Finite Group Databasebeta

← #30648680

Subgroups — #30648680

Order 16256.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16861189C211yesmaximal, characteristic
2#117×D127C211yesmaximal, characteristic
3#22550220C212yesmaximal, complemented
4#22550219C212yesmaximal, complemented
5#22550213C212yesmaximal, complemented
6#22550208C212yesmaximal, complemented
7#22549916C211yesmaximal, characteristic, complemented
8#22549886C211yesmaximal, characteristic, complemented
9#22549884C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic, complemented
12C22×Dic254C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861189—21no—
16#16861189—21no—
17#16861189C2212yes—
18#16861189C2212yes—
19#16861168C2211yescharacteristic
20#16861167C2211yescharacteristic
21#16861167C2211yescharacteristic
22#16861166—21no—
23#16861165—21no—
24C2×C4⋊Dic127—21no—
25C4⋊C4×D127C2211yescharacteristic
26C4⋊C4×D127C2211yescharacteristic
27C2×(C2×D127)⋊C4—21no—
28C22⋊C4×D127—21no—
29C42×D127—21no—
30#16861142C2212yescomplemented
31#16861140C2212yescomplemented
32#16861134C2212yescomplemented
33#16861134C2212yescomplemented
34#16861132C2212yescomplemented
35#16861132C2212yescomplemented
36(C2×D127)⋊C8C2212yes—
37(C2×D127)⋊C8C2212yes—
38C2×M16×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#107×C127C2212yescomplemented
42#106×C127C2212yescomplemented
43C23×D127D411yescharacteristic
44C2×Dic254—21no—
45C2×Dic254—22no—
46C2×Dic254—22no—
47C2×Dic254C2312yes—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C2×C4×D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127D411yescharacteristic
55C2×C4×D127D411yescharacteristic
56C2×C4×D127C2311yescharacteristic
57C2×C4×D127C2311yescharacteristic
58#14523232—22no—
59C2×C127⋊C8C2312yes—
60C4⋊Dic127—22no—
61C4⋊Dic127—22no—
62C4⋊Dic127C2311yescharacteristic
63C4⋊Dic127C2311yescharacteristic
64Dic127⋊C4—21no—
65Dic127⋊C4—21no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—21no—
71C4×Dic127—21no—
72C4×Dic127—21no—
73C4×Dic127—21no—
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76M16×C127—22no—
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2312yes—
80C4⋊C4×C127C2311yescharacteristic
81C4⋊C4×C127C2311yescharacteristic
82C22⋊C4×C127—21no—
83C42×C127—21no—
84C22×D127—22no—
85C22×D127—22no—
86C22×D127—41no—
87C22×D127C2×D411yescharacteristic
88C22×D127C2×D411yescharacteristic
89Dic254—22no—
90Dic254—22no—
91Dic254—42no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—42no—
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—21no—
104C4×D127—21no—
105C4×D127—42no—
106C4×D127—42no—
107C127⋊C8—22no—
108C23×C127C2×D411yescharacteristic
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—42no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127#4112yes—
115C2×C4×C127#4112yes—
116C2×C4×C127C2411yescharacteristic
117C8×C127—22no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127#12312yes—
121C22×C127C22×D411yescharacteristic
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—21no—
125C2×D127—21no—
126C2×D127—21no—
127C2×D127—44no—
128C2×D127#13712yes—
129C4×C127—41no—
130C4×C127—42no—
131C4×C127—41no—
132C4×C127#12312yes—
133C4×C127C2×#4111yescharacteristic
134C4×C127C2×#4111yescharacteristic
135Dic127—21no—
136Dic127—21no—
137Dic127—42no—
138Dic127—41no—
139Dic127—41no—
140C2×C127—22no—
141C2×C127C2×#12311yescharacteristic
142C2×C127C2×#13711yescharacteristic
143C2×C127#57511yescharacteristic
144D127—22no—
145D127—41no—
146#3095—1271nomaximal, Sylow
