Finite Group Databasebeta

← #30641488

Subgroups — #30641488

Order 16256.

484 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 217 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
1#22549783C212yesmaximal, complemented
2#22549769C212yesmaximal
3#22549769C212yesmaximal
4#22549762C212yesmaximal, complemented
5#22549762C212yesmaximal, complemented
6#22549739C211yesmaximal, characteristic
7C4⋊C4×Dic127C212yesmaximal, complemented
8C4×C22⋊Dic127C211yesmaximal, characteristic
9C2×C22⋊Q8×C127C211yesmaximal, characteristic
10C2×C4⋊Dic127C2212yes—
11C2×C4⋊Dic127C2212yes—
12C2×C22⋊Dic127C2212yes—
13C2×C22⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127C2211yescharacteristic
15C2×C4×Dic127C2212yes—
16C2×C4×Dic127C2211yescharacteristic
17C2×C4×Dic127C2212yes—
18C2×C4×Dic127C2211yescharacteristic
19C127⋊#95C2212yes—
20C127⋊#95C2212yes—
21C127⋊#95C2212yes—
22C127⋊#95C2212yes—
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C22×Q8×C127C2211yescharacteristic
27C23×C4×C127C2211yescharacteristic
28C22⋊Q8×C127C418yescomplemented
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C4⋊C4×C127C2212yes—
31C2×C22⋊C4×C127C2212yes—
32C22×Dic127C2312yes—
33C22×Dic127C2312yes—
34C22×Dic127C2312yes—
35C22×Dic127C2312yes—
36C4⋊Dic127—24no—
37C4⋊Dic127—42no—
38C22⋊Dic127—24no—
39C22⋊Dic127—41no—
40C22⋊Dic127—41no—
41C4×Dic127—24no—
42C4×Dic127—41no—
43C4×Dic127—41no—
44C4×Dic127—42no—
45C24×C127C2311yescharacteristic
46C2×Q8×C127—41no—
47C2×Q8×C127—41no—
48C2×Q8×C127C2×C414yes—
49C22×C4×C127—41no—
50C22×C4×C127—41no—
51C22×C4×C127C2×C414yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2311yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127C2×C414yes—
57C4⋊C4×C127C2×C418yes—
58C22⋊C4×C127C2×C418yes—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127#4114yes—
71C23×C127—22no—
72C23×C127—21no—
73C23×C127—21no—
74C23×C127—22no—
75C23×C127C22×C412yes—
76C23×C127C2411yescharacteristic
77Q8×C127—44no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—44no—
87C2×C4×C127C22×C414yes—
88C2×C4×C127C22×C414yes—
89C2×C4×C127C22×C412yes—
90C2×C4×C127C22×C412yes—
91C22×C127—22no—
92C22×C127—22no—
93C22×C127—22no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127C23×C411yescharacteristic
97C22×C127C2×#4112yes—
98C22×C127H(5,𝔽2)11yescharacteristic
99C22×C127H(5,𝔽2)11yescharacteristic
100C22×C1272−1+412yes—
101C4×C127—24no—
102C4×C127—22no—
103C4×C127—22no—
104C4×C127—24no—
105Dic127—24no—
106Dic127—42no—
107Dic127—42no—
108Dic127—42no—
109C2×C127—42no—
110C2×C127C4×#4111yescharacteristic
111C2×C127#51811yescharacteristic
112C2×C127#51911yescharacteristic
113C2×C127#53912yes—
114C2×C127#54212yes—
115#2332—1271nomaximal, Sylow
116C127#233211yescharacteristic, complemented, Sylow
117C2×C22⋊Q8D12711yescharacteristic
118#391—1272no—
119#388—1272no—
120#388—1272no—
121#387—1272no—
122#387—1272no—
123#381—1271no—
124C4×C4⋊C4—1272no—
125C4×C22⋊C4—1271no—
126C22×Q8C2×D12711yescharacteristic
127C23×C4C2×D12711yescharacteristic
128C22⋊Q8Dic12718yescomplemented
129C2×C4⋊C4—1272no—
130C2×C4⋊C4—1272no—
131C2×C4⋊C4C2×D12712yes—
132C2×C4⋊C4C2×D12711yescharacteristic
133C2×C22⋊C4—1271no—
134C2×C22⋊C4—1271no—
135C2×C22⋊C4—1272no—
136C2×C22⋊C4C2×D12712yes—
137C2×C42—1271no—
138C2×C42—1271no—
139C2×C42—1272no—
140C2×C42—1272no—
141#95—1272no—
142#95—1272no—
143#95—1272no—
144#95—1272no—
145#95—1272no—
146#95—1272no—
147#95—1272no—
148C24C22×D12711yescharacteristic
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8C2×Dic12714yes—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—1272no—
155C22×C4—1272no—
156C22×C4—1272no—
157C22×C4—1272no—
158C22×C4C2×Dic12714yes—
159C22×C4C22×D12711yescharacteristic
160C22×C4C22×D12711yescharacteristic
161C22×C4C22×D12712yes—
162C22×C4C22×D12712yes—
163C4⋊C4—5082no—
164C4⋊C4—2544no—
165C4⋊C4C2×Dic12714yes—
166C4⋊C4C2×Dic12718yes—
167C22⋊C4—5081no—
168C22⋊C4—5081no—
169C22⋊C4—2544no—
170C22⋊C4C2×Dic12718yes—
171C42—5081no—
172C42—5081no—
173C42—5082no—
174C42—2544no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C22×Dic12712yes—
180C23C23×D12711yescharacteristic
181Q8—44no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—1274no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4—2542no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4—2542no—
203C2×C4C22×Dic12714yes—
204C2×C4C22×Dic12714yes—
205C2×C4C22×Dic12712yes—
206C2×C4C22×Dic12712yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C23×Dic12711yescharacteristic
213C22#41×D12712yes—
214C22#1686125411yescharacteristic
215C22#1686125411yescharacteristic
216C22#1686125712yes—
217C4—24no—
218C4—22no—
219C4—22no—
220C4—24no—
221C4—5082no—
222C4—5082no—
223C4—5082no—
224C4—2544no—
225C2—42no—
226C2#41×Dic12711yescharacteristic
227C2#2255035111yescharacteristic
228C2#2255035411yescharacteristic
229C2#2255047212yes—
230C2#2255049612yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×C22⋊Q8×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenterC2×C22⋊Q8D127
hypocenterC127#2332
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22549783C2semidirect
4. #22549762C2semidirect
5. #22549762C2semidirect
7. C4⋊C4×Dic127C2semidirect
28. C22⋊Q8×C127C4semidirect
116. C127#2332semidirect
128. C22⋊Q8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497832
#225497692
#225497622
#225497392
C4⋊C4×Dic1272
C4×C22⋊Dic1272
C2×C22⋊Q8×C1272
#2332127