Finite Group Databasebeta

← #30639096

Subgroups — #30639096

Order 16256.

334 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861240C211yesmaximal, characteristic
2#22550356C211yesmaximal, characteristic
3C2×#16861135C211yesmaximal, characteristic
4#22549783C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240C414yescomplemented
11C2×#14523238—21no—
12C2×#14523235—21no—
13C2×#14523232C2211yescharacteristic
14C4×Dic254—21no—
15C4×D508—21no—
16(C22⋊D127)⋊C4—22no—
17#16861167—21no—
18C2×C4⋊Dic127C2211yescharacteristic
19C4⋊C4×D127—21no—
20C2×C4×Dic127C2211yescharacteristic
21#16861135—22no—
22C127⋊#95—22no—
23C22×Q8×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26#103×C127—21no—
27#103×C127—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248C2×C412yes—
31#14523248D412yes—
32C2×Dic254D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C22×Dic127D412yes—
37C2×C4×D127—22no—
38C2×C4×D127—21no—
39#14523238—24no—
40#14523235—24no—
41#14523232—21no—
42#14523232C2×C412yescomplemented
43C2×C127⋊C8—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127—21no—
46Dic127⋊C4—22no—
47(C2×D127)⋊C4—22no—
48C4×Dic127—22no—
49C4×Dic127—21no—
50C4×Dic127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127C2×C412yescomplemented
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59M16×C127—21no—
60M16×C127—21no—
61C2×C8×C127—21no—
62C4⋊C4×C127D411yescharacteristic
63C4⋊C4×C127D411yescharacteristic
64C22⋊C4×C127—21no—
65C42×C127—21no—
66C22×D127—21no—
67Dic254—21no—
68Dic254C22⋊C412yes—
69D508—21no—
70D508C22⋊C412yes—
71C22⋊D127—22no—
72C22⋊D127—22no—
73C2×Dic127—21no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127—42no—
79C4×D127—22no—
80C4×D127—22no—
81C4×D127—82no—
82C127⋊C8—22no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—22no—
85Q8×C127—22no—
86Q8×C127—41no—
87Q8×C127—41no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C22×C411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C8×C127—41no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127C4×D411yescharacteristic
103C22×C127#12011yescharacteristic
104C22×C127#12111yescharacteristic
105C2×D127—22no—
106C2×D127—41no—
107C2×D127—41no—
108C4×C127—42no—
109C4×C127—41no—
110C4×C127—41no—
111C4×C127C2×C22⋊C411yescharacteristic
112C4×C127C4×D411yescharacteristic
113C4×C127#12111yescharacteristic
114C4×C127#12311yescharacteristic
115Dic127—22no—
116Dic127—42no—
117Dic127—42no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127#38611yescharacteristic
121C2×C127#45511yescharacteristic
122C2×C127#45611yescharacteristic
123D127—42no—
124#1882—1271nomaximal, Sylow
125C127#188211yescharacteristic, complemented, Sylow
126C2×#137—1271no—
127#520—1271no—
128C2×#104—1271no—
129#419D12711yescharacteristic, complemented
130#391—1271no—
131#343—1271no—
132#339—1271no—
133C2×#41—1271no—
134C22×Q8C2×D12711yescharacteristic
135#137—1274no—
136#137—2542no—
137C2×Q16—2541no—
138C2×SD16—2541no—
139C2×M16—1271no—
140C2×M16C2×D12711yescharacteristic
141C4×Q8—2541no—
142C4×D4—2541no—
143C4×D4—2542no—
144#117—2541no—
145#117C2×D12711yescharacteristic
146C2×C4⋊C4—1271no—
147C2×C4⋊C4—2541no—
148C2×C42—1271no—
149#104—2542no—
150#103—21no—
151#103—21no—
152#95—2542no—
153#41—1272no—
154#41—1272no—
155#41—2541no—
156#41—2541no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—1271no—
163C2×Q8C4×D12712yescomplemented
164C2×D4—1271no—
165C2×D4—2541no—
166C22×C4—21no—
167C22×C4—1272no—
168C22×C4—2542no—
169C22×C4—2541no—
170C22×C4—2541no—
171C22×C4C22×D12711yescharacteristic
172Q16—2544no—
173SD16—2544no—
174M16—21no—
175M16—21no—
176M16—1272no—
177M16—2541no—
178C2×C8—21no—
179C2×C8—2541no—
180C4⋊C4—2542no—
181C4⋊C4—2541no—
182C4⋊C4—2541no—
183C4⋊C4C22⋊D12711yescharacteristic
184C4⋊C4C22⋊D12711yescharacteristic
185C22⋊C4—21no—
186C22⋊C4—2542no—
187C42—21no—
188C42—2542no—
189C42—2541no—
190C42—2541no—
191C23—2541no—
192C23D4×D12711yescharacteristic
193Q8—22no—
194Q8—22no—
195Q8—41no—
196Q8—41no—
197Q8—1272no—
198Q8—2541no—
199D4—1272no—
200D4—2542no—
201D4—2542no—
202D4—2541no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—10162no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—5082no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2541no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4C2×C4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4D4×D12711yescharacteristic
220C2×C4C2×C22⋊D12711yescharacteristic
221C2×C4#1452324711yescharacteristic
222C2×C4#1452324811yescharacteristic
223C8—41no—
224C8—2542no—
225C22—21no—
226C22—21no—
227C22—5081no—
228C22—5081no—
229C22—2542no—
230C22(C22⋊D127)⋊C411yescharacteristic
231C22#1686117711yescharacteristic
232C22#1686118211yescharacteristic
233C4—41no—
234C4—41no—
235C4—42no—
236C4—5082no—
237C4—5082no—
238C4—2542no—
239C4C22⋊C4×D12711yescharacteristic
240C4C4×C22⋊D12711yescharacteristic
241C4#1686118011yescharacteristic
242C4#1686119111yescharacteristic
243C2—21no—
244C2—21no—
245C2—5082no—
246C2#2254975211yescharacteristic
247C2#2255005211yescharacteristic
248C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#120
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1882
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
4. #22549783C2semidirect
5. #22549644C2semidirect
6. #22549635C2semidirect
7. #419×C127C2semidirect
10. #16861240C4semidirect
42. #14523232C2×C4semidirect
56. C2×Q8×C127C2×C4semidirect
125. C127#1882semidirect
129. #419D127semidirect
163. C2×Q8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225503562
C2×#168611352
#225497832
#225496442
#225496352
#419×C1272
#1882127