Finite Group Databasebeta

← #30637820

Subgroups — #30637820

Order 16256.

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

normal subgroups only

This group has 256 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#22550529C211yesmaximal, characteristic, complemented
2#22550224C211yesmaximal, characteristic, complemented
3#22550072C211yesmaximal, characteristic, complemented
4#22550072C211yesmaximal, characteristic, complemented
5#22549603C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9C2×D1016—21no—
10C2×D1016—21no—
11#16861206C2211yescharacteristic
12#16861194—22no—
13#16861185—21no—
14#16861184—21no—
15C4×D508C2211yescharacteristic
16C4×D508C2211yescharacteristic
17C4⋊C4×D127—21no—
18#16861131—21no—
19#16861131—21no—
20#16861129—22no—
21C4×C127⋊C8C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523249—42no—
26#14523249—42no—
27C2×D508—22no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C4×D127—22no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36D1016—42no—
37D1016—42no—
38C2×C127⋊C8—21no—
39C4⋊Dic127D411yescharacteristic
40C4⋊Dic127D411yescharacteristic
41Dic127⋊C4—22no—
42(C2×D127)⋊C4—22no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45C2×Q8×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127D412yes—
50C42×C127C2311yescharacteristic
51C22×D127—21no—
52C22×D127—21no—
53C22×D127—41no—
54D508—21no—
55D508—21no—
56D508—21no—
57D508—42no—
58D508—44no—
59D508—41no—
60D508—41no—
61D508—41no—
62D508—41no—
63D508D812yescomplemented
64C2×Dic127—21no—
65C2×Dic127—21no—
66C4×D127—44no—
67C4×D127—41no—
68C4×D127—41no—
69C4×D127—41no—
70C4×D127—41no—
71C127⋊C8—41no—
72Q8×C127—41no—
73Q8×C127—41no—
74C2×C4×C127—22no—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C8×C127—41no—
79C8×C127—41no—
80C22×C127#12011yescharacteristic
81C2×D127—22no—
82C2×D127—41no—
83C2×D127—41no—
84C2×D127—41no—
85C2×D127—41no—
86C2×D127—41no—
87C2×D127—81no—
88C2×D127—81no—
89C2×D127—81no—
90C4×C127—21no—
91C4×C127—42no—
92C4×C127C2×D811yescharacteristic
93C4×C127#13511yescharacteristic
94C4×C127AGL(1,ℤ/8)11yescharacteristic
95C4×C127AGL(1,ℤ/8)11yescharacteristic
96Dic127—41no—
97Dic127—41no—
98C2×C127#44711yescharacteristic
99C2×C127#44911yescharacteristic
100C2×C127#45411yescharacteristic
101D127—42no—
102D127—81no—
103D127—161no—
104#1625—1271nomaximal, Sylow
105C127#162511yescharacteristic, complemented, Sylow
106#547—1271no—
107#486—1271no—
108#459—1271no—
109#459—1271no—
110#332D12711yescharacteristic, complemented
111D4⋊C8—1271no—
112D4⋊C8—1271no—
113C2×#41—2541no—
114C2×D8—2541no—
115C2×D8—2541no—
116C4⋊Q8C2×D12711yescharacteristic
117#127—1271no—
118#123—2542no—
119C22⋊Q8—2541no—
120#121—2541no—
121C4×D4—1271no—
122C4×D4—1271no—
123C2×C4⋊C4—2541no—
124C4⋊C8C2×D12711yescharacteristic
125C4⋊C8C2×D12711yescharacteristic
126#102—2541no—
127#102—2541no—
128#102—2542no—
129C4×C8—1271no—
130#41—5082no—
131#41—5082no—
132C2×Q8—21no—
133C2×D4—1271no—
134C2×D4—1271no—
135C2×D4—2542no—
136C2×D4—2541no—
137C2×D4—2541no—
138C2×D4—2541no—
139C22×C4—2542no—
140C22×C4—2541no—
141C22×C4—2541no—
142D8—5082no—
143D8—5082no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—2541no—
147C4⋊C4—21no—
148C4⋊C4—1271no—
149C4⋊C4—1271no—
150C4⋊C4—2542no—
151C4⋊C4D50812yes—
152C22⋊C4—2542no—
153C22⋊C4—2541no—
154C22⋊C4—2541no—
155C42C22×D12711yescharacteristic
156C23—5081no—
157C23—2541no—
158C23—2541no—
159Q8—41no—
160Q8—41no—
161D4—1272no—
162D4—5082no—
163D4—5084no—
164D4—5081no—
165D4—5081no—
166D4—5081no—
167D4—5081no—
168D4—2541no—
169D4—2541no—
170D4—2541no—
171C2×C4—22no—
172C2×C4—5084no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—5081no—
176C2×C4—5081no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4D4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4C2×D50811yescharacteristic
182C8—41no—
183C8—41no—
184C8—5081no—
185C22—10161no—
186C22—10161no—
187C22—10161no—
188C22—5081no—
189C22—5081no—
190C22—5081no—
191C22—5081no—
192C22—5081no—
193C22—2542no—
194C22#1686117911yescharacteristic
195C4—21no—
196C4—42no—
197C4—5081no—
198C4—5081no—
199C4D8×D12711yescharacteristic
200C4#1686123211yescharacteristic
201C4#1686123611yescharacteristic
202C4#1686123611yescharacteristic
203C2—20321no—
204C2—10161no—
205C2—5082no—
206C2#2255001111yescharacteristic
207C2#2255002411yescharacteristic
208C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127#120
hypercenter#332D127
hypocenterC127#1625
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #22550529C2semidirect
2. #22550224C2semidirect
3. #22550072C2semidirect
4. #22550072C2semidirect
5. #22549603C2semidirect
6. #22549603C2semidirect
7. #332×C127C2semidirect
63. D508D8semidirect
105. C127#1625semidirect
110. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225505292
#225502242
#225500722
#225496032
#332×C1272
#1625127