Finite Group Databasebeta

← #30637834

Subgroups — #30637834

Order 16256.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550541C211yesmaximal, characteristic, complemented
2#22550224C211yesmaximal, characteristic, complemented
3#22550077C211yesmaximal, characteristic, complemented
4#22550072C211yesmaximal, characteristic
5#22549606C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×Q8×D127—21no—
9C2×#14523237—21no—
10C2×D1016—21no—
11#16861208—21no—
12#16861206C2211yescharacteristic
13#16861187—22no—
14#16861185—21no—
15C4×Dic254C2211yescharacteristic
16C4×D508C2211yescharacteristic
17#16861171—21no—
18#16861131—21no—
19#16861131—21no—
20#16861129—22no—
21C4×C127⋊C8C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254D411yescharacteristic
27Q8×D127—42no—
28Q8×D127—42no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508D411yescharacteristic
33C2×C4×D127—22no—
34C2×C4×D127—21no—
35#14523237—42no—
36D1016—42no—
37C2×C127⋊C8—21no—
38C4⋊Dic127D411yescharacteristic
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—22no—
41Dic127⋊C4—21no—
42(C2×D127)⋊C4—22no—
43(C2×D127)⋊C4—21no—
44C4×Dic127—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—41no—
53Dic254—21no—
54Dic254—44no—
55Dic254D812yescomplemented
56D508—21no—
57D508—42no—
58D508—41no—
59D508—41no—
60D508—41no—
61D508—41no—
62D508SD1612yescomplemented
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C4×D127—44no—
67C4×D127—41no—
68C4×D127—41no—
69C127⋊C8—41no—
70Q8×C127—41no—
71Q8×C127—41no—
72C2×C4×C127—22no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—41no—
77C8×C127—41no—
78C22×C127#12011yescharacteristic
79C2×D127—22no—
80C2×D127—41no—
81C2×D127—41no—
82C2×D127—81no—
83C2×D127—81no—
84C2×D127—81no—
85C4×C127—21no—
86C4×C127—42no—
87C4×C127C2×D811yescharacteristic
88C4×C127C2×SD1611yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C4×C127#13711yescharacteristic
91Dic127—22no—
92Dic127—41no—
93Dic127—41no—
94C2×C127#44711yescharacteristic
95C2×C127#44811yescharacteristic
96C2×C127#45411yescharacteristic
97D127—42no—
98D127—161no—
99#1627—1271nomaximal, Sylow
100C127#162711yescharacteristic, complemented, Sylow
101D4×Q8—1271no—
102#486—1271no—
103#460—1271no—
104#459—1271no—
105#332D12711yescharacteristic, complemented
106Q8⋊C8—1271no—
107D4⋊C8—1271no—
108C22×Q8—2541no—
109C2×SD16—2541no—
110C2×D8—2541no—
111C4⋊Q8—2541no—
112C4⋊Q8C2×D12711yescharacteristic
113#127—1271no—
114C22⋊Q8—2541no—
115C22⋊Q8—2542no—
116C4×Q8—1271no—
117C4×D4—1271no—
118C4×D4—2541no—
119C4⋊C8C2×D12711yescharacteristic
120C4⋊C8C2×D12711yescharacteristic
121#102—2541no—
122#102—2541no—
123#102—2542no—
124C4×C8—1271no—
125C2×Q8—21no—
126C2×Q8—1271no—
127C2×Q8—5082no—
128C2×Q8—5082no—
129C2×Q8—2542no—
130C2×D4—1271no—
131C2×D4—2541no—
132C2×D4—2541no—
133C2×D4—2541no—
134C22×C4—2542no—
135C22×C4—2541no—
136SD16—5082no—
137D8—5082no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2541no—
141C4⋊C4—21no—
142C4⋊C4—1271no—
143C4⋊C4—1271no—
144C4⋊C4—2542no—
145C4⋊C4—2541no—
146C4⋊C4D50812yes—
147C22⋊C4—2542no—
148C22⋊C4—2541no—
149C42—2541no—
150C42C22×D12711yescharacteristic
151C23—5081no—
152C23—2541no—
153Q8—41no—
154Q8—41no—
155Q8—1272no—
156Q8—5084no—
157Q8—2541no—
158D4—1272no—
159D4—5082no—
160D4—5081no—
161D4—5081no—
162D4—5081no—
163D4—5081no—
164D4—2541no—
165C2×C4—22no—
166C2×C4—5084no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—2541no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×D50811yescharacteristic
175C8—41no—
176C8—41no—
177C8—5081no—
178C22—10161no—
179C22—10161no—
180C22—10161no—
181C22—5081no—
182C22—5081no—
183C22—2542no—
184C22#1686117911yescharacteristic
185C4—21no—
186C4—42no—
187C4—5081no—
188C4—5081no—
189C4—2542no—
190C4D8×D12711yescharacteristic
191C4SD16×D12711yescharacteristic
192C4#1686123611yescharacteristic
193C4#1686124111yescharacteristic
194C2—20321no—
195C2—5082no—
196C2#2255001111yescharacteristic
197C2#2255001611yescharacteristic
198C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127#120
prime radical#22550072C2
hypercenter#332D127
hypocenterC127#1627
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. #22550541C2semidirect
2. #22550224C2semidirect
3. #22550077C2semidirect
5. #22549606C2semidirect
6. #22549603C2semidirect
7. #332×C127C2semidirect
55. Dic254D8semidirect
62. D508SD16semidirect
100. C127#1627semidirect
105. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225505412
#225502242
#225500772
#225500722
#225496062
#225496032
#332×C1272
#1627127