Finite Group Databasebeta

← #45663544

Subgroups — #45663544

Order 32896.

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

normal subgroups only

This group has 200 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×Dic514C211yesmaximal, characteristic
2#30657088C211yesmaximal, characteristic
3#30657022C211yesmaximal, characteristic
4#30656948C211yesmaximal, characteristic
5#30656485C211yesmaximal, characteristic
6#30656469C211yesmaximal, characteristic
7Q8⋊C8×C257C211yesmaximal, characteristic
8C2×Dic1028—21no—
9#22551592—21no—
10#22551592C2211yescharacteristic
11#22551592C2211yescharacteristic
12#22551591—22no—
13#22551591—21no—
14Q8×Dic257—21no—
15C4×Dic514—21no—
16C4×Dic514C2211yescharacteristic
17#22551523—21no—
18C4⋊#14523339C2211yescharacteristic
19#22551517—22no—
20#22551517—21no—
21#22551516—21no—
22C4×Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—22no—
26C2×Dic514—21no—
27C2×Dic514D411yescharacteristic
28C2×Dic514D412yes—
29Dic1028—42no—
30C2×#14523339—21no—
31C4⋊Dic257—22no—
32C4⋊Dic257—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257D411yescharacteristic
37#16861597—22no—
38#16861597—21no—
39C4×Dic257—22no—
40C4×Dic257—21no—
41C2×Q8×C257D411yescharacteristic
42C2×C8×C257—21no—
43C2×C8×C257—21no—
44C4⋊C4×C257—21no—
45C4⋊C4×C257D411yescharacteristic
46C42×C257—21no—
47C42×C257C2311yescharacteristic
48Dic514—21no—
49Dic514—42no—
50Dic514—41no—
51Dic514—41no—
52Dic514—82no—
53Dic514SD1612yes—
54C2×Dic257—21no—
55C2×Dic257—22no—
56C2×Dic257—21no—
57C2×Dic257—41no—
58#14523339—41no—
59Q8×C257—21no—
60Q8×C257Q1612yescomplemented
61C2×C4×C257—21no—
62C2×C4×C257—21no—
63C2×C4×C257C2×D411yescharacteristic
64C2×C4×C257C2×D411yescharacteristic
65C2×C4×C257C2×D411yescharacteristic
66C8×C257—41no—
67C8×C257—41no—
68C22×C257#12011yescharacteristic
69C4×C257—22no—
70C4×C257—21no—
71C4×C257—41no—
72C4×C257C2×SD1611yescharacteristic
73C4×C257C2×Q1611yescharacteristic
74C4×C257#13711yescharacteristic
75C4×C257#13711yescharacteristic
76Dic257—22no—
77Dic257—42no—
78Dic257—41no—
79Dic257—81no—
80C2×C257#44811yescharacteristic
81C2×C257#45111yescharacteristic
82C2×C257#45511yescharacteristic
83C257#164511yescharacteristic, complemented, Sylow
84#1645—2571nomaximal, Sylow
85Q82—2571no—
86#487—2571no—
87#475—2571no—
88#462—2571no—
89#333—2571no—
90Q8⋊C8—2571no—
91Q8⋊C8D25711yescharacteristic
92C2×Q16—5141no—
93C4⋊Q8—2571no—
94C4⋊Q8—2571no—
95C4⋊Q8—5141no—
96C4⋊Q8—5142no—
97C4⋊Q8—5141no—
98C4×Q8—2571no—
99C4×Q8—5141no—
100C4×Q8—5141no—
101C4×Q8C2×D25711yescharacteristic
102#106—5141no—
103C4⋊C8—2571no—
104C4⋊C8C2×D25711yescharacteristic
105#103—5141no—
106#103—5142no—
107#103—5141no—
108C4×C8C2×D25711yescharacteristic
109C2×Q8—2571no—
110C2×Q8—2572no—
111C2×Q8—5142no—
112C2×Q8—5141no—
113C2×Q8D102811yescharacteristic
114Q16—10282no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—5141no—
118C4⋊C4—2571no—
119C4⋊C4—21no—
120C4⋊C4—5142no—
121C4⋊C4—5142no—
122C4⋊C4—5141no—
123C4⋊C4—5141no—
124C4⋊C4—5141no—
125C4⋊C4—5141no—
126C4⋊C4—5141no—
127C4⋊C4D102811yescharacteristic
128C42—21no—
129C42—5142no—
130C42—5141no—
131C42C22×D25711yescharacteristic
132Q8—2572no—
133Q8—21no—
134Q8—5141no—
135Q8—10282no—
136Q8—10281no—
137Q8—10281no—
138Q8—20562no—
139Q8Dic102812yescomplemented
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—5141no—
143C2×C4—5141no—
144C2×C4—5142no—
145C2×C4—10281no—
146C2×C4D4×D25711yescharacteristic
147C2×C4D4×D25711yescharacteristic
148C2×C4C2×D102811yescharacteristic
149C8—41no—
150C8—41no—
151C8—10281no—
152C22#2255156211yescharacteristic
153C4—22no—
154C4—21no—
155C4—5142no—
156C4—41no—
157C4—10282no—
158C4—10281no—
159C4—20561no—
160C4SD16×D25711yescharacteristic
161C4C2×Dic102811yescharacteristic
162C4#2255162411yescharacteristic
163C4#2255162511yescharacteristic
164C2#3065687711yescharacteristic
165C2#3065689511yescharacteristic
166C2#3065691611yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingQ8⋊C8×C257C2
socle, prime radicalC22×C257#120
hypercenterQ8⋊C8D257
hypocenterC257#1645
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
60. Q8×C257Q16semidirect
83. C257#1645semidirect
139. Q8Dic1028semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic5142
#306570882
#306570222
#306569482
#306564852
#306564692
Q8⋊C8×C2572
#1645257