147C127#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—1271no—
149C2×#117—1271no—
150#485—1272no—
151#485—1272no—
152#484—1272no—
153#483—1272no—
154#428—1271no—
155#428D12711yescharacteristic, complemented
156#419—1271no—
157#419—1271no—
158#407—1271no—
159C22×Q8—1271no—
160C23×C4—1271no—
161C2×M16—1271no—
162C2×M16C2×D12711yescharacteristic
163C22⋊Q8—1272no—
164C22⋊Q8—1272no—
165C22⋊Q8—2541no—
166C22⋊Q8—2541no—
167#117—1271no—
168#117—1271no—
169#117—1271no—
170#117—2541no—
171#117—2541no—
172#117C2×D12711yescharacteristic
173C2×C4⋊C4—1271no—
174C2×C4⋊C4—1271no—
175C2×C4⋊C4—2541no—
176C2×C4⋊C4C2×D12711yescharacteristic
177C2×C22⋊C4—2541no—
178C2×C22⋊C4—2541no—
179C2×C42—2541no—
180#107—1272no—
181#107C2×D12712yescomplemented
182#106—1272no—
183#106C2×D12712yescomplemented
184#103—1272no—
185#103—1272no—
186#103—1272no—
187#103—1272no—
188C22⋊C8—1272no—
189C22⋊C8—1272no—
190C24—1271no—
191C2×Q8—1272no—
192C2×Q8—2542no—
193C2×Q8—2542no—
194C2×Q8—2541no—
195C22×C4—21no—
196C22×C4—1271no—
197C22×C4—1271no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—2541no—
202C22×C4—2541no—
203C22×C4—2541no—
204C22×C4—2542no—
205C22×C4—2542no—
206C22×C4C22×D12711yescharacteristic
207M16—22no—
208M16—2542no—
209C2×C8—1272no—
210C2×C8C22×D12712yes—
211C4⋊C4—21no—
212C4⋊C4—1271no—
213C4⋊C4—1271no—
214C4⋊C4—2541no—
215C4⋊C4—2541no—
216C4⋊C4—2542no—
217C4⋊C4—2542no—
218C4⋊C4C22×D12711yescharacteristic
219C4⋊C4C22×D12711yescharacteristic
220C4⋊C4C22×D12712yes—
221C22⋊C4—21no—
222C22⋊C4—2541no—
223C22⋊C4—2541no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2541no—
227C42—21no—
228C42—2541no—
229C42—2541no—
230C42—2541no—
231C23—1271no—
232C23—1271no—
233C23—5081no—
234C23—2542no—
235C23—2542no—
236C23D4×D12711yescharacteristic
237Q8—5082no—
238Q8—2542no—
239Q8—2542no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1271no—
245C2×C4—1271no—
246C2×C4—5082no—
247C2×C4—5082no—
248C2×C4—5082no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2541no—
256C2×C4—2541no—
257C2×C4—2541no—
258C2×C4—2541no—
259C2×C4D4×D12711yescharacteristic
260C2×C4#1452324812yes—
261C2×C4#1452324912yes—
262C2×C4C23×D12711yescharacteristic
263C8—22no—
264C8—2542no—
265C22—21no—
266C22—21no—
267C22—1272no—
268C22—5084no—
269C22—2541no—
270C22—2541no—
271C22—2541no—
272C22—2541no—
273C22—2541no—
274C22#1686119412yes—
275C22C2×D4×D12711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—5082no—
280C4—5081no—
281C4—5081no—
282C4—2541no—
283C4—2541no—
284C4#1686119412yes—
285C4C2×#1452324811yescharacteristic
286C4C2×#1452324911yescharacteristic
287C2—22no—
288C2—5081no—
289C2—2542no—
290C2C2×#1686119411yescharacteristic
291C2#137×D12711yescharacteristic
292C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC23×D127D4
hypercenter#428D127
hypocenterC127#3095
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #22550220C2semidirect
4. #22550219C2semidirect
5. #22550213C2semidirect
6. #22550208C2semidirect
7. #22549916C2semidirect
8. #22549886C2semidirect
9. #22549884C2semidirect
11. #428×C127C2semidirect
30. #16861142C22semidirect
31. #16861140C22semidirect
32. #16861134C22semidirect
33. #16861134C22semidirect
34. #16861132C22semidirect
35. #16861132C22semidirect
41. #107×C127C22semidirect
42. #106×C127C22semidirect
147. C127#3095semidirect
155. #428D127semidirect
181. #107C2×D127semidirect
183. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611892
#117×D1272
#225502202
#225502192
#225502132
#225502082
#225499162
#225498862
#225498842
#225498392
#428×C1272
#3095127