Finite Group Databasebeta

← #30637802

Subgroups — #30637802

Order 16256.

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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#22550250C211yesmaximal, characteristic, complemented
2#22550076C211yesmaximal, characteristic, complemented
3#22550071C211yesmaximal, characteristic
4#22549625C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#554×C127C211yesmaximal, characteristic, complemented
8C2×#14523235—21no—
9C2×#14523235—22no—
10C2×#14523233—21no—
11#16861206C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13C4⋊(C127⋊C8)C2211yescharacteristic
14#16861129—21no—
15#16861129—21no—
16C4×C127⋊C8C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#125×C127—21no—
19C22⋊Q8×C127—21no—
20C22⋊Q8×C127—22no—
21C4×Q8×C127C2211yescharacteristic
22C4×D4×C127—21no—
23C4×D4×C127C2211yescharacteristic
24C2×C4⋊C4×C127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28#14523235—42no—
29#14523235—44no—
30#14523233—42no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×Q8×C127D411yescharacteristic
35C2×Q8×C127D412yes—
36C2×D4×C127D411yescharacteristic
37C22×C4×C127—22no—
38C22×C4×C127—21no—
39C4⋊C4×C127—22no—
40C4⋊C4×C127—22no—
41C4⋊C4×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127D411yescharacteristic
45C4⋊C4×C127D411yescharacteristic
46C22⋊C4×C127—22no—
47C22⋊C4×C127—21no—
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—41no—
51D508—42no—
52D508—41no—
53D508—41no—
54D508—41no—
55D508—41no—
56C127⋊C8—22no—
57C127⋊C8—41no—
58C127⋊C8—41no—
59C23×C127—21no—
60Q8×C127—21no—
61Q8×C127—21no—
62Q8×C127—42no—
63D4×C127—21no—
64D4×C127SD1612yescomplemented
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—22no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—82no—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C22×C127—22no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127#12011yescharacteristic
79C2×D127—81no—
80C2×D127—81no—
81C2×D127—81no—
82C4×C127—21no—
83C4×C127—41no—
84C4×C127—41no—
85C4×C127—41no—
86C4×C127—42no—
87C4×C127C2×SD1611yescharacteristic
88C4×C127#13511yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C4×C127#13711yescharacteristic
91C2×C127—42no—
92C2×C127#44811yescharacteristic
93C2×C127#44911yescharacteristic
94C2×C127#45311yescharacteristic
95D127—161no—
96#1622—1271nomaximal, Sylow
97C127#162211yescharacteristic, complemented, Sylow
98#554D12711yescharacteristic, complemented
99#492—1271no—
100#460—1271no—
101#459—1271no—
102#333—1271no—
103Q8⋊C8—1271no—
104D4⋊C8—1271no—
105C2×SD16—2541no—
106C2×SD16—2542no—
107C2×D8—2541no—
108C4⋊Q8C2×D12711yescharacteristic
109#127—1271no—
110#125—21no—
111C22⋊Q8—21no—
112C22⋊Q8—22no—
113C4×Q8C2×D12711yescharacteristic
114C4×D4—21no—
115C4×D4C2×D12711yescharacteristic
116C2×C4⋊C4—21no—
117C4⋊C8—1271no—
118C4⋊C8—1271no—
119#102—2541no—
120#102—2541no—
121C4×C8—1271no—
122C2×Q8C22⋊D12711yescharacteristic
123C2×Q8C22⋊D12712yes—
124C2×D4—2541no—
125C2×D4—2541no—
126C2×D4—2541no—
127C2×D4C22⋊D12711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130SD16—5082no—
131SD16—5084no—
132D8—5082no—
133C2×C8—2541no—
134C2×C8—2541no—
135C2×C8—2541no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4C22⋊D12711yescharacteristic
142C4⋊C4C22⋊D12711yescharacteristic
143C22⋊C4—22no—
144C22⋊C4—21no—
145C42—21no—
146C42C22×D12711yescharacteristic
147C23—21no—
148C23—5081no—
149Q8—21no—
150Q8—21no—
151Q8—42no—
152D4—21no—
153D4—5082no—
154D4—5081no—
155D4—5081no—
156D4—5081no—
157D4—5081no—
158D4#1452323512yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4C2×C22⋊D12711yescharacteristic
167C2×C4C2×C22⋊D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C8—5081no—
170C8—5081no—
171C8—2542no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—10161no—
176C22—10161no—
177C22—10161no—
178C22#1686117811yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—42no—
184C4C2×#1452323511yescharacteristic
185C4#1686122911yescharacteristic
186C4#1686123411yescharacteristic
187C4#1686124011yescharacteristic
188C2—42no—
189C2—20321no—
190C2#2255001511yescharacteristic
191C2#2255002111yescharacteristic
192C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#554×C127C2
socleC22×C127#120
prime radical#22550071C2
hypercenter#554D127
hypocenterC127#1622
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #22550250C2semidirect
2. #22550076C2semidirect
4. #22549625C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
7. #554×C127C2semidirect
64. D4×C127SD16semidirect
97. C127#1622semidirect
98. #554D127semidirect
158. D4#14523235semidirect

Maximal subgroups

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

subgroupindex
#225502502
#225500762
#225500712
#225496252
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
#554×C1272
#1622127