Finite Group Databasebeta

← #30637805

Subgroups — #30637805

Order 16256.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550586C211yesmaximal, characteristic, complemented
2#22550249C211yesmaximal, characteristic, complemented
3#22550076C211yesmaximal, characteristic, complemented
4#22550072C211yesmaximal, characteristic
5#22549624C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523237—22no—
9C2×#14523235—21no—
10C2×D1016—21no—
11#16861209C2211yescharacteristic
12#16861206C2211yescharacteristic
13#16861201—21no—
14#16861187—22no—
15#16861185—21no—
16C4×D508—21no—
17C4×D508C2211yescharacteristic
18C4⋊C4×D127—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861131—21no—
21#16861129—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254D412yes—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508D411yescharacteristic
30C2×C4×D127—22no—
31C2×C4×D127—21no—
32#14523237—44no—
33#14523235—42no—
34D1016—42no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—22no—
40Dic127⋊C4—22no—
41(C2×D127)⋊C4—22no—
42(C2×D127)⋊C4—21no—
43C2×Q8×C127D411yescharacteristic
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51C22×D127—41no—
52Dic254—42no—
53D508—21no—
54D508—42no—
55D508—41no—
56D508—41no—
57D508—41no—
58D508—41no—
59D508SD1612yescomplemented
60C2×Dic127—21no—
61C2×Dic127—22no—
62C4×D127—41no—
63C4×D127—41no—
64C4×D127—82no—
65C127⋊C8—41no—
66Q8×C127—21no—
67Q8×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—22no—
74C8×C127—41no—
75C22×C127#12011yescharacteristic
76C2×D127—22no—
77C2×D127—41no—
78C2×D127—41no—
79C2×D127—81no—
80C2×D127—81no—
81C2×D127—81no—
82C4×C127—21no—
83C4×C127—41no—
84C4×C127—41no—
85C4×C127C2×SD1611yescharacteristic
86C4×C127#13511yescharacteristic
87C4×C127AGL(1,ℤ/8)11yescharacteristic
88C4×C127#13711yescharacteristic
89Dic127—41no—
90Dic127—42no—
91C2×C127#44811yescharacteristic
92C2×C127#44911yescharacteristic
93C2×C127#45311yescharacteristic
94D127—42no—
95D127—161no—
96#1622—1271nomaximal, Sylow
97C127#162211yescharacteristic, complemented, Sylow
98#554—1271no—
99#492—1271no—
100#460—1271no—
101#459—1271no—
102#333—1271no—
103Q8⋊C8D12711yescharacteristic, complemented
104D4⋊C8—1271no—
105C2×SD16—2542no—
106C2×SD16—2541no—
107C2×D8—2541no—
108C4⋊Q8—1271no—
109#127—1271no—
110#125—2541no—
111C22⋊Q8—2542no—
112C22⋊Q8—2541no—
113C4×Q8C2×D12711yescharacteristic
114C4×D4—1271no—
115C4×D4—2541no—
116C2×C4⋊C4—2541no—
117C4⋊C8—1271no—
118C4⋊C8C2×D12711yescharacteristic
119#102—2541no—
120#102—2541no—
121C4×C8C2×D12711yescharacteristic
122C2×Q8—1272no—
123C2×Q8D50811yescharacteristic
124C2×D4—1271no—
125C2×D4—2541no—
126C2×D4—2541no—
127C2×D4—2541no—
128C22×C4—2542no—
129C22×C4—2541no—
130SD16—5084no—
131SD16—5082no—
132D8—5082no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2541no—
136C4⋊C4—21no—
137C4⋊C4—1271no—
138C4⋊C4—2542no—
139C4⋊C4—2542no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4D50811yescharacteristic
143C22⋊C4—2542no—
144C22⋊C4—2541no—
145C42—21no—
146C42C22×D12711yescharacteristic
147C23—5081no—
148C23—2541no—
149Q8—21no—
150Q8—21no—
151Q8—5082no—
152D4—1272no—
153D4—5082no—
154D4—5081no—
155D4—5081no—
156D4—5081no—
157D4—5081no—
158D4—2541no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—10162no—
162C2×C4—5081no—
163C2×C4—5081no—
164C2×C4—2541no—
165C2×C4—2542no—
166C2×C4D4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C2×C4C2×D50811yescharacteristic
169C8—22no—
170C8—41no—
171C8—5081no—
172C22—10161no—
173C22—10161no—
174C22—10161no—
175C22—5081no—
176C22—5081no—
177C22—2542no—
178C22#1686117911yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—5081no—
183C4—5082no—
184C4SD16×D12711yescharacteristic
185C4#1686123111yescharacteristic
186C4#1686123711yescharacteristic
187C4#1686124111yescharacteristic
188C2—20321no—
189C2—5082no—
190C2#2255001611yescharacteristic
191C2#2255002311yescharacteristic
192C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127#120
prime radical#22550072C2
hypercenterQ8⋊C8D127
hypocenterC127#1622
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #22550586C2semidirect
2. #22550249C2semidirect
3. #22550076C2semidirect
5. #22549624C2semidirect
6. #22549605C2semidirect
7. Q8⋊C8×C127C2semidirect
59. D508SD16semidirect
97. C127#1622semidirect
103. Q8⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502492
#225500762
#225500722
#225496242
#225496052
Q8⋊C8×C1272
#1622127