Finite Group Databasebeta

← #30640761

Subgroups — #30640761

Order 16256.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550606C211yesmaximal, characteristic, complemented
2#22550061C212yesmaximal
3#22550058C211yesmaximal, characteristic
4#22549674C212yesmaximal, complemented
5#356×C127C211yesmaximal, characteristic, complemented
6#16861257C2211yescharacteristic
7#16861242—22no—
8#16861240—21no—
9#16861208—22no—
10#16861207—21no—
11#16861207C2212yes—
12C4×Dic254—21no—
13#16861174—22no—
14#16861137—21no—
15#16861135—22no—
16#16861128C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#101×C127C2212yes—
19#14523249—41no—
20#14523248—21no—
21#14523248—22no—
22C2×Dic254—21no—
23C2×Dic254—21no—
24C2×Dic254D412yes—
25Q8×D127—41no—
26Dic508—42no—
27#14523238—41no—
28#14523237—42no—
29#14523235—41no—
30#14523232—21no—
31C4⋊Dic127—21no—
32Dic127⋊C4—22no—
33Dic127⋊C4—22no—
34Dic127⋊C4—22no—
35C4×Dic127—21no—
36C4×Dic127—22no—
37C4×Dic127D412yes—
38C2×Q8×C127D411yescharacteristic
39C2×Q8×C127C2311yescharacteristic
40M16×C127—22no—
41C4⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43Dic254—24no—
44Dic254—21no—
45Dic254—22no—
46Dic254—42no—
47Dic254—81no—
48D508—21no—
49D508—22no—
50C22⋊D127—41no—
51C2×Dic127—22no—
52C2×Dic127—22no—
53C4×D127—42no—
54C4×D127—41no—
55C127⋊C8—41no—
56Q8×C127—41no—
57Q8×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—42no—
63C22×C127#12011yescharacteristic
64C2×D127—41no—
65C4×C127—22no—
66C4×C127—22no—
67C4×C127—21no—
68C4×C127—41no—
69Dic127—24no—
70Dic127—41no—
71Dic127—42no—
72C2×C127—21no—
73C2×C127B(4,𝔽2)11yescharacteristic
74D127—81no—
75#2205—1271nomaximal, Sylow
76C127#220511yescharacteristic, complemented, Sylow
77Q82—1271no—
78#456—1271no—
79#456—1272no—
80#356—1272no—
81#356D12711yescharacteristic, complemented
822−1+4—1271no—
83#137—2542no—
84#137—2541no—
85C4⋊Q8—1272no—
86C4⋊Q8—2542no—
87C4⋊Q8—2541no—
88C4⋊Q8C2×D12711yescharacteristic
89C4×Q8—2542no—
90C4×Q8—2541no—
91#104—2542no—
92#104—2541no—
93#101—1271no—
94#101C2×D12712yes—
95#41—5081no—
96#41—2542no—
97#41—2541no—
98C2×Q8—1272no—
99C2×Q8—5081no—
100C2×Q8—2541no—
101C2×Q8—2541no—
102C2×Q8D50811yescharacteristic
103C2×Q8C22×D12711yescharacteristic
104Q16—5082no—
105Q16—5081no—
106SD16—5082no—
107SD16—5081no—
108M16—22no—
109M16—2541no—
110C4⋊C4—22no—
111C4⋊C4—2542no—
112C4⋊C4—2541no—
113C4⋊C4—2542no—
114C4⋊C4—2542no—
115C42—1272no—
116C42—2542no—
117C42—2541no—
118C42D50811yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—10161no—
122Q8—5082no—
123Q8—2542no—
124Q8—2541no—
125Q8—2544no—
126D4—5081no—
127D4—2542no—
128D4—2541no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—5082no—
132C2×C4—5081no—
133C2×C4—2542no—
134C2×C4—2542no—
135C2×C4D4×D12712yes—
136C2×C4C2×D50811yescharacteristic
137C8—42no—
138C8—5081no—
139C22—5081no—
140C22#1686117911yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—5081no—
146C4—5082no—
147C4—2544no—
148C2—21no—
149C2—10161no—
150C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#356×C127C2
socleC2×C127B(4,𝔽2)
prime radical#16861257C22
hypercenter#356D127
hypocenterC127#2205
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #22550606C2semidirect
4. #22549674C2semidirect
5. #356×C127C2semidirect
76. C127#2205semidirect
81. #356D127semidirect

Maximal subgroups

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

subgroupindex
#225506062
#225500612
#225500582
#225496742
#356×C1272
#2205127