Finite Group Databasebeta

← #30637199

Subgroups — #30637199

Order 16256.

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

normal subgroups only

This group has 246 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#22549933C211yesmaximal, characteristic, complemented
2#22549929C211yesmaximal, characteristic, complemented
3D4⋊(C127⋊C8)C214yesmaximal, complemented
4#532×C127C211yesmaximal, characteristic
5C4⋊(C127⋊C8)—21no—
6C4⋊(C127⋊C8)C2212yes—
7C22⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)—21no—
9#16861120—21no—
10C4×C127⋊C8C2212yes—
11C2×#41×C127—21no—
12C4⋊Q8×C127C411yescharacteristic
13#127×C127C411yescharacteristic
14#124×C127—21no—
15#121×C127—21no—
16#121×C127C412yes—
17C4×D4×C127—21no—
18C4×D4×C127C2212yes—
19C2×C42×C127C2211yescharacteristic
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22#41×C127—44no—
23C2×Q8×C127—21no—
24C2×D4×C127—22no—
25C2×D4×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127C2×C412yes—
28C22×C4×C127—22no—
29C22×C4×C127C2×C411yescharacteristic
30C22×C4×C127D411yescharacteristic
31C22×C4×C127D411yescharacteristic
32C4⋊C4×C127—21no—
33C4⋊C4×C127C2×C412yes—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—22no—
36C42×C127C2×C411yescharacteristic
37C42×C127D411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127C2311yescharacteristic
40C127⋊C8—42no—
41C127⋊C8—42no—
42C23×C127—22no—
43C23×C127C22⋊C411yescharacteristic
44Q8×C127—42no—
45D4×C127—24no—
46D4×C127—42no—
47D4×C127—42no—
48D4×C127—44no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—41no—
53C2×C4×C127—44no—
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C22×C127—21no—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127—42no—
64C22×C127—42no—
65C22×C127—42no—
66C22×C127C2×C22⋊C411yescharacteristic
67C4×C127—21no—
68C4×C127—21no—
69C4×C127—21no—
70C4×C127—21no—
71C4×C127—42no—
72C4×C127#13112yes—
73C4×C127AGL(1,ℤ/8)12yes—
74C2×C127—41no—
75C2×C127—82no—
76C2×C127#40811yescharacteristic
77C2×C127#41811yescharacteristic
78C2×C127#42111yescharacteristic
79#1496—1271nomaximal, Sylow
80C127#149611yescharacteristic, complemented, Sylow
81#532D12711yescharacteristic
82#432—1271no—
83#431—1271no—
84D4⋊C8—1274no—
85C2×#41—21no—
86C4⋊Q8Dic12711yescharacteristic
87#127Dic12711yescharacteristic
88#124—21no—
89#121—21no—
90#121Dic12712yes—
91C4×D4—21no—
92C4×D4C2×D12712yes—
93C2×C42C2×D12711yescharacteristic
94C4⋊C8—1272no—
95C4⋊C8—2541no—
96C22⋊C8—2541no—
97C22⋊C8—2541no—
98#97—2541no—
99C4×C8—1272no—
100#41—44no—
101C2×Q8—21no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4C2×Dic12712yes—
106C22×C4—22no—
107C22×C4C2×Dic12711yescharacteristic
108C22×C4C22⋊D12711yescharacteristic
109C22×C4C22⋊D12711yescharacteristic
110C2×C8—2542no—
111C2×C8—2542no—
112C4⋊C4—21no—
113C4⋊C4C2×Dic12712yes—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C42C2×Dic12711yescharacteristic
117C42C22⋊D12711yescharacteristic
118C42C22⋊D12711yescharacteristic
119C42C22×D12711yescharacteristic
120C23—22no—
121C23C22⋊Dic12711yescharacteristic
122Q8—42no—
123D4—24no—
124D4—42no—
125D4—42no—
126D4—44no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—41no—
131C2×C4—44no—
132C2×C4C22⋊Dic12711yescharacteristic
133C2×C4C22⋊Dic12711yescharacteristic
134C2×C4C22⋊Dic12711yescharacteristic
135C2×C4C22×Dic12711yescharacteristic
136C2×C4C2×C22⋊D12711yescharacteristic
137C2×C4C2×C22⋊D12711yescharacteristic
138C8—5082no—
139C8—5082no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—42no—
144C22—42no—
145C22—42no—
146C22C2×C22⋊Dic12711yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—21no—
151C4—42no—
152C4#1686121612yes—
153C4#1686123512yes—
154C2—41no—
155C2—82no—
156C2#2254984311yescharacteristic
157C2#2254988111yescharacteristic
158C2#2254989311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting, prime radical#532×C127C2
socleC22×C127C2×C22⋊C4
cosocleC42×C127C23
hypercenter#532D127
hypocenterC127#1496
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549929C2semidirect
3. D4⋊(C127⋊C8)C2semidirect
80. C127#1496semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225499292
D4⋊(C127⋊C8)2
#532×C1272
#1496127