Finite Group Databasebeta

← #30637872

Subgroups — #30637872

Order 16256.

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

normal subgroups only

This group has 206 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
1Q8×Dic254C211yesmaximal, characteristic
2#22550256C211yesmaximal, characteristic
3#22550172C211yesmaximal, characteristic
4#22550171C211yesmaximal, characteristic
5#22549624C211yesmaximal, characteristic
6#22549608C211yesmaximal, characteristic
7Q8⋊C8×C127C211yesmaximal, characteristic
8C2×Dic508—22no—
9#16861209—21no—
10#16861209C2211yescharacteristic
11#16861209C2211yescharacteristic
12#16861208—22no—
13#16861208—21no—
14Q8×Dic127—21no—
15C4×Dic254—21no—
16C4×Dic254C2211yescharacteristic
17#16861143—21no—
18#16861142—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861134—21no—
21#16861133—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254D411yescharacteristic
28C2×Dic254D412yes—
29Dic508—44no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38Dic127⋊C4—21no—
39C4×Dic127—22no—
40C4×Dic127—21no—
41C2×Q8×C127D411yescharacteristic
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—42no—
50Dic254—42no—
51Dic254—82no—
52Dic254Q1612yescomplemented
53C2×Dic127—21no—
54C2×Dic127—22no—
55C2×Dic127—21no—
56C2×Dic127—41no—
57C127⋊C8—41no—
58Q8×C127—21no—
59Q8×C127Q1612yescomplemented
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—22no—
66C8×C127—41no—
67C22×C127#12011yescharacteristic
68C4×C127—22no—
69C4×C127—21no—
70C4×C127—41no—
71C4×C127C2×Q1611yescharacteristic
72C4×C127C2×Q1611yescharacteristic
73C4×C127#13711yescharacteristic
74C4×C127#13711yescharacteristic
75Dic127—22no—
76Dic127—42no—
77Dic127—41no—
78Dic127—81no—
79C2×C127#45111yescharacteristic
80C2×C127#45111yescharacteristic
81C2×C127#45311yescharacteristic
82#1633—1271nomaximal, Sylow
83C127#163311yescharacteristic, complemented, Sylow
84Q82—1271no—
85#494—1271no—
86#477—1271no—
87#477—1271no—
88#333—1271no—
89Q8⋊C8—1271no—
90Q8⋊C8D12711yescharacteristic
91C2×Q16—2542no—
92C4⋊Q8—1271no—
93C4⋊Q8—1271no—
94C4⋊Q8—2541no—
95C4⋊Q8—2542no—
96C4⋊Q8—2541no—
97C4×Q8—1271no—
98C4×Q8—2541no—
99C4×Q8—2541no—
100C4×Q8C2×D12711yescharacteristic
101#107—2541no—
102#107—2541no—
103C4⋊C8—1271no—
104C4⋊C8C2×D12711yescharacteristic
105#103—2541no—
106#103—2541no—
107C4×C8C2×D12711yescharacteristic
108C2×Q8—1271no—
109C2×Q8—1272no—
110C2×Q8—2542no—
111C2×Q8—2541no—
112C2×Q8D50811yescharacteristic
113Q16—5084no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—2541no—
117C4⋊C4—21no—
118C4⋊C4—1271no—
119C4⋊C4—2542no—
120C4⋊C4—2542no—
121C4⋊C4—2541no—
122C4⋊C4—2541no—
123C4⋊C4—2541no—
124C4⋊C4—2541no—
125C4⋊C4—2541no—
126C4⋊C4D50811yescharacteristic
127C42—21no—
128C42—2542no—
129C42—2541no—
130C42C22×D12711yescharacteristic
131Q8—21no—
132Q8—1272no—
133Q8—10162no—
134Q8—5082no—
135Q8—5082no—
136Q8—2541no—
137Q8Dic50812yescomplemented
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—5081no—
141C2×C4—2541no—
142C2×C4—2542no—
143C2×C4—2541no—
144C2×C4D4×D12711yescharacteristic
145C2×C4D4×D12711yescharacteristic
146C2×C4C2×D50811yescharacteristic
147C8—22no—
148C8—41no—
149C8—5081no—
150C22#1686117911yescharacteristic
151C4—22no—
152C4—21no—
153C4—41no—
154C4—10161no—
155C4—5081no—
156C4—5082no—
157C4—2542no—
158C4Q16×D12711yescharacteristic
159C4C2×Dic50811yescharacteristic
160C4#1686123811yescharacteristic
161C4#1686124211yescharacteristic
162C2#2255003311yescharacteristic
163C2#2255003411yescharacteristic
164C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socle, prime radicalC22×C127#120
hypercenterQ8⋊C8D127
hypocenterC127#1633
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
52. Dic254Q16semidirect
59. Q8×C127Q16semidirect
83. C127#1633semidirect
137. Q8Dic508semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225502562
#225501722
#225501712
#225496242
#225496082
Q8⋊C8×C1272
#1633127