Finite Group Databasebeta

← #30648674

Subgroups — #30648674

Order 16256.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861168C211yesmaximal, characteristic
2#22550214C214yesmaximal, complemented
3#22550209C212yesmaximal, complemented
4#22550204C212yesmaximal, complemented
5#22549916C212yesmaximal, complemented
6#22549885C212yesmaximal, complemented
7#22549840C211yesmaximal, characteristic
8C2×C22⋊Q8×C127C211yesmaximal, characteristic
9C2×#14523232C2212yes—
10#16861168—21no—
11#16861168—21no—
12#16861168C2212yes—
13#16861168C2212yes—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127—21no—
17C2×C4×Dic127—21no—
18#16861142C2214yescomplemented
19#16861140C2214yescomplemented
20#16861133C2214yescomplemented
21#16861133C2214yescomplemented
22C22⋊(C127⋊C8)C2212yes—
23C22⋊(C127⋊C8)C2212yes—
24C22×Q8×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26C22⋊Q8×C127—21no—
27C22⋊Q8×C127—21no—
28C22⋊Q8×C127C2212yes—
29C22⋊Q8×C127C2212yes—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127C2211yescharacteristic
32C2×C22⋊C4×C127—21no—
33C22×Dic127—21no—
34C22×Dic127—21no—
35#14523232—24no—
36C2×C127⋊C8C2314yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127C2312yes—
40C4⋊Dic127C2312yes—
41C22⋊Dic127—22no—
42C22⋊Dic127—22no—
43C4×Dic127—22no—
44C4×Dic127—22no—
45C24×C127D411yescharacteristic
46C2×Q8×C127—21no—
47C2×Q8×C127—24no—
48C2×Q8×C127C2312yes—
49C22×C4×C127—24no—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127D411yescharacteristic
53C22×C4×C127D411yescharacteristic
54C22×C4×C127D411yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C4⋊C4×C127—22no—
59C4⋊C4×C127—22no—
60C4⋊C4×C127—21no—
61C4⋊C4×C127C2312yes—
62C22⋊C4×C127—22no—
63C22⋊C4×C127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—42no—
67C2×Dic127—42no—
68C127⋊C8—24no—
69C23×C127—22no—
70C23×C127—22no—
71C23×C127—41no—
72C23×C127C2×D411yescharacteristic
73C23×C127C2×D411yescharacteristic
74C23×C127C2×D411yescharacteristic
75Q8×C127—24no—
76Q8×C127—42no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—42no—
85C2×C4×C127—42no—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127#4114yes—
90C2×C4×C127C2411yescharacteristic
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—44no—
99C22×C127#12312yes—
100C22×C127#13712yes—
101C22×C127C22×D411yescharacteristic
102C4×C127—21no—
103C4×C127—21no—
104C4×C127—42no—
105C4×C127—42no—
106C4×C127#12312yes—
107C4×C127C2×#4112yes—
108Dic127—42no—
109Dic127—42no—
110C2×C127—22no—
111C2×C127—22no—
112C2×C127—41no—
113C2×C127C2×#12311yescharacteristic
114C2×C127C2×#13711yescharacteristic
115C2×C127#57511yescharacteristic
116#3095—1271nomaximal, Sylow
117C127#309511yescharacteristic, complemented, Sylow
118C2×C22⋊Q8D12711yescharacteristic
119C2×#117—1271no—
120#485—1274no—
121#484—1272no—
122#483—1272no—
123#428—1272no—
124#419—1272no—
125#407—1271no—
126C22×Q8C2×D12711yescharacteristic
127C23×C4C2×D12711yescharacteristic
128C2×M16—1272no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8C2×D12712yes—
132C22⋊Q8C2×D12712yes—
133#117—1272no—
134#117—1272no—
135#117—2541no—
136#117—2541no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—1271no—
139C2×C4⋊C4—1271no—
140C2×C4⋊C4C2×D12711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—2541no—
143C2×C42—2541no—
144#107—1274no—
145#106—1274no—
146#103—1274no—
147#103—1274no—
148C22⋊C8—1272no—
149C22⋊C8—1272no—
150C24C22⋊D12711yescharacteristic
151C2×Q8—21no—
152C2×Q8—24no—
153C2×Q8C22×D12712yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—2541no—
158C22×C4—2541no—
159C22×C4C22⋊D12711yescharacteristic
160C22×C4C22⋊D12711yescharacteristic
161C22×C4C22⋊D12711yescharacteristic
162C22×C4C22×D12711yescharacteristic
163C22×C4C22×D12711yescharacteristic
164C22×C4C22×D12711yescharacteristic
165M16—2544no—
166C2×C8—1274no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—1272no—
171C4⋊C4—1272no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C4⋊C4C22×D12712yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—2542no—
178C22⋊C4—2542no—
179C42—2542no—
180C42—2542no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×C22⋊D12711yescharacteristic
185C23C2×C22⋊D12711yescharacteristic
186C23C2×C22⋊D12711yescharacteristic
187Q8—24no—
188Q8—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—5082no—
199C2×C4—5082no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4C2×C22⋊D12711yescharacteristic
204C2×C4C2×C22⋊D12711yescharacteristic
205C2×C4#1452324714yes—
206C2×C4C23×D12711yescharacteristic
207C8—2544no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1686119012yes—
217C22#1686124012yes—
218C22C22×C22⋊D12711yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—5082no—
224C4—5082no—
225C4#1686119012yes—
226C4C2×#1452324712yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1686119011yescharacteristic
231C2C2×#1686124011yescharacteristic
232C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC22×C127C22×D4
prime radicalC24×C127D4
hypercenterC2×C22⋊Q8D127
hypocenterC127#3095
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22550214C2semidirect
3. #22550209C2semidirect
4. #22550204C2semidirect
5. #22549916C2semidirect
6. #22549885C2semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. #16861133C22semidirect
21. #16861133C22semidirect
117. C127#3095semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225502142
#225502092
#225502042
#225499162
#225498852
#225498402
C2×C22⋊Q8×C1272
#3095127