Finite Group Databasebeta

← #30640758

Subgroups — #30640758

Order 16256.

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

normal subgroups only

This group has 236 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#22550592C211yesmaximal, characteristic, complemented
2#22550058C211yesmaximal, characteristic, complemented
3#22550051C212yesmaximal, complemented
4#22549672C212yesmaximal, complemented
5#356×C127C211yesmaximal, characteristic, complemented
6#16861257C2211yescharacteristic
7#16861240—21no—
8#16861237—22no—
9#16861198—21no—
10#16861198C2212yes—
11#16861184—22no—
12C4×D508—21no—
13#16861167—22no—
14#16861137—21no—
15#16861135—22no—
16#16861128C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#101×C127C2212yes—
19#14523249—41no—
20#14523248—22no—
21#14523248—21no—
22Q8×D127—41no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508D412yes—
26C2×C4×D127—22no—
27#14523238—41no—
28#14523237—42no—
29#14523235—41no—
30D1016—42no—
31#14523232—21no—
32C4⋊Dic127—21no—
33(C2×D127)⋊C4—22no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36C4×Dic127—21no—
37C4×Dic127D412yes—
38C2×Q8×C127D411yescharacteristic
39C2×Q8×C127C2311yescharacteristic
40M16×C127—22no—
41C4⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43C22×D127—22no—
44Dic254—22no—
45Dic254—21no—
46D508—22no—
47D508—21no—
48D508—42no—
49D508—42no—
50D508—81no—
51C22⋊D127—41no—
52C2×Dic127—22no—
53C4×D127—44no—
54C4×D127—42no—
55C4×D127—41no—
56C127⋊C8—41no—
57Q8×C127—41no—
58Q8×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127C2×D412yes—
62C2×C4×C127C2×D411yescharacteristic
63C8×C127—42no—
64C22×C127#12011yescharacteristic
65C2×D127—42no—
66C2×D127—42no—
67C2×D127—41no—
68C2×D127—42no—
69C4×C127—22no—
70C4×C127—22no—
71C4×C127—21no—
72C4×C127—41no—
73Dic127—42no—
74Dic127—41no—
75C2×C127—21no—
76C2×C127B(4,𝔽2)11yescharacteristic
77D127—81no—
78D127—82no—
79#2204—1271nomaximal, Sylow
80C127#220411yescharacteristic, complemented, Sylow
81#555—1271no—
82#456—1271no—
83#454—1272no—
84#356D12711yescharacteristic, complemented
85#355—1272no—
862−1+4—1271no—
87#137—2541no—
88AGL(1,ℤ/8)—2542no—
89C4⋊Q8C2×D12711yescharacteristic
90#124—1272no—
91#124—2541no—
92#121—2542no—
93C4×D4—2541no—
94#117—2542no—
95#104—2542no—
96#104—2541no—
97#101—1271no—
98#101C2×D12712yes—
99#41—5081no—
100#41—2542no—
101#41—2541no—
102C2×Q8—5081no—
103C2×Q8D50811yescharacteristic
104C2×Q8C22×D12711yescharacteristic
105C2×D4—1272no—
106C2×D4—2541no—
107C2×D4—2541no—
108C22×C4—2542no—
109Q16—5081no—
110SD16—5082no—
111SD16—5081no—
112D8—5082no—
113M16—22no—
114M16—2541no—
115C4⋊C4—22no—
116C4⋊C4—2541no—
117C22⋊C4—2542no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C42—1272no—
121C42—2541no—
122C42D50811yescharacteristic
123C23—2542no—
124Q8—41no—
125Q8—41no—
126Q8—2542no—
127Q8—2541no—
128D4—10161no—
129D4—5082no—
130D4—5082no—
131D4—5081no—
132D4—2542no—
133D4—2541no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—5084no—
137C2×C4—5082no—
138C2×C4—5081no—
139C2×C4—2542no—
140C2×C4D4×D12712yes—
141C2×C4C2×D50811yescharacteristic
142C8—42no—
143C8—5081no—
144C22—5082no—
145C22—5082no—
146C22—5081no—
147C22—5082no—
148C22#1686117911yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—5082no—
154C4—5081no—
155C2—21no—
156C2—10162no—
157C2—10161no—
158C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#356×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#356D127
hypocenterC127#2204
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. #22550592C2semidirect
2. #22550058C2semidirect
3. #22550051C2semidirect
4. #22549672C2semidirect
5. #356×C127C2semidirect
80. C127#2204semidirect
84. #356D127semidirect

Maximal subgroups

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

subgroupindex
#225505922
#225500582
#225500512
#225496722
#356×C1272
#2204127