Finite Group Databasebeta

← #30638208

Subgroups — #30638208

Order 16256.

198 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550252C211yesmaximal, characteristic, complemented
2#22550072C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic, complemented
4#22549953C211yesmaximal, characteristic, complemented
5#22549626C211yesmaximal, characteristic, complemented
6#22549619C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×D1016—21no—
9C2×#14523233—21no—
10C2×#14523233—21no—
11C2×#14523233—21no—
12C2×#14523231—21no—
13#16861206C2211yescharacteristic
14C4×D508C2211yescharacteristic
15C4⋊(C127⋊C8)C2211yescharacteristic
16#16861131—21no—
17#16861130—21no—
18(C2×D127)⋊C8—21no—
19C4×C127⋊C8C2211yescharacteristic
20#127×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22C4⋊C8×C127C2211yescharacteristic
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508D411yescharacteristic
27C2×C4×D127—21no—
28D1016—42no—
29#14523233—42no—
30#14523233—42no—
31#14523233—42no—
32#14523231—42no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8D411yescharacteristic
35C2×C127⋊C8D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37(C2×D127)⋊C4—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C2×C8×C127—21no—
42C2×C8×C127—21no—
43C42×C127C2311yescharacteristic
44C22×D127—21no—
45C22×D127—41no—
46D508—21no—
47D508—21no—
48D508—41no—
49D508—41no—
50D508—41no—
51D508—41no—
52D508—81no—
53C2×Dic127—21no—
54C4×D127—41no—
55C4×D127—41no—
56C127⋊C8—21no—
57C127⋊C8—41no—
58C127⋊C8D812yescomplemented
59C23×C127—41no—
60D4×C127—41no—
61D4×C127—41no—
62D4×C127—41no—
63D4×C127—41no—
64D4×C127—81no—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127#4111yescharacteristic
68C8×C127—41no—
69C8×C127—41no—
70C22×C127—81no—
71C22×C127—81no—
72C22×C127—81no—
73C22×C127#12111yescharacteristic
74C2×D127—41no—
75C2×D127—41no—
76C2×D127—41no—
77C2×D127—81no—
78C2×D127—81no—
79C2×D127—81no—
80C4×C127—21no—
81C4×C127C2×D811yescharacteristic
82C4×C127AGL(1,ℤ/8)11yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127AGL(1,ℤ/8)11yescharacteristic
85Dic127—41no—
86C2×C127—161no—
87C2×C127#45911yescharacteristic
88C2×C127#46911yescharacteristic
89C2×C127#47211yescharacteristic
90D127—81no—
91D127—161no—
92#1687—1271nomaximal, Sylow
93C127#168711yescharacteristic, complemented, Sylow
94#493—1271no—
95#459—1271no—
96#459—1271no—
97#436—1271no—
98#334—1271no—
99#331—1271no—
100#331D12711yescharacteristic, complemented
101C2×D8—2541no—
102C2×D8—2541no—
103C2×D8—2541no—
104C2×D8—2541no—
105C2×M16—2541no—
106#127—1271no—
107#127C2×D12711yescharacteristic
108C4×D4—1271no—
109C4⋊C8—1271no—
110C4⋊C8C2×D12711yescharacteristic
111C4⋊C8C2×D12711yescharacteristic
112#102—2541no—
113#102—2541no—
114C22⋊C8—2541no—
115C4×C8—1271no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—1271no—
120C2×D4—2541no—
121C2×D4—2541no—
122C2×D4—2541no—
123C22×C4—2541no—
124D8—5082no—
125D8—5082no—
126D8—5082no—
127D8—5082no—
128M16—5082no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—1271no—
132C2×C8—1271no—
133C2×C8—2541no—
134C4⋊C4—1271no—
135C22⋊C4—2541no—
136C42C22×D12711yescharacteristic
137C23—41no—
138C23—5081no—
139C23—2541no—
140D4—41no—
141D4—41no—
142D4—41no—
143D4—41no—
144D4—81no—
145D4—10161no—
146D4—5081no—
147D4—5081no—
148D4—5081no—
149D4—5081no—
150D4—2541no—
151D4—2541no—
152C2×C4—5081no—
153C2×C4—5081no—
154C2×C4—2541no—
155C2×C4D4×D12711yescharacteristic
156C2×C4D4×D12711yescharacteristic
157C2×C4#1452324811yescharacteristic
158C8—41no—
159C8—41no—
160C8—1272no—
161C8—5081no—
162C8—2541no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—10161no—
167C22—10161no—
168C22—10161no—
169C22—5081no—
170C22—5081no—
171C22—5081no—
172C22#1686118211yescharacteristic
173C4—21no—
174C4—5081no—
175C4D8×D12711yescharacteristic
176C4#1686123311yescharacteristic
177C4#1686123611yescharacteristic
178C4#1686123611yescharacteristic
179C2—161no—
180C2—20321no—
181C2—10161no—
182C2#2255007011yescharacteristic
183C2#2255012811yescharacteristic
184C2#2255014511yescharacteristic
subgroupG/H
centerC22#16861182
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#121
hypercenter#331D127
hypocenterC127#1687
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #22550252C2semidirect
2. #22550072C2semidirect
3. #22550068C2semidirect
4. #22549953C2semidirect
5. #22549626C2semidirect
6. #22549619C2semidirect
7. #331×C127C2semidirect
58. C127⋊C8D8semidirect
93. C127#1687semidirect
100. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225502522
#225500722
#225500682
#225499532
#225496262
#225496192
#331×C1272
#1687127