Finite Group Databasebeta

← #30637916

Subgroups — #30637916

Order 16256.

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

normal subgroups only

This group has 212 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#22550222C211yesmaximal, characteristic
2#22550161C211yesmaximal, characteristic, complemented
3#22550086C211yesmaximal, characteristic, complemented
4#22549620C211yesmaximal, characteristic
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#559×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861209C2211yescharacteristic
10#16861140—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861133—21no—
14#16861132—21no—
15#16861130—22no—
16C4×C127⋊C8C2211yescharacteristic
17#127×C127C2211yescharacteristic
18#124×C127—21no—
19#124×C127—21no—
20#123×C127—22no—
21C4×Q8×C127C2211yescharacteristic
22C4×Q8×C127C2211yescharacteristic
23C4×D4×C127—21no—
24C4×D4×C127—21no—
25C2×Dic254—21no—
26#14523238—42no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×Q8×C127D411yescharacteristic
33C2×Q8×C127D411yescharacteristic
34C2×D4×C127—21no—
35C2×D4×C127—21no—
36C2×D4×C127D412yes—
37C22×C4×C127—22no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127—21no—
40C4⋊C4×C127D411yescharacteristic
41C4⋊C4×C127D411yescharacteristic
42C22⋊C4×C127—22no—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—22no—
45C42×C127—21no—
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—41no—
49Dic254—41no—
50C2×Dic127—41no—
51C127⋊C8—41no—
52C127⋊C8—41no—
53C127⋊C8—41no—
54C23×C127—22no—
55Q8×C127—21no—
56Q8×C127—21no—
57Q8×C127—21no—
58Q8×C127—21no—
59D4×C127—42no—
60D4×C127—41no—
61D4×C127—41no—
62D4×C127—41no—
63D4×C127—41no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—82no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C22×C127—42no—
73C22×C127—42no—
74C22×C127—42no—
75C22×C127#12011yescharacteristic
76C4×C127—21no—
77C4×C127—41no—
78C4×C127—41no—
79C4×C127—41no—
80C4×C127—41no—
81C4×C127#13511yescharacteristic
82C4×C127#13511yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127#13711yescharacteristic
85Dic127—81no—
86C2×C127—82no—
87C2×C127#44911yescharacteristic
88C2×C127#45211yescharacteristic
89C2×C127#45511yescharacteristic
90#1640—1271nomaximal, Sylow
91C127#164011yescharacteristic, complemented, Sylow
92#559D12711yescharacteristic
93#486—1271no—
94#475—1271no—
95#462—1271no—
96#331—1271no—
97Q8⋊C8—1271no—
98Q8⋊C8—1271no—
99C2×Q16—2541no—
100C4⋊Q8—1271no—
101#127C2×D12711yescharacteristic
102#124—21no—
103#124—21no—
104#123—22no—
105C4×Q8C2×D12711yescharacteristic
106C4×Q8C2×D12711yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109#106—2541no—
110C4⋊C8—1271no—
111C4⋊C8—1271no—
112#103—2541no—
113#103—2541no—
114#102—2542no—
115C4×C8—1271no—
116C2×Q8—2541no—
117C2×Q8C22⋊D12711yescharacteristic
118C2×Q8C22⋊D12711yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D12712yes—
122C22×C4—22no—
123Q16—5082no—
124C2×C8—2541no—
125C2×C8—2541no—
126C2×C8—2541no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4C22⋊D12711yescharacteristic
132C4⋊C4C22⋊D12711yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C22×D12711yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—5081no—
145Q8—5081no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—82no—
156C2×C4—5081no—
157C2×C4C2×C22⋊D12711yescharacteristic
158C2×C4C2×C22⋊D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—5081no—
161C8—5081no—
162C8—5081no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1686117811yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—10161no—
173C4#1686122911yescharacteristic
174C4#1686122911yescharacteristic
175C4#1686123411yescharacteristic
176C4#1686123911yescharacteristic
177C2—82no—
178C2#2255002111yescharacteristic
179C2#2255003811yescharacteristic
180C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#559×C127C2
socleC22×C127#120
prime radical#127×C127C22
hypercenter#559D127
hypocenterC127#1640
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #22550161C2semidirect
3. #22550086C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. Q8⋊(C127⋊C8)C2semidirect
91. C127#1640semidirect

Maximal subgroups

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

subgroupindex
#225502222
#225501612
#225500862
#225496202
Q8⋊(C127⋊C8)2
#559×C1272
#1640127