Finite Group Databasebeta

← #45663570

Subgroups — #45663570

Order 32896.

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

normal subgroups only

This group has 234 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#30657402C211yesmaximal, characteristic
2#30657140C211yesmaximal, characteristic, complemented
3#30657018C211yesmaximal, characteristic
4#30656948C211yesmaximal, characteristic, complemented
5#30656467C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic
7#332×C257C211yesmaximal, characteristic, complemented
8C2×Q8×D257—21no—
9C2×Dic1028—21no—
10#22551592C2211yescharacteristic
11#22551591—21no—
12#22551570—22no—
13#22551568—21no—
14C4×Dic514C2211yescharacteristic
15C4×D1028C2211yescharacteristic
16#22551554—21no—
17#22551526—21no—
18#22551525—22no—
19#22551517—21no—
20#22551514—21no—
21C4×#14523339C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514—22no—
26C2×Dic514—21no—
27C2×Dic514D411yescharacteristic
28Q8×D257—42no—
29Q8×D257—42no—
30C2×D1028D411yescharacteristic
31C2×C4×D257—22no—
32C2×C4×D257—21no—
33Dic1028—42no—
34C2×#14523339—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257—21no—
37C4⋊Dic257D411yescharacteristic
38C4⋊Dic257D411yescharacteristic
39#16861597—22no—
40#16861597—21no—
41#16861595—22no—
42#16861595—21no—
43C4×Dic257—21no—
44C2×Q8×C257—21no—
45C2×C8×C257—21no—
46C2×C8×C257—21no—
47C4⋊C4×C257—21no—
48C4⋊C4×C257D412yes—
49C42×C257C2311yescharacteristic
50C22×D257—21no—
51Dic514—21no—
52Dic514—44no—
53Dic514—41no—
54Dic514—41no—
55Dic514D812yes—
56D1028—21no—
57D1028Q1612yescomplemented
58C2×Dic257—21no—
59C2×Dic257—21no—
60C2×Dic257—21no—
61C2×Dic257—41no—
62C4×D257—44no—
63C4×D257—41no—
64C4×D257—41no—
65#14523339—22no—
66Q8×C257—41no—
67Q8×C257—41no—
68C2×C4×C257—22no—
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C8×C257—41no—
73C8×C257—41no—
74C22×C257#12011yescharacteristic
75C2×D257—22no—
76C2×D257—41no—
77C2×D257—41no—
78C4×C257—21no—
79C4×C257—42no—
80C4×C257C2×D811yescharacteristic
81C4×C257C2×Q1611yescharacteristic
82C4×C257AGL(1,ℤ/8)11yescharacteristic
83C4×C257#13711yescharacteristic
84Dic257—22no—
85Dic257—41no—
86Dic257—41no—
87Dic257—81no—
88C2×C257#44711yescharacteristic
89C2×C257#45111yescharacteristic
90C2×C257#45611yescharacteristic
91D257—42no—
92C257#164911yescharacteristic, complemented, Sylow
93#1649—2571nomaximal, Sylow
94D4×Q8—2571no—
95#500—2571no—
96#474—2571no—
97#462—2571no—
98#332D25711yescharacteristic, complemented
99Q8⋊C8—2571no—
100D4⋊C8—2571no—
101C22×Q8—5141no—
102C2×Q16—5141no—
103C4⋊Q8—2571no—
104C4⋊Q8—5141no—
105C4⋊Q8C2×D25711yescharacteristic
106C22⋊Q8—5141no—
107C22⋊Q8—5142no—
108C4×Q8—2571no—
109C4×D4—2571no—
110C4×D4—5141no—
111#107—5141no—
112#107—5142no—
113C4⋊C8C2×D25711yescharacteristic
114C4⋊C8C2×D25711yescharacteristic
115#103—5141no—
116#102—5141no—
117C4×C8—2571no—
118C2×Q8—2571no—
119C2×Q8—21no—
120C2×Q8—5142no—
121C2×Q8—5141no—
122C2×Q8—10282no—
123C2×Q8—10282no—
124C2×D4—2571no—
125C22×C4—5142no—
126C22×C4—5141no—
127Q16—10282no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—5141no—
131C4⋊C4—2571no—
132C4⋊C4—2571no—
133C4⋊C4—21no—
134C4⋊C4—5142no—
135C4⋊C4—5141no—
136C4⋊C4—5141no—
137C4⋊C4—5141no—
138C4⋊C4D102812yes—
139C22⋊C4—5142no—
140C22⋊C4—5141no—
141C42—5141no—
142C42C22×D25711yescharacteristic
143C23—5141no—
144Q8—2572no—
145Q8—5141no—
146Q8—41no—
147Q8—41no—
148Q8—10284no—
149Q8—10281no—
150Q8—10281no—
151D4—2572no—
152D4—5141no—
153C2×C4—22no—
154C2×C4—5141no—
155C2×C4—5141no—
156C2×C4—5141no—
157C2×C4—10284no—
158C2×C4—10281no—
159C2×C4—10281no—
160C2×C4—10281no—
161C2×C4D4×D25711yescharacteristic
162C2×C4D4×D25711yescharacteristic
163C2×C4C2×D102811yescharacteristic
164C8—5142no—
165C8—41no—
166C8—41no—
167C22—5142no—
168C22—10281no—
169C22—10281no—
170C22#2255156211yescharacteristic
171C4—21no—
172C4—5142no—
173C4—42no—
174C4—10281no—
175C4—10281no—
176C4—20561no—
177C4D8×D25711yescharacteristic
178C4Q16×D25711yescharacteristic
179C4#2255161911yescharacteristic
180C4#2255162111yescharacteristic
181C2—10282no—
182C2#3065687211yescharacteristic
183C2#3065689411yescharacteristic
184C2#3065692211yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#332×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#332D257
hypocenterC257#1649
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #30657140C2semidirect
4. #30656948C2semidirect
5. #30656467C2semidirect
7. #332×C257C2semidirect
57. D1028Q16semidirect
92. C257#1649semidirect
98. #332D257semidirect

Maximal subgroups

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

subgroupindex
#306574022
#306571402
#306570182
#306569482
#306564672
#306564642
#332×C2572
#1649257