Finite Group Databasebeta

← #30640385

Subgroups — #30640385

Order 16256.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550447C211yesmaximal, characteristic
2#22549850C211yesmaximal, characteristic
3#22549667C214yesmaximal, complemented
4#413×C127C211yesmaximal, characteristic
5C22×Dic254C411yescharacteristic, complemented
6#16861207—21no—
7#16861196—21no—
8#16861190C412yescomplemented
9#16861190C2212yes—
10#16861188—21no—
11#16861188—21no—
12#16861168—21no—
13#16861168C411yescharacteristic, complemented
14#16861124—21no—
15#16861124C2212yes—
16C2×#41×C127C2211yescharacteristic
17C2×M16×C127—21no—
18#101×C127—21no—
19#100×C127C2212yes—
20C2×Dic254—21no—
21C2×Dic254—21no—
22C2×Dic254—41no—
23C2×Dic254—41no—
24C22×Dic127C2×C412yes—
25C4⋊Dic127—21no—
26C4⋊Dic127—41no—
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C22⋊Dic127—22no—
30C22⋊Dic127—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127C2×C412yes—
33C4×Dic127—21no—
34C4×Dic127—41no—
35#41×C127—41no—
36#41×C127—41no—
37C2×Q8×C127D411yescharacteristic
38C2×D4×C127C2×C411yescharacteristic
39C2×D4×C127D411yescharacteristic
40C2×D4×C127C2311yescharacteristic
41C22×C4×C127C2×C411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44M16×C127—22no—
45M16×C127—41no—
46C2×C8×C127—41no—
47Dic254—42no—
48Dic254—41no—
49Dic254—41no—
50C2×Dic127—22no—
51C2×Dic127—22no—
52C2×Dic127—42no—
53C2×Dic127—42no—
54C23×C127C22⋊C411yescharacteristic
55C23×C127C22×C411yescharacteristic
56C23×C127C2×D411yescharacteristic
57Q8×C127—41no—
58D4×C127—41no—
59D4×C127—41no—
60D4×C127—41no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22⋊C411yescharacteristic
69C2×C4×C127C22⋊C411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—42no—
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#9911yescharacteristic
83C4×C127#9911yescharacteristic
84Dic127—42no—
85Dic127—42no—
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#536—1271no—
95#413D12711yescharacteristic
96#410—1271no—
97#352—1274no—
98C2×#41C2×D12711yescharacteristic
99C22×Q8—1271no—
100C2×M16—21no—
101C4⋊Q8—2541no—
102#124—2541no—
103#123—1272no—
104#123—1272no—
105C22⋊Q8—2541no—
106C22⋊Q8—2541no—
107#117—1271no—
108#117—2541no—
109#101—21no—
110#100C2×D12712yes—
111#99—1272no—
112#99—2541no—
113#41—41no—
114#41—41no—
115C2×Q8—5081no—
116C2×Q8—5081no—
117C2×Q8—2541no—
118C2×Q8—2541no—
119C2×Q8C22⋊D12711yescharacteristic
120C2×D4C4×D12711yescharacteristic
121C2×D4D50811yescharacteristic
122C2×D4C22×D12711yescharacteristic
123C22×C4—1272no—
124C22×C4C4×D12711yescharacteristic
125C22×C4C22⋊D12711yescharacteristic
126C22×C4D50811yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—5081no—
131C4⋊C4—2542no—
132C4⋊C4—2541no—
133C4⋊C4—2542no—
134C22⋊C4—1272no—
135C22⋊C4—2542no—
136C22⋊C4—2542no—
137C22⋊C4—2542no—
138C42—5081no—
139C42—2541no—
140C23(C2×D127)⋊C411yescharacteristic
141C23C2×C4×D12711yescharacteristic
142C23C2×D50811yescharacteristic
143Q8—41no—
144Q8—5082no—
145Q8—5081no—
146Q8—5081no—
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—5082no—
157C2×C4—5082no—
158C2×C4—2542no—
159C2×C4—2542no—
160C2×C4(C2×D127)⋊C411yescharacteristic
161C2×C4(C2×D127)⋊C411yescharacteristic
162C2×C4(C2×D127)⋊C411yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D127)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—10162no—
176C4—5082no—
177C4—5082no—
178C4#1686112311yescharacteristic
179C4#1686112311yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1686112311yescharacteristic
subgroupG/H
centerC2C2×#16861123
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#413×C127C2
socleC2×C127C2×#99
prime radicalC2×#41×C127C22
cosocleC2×D4×C127C23
hypercenter#413D127
hypocenterC127#2123
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #22549667C2semidirect
5. C22×Dic254C4semidirect
8. #16861190C4semidirect
13. #16861168C4semidirect
93. C127#2123semidirect

Maximal subgroups

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

subgroupindex
#225504472
#225498502
#225496672
#413×C1272
#2123127