Finite Group Databasebeta

← #30640384

Subgroups — #30640384

Order 16256.

234 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 73 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#22549860C211yesmaximal, characteristic
2#22549850C211yesmaximal, characteristic
3#22549666C214yesmaximal, complemented
4#536×C127C211yesmaximal, characteristic
5C2×#14523232—21no—
6#16861168—21no—
7#16861128—21no—
8#16861126C2212yes—
9#16861124—21no—
10#16861124C2212yes—
11C2×#41×C127C2211yescharacteristic
12C22×Q8×C127C411yescharacteristic, complemented
13C4⋊Q8×C127—21no—
14#124×C127—21no—
15#123×C127C412yescomplemented
16#123×C127C2212yes—
17C22⋊Q8×C127—21no—
18C22⋊Q8×C127—21no—
19#117×C127C411yescharacteristic, complemented
20#14523232—22no—
21#14523232—41no—
22C2×C127⋊C8—41no—
23C4⋊Dic127—41no—
24C22⋊Dic127—22no—
25C4×Dic127—41no—
26#41×C127—41no—
27#41×C127—41no—
28C2×Q8×C127—21no—
29C2×Q8×C127—21no—
30C2×Q8×C127—41no—
31C2×Q8×C127—41no—
32C2×Q8×C127D411yescharacteristic
33C2×D4×C127C2×C411yescharacteristic
34C2×D4×C127D411yescharacteristic
35C2×D4×C127C2311yescharacteristic
36C22×C4×C127C2×C411yescharacteristic
37C22×C4×C127C2×C412yes—
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C4⋊C4×C127—21no—
41C4⋊C4×C127—22no—
42C4⋊C4×C127—22no—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—22no—
45C22⋊C4×C127C2×C412yes—
46C42×C127—21no—
47C2×Dic127—42no—
48C127⋊C8—42no—
49C23×C127C22⋊C411yescharacteristic
50C23×C127C22×C411yescharacteristic
51C23×C127C2×D411yescharacteristic
52Q8×C127—41no—
53Q8×C127—42no—
54Q8×C127—41no—
55Q8×C127—41no—
56D4×C127—41no—
57D4×C127—41no—
58D4×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—22no—
66C2×C4×C127—22no—
67C2×C4×C127—42no—
68C2×C4×C127C22⋊C411yescharacteristic
69C2×C4×C127C22⋊C411yescharacteristic
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127C2×C22⋊C411yescharacteristic
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—42no—
83C4×C127—42no—
84C4×C127#9911yescharacteristic
85C4×C127#9911yescharacteristic
86Dic127—82no—
87C2×C127—21no—
88C2×C127—41no—
89C2×C127—41no—
90C2×C127—41no—
91C2×C127C2×#9911yescharacteristic
92#2123—1271nomaximal, Sylow
93C127#212311yescharacteristic, complemented, Sylow
94#536D12711yescharacteristic
95#413—1271no—
96#410—1271no—
97#352—1274no—
98C2×#41C2×D12711yescharacteristic
99C22×Q8Dic12711yescharacteristic, complemented
100C2×M16—2541no—
101C4⋊Q8—21no—
102#124—21no—
103#123Dic12712yescomplemented
104#123C2×D12712yes—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107#117—2541no—
108#117Dic12711yescharacteristic, complemented
109#101—2541no—
110#100—1272no—
111#99—1272no—
112#99—2541no—
113#41—41no—
114#41—41no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8C22⋊D12711yescharacteristic
120C2×D4C2×Dic12711yescharacteristic
121C2×D4C22⋊D12711yescharacteristic
122C2×D4C22×D12711yescharacteristic
123C22×C4C2×Dic12711yescharacteristic
124C22×C4C2×Dic12712yes—
125C22×C4C22⋊D12711yescharacteristic
126C22×C4C22⋊D12711yescharacteristic
127M16—5081no—
128M16—2542no—
129C2×C8—5081no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—5081no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—2542no—
137C22⋊C4C2×Dic12712yes—
138C42—21no—
139C42—5081no—
140C23C22⋊Dic12711yescharacteristic
141C23C22×Dic12711yescharacteristic
142C23C2×C22⋊D12711yescharacteristic
143Q8—41no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—5082no—
160C2×C4C22⋊Dic12711yescharacteristic
161C2×C4C22⋊Dic12711yescharacteristic
162C2×C4C22⋊Dic12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊Dic12711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—42no—
176C4—42no—
177C4—10162no—
178C4#1686112411yescharacteristic
179C4#1686112411yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1686112411yescharacteristic
subgroupG/H
centerC2C2×#16861124
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#536×C127C2
socleC2×C127C2×#99
prime radicalC2×#41×C127C22
cosocleC2×D4×C127C23
hypercenter#536D127
hypocenterC127#2123
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
3. #22549666C2semidirect
12. C22×Q8×C127C4semidirect
15. #123×C127C4semidirect
19. #117×C127C4semidirect
93. C127#2123semidirect
99. C22×Q8Dic127semidirect
103. #123Dic127semidirect
108. #117Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498602
#225498502
#225496662
#536×C1272
#2123127