Finite Group Databasebeta

← #30637877

Subgroups — #30637877

Order 16256.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550591C211yesmaximal, characteristic
2#22550256C211yesmaximal, characteristic, complemented
3#22550167C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×Dic508—22no—
9#16861209C2211yescharacteristic
10#16861209C2211yescharacteristic
11#16861195—22no—
12#16861183—21no—
13#16861180—21no—
14C4×D508C2211yescharacteristic
15#16861168—21no—
16#16861167—21no—
17#16861141—21no—
18#16861140—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861131—21no—
21#16861130—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254D412yes—
27C2×D508D411yescharacteristic
28C2×C22⋊D127—22no—
29C2×C4×D127—21no—
30Dic508—44no—
31C2×C127⋊C8—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37C22⋊Dic127—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—22no—
41C2×D4×C127D411yescharacteristic
42C22×C4×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C22⋊C4×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—42no—
50Dic254—42no—
51D508—21no—
52D508—21no—
53C22⋊D127—82no—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C2×Dic127—41no—
57C4×D127—41no—
58C4×D127—41no—
59C127⋊C8—41no—
60C23×C127—21no—
61D4×C127—21no—
62D4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—22no—
70C8×C127—41no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127—41no—
74C22×C127#12011yescharacteristic
75C2×D127—41no—
76C2×D127—41no—
77C2×D127—41no—
78C4×C127—21no—
79C4×C127—41no—
80C4×C127#13511yescharacteristic
81C4×C127#13511yescharacteristic
82C4×C127#13711yescharacteristic
83C4×C127#13711yescharacteristic
84Dic127—41no—
85Dic127—42no—
86Dic127—81no—
87C2×C127—81no—
88C2×C127#45211yescharacteristic
89C2×C127#45211yescharacteristic
90C2×C127#45311yescharacteristic
91D127—81no—
92#1634—1271nomaximal, Sylow
93C127#163411yescharacteristic, complemented, Sylow
94#555—1271no—
95#494—1271no—
96#476—1271no—
97#476—1271no—
98#333—1271no—
99D4⋊C8—1271no—
100D4⋊C8D12711yescharacteristic, complemented
101C2×Q16—2542no—
102C4⋊Q8—1271no—
103C4⋊Q8—1271no—
104#124—2542no—
105#121—2541no—
106#121—2541no—
107C4×D4—1271no—
108C4×D4C2×D12711yescharacteristic
109#117—2541no—
110#117—2541no—
111#106—2541no—
112#106—2541no—
113C4⋊C8—1271no—
114C4⋊C8C2×D12711yescharacteristic
115#102—2541no—
116#102—2541no—
117C4×C8C2×D12711yescharacteristic
118C2×Q8—1272no—
119C2×Q8—2541no—
120C2×D4—1271no—
121C2×D4—2542no—
122C2×D4D50811yescharacteristic
123C22×C4—21no—
124C22×C4—2541no—
125Q16—5084no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—2541no—
129C4⋊C4—1271no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C4⋊C4—2541no—
134C4⋊C4D50811yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—2542no—
137C22⋊C4—2542no—
138C22⋊C4—2541no—
139C42—2542no—
140C42C22×D12711yescharacteristic
141C23—21no—
142C23—2541no—
143Q8—5082no—
144Q8—5082no—
145D4—21no—
146D4—21no—
147D4—10162no—
148D4—2541no—
149D4—2541no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—5081no—
154C2×C4—5081no—
155C2×C4—5081no—
156C2×C4—2541no—
157C2×C4—2542no—
158C2×C4D4×D12711yescharacteristic
159C2×C4D4×D12711yescharacteristic
160C2×C4C2×D50811yescharacteristic
161C8—22no—
162C8—41no—
163C8—5081no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—5081no—
168C22—5081no—
169C22—5081no—
170C22#1686117911yescharacteristic
171C4—21no—
172C4—41no—
173C4—10161no—
174C4—5081no—
175C4—5082no—
176C4#1686122811yescharacteristic
177C4#1686123111yescharacteristic
178C4#1686123811yescharacteristic
179C4#1686124211yescharacteristic
180C2—81no—
181C2—10161no—
182C2#2255003911yescharacteristic
183C2#2255004011yescharacteristic
184C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
prime radical#22550591C2
hypercenterD4⋊C8D127
hypocenterC127#1634
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
2. #22550256C2semidirect
3. #22550167C2semidirect
4. #22550166C2semidirect
5. #22549624C2semidirect
6. #22549605C2semidirect
7. D4⋊C8×C127C2semidirect
93. C127#1634semidirect
100. D4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225502562
#225501672
#225501662
#225496242
#225496052
D4⋊C8×C1272
#1634127