Finite Group Databasebeta

← #30641896

Subgroups — #30641896

Order 16256.

464 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 213 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#22549775C212yesmaximal, complemented
3#22549769C212yesmaximal
4#22549769C212yesmaximal
5#22549762C212yesmaximal, complemented
6#22549762C212yesmaximal, complemented
7#22549739C211yesmaximal, characteristic
8#22549736C211yesmaximal, characteristic
9C2×C22⋊Q8×C127C211yesmaximal, characteristic
10C2×C4⋊Dic127C2212yes—
11C2×C4⋊Dic127C2212yes—
12C2×C4⋊Dic127C2212yes—
13C2×C22⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127C2211yescharacteristic
15C2×C22⋊Dic127C2212yes—
16C2×C4×Dic127C2212yes—
17C2×C4×Dic127C2212yes—
18C127⋊#95C2212yes—
19C127⋊#95C2212yes—
20C127⋊#95C2211yescharacteristic
21C127⋊#95C2212yes—
22C127⋊#95C2212yes—
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2211yescharacteristic
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—42no—
37C4⋊Dic127—42no—
38C4⋊Dic127—42no—
39C22⋊Dic127—42no—
40C22⋊Dic127—41no—
41C22⋊Dic127—41no—
42C4×Dic127—42no—
43C4×Dic127—42no—
44C24×C127C2311yescharacteristic
45C2×Q8×C127—42no—
46C2×Q8×C127C2×C414yes—
47C22×C4×C127—42no—
48C22×C4×C127C2×C414yes—
49C22×C4×C127C2312yes—
50C22×C4×C127C2312yes—
51C22×C4×C127C2311yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127C2×C414yes—
54C4⋊C4×C127C2×C418yes—
55C22⋊C4×C127C2×C418yes—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—22no—
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—
68C23×C127—22no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—22no—
72C23×C127C22×C412yes—
73C23×C127C2411yescharacteristic
74Q8×C127—44no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—44no—
82C2×C4×C127C22×C414yes—
83C2×C4×C127C22×C414yes—
84C2×C4×C127C22×C412yes—
85C2×C4×C127C22×C412yes—
86C22×C127—22no—
87C22×C127—22no—
88C22×C127—22no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127C23×C411yescharacteristic
92C22×C127H(5,𝔽2)12yes—
93C22×C127H(5,𝔽2)11yescharacteristic
94C22×C1272−1+411yescharacteristic
95C22×C1272−1+412yes—
96C4×C127—22no—
97C4×C127—22no—
98C4×C127—24no—
99C4×C127—24no—
100Dic127—42no—
101Dic127—42no—
102Dic127—42no—
103Dic127—42no—
104C2×C127—42no—
105C2×C127#51811yescharacteristic
106C2×C127#51911yescharacteristic
107C2×C127#52011yescharacteristic
108C2×C127#56212yes—
109C2×C127#56312yes—
110#2378—1271nomaximal, Sylow
111C127#237811yescharacteristic, complemented, Sylow
112C2×C22⋊Q8D12711yescharacteristic
113#391—1272no—
114#389—1272no—
115#388—1272no—
116#388—1272no—
117#387—1272no—
118#387—1272no—
119#381—1271no—
120#380—1271no—
121C22×Q8C2×D12711yescharacteristic
122C23×C4C2×D12711yescharacteristic
123C22⋊Q8Dic12718yescomplemented
124C2×C4⋊C4—1272no—
125C2×C4⋊C4—1272no—
126C2×C4⋊C4—1272no—
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C4⋊C4C2×D12712yes—
129C2×C22⋊C4—1271no—
130C2×C22⋊C4—1271no—
131C2×C22⋊C4—1272no—
132C2×C22⋊C4C2×D12712yes—
133C2×C42—1272no—
134C2×C42—1272no—
135#95—1272no—
136#95—1271no—
137#95—1272no—
138#95—1271no—
139#95—1272no—
140#95—1272no—
141#95—1272no—
142#95—1272no—
143C24C22×D12711yescharacteristic
144C2×Q8—42no—
145C2×Q8C2×Dic12714yes—
146C22×C4—42no—
147C22×C4—1272no—
148C22×C4—1272no—
149C22×C4—1272no—
150C22×C4—1272no—
151C22×C4C2×Dic12714yes—
152C22×C4C22×D12711yescharacteristic
153C22×C4C22×D12712yes—
154C22×C4C22×D12712yes—
155C22×C4C22×D12711yescharacteristic
156C4⋊C4—5082no—
157C4⋊C4—5082no—
158C4⋊C4—5082no—
159C4⋊C4C2×Dic12714yes—
160C4⋊C4C2×Dic12718yes—
161C22⋊C4—5082no—
162C22⋊C4—5081no—
163C22⋊C4—5081no—
164C22⋊C4C2×Dic12718yes—
165C42—5082no—
166C42—5082no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23C22×Dic12712yes—
172C23C23×D12711yescharacteristic
173Q8—44no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4C22×Dic12714yes—
194C2×C4C22×Dic12714yes—
195C2×C4C22×Dic12712yes—
196C2×C4C22×Dic12712yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C23×Dic12711yescharacteristic
203C22#1686125412yes—
204C22#1686125511yescharacteristic
205C22#1686125611yescharacteristic
206C22#1686125712yes—
207C4—22no—
208C4—22no—
209C4—24no—
210C4—24no—
211C4—5082no—
212C4—5082no—
213C4—5082no—
214C4—5082no—
215C2—42no—
216C2#2255035111yescharacteristic
217C2#2255035411yescharacteristic
218C2#2255035911yescharacteristic
219C2#2255062212yes—
220C2#2255062712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×C22⋊Q8×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenterC2×C22⋊Q8D127
hypocenterC127#2378
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
2. #22549775C2semidirect
5. #22549762C2semidirect
6. #22549762C2semidirect
28. C22⋊Q8×C127C4semidirect
111. C127#2378semidirect
123. C22⋊Q8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497832
#225497752
#225497692
#225497622
#225497392
#225497362
C2×C22⋊Q8×C1272
#2378127