Finite Group Databasebeta

← #30637195

Subgroups — #30637195

Order 16256.

216 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 81 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#22549929C211yesmaximal, characteristic, complemented
2#22549900C211yesmaximal, characteristic, complemented
3Q8⋊(C127⋊C8)C212yesmaximal, complemented
4D4⋊(C127⋊C8)C212yesmaximal, complemented
5#529×C127C211yesmaximal, characteristic
6C2×#14523232—21no—
7C4⋊(C127⋊C8)—21no—
8C4⋊(C127⋊C8)—21no—
9C4⋊(C127⋊C8)C2212yes—
10C22⋊(C127⋊C8)—21no—
11C4×C127⋊C8C2212yes—
12#126×C127—21no—
13#125×C127C411yescharacteristic
14#124×C127C411yescharacteristic
15#123×C127—21no—
16C22⋊Q8×C127C411yescharacteristic
17#121×C127C411yescharacteristic
18C4×Q8×C127C2211yescharacteristic
19C4×D4×C127—21no—
20C4×D4×C127C2211yescharacteristic
21#117×C127—21no—
22C2×C42×C127C2211yescharacteristic
23#14523232—42no—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—22no—
26C2×Q8×C127C2×C411yescharacteristic
27C2×D4×C127—21no—
28C2×D4×C127C2×C411yescharacteristic
29C22×C4×C127—21no—
30C22×C4×C127C2×C411yescharacteristic
31C22×C4×C127D411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C4⋊C4×C127—21no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127C2×C411yescharacteristic
38C4⋊C4×C127C2×C411yescharacteristic
39C22⋊C4×C127—21no—
40C22⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—21no—
43C22⋊C4×C127—21no—
44C42×C127—21no—
45C42×C127C2×C411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127C2311yescharacteristic
49C127⋊C8—42no—
50C127⋊C8—42no—
51C23×C127—21no—
52C23×C127C22⋊C411yescharacteristic
53Q8×C127—22no—
54D4×C127—22no—
55D4×C127—42no—
56C2×C4×C127—22no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C22×C127—21no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—41no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127C2×C22⋊C411yescharacteristic
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127—41no—
84C4×C127—41no—
85C4×C127#13112yes—
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127#13711yescharacteristic
88C2×C127—41no—
89C2×C127—81no—
90C2×C127#40811yescharacteristic
91C2×C127#41711yescharacteristic
92C2×C127#42111yescharacteristic
93#1495—1271nomaximal, Sylow
94C127#149511yescharacteristic, complemented, Sylow
95#529D12711yescharacteristic
96#431—1271no—
97#423—1271no—
98Q8⋊C8—1272no—
99D4⋊C8—1272no—
100C2×M16—2541no—
101#126—21no—
102#125Dic12711yescharacteristic
103#124Dic12711yescharacteristic
104#123—21no—
105C22⋊Q8Dic12711yescharacteristic
106#121Dic12711yescharacteristic
107C4×Q8C2×D12711yescharacteristic
108C4×D4—21no—
109C4×D4C2×D12711yescharacteristic
110#117—21no—
111C2×C42C2×D12711yescharacteristic
112C4⋊C8—1272no—
113C4⋊C8—2541no—
114C4⋊C8—2541no—
115C22⋊C8—2541no—
116C4×C8—1272no—
117C2×Q8C2×Dic12711yescharacteristic
118C2×D4—21no—
119C2×D4C2×Dic12711yescharacteristic
120C22×C4—21no—
121C22×C4C2×Dic12711yescharacteristic
122C22×C4C22⋊D12711yescharacteristic
123C22×C4C22⋊D12711yescharacteristic
124M16—5082no—
125C2×C8—2542no—
126C2×C8—2542no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4C2×Dic12711yescharacteristic
132C4⋊C4C2×Dic12711yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—21no—
138C42—21no—
139C42C2×Dic12711yescharacteristic
140C42C22⋊D12711yescharacteristic
141C42C22⋊D12711yescharacteristic
142C42C22×D12711yescharacteristic
143C23—21no—
144C23C22⋊Dic12711yescharacteristic
145Q8—22no—
146D4—22no—
147D4—42no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4C22⋊Dic12711yescharacteristic
158C2×C4C22⋊Dic12711yescharacteristic
159C2×C4C22⋊Dic12711yescharacteristic
160C2×C4C22×Dic12711yescharacteristic
161C2×C4C2×C22⋊D12711yescharacteristic
162C2×C4C2×C22⋊D12711yescharacteristic
163C8—5082no—
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊Dic12711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4#1686121612yes—
180C4#1686123411yescharacteristic
181C4#1686123911yescharacteristic
182C2—41no—
183C2—81no—
184C2#2254984311yescharacteristic
185C2#2254987711yescharacteristic
186C2#2254989311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#529×C127C2
socleC22×C127C2×C22⋊C4
prime radical#121×C127C4
cosocleC42×C127C23
hypercenter#529D127
hypocenterC127#1495
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #22549929C2semidirect
2. #22549900C2semidirect
3. Q8⋊(C127⋊C8)C2semidirect
4. D4⋊(C127⋊C8)C2semidirect
94. C127#1495semidirect

Maximal subgroups

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

subgroupindex
#225499292
#225499002
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
#529×C1272
#1495127