D127C2311yescharacteristic
74#14523235—24no—
75#14523235—24no—
76D1016—24no—
77C2×C127⋊C8C2312yes—
78C8×D127—24no—
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85C4×Dic127—21no—
86C2×Q8×C127—24no—
87C2×Q8×C127—21no—
88C2×Q8×C127C2311yescharacteristic
89C2×Q8×C127C2312yes—
90C22×C4×C127—21no—
91C22×C4×C127C2311yescharacteristic
92Q16×C127—24no—
93C2×C8×C127C2312yes—
94C4⋊C4×C127—21no—
95C4⋊C4×C127C2311yescharacteristic
96C22⋊C4×C127—21no—
97C22×D127—24no—
98C22×D127—22no—
99C22×D127—22no—
100C22×D127—22no—
101C22×D127—21no—
102C22×D127—41no—
103C22×D127—42no—
104C22×D127—42no—
105C22×D127C2×D411yescharacteristic
106C22×D127C2×D411yescharacteristic
107D508—28no—
108D508—24no—
109D508—28no—
110D508—22no—
111D508—22no—
112D508—42no—
113D508—42no—
114D508C2×D414yes—
115C22⋊D127—42no—
116C2×Dic127—22no—
117C2×Dic127—21no—
118C2×Dic127C2×D411yescharacteristic
119C2×Dic127C2×D411yescharacteristic
120C4×D127—28no—
121C4×D127—22no—
122C4×D127—22no—
123C4×D127—28no—
124C4×D127—22no—
125C4×D127—42no—
126C4×D127C2×D414yes—
127C127⋊C8—22no—
128C23×C127C2×D411yescharacteristic
129Q8×C127—24no—
130Q8×C127—22no—
131Q8×C127—22no—
132Q8×C127C2×D414yes—
133C2×C4×C127—24no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2411yescharacteristic
140C8×C127—22no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127#13512yescomplemented
144C22×C127C22×D411yescharacteristic
145C2×D127—21no—
146C2×D127—21no—
147C2×D127—21no—
148C2×D127—24no—
149C2×D127—21no—
150C2×D127—21no—
151C2×D127—44no—
152C2×D127—48no—
153C2×D127—42no—
154C2×D127—42no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127#12012yes—
159C4×C127—24no—
160C4×C127—21no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127C22×D412yes—
164Dic127—21no—
165Dic127—41no—
166Dic127#12012yes—
167C2×C127—22no—
168C2×C127C2×#12011yescharacteristic
169C2×C127C2×#13511yescharacteristic
170C2×C127#57511yescharacteristic
171D127—22no—
172D127—41no—
173D127—44no—
174D127—81no—
175#3002—1271nomaximal, Sylow
176C127#300211yescharacteristic, complemented, Sylow
177C22×#41—1271no—
178C2×#135—1272no—
179#525—1271no—
180#452—1272no—
181#451D12711yescharacteristic, complemented
182#450—1271no—
183#449—1272no—
184#448—1271no—
185#447—1271no—
186#416—1272no—
187C2×C22⋊C8—1271no—
188C2×#41—1272no—
189C2×#41—1271no—
190C2×#41—1272no—
191C2×#41—2541no—
192C2×#41—2541no—
193C2×#41—2548no—
194C22×Q8C2×D12711yescharacteristic
195C22×D4—1271no—
196C22×D4—2541no—
197C23×C4—1271no—
198C23×C4—2541no—
199#135—2548no—
200C2×Q16C2×D12712yescomplemented
201C2×SD16—1272no—
202C2×SD16—1272no—
203C2×D8—1272no—
204C22×C8—1272no—
205#123—2541no—
206C22⋊Q8—1271no—
207C22⋊Q8C2×D12711yescharacteristic
208#121—1271no—
209#121—1271no—
210#120—2541no—
211C4×D4—2541no—
212C4×D4—2541no—
213#117—1271no—
214#103—1272no—
215#103C2×D12712yescomplemented
216#102—1272no—
217#102—1272no—
218C22⋊C8—1271no—
219C22⋊C8—1272no—
220C22⋊C8C2×D12711yescharacteristic
221C24—1271no—
222C24—2541no—
223#41—1278no—
224#41—2544no—
225#41—2544no—
226#41—2544no—
227#41—2548no—
228#41—2548no—
229#41—2544no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8C22×D12711yescharacteristic
233C2×Q8C22×D12712yes—
