Finite Group Databasebeta

← #30637869

Subgroups — #30637869

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#22550255C211yesmaximal, characteristic
3#22550168C211yesmaximal, characteristic
4#22549624C211yesmaximal, characteristic
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic
6#22549606C211yesmaximal, characteristic
7#477×C127C211yesmaximal, characteristic
8C2×#14523238—22no—
9#16861209—21no—
10#16861209C2211yescharacteristic
11#16861208—22no—
12#16861208—21no—
13Q8×Dic127—21no—
14C4×Dic254—21no—
15C4×Dic254C2211yescharacteristic
16#16861142—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861132—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254—22no—
26C2×Dic254D411yescharacteristic
27C2×Dic254D412yes—
28#14523238—44no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—22no—
36Dic127⋊C4—21no—
37C4×Dic127—22no—
38C4×Dic127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127D411yescharacteristic
41C2×C8×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—42no—
50Dic254—82no—
51Dic254Q1612yescomplemented
52C2×Dic127—21no—
53C2×Dic127—22no—
54C2×Dic127—21no—
55C127⋊C8—22no—
56C127⋊C8—41no—
57Q8×C127—21no—
58Q8×C127—42no—
59Q8×C127Q1612yescomplemented
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—41no—
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C8×C127—41no—
67C22×C127#12011yescharacteristic
68C4×C127—22no—
69C4×C127—21no—
70C4×C127—41no—
71C4×C127—81no—
72C4×C127C2×Q1611yescharacteristic
73C4×C127C2×Q1611yescharacteristic
74C4×C127#13711yescharacteristic
75C4×C127#13711yescharacteristic
76Dic127—22no—
77Dic127—42no—
78Dic127—41no—
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#477D12711yescharacteristic
88#333—1271no—
89Q8⋊C8—1271no—
90Q8⋊C8—1271no—
91C2×Q16—2542no—
92C4⋊Q8—1271no—
93C4⋊Q8—2541no—
94C4⋊Q8—2542no—
95C4⋊Q8—2541no—
96C4⋊Q8C2×D12711yescharacteristic
97C4×Q8—1271no—
98C4×Q8—2541no—
99C4×Q8—2541no—
100C4×Q8C2×D12711yescharacteristic
101#107—21no—
102#107—2541no—
103C4⋊C8—1271no—
104C4⋊C8C2×D12711yescharacteristic
105#103—21no—
106#103—2541no—
107C4×C8—1271no—
108C2×Q8—21no—
109C2×Q8—1271no—
110C2×Q8—1272no—
111C2×Q8—2542no—
112C2×Q8C22⋊D12711yescharacteristic
113Q16—5084no—
114C2×C8—21no—
115C2×C8—2541no—
116C2×C8—2541no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—1271no—
121C4⋊C4—2542no—
122C4⋊C4—2542no—
123C4⋊C4—2541no—
124C4⋊C4—2541no—
125C4⋊C4—2541no—
126C4⋊C4C22⋊D12711yescharacteristic
127C42—21no—
128C42—2542no—
129C42—2541no—
130C42C22×D12711yescharacteristic
131Q8—21no—
132Q8—42no—
133Q8—1272no—
134Q8—10162no—
135Q8—5082no—
136Q8—2541no—
137Q8#1452323812yescomplemented
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—2541no—
142C2×C4—2542no—
143C2×C4—2541no—
144C2×C4D4×D12711yescharacteristic
145C2×C4D4×D12711yescharacteristic
146C2×C4C2×C22⋊D12711yescharacteristic
147C8—41no—
148C8—5081no—
149C8—2542no—
150C22#1686117711yescharacteristic
151C4—22no—
152C4—21no—
153C4—41no—
154C4—81no—
155C4—5081no—
156C4—5082no—
157C4—2542no—
158C4Q16×D12711yescharacteristic
159C4C2×#1452323811yescharacteristic
160C4#1686123911yescharacteristic
161C4#1686124111yescharacteristic
162C2#2255003011yescharacteristic
163C2#2255003111yescharacteristic
164C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#477×C127C2
socle, prime radicalC22×C127#120
hypercenter#477D127
hypocenterC127#1633
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
51. Dic254Q16semidirect
59. Q8×C127Q16semidirect
83. C127#1633semidirect
137. Q8#14523238semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225502552
#225501682
#225496242
Q8⋊(C127⋊C8)2
#225496062
#477×C1272
#1633127