Finite Group Databasebeta

← #30639102

Subgroups — #30639102

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#22550357C211yesmaximal, characteristic
3C2×#16861137C211yesmaximal, characteristic
4#22549885C211yesmaximal, characteristic, complemented
5#22549645C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#391×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240C414yescomplemented
11C2×#14523238—21no—
12C2×#14523235—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861174—21no—
16#16861171—21no—
17C4×C22⋊D127—22no—
18#16861168C2211yescharacteristic
19#16861167—21no—
20C4⋊C4×D127—21no—
21#16861137—22no—
22#16861133—21no—
23#16861133—21no—
24C22×Q8×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27#95×C127—22no—
28#14523248—21no—
29#14523248—21no—
30#14523248C2×C412yes—
31#14523248D412yes—
32C2×Dic254D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C2×C4×D127—22no—
36C2×C4×D127—21no—
37#14523238—24no—
38#14523235—24no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232C2×C412yescomplemented
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127D411yescharacteristic
46C4⋊Dic127D411yescharacteristic
47Dic127⋊C4—22no—
48C22⋊Dic127—21no—
49(C2×D127)⋊C4—22no—
50C4×Dic127—22no—
51C4×Dic127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127C2×C412yescomplemented
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127D412yes—
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C42×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—21no—
75C2×Dic127—21no—
76C4×D127—22no—
77C4×D127—22no—
78C4×D127—82no—
79C127⋊C8—22no—
80C127⋊C8—41no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—22no—
83Q8×C127—22no—
84Q8×C127—41no—
85Q8×C127—41no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—42no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
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—42no—
110C4×C127—42no—
111C4×C127C2×C22⋊C411yescharacteristic
112C4×C127C4×D411yescharacteristic
113C4×C127#12111yescharacteristic
114C4×C127#12311yescharacteristic
115Dic127—22no—
116Dic127—41no—
117Dic127—41no—
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#419—1271no—
130#391D12711yescharacteristic, complemented
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×M16—1271no—
141C4×Q8—2541no—
142C4×D4—2541no—
143C4×D4—2542no—
144#117—1271no—
145#117—2541no—
146C2×C4⋊C4—2541no—
147C2×C4⋊C4C2×D12711yescharacteristic
148C2×C42C2×D12711yescharacteristic
149#104—2542no—
150#103—2541no—
151#103—2541no—
152#95—22no—
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—21no—
168C22×C4—2542no—
169C22×C4—2541no—
170C22×C4C22⋊D12712yes—
171C22×C4C22×D12711yescharacteristic
172Q16—2544no—
173SD16—2544no—
174M16—1272no—
175M16—2541no—
176M16—2541no—
177M16—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—1271no—
183C4⋊C4—1271no—
184C4⋊C4—2542no—
185C22⋊C4—2542no—
186C22⋊C4—2541no—
187C42—21no—
188C42—21no—
189C42—2542no—
190C42—2541no—
191C23—2541no—
192C23C2×C22⋊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—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—10162no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4C2×C4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4D4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4#1452324711yescharacteristic
222C2×C4#1452324811yescharacteristic
223C8—5081no—
224C8—2542no—
225C22—21no—
226C22—21no—
227C22—5081no—
228C22—5081no—
229C22—2542no—
230C22C4×C22⋊D12711yescharacteristic
231C22#1686117711yescharacteristic
232C22#1686118011yescharacteristic
233C4—42no—
234C4—42no—
235C4—42no—
236C4—5081no—
237C4—5081no—
238C4—2542no—
239C4C22⋊C4×D12711yescharacteristic
240C4(C22⋊D127)⋊C411yescharacteristic
241C4#1686118211yescharacteristic
242C4#1686119111yescharacteristic
243C2—21no—
244C2—21no—
245C2—5082no—
246C2#2254975211yescharacteristic
247C2#2255005311yescharacteristic
248C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC22×C127#120
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#391D127
hypocenterC127#1882
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #22549885C2semidirect
5. #22549645C2semidirect
6. #22549636C2semidirect
7. #391×C127C2semidirect
10. #16861240C4semidirect
42. #14523232C2×C4semidirect
57. C2×Q8×C127C2×C4semidirect
125. C127#1882semidirect
130. #391D127semidirect
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
#225503572
C2×#168611372
#225498852
#225496452
#225496362
#391×C1272
#1882127