Finite Group Databasebeta

← #45663480

Subgroups — #45663480

Order 32896.

188 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 55 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#30657465C211yesmaximal, characteristic
2#30657088C211yesmaximal, characteristic
3#30657033C211yesmaximal, characteristic
4#30657032C211yesmaximal, characteristic
5#30656482C211yesmaximal, characteristic
6#30656469C211yesmaximal, characteristic
7Q8⋊C8×C257C211yesmaximal, characteristic
8#22551592C2211yescharacteristic
9#22551592C2211yescharacteristic
10#22551590—21no—
11#22551584—21no—
12#22551582—22no—
13C4×Dic514—21no—
14C4×Dic514C2211yescharacteristic
15#22551557—21no—
16#22551526—21no—
17#22551525—21no—
18C4⋊#14523339C2211yescharacteristic
19#22551517—21no—
20#22551517—22no—
21#22551516—21no—
22C4×Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×Dic514—21no—
27C2×Dic514D411yescharacteristic
28C2×#14523339—21no—
29C4⋊Dic257—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32C4⋊Dic257D411yescharacteristic
33C4⋊Dic257D412yes—
34#16861597—22no—
35#16861597—22no—
36#16861597—22no—
37#16861597—21no—
38C4×Dic257—22no—
39C4×Dic257—21no—
40C2×Q8×C257D411yescharacteristic
41C2×C8×C257—21no—
42C2×C8×C257—21no—
43C4⋊C4×C257—21no—
44C4⋊C4×C257D411yescharacteristic
45C42×C257—21no—
46C42×C257C2311yescharacteristic
47Dic514—21no—
48Dic514—41no—
49Dic514—41no—
50Dic514—42no—
51Dic514Q1612yes—
52C2×Dic257—21no—
53C2×Dic257—21no—
54C2×Dic257—22no—
55C2×Dic257—41no—
56#14523339—41no—
57Q8×C257—21no—
58Q8×C257—21no—
59C2×C4×C257—21no—
60C2×C4×C257—21no—
61C2×C4×C257C2×D411yescharacteristic
62C2×C4×C257C2×D411yescharacteristic
63C2×C4×C257C2×D411yescharacteristic
64C8×C257—41no—
65C8×C257—41no—
66C22×C257#12011yescharacteristic
67C4×C257—21no—
68C4×C257—41no—
69C4×C257—41no—
70C4×C257C2×Q1611yescharacteristic
71C4×C257#13511yescharacteristic
72C4×C257#13711yescharacteristic
73C4×C257#13711yescharacteristic
74Dic257—22no—
75Dic257—42no—
76Dic257—41no—
77Dic257—81no—
78C2×C257#45111yescharacteristic
79C2×C257#45211yescharacteristic
80C2×C257#45411yescharacteristic
81C257#163611yescharacteristic, complemented, Sylow
82#1636—2571nomaximal, Sylow
83#557—2571no—
84#487—2571no—
85#477—2571no—
86#477—2571no—
87#332—2571no—
88Q8⋊C8—2571no—
89Q8⋊C8D25711yescharacteristic
90C4⋊Q8—2571no—
91C4⋊Q8—2571no—
92C4⋊Q8—5141no—
93#125—5141no—
94#125—5142no—
95C4×Q8—2571no—
96C4×Q8—5141no—
97C4×Q8—5141no—
98C4×Q8C2×D25711yescharacteristic
99#107—5141no—
100#107—5141no—
101C4⋊C8—2571no—
102C4⋊C8C2×D25711yescharacteristic
103#103—5142no—
104#103—5141no—
105#103—5141no—
106C4×C8C2×D25711yescharacteristic
107C2×Q8—2571no—
108C2×Q8—5141no—
109C2×Q8—5141no—
110C2×Q8D102811yescharacteristic
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—5141no—
114C4⋊C4—2572no—
115C4⋊C4—2571no—
116C4⋊C4—21no—
117C4⋊C4—5142no—
118C4⋊C4—5142no—
119C4⋊C4—5142no—
120C4⋊C4—5141no—
121C4⋊C4—5141no—
122C4⋊C4—5141no—
123C4⋊C4—5141no—
124C4⋊C4D102811yescharacteristic
125C42—21no—
126C42—5142no—
127C42—5141no—
128C42C22×D25711yescharacteristic
129Q8—2572no—
130Q8—21no—
131Q8—21no—
132Q8—5141no—
133Q8—10282no—
134Q8—10281no—
135Q8—10281no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—5141no—
139C2×C4—5141no—
140C2×C4—5142no—
141C2×C4—10281no—
142C2×C4D4×D25711yescharacteristic
143C2×C4D4×D25711yescharacteristic
144C2×C4C2×D102811yescharacteristic
145C8—41no—
146C8—41no—
147C8—10281no—
148C22#2255156211yescharacteristic
149C4—21no—
150C4—5142no—
151C4—41no—
152C4—41no—
153C4—10282no—
154C4—10281no—
155C4—20561no—
156C4Q16×D25711yescharacteristic
157C4#2255161411yescharacteristic
158C4#2255162111yescharacteristic
159C4#2255162511yescharacteristic
160C2#3065689411yescharacteristic
161C2#3065690111yescharacteristic
162C2#3065691111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingQ8⋊C8×C257C2
socle, prime radicalC22×C257#120
hypercenterQ8⋊C8D257
hypocenterC257#1636
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
81. C257#1636semidirect

Maximal subgroups

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

subgroupindex
#306574652
#306570882
#306570332
#306570322
#306564822
#306564692
Q8⋊C8×C2572
#1636257