Finite Group Databasebeta

← #30641909

Subgroups — #30641909

Order 16256.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16861207C211yesmaximal, characteristic
2#22549783C211yesmaximal, characteristic
3#22549780C212yesmaximal
4#22549763C212yesmaximal
5#22549760C212yesmaximal
6#22549758C212yesmaximal
7#22549757C212yesmaximal
8#22549743C212yesmaximal
9#391×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861207C418yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4C2212yes—
14C2×Dic127⋊C4C2212yes—
15C2×Dic127⋊C4C2212yes—
16C2×C4×Dic127C2212yes—
17C2×C4×Dic127C2211yescharacteristic
18C2×C4×Dic127C2211yescharacteristic
19C127⋊#95C2212yes—
20C127⋊#95C2212yes—
21C127⋊#95C2212yes—
22C127⋊#95C2212yes—
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C22×Q8×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29C2×C42×C127C2211yescharacteristic
30#95×C127C2212yes—
31#95×C127C2212yes—
32C2×Dic254—41no—
33C2×Dic254—41no—
34C2×Dic254C2×C414yes—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C4⋊Dic127—41no—
40Dic127⋊C4—42no—
41Dic127⋊C4C2×C418yes—
42Dic127⋊C4C2×C418yes—
43C4×Dic127—41no—
44C4×Dic127—42no—
45C4×Dic127C2×C414yes—
46C2×Q8×C127—41no—
47C2×Q8×C127—41no—
48C2×Q8×C127C2×C414yes—
49C22×C4×C127C2312yes—
50C22×C4×C127C2312yes—
51C22×C4×C127C2311yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—41no—
55C42×C127—41no—
56Dic254—44no—
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—24no—
66C2×Dic127C22×C414yes—
67C2×Dic127C22×C414yes—
68C23×C127C2411yescharacteristic
69Q8×C127—44no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—24no—
80C2×C4×C127—22no—
81C2×C4×C127C22×C412yes—
82C2×C4×C127C22×C412yes—
83C2×C4×C127C22×C412yes—
84C22×C127C23×C411yescharacteristic
85C22×C127H(5,𝔽2)11yescharacteristic
86C22×C127H(5,𝔽2)11yescharacteristic
87C22×C1272−1+412yes—
88C22×C1272−1+412yes—
89C4×C127—22no—
90C4×C127—22no—
91C4×C127—22no—
92C4×C127—42no—
93C4×C127—42no—
94Dic127—24no—
95Dic127—24no—
96Dic127—42no—
97Dic127—42no—
98C2×C127#51811yescharacteristic
99C2×C127#51912yes—
100C2×C127#56311yescharacteristic
101C2×C127#56311yescharacteristic
102C2×C127#56312yes—
103#2381—1271nomaximal, Sylow
104C127#238111yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—1271no—
106#391—1271no—
107#391—1272no—
108#391D12711yescharacteristic
109#387—1272no—
110#387—1272no—
111#387—1272no—
112#387—1272no—
113#383—1272no—
114C22×Q8—1271no—
115C22×Q8C2×D12711yescharacteristic
116C4⋊Q8—1278no—
117C2×C4⋊C4—1272no—
118C2×C4⋊C4—1272no—
119C2×C4⋊C4—1272no—
120C2×C4⋊C4—1271no—
121C2×C4⋊C4C2×D12711yescharacteristic
122C2×C42—1271no—
123C2×C42—1272no—
124C2×C42—1271no—
125C2×C42C2×D12711yescharacteristic
126#95—1272no—
127#95—1272no—
128#95—1272no—
129#95—1272no—
130#95—1272no—
131#95—1272no—
132#95—1272no—
133#95—1272no—
134#95C2×D12712yes—
135#95C2×D12712yes—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—1274no—
139C2×Q8—5081no—
140C2×Q8—5081no—
141C2×Q8C4×D12714yes—
142C22×C4—1272no—
143C22×C4—1272no—
144C22×C4—1272no—
145C22×C4—1272no—
146C22×C4C22×D12712yes—
147C22×C4C22×D12712yes—
148C22×C4C22×D12711yescharacteristic
149C22×C4C22×D12711yescharacteristic
150C22×C4C22×D12711yescharacteristic
151C4⋊C4—41no—
152C4⋊C4—1278no—
153C4⋊C4—1278no—
154C4⋊C4—5082no—
155C4⋊C4—5081no—
156C42—41no—
157C42—1274no—
158C42—5081no—
159C42—5082no—
160C23C23×D12711yescharacteristic
161Q8—44no—
162Q8—5084no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—1274no—
175C2×C4—1274no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4—2542no—
179C2×C4—2544no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4C2×C4×D12712yes—
186C2×C4C2×C4×D12712yes—
187C2×C4C2×C4×D12712yes—
188C22C22×C4×D12711yescharacteristic
189C22#1686125411yescharacteristic
190C22#1686125411yescharacteristic
191C22#1686125612yes—
192C22#1686125712yes—
193C4—22no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—5082no—
199C4—5082no—
200C4—2544no—
201C4—2544no—
202C2#2255034911yescharacteristic
203C2#2255035212yes—
204C2#2255062511yescharacteristic
205C2#2255062611yescharacteristic
206C2#2255062712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#391×C127C2
socle, prime radical, cosocleC23×C127C24
hypercenter#391D127
hypocenterC127#2381
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #16861207C4semidirect
104. C127#2381semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
#225497832
#225497802
#225497632
#225497602
#225497582
#225497572
#225497432
#391×C1272
#2381127