234C2×D4—1272no—
235C2×D4—1271no—
236C2×D4—2544no—
237C2×D4—2541no—
238C2×D4—2541no—
239C2×D4—2542no—
240C2×D4—2542no—
241C2×D4—2548no—
242C2×D4—2541no—
243C2×D4—2541no—
244C22×C4—21no—
245C22×C4—1271no—
246C22×C4—1271no—
247C22×C4—1271no—
248C22×C4—1271no—
249C22×C4—1271no—
250C22×C4—2541no—
251C22×C4—2544no—
252C22×C4—2542no—
253C22×C4—2542no—
254C22×C4—2548no—
255C22×C4C22×D12711yescharacteristic
256Q16—24no—
257SD16—2544no—
258SD16—2544no—
259D8—2544no—
260C2×C8—1272no—
261C2×C8—2544no—
262C2×C8C22×D12712yes—
263C4⋊C4—21no—
264C4⋊C4—1271no—
265C4⋊C4—2541no—
266C4⋊C4C22×D12711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—2541no—
269C22⋊C4—2541no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C42—2541no—
273C23—1271no—
274C23—1271no—
275C23—5082no—
276C23—5081no—
277C23—5082no—
278C23—2542no—
279C23—2544no—
280C23—2541no—
281C23—2542no—
282C23—2542no—
283C23D4×D12711yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—22no—
287Q8D4×D12714yes—
288D4—1274no—
289D4—5082no—
290D4—5082no—
291D4—5082no—
292D4—2548no—
293D4—2548no—
294D4—2544no—
295D4—2542no—
296D4—2542no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1271no—
303C2×C4—1274no—
304C2×C4—1271no—
305C2×C4—5082no—
306C2×C4—2542no—
307C2×C4—2548no—
308C2×C4—2542no—
309C2×C4—2548no—
310C2×C4—2542no—
311C2×C4—2541no—
312C2×C4—2542no—
313C2×C4D4×D12711yescharacteristic
314C2×C4C23×D12711yescharacteristic
315C8—22no—
316C8—2542no—
317C22—21no—
318C22—21no—
319C22—1272no—
320C22—5088no—
321C22—5084no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22—5082no—
326C22—5082no—
327C22—2544no—
328C22—2541no—
329C22—2541no—
330C22—2541no—
331C22—2541no—
332C22—2541no—
333C22#1686123212yescomplemented
334C22C2×D4×D12711yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—1272no—
340C4—5081no—
341C4—2541no—
342C4C2×D4×D12712yes—
343C2—22no—
344C2—10161no—
345C2—5084no—
346C2—5081no—
347C2—2542no—
348C2#120×D12711yescharacteristic
349C2C2×#1686123211yescharacteristic
350C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#451×C127C2
socleC22×C127C22×D4
hypercenter#451D127
hypocenterC127#3002
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C2×#16861232C2semidirect
3. #22550389C2semidirect
4. #22550037C2semidirect
5. #22550026C2semidirect
6. #22550024C2semidirect
7. #22550015C2semidirect
8. #22550012C2semidirect
9. #22549871C2semidirect
10. C22⋊C8×D127C2semidirect
11. #451×C127C2semidirect
17. C2×#14523249C22semidirect
23. C2×#14523235C22semidirect
24. C2×#14523235C22semidirect
25. C2×D1016C22semidirect
26. C2×C8×D127C22semidirect
28. #16861185C22semidirect
29. #16861184C22semidirect
34. #16861167C22semidirect
35. #16861133C22semidirect
36. #16861131C22semidirect
37. #16861129C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C2×Q16×C127C22semidirect
43. #103×C127C22semidirect
53. #14523249D4semidirect
143. C22×C127#135semidirect
176. C127#3002semidirect
181. #451D127semidirect
200. C2×Q16C2×D127semidirect
215. #103C2×D127semidirect
333. C22#16861232semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168612322
#225503892
#225500372
#225500262
#225500242
#225500152
#225500122
#225498712
C22⋊C8×D1272
#451×C1272
#3002127