Finite Group Databasebeta

← #30640756

Subgroups — #30640756

Order 16256.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550058C211yesmaximal, characteristic, complemented
2#22550048C212yesmaximal, complemented
3#22549675C211yesmaximal, characteristic, complemented
4#22549673C212yesmaximal, complemented
5#555×C127C211yesmaximal, characteristic, complemented
6#16861257C2211yescharacteristic
7#16861240—21no—
8#16861235—22no—
9#16861137—22no—
10#16861137—21no—
11#16861128C2212yes—
12#16861128C2211yescharacteristic
13C4⋊Q8×C127C2211yescharacteristic
14#124×C127—21no—
15#124×C127C2212yes—
16#121×C127—22no—
17C4×D4×C127—21no—
18#117×C127—22no—
19#14523249—41no—
20#14523248—22no—
21#14523248—21no—
22Q8×D127—41no—
23#14523238—41no—
24#14523236—42no—
25#14523235—41no—
26#14523233—42no—
27#14523232—22no—
28#14523232—21no—
29C2×Q8×C127D411yescharacteristic
30C2×Q8×C127C2311yescharacteristic
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127D412yes—
34C22×C4×C127—22no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C42×C127—21no—
41C42×C127D412yes—
42C42×C127D411yescharacteristic
43Dic254—22no—
44Dic254—21no—
45D508—22no—
46D508—21no—
47C22⋊D127—41no—
48C4×D127—42no—
49C4×D127—41no—
50C127⋊C8—42no—
51C127⋊C8—41no—
52C23×C127—22no—
53Q8×C127—41no—
54Q8×C127—41no—
55D4×C127—42no—
56D4×C127—42no—
57D4×C127—81no—
58C2×C4×C127—21no—
59C2×C4×C127—22no—
60C2×C4×C127—21no—
61C2×C4×C127—44no—
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127C2×D412yes—
64C22×C127—42no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127#12011yescharacteristic
68C2×D127—41no—
69C4×C127—22no—
70C4×C127—22no—
71C4×C127—21no—
72C4×C127—42no—
73C4×C127—41no—
74Dic127—41no—
75C2×C127—21no—
76C2×C127—82no—
77C2×C127B(4,𝔽2)11yescharacteristic
78D127—81no—
79#2204—1271nomaximal, Sylow
80C127#220411yescharacteristic, complemented, Sylow
81#555D12711yescharacteristic, complemented
82#456—1271no—
83#454—1272no—
84#356—1271no—
85#355—1272no—
862−1+4—1271no—
87#137—2541no—
88AGL(1,ℤ/8)—2542no—
89C4⋊Q8C2×D12711yescharacteristic
90#124—21no—
91#124C2×D12712yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—2542no—
96#104—2541no—
97#101—1271no—
98#101—1272no—
99#41—5081no—
100#41—2542no—
101#41—2541no—
102C2×Q8—5081no—
103C2×Q8C22⋊D12711yescharacteristic
104C2×Q8C22×D12711yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D12712yes—
108C22×C4—22no—
109Q16—5081no—
110SD16—5082no—
111SD16—5081no—
112D8—5082no—
113M16—2542no—
114M16—2541no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D12712yes—
122C42C22⋊D12711yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—2542no—
127Q8—2541no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—5081no—
132D4—2542no—
133D4—2541no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—5082no—
139C2×C4—5081no—
140C2×C4C2×C22⋊D12711yescharacteristic
141C2×C4C2×C22⋊D12712yes—
142C8—5082no—
143C8—5081no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—5081no—
148C22#1686117811yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—5081no—
155C2—21no—
156C2—82no—
157C2—10161no—
158C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#555×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#555D127
hypocenterC127#2204
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22550058C2semidirect
2. #22550048C2semidirect
3. #22549675C2semidirect
4. #22549673C2semidirect
5. #555×C127C2semidirect
80. C127#2204semidirect
81. #555D127semidirect

Maximal subgroups

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

subgroupindex
#225500582
#225500482
#225496752
#225496732
#555×C1272
#2204127