Finite Group Databasebeta

← #30637041

Subgroups — #30637041

Order 16256.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861165C211yesmaximal, characteristic
2#22549926C212yesmaximal, complemented
3#22549601C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic
5#323×C127C211yesmaximal, characteristic
6C23×Dic127C411yescharacteristic
7#16861165—22no—
8#16861165—22no—
9C2×Dic127⋊C4C411yescharacteristic
10C2×Dic127⋊C4C2211yescharacteristic
11C2×C22⋊Dic127C411yescharacteristic
12C2×C4×Dic127C411yescharacteristic
13C2×C4×Dic127C2211yescharacteristic
14Dic127⋊C8—22no—
15C22⋊(C127⋊C8)C2212yes—
16C8×Dic127—22no—
17C2×C22⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C22×Dic127—21no—
20C22×Dic127—21no—
21C22×Dic127—22no—
22C22×Dic127C814yescomplemented
23C22×Dic127C2×C411yescharacteristic
24C22×Dic127C2×C411yescharacteristic
25C22×Dic127C2×C411yescharacteristic
26C22×Dic127C2×C411yescharacteristic
27C2×C127⋊C8—22no—
28Dic127⋊C4—22no—
29Dic127⋊C4—22no—
30C22⋊Dic127—22no—
31C4×Dic127—21no—
32C4×Dic127—21no—
33C4×Dic127D412yes—
34C4×Dic127D412yes—
35C24×C127C2×C411yescharacteristic
36C22×C4×C127C2×C411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C2×C8×C127—22no—
39C22⋊C4×C127—22no—
40C2×Dic127—21no—
41C2×Dic127—21no—
42C2×Dic127—24no—
43C2×Dic127—24no—
44C2×Dic127—21no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C2×Dic127C22⋊C412yes—
48C2×Dic127C22⋊C411yescharacteristic
49C2×Dic127C22⋊C411yescharacteristic
50C2×Dic127C2×C812yes—
51C2×Dic127C2×C812yes—
52C2×Dic127M1612yes—
53C127⋊C8—42no—
54C23×C127—21no—
55C23×C127—21no—
56C23×C127—41no—
57C23×C127—41no—
58C23×C127C2×C812yes—
59C23×C127C22×C411yescharacteristic
60C2×C4×C127—21no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127C2×D412yes—
64C8×C127—42no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—41no—
71C22×C127—41no—
72C22×C127C2×C22⋊C411yescharacteristic
73C22×C127C22×C811yescharacteristic
74C22×C127C2×M1611yescharacteristic
75C4×C127—22no—
76C4×C127—41no—
77Dic127—22no—
78Dic127—22no—
79Dic127—41no—
80Dic127C22⋊C812yes—
81Dic127C22⋊C812yes—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127—42no—
85C2×C127C2×C22⋊C811yescharacteristic
86C2×C127#41011yescharacteristic
87C2×C127#41311yescharacteristic
88#1458—1271nomaximal, Sylow
89C127#145811yescharacteristic, complemented, Sylow
90C2×#117—1271no—
91#431—1272no—
92#324—1272no—
93#323—1271no—
94#323D12711yescharacteristic
95C23×C4—1271no—
96#117—2542no—
97#117—2542no—
98C2×C4⋊C4—1271no—
99C2×C4⋊C4—1271no—
100C2×C22⋊C4—1271no—
101C2×C22⋊C4C2×D12711yescharacteristic
102C2×C42—1271no—
103C2×C42—1271no—
104C4⋊C8—2542no—
105C22⋊C8—1272no—
106C22⋊C8C2×D12712yes—
107C4×C8—2542no—
108C24C4×D12711yescharacteristic
109C22×C4—1271no—
110C22×C4—1271no—
111C22×C4—1274no—
112C22×C4—1271no—
113C22×C4—1271no—
114C22×C4—2541no—
115C22×C4—2541no—
116C22×C4—2542no—
117C22×C4C4×D12711yescharacteristic
118C22×C4C22×D12711yescharacteristic
119C2×C8—22no—
120C2×C8—2542no—
121C4⋊C4—2542no—
122C4⋊C4—2542no—
123C22⋊C4—22no—
124C22⋊C4—2542no—
125C42—1272no—
126C42—1272no—
127C42—2541no—
128C42—2541no—
129C23—21no—
130C23—21no—
131C23—41no—
132C23—41no—
133C23C8×D12712yes—
134C23C2×C4×D12711yescharacteristic
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—1272no—
139C2×C4—1272no—
140C2×C4—1271no—
141C2×C4—1271no—
142C2×C4—1272no—
143C2×C4—1272no—
144C2×C4—5081no—
145C2×C4—5081no—
146C2×C4—2544no—
147C2×C4—2544no—
148C2×C4—2541no—
149C2×C4—2541no—
150C2×C4—2541no—
151C2×C4D4×D12712yes—
152C8—42no—
153C8—5082no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22C22⋊C4×D12711yescharacteristic
162C22C2×C8×D12711yescharacteristic
163C22M16×D12711yescharacteristic
164C4—22no—
165C4—41no—
166C4—1272no—
167C4—1272no—
168C4—5081no—
169C4—2542no—
170C4—2542no—
171C2—21no—
172C2—21no—
173C2—42no—
174C2C22⋊C8×D12711yescharacteristic
175C2#2254984811yescharacteristic
176C2#2254985711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC22×C127C22×C8
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenter#323D127
hypocenterC127#1458
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22549926C2semidirect
3. #22549601C2semidirect
22. C22×Dic127C8semidirect
89. C127#1458semidirect

Maximal subgroups

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

subgroupindex
C2×#168611652
#225499262
#225496012
#225496002
#323×C1272
#1458127