Finite Group Databasebeta

← #30637836

Subgroups — #30637836

Order 16256.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic254C211yesmaximal, characteristic, complemented
2#22550223C211yesmaximal, characteristic, complemented
3#22550077C211yesmaximal, characteristic, complemented
4#22550071C211yesmaximal, characteristic
5#22549623C211yesmaximal, characteristic, complemented
6#22549608C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C22×Dic254—21no—
9C2×#14523237—21no—
10C2×#14523233—21no—
11#16861209C2211yescharacteristic
12#16861208—21no—
13#16861206C2211yescharacteristic
14#16861188—21no—
15#16861186—22no—
16C4×Dic254C2211yescharacteristic
17D4×Dic127—21no—
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861131—21no—
20#16861131—22no—
21#16861129—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254—42no—
28C2×Dic254—42no—
29C2×Dic254D411yescharacteristic
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C22×Dic127—22no—
34#14523237—42no—
35#14523233—42no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D412yes—
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—22no—
41Dic127⋊C4—21no—
42C22⋊Dic127—22no—
43C4×Dic127—21no—
44C2×D4×C127D411yescharacteristic
45C22×C4×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127—21no—
48C4⋊C4×C127D411yescharacteristic
49C22⋊C4×C127—21no—
50C42×C127C2311yescharacteristic
51C22×D127—41no—
52Dic254—21no—
53Dic254—44no—
54Dic254—41no—
55Dic254—41no—
56Dic254D812yescomplemented
57D508—42no—
58D508—41no—
59D508—41no—
60D508—41no—
61D508—41no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—22no—
65C2×Dic127—44no—
66C127⋊C8—41no—
67C23×C127—21no—
68D4×C127—21no—
69D4×C127SD1612yescomplemented
70C2×C4×C127—21no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—41no—
77C8×C127—41no—
78C22×C127—22no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127#12011yescharacteristic
82C2×D127—81no—
83C2×D127—81no—
84C2×D127—81no—
85C4×C127—21no—
86C4×C127—41no—
87C4×C127C2×D811yescharacteristic
88C4×C127C2×SD1611yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C4×C127#13711yescharacteristic
91Dic127—22no—
92Dic127—42no—
93Dic127—41no—
94C2×C127—42no—
95C2×C127#44711yescharacteristic
96C2×C127#44811yescharacteristic
97C2×C127#45411yescharacteristic
98D127—161no—
99#1627—1271nomaximal, Sylow
100C127#162711yescharacteristic, complemented, Sylow
101D4×Q8—1271no—
102#486—1271no—
103#460—1271no—
104#459—1271no—
105#332—1271no—
106Q8⋊C8—1271no—
107D4⋊C8D12711yescharacteristic, complemented
108C22×Q8—2541no—
109C2×SD16—2541no—
110C2×D8—2541no—
111C4⋊Q8—1271no—
112C4⋊Q8—2541no—
113#127—1271no—
114C22⋊Q8—2542no—
115C22⋊Q8—2541no—
116C4×Q8—1271no—
117C4×D4—2541no—
118C4×D4C2×D12711yescharacteristic
119C4⋊C8—1271no—
120C4⋊C8C2×D12711yescharacteristic
121#102—2541no—
122#102—2542no—
123#102—2541no—
124C4×C8C2×D12711yescharacteristic
125C2×Q8—1271no—
126C2×Q8—5082no—
127C2×Q8—5082no—
128C2×Q8—2542no—
129C2×Q8—2541no—
130C2×D4—2541no—
131C2×D4—2541no—
132C2×D4—2541no—
133C2×D4D50811yescharacteristic
134C22×C4—21no—
135C22×C4—2542no—
136SD16—5082no—
137D8—5082no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2541no—
141C4⋊C4—1271no—
142C4⋊C4—1272no—
143C4⋊C4—2542no—
144C4⋊C4—2541no—
145C4⋊C4—2541no—
146C4⋊C4D50811yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—2542no—
149C42—2541no—
150C42C22×D12711yescharacteristic
151C23—21no—
152C23—5081no—
153Q8—1272no—
154Q8—5084no—
155Q8—5081no—
156Q8—5081no—
157Q8—2541no—
158D4—21no—
159D4—5082no—
160D4—5081no—
161D4—5081no—
162D4—5081no—
163D4—5081no—
164D4#1452323712yescomplemented
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—5084no—
169C2×C4—2541no—
170C2×C4—2542no—
171C2×C4—2541no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×D50811yescharacteristic
175C8—41no—
176C8—41no—
177C8—5081no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—10161no—
182C22—10161no—
183C22—10161no—
184C22#1686117911yescharacteristic
185C4—21no—
186C4—41no—
187C4—5081no—
188C4—5082no—
189C4—2542no—
190C4D8×D12711yescharacteristic
191C4C2×#1452323711yescharacteristic
192C4#1686123611yescharacteristic
193C4#1686124211yescharacteristic
194C2—42no—
195C2—20321no—
196C2#2255001111yescharacteristic
197C2#2255001711yescharacteristic
198C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
prime radical#22550071C2
hypercenterD4⋊C8D127
hypocenterC127#1627
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. D4×Dic254C2semidirect
2. #22550223C2semidirect
3. #22550077C2semidirect
5. #22549623C2semidirect
6. #22549608C2semidirect
7. D4⋊C8×C127C2semidirect
56. Dic254D8semidirect
69. D4×C127SD16semidirect
100. C127#1627semidirect
107. D4⋊C8D127semidirect
164. D4#14523237semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2542
#225502232
#225500772
#225500712
#225496232
#225496082
D4⋊C8×C1272
#1627127