Finite Group Databasebeta

← #45663566

Subgroups — #45663566

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#30657014C211yesmaximal, characteristic
3#30656944C211yesmaximal, characteristic, complemented
4#30656483C211yesmaximal, characteristic, complemented
5#30656469C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic
7#500×C257C211yesmaximal, characteristic, complemented
8C2×Q8×D257—21no—
9C2×#16861608—21no—
10#22551591—21no—
11#22551570—22no—
12#22551568—21no—
13C4×Dic514C2211yescharacteristic
14C4×D1028C2211yescharacteristic
15#22551554—21no—
16#22551525—21no—
17C4⋊#14523339C2211yescharacteristic
18C4⋊#14523339C2211yescharacteristic
19#22551516—21no—
20#22551512—21no—
21C4⋊Q8×C257C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23#107×C257—22no—
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—22no—
26C2×Dic514D411yescharacteristic
27Q8×D257—42no—
28Q8×D257—42no—
29C2×D1028D411yescharacteristic
30C2×C4×D257—22no—
31C2×C4×D257—21no—
32#16861608—42no—
33C2×#14523339—21no—
34C2×#14523339—21no—
35C4⋊Dic257D411yescharacteristic
36C4⋊Dic257D411yescharacteristic
37#16861597—22no—
38#16861597—21no—
39#16861595—22no—
40#16861595—21no—
41C4×Dic257—21no—
42C2×Q8×C257—21no—
43C2×Q8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257—21no—
48C4⋊C4×C257D412yes—
49C42×C257C2311yescharacteristic
50C22×D257—21no—
51Dic514—21no—
52Dic514—44no—
53Dic514D812yes—
54D1028—21no—
55D1028Q1612yescomplemented
56C2×Dic257—21no—
57C2×Dic257—21no—
58C2×Dic257—21no—
59C4×D257—44no—
60C4×D257—41no—
61C4×D257—41no—
62#14523339—41no—
63#14523339—41no—
64Q8×C257—41no—
65Q8×C257—41no—
66Q8×C257—41no—
67Q8×C257—41no—
68C2×C4×C257—22no—
69C2×C4×C257—41no—
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C2×C4×C257C2×D411yescharacteristic
73C8×C257—22no—
74C22×C257#12011yescharacteristic
75C2×D257—22no—
76C2×D257—41no—
77C2×D257—41no—
78C4×C257—21no—
79C4×C257—42no—
80C4×C257—81no—
81C4×C257C2×D811yescharacteristic
82C4×C257C2×Q1611yescharacteristic
83C4×C257AGL(1,ℤ/8)11yescharacteristic
84C4×C257#13711yescharacteristic
85Dic257—22no—
86Dic257—41no—
87Dic257—41no—
88C2×C257#44711yescharacteristic
89C2×C257#45111yescharacteristic
90C2×C257#45611yescharacteristic
91D257—42no—
92C257#164911yescharacteristic, complemented, Sylow
93#1649—2571nomaximal, Sylow
94D4×Q8—2571no—
95#500D25711yescharacteristic, complemented
96#474—2571no—
97#462—2571no—
98#332—2571no—
99Q8⋊C8—2571no—
100D4⋊C8—2571no—
101C22×Q8—5141no—
102C2×Q16—5141no—
103C4⋊Q8—5141no—
104C4⋊Q8C2×D25711yescharacteristic
105C4⋊Q8C2×D25711yescharacteristic
106C22⋊Q8—5141no—
107C22⋊Q8—5142no—
108C4×Q8—2571no—
109C4×D4—2571no—
110C4×D4—5141no—
111#107—22no—
112#107—5141no—
113C4⋊C8—2571no—
114C4⋊C8—2571no—
115#103—5141no—
116#102—5141no—
117C4×C8C2×D25711yescharacteristic
118C2×Q8—2571no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—5142no—
122C2×Q8—10282no—
123C2×Q8—10282no—
124C2×D4—2571no—
125C22×C4—5142no—
126C22×C4—5141no—
127Q16—10282no—
128C2×C8—21no—
129C2×C8—5141no—
130C2×C8—5141no—
131C4⋊C4—2571no—
132C4⋊C4—2571no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—5142no—
137C4⋊C4—5141no—
138C4⋊C4C22⋊D25712yes—
139C22⋊C4—5142no—
140C22⋊C4—5141no—
141C42—5141no—
142C42C22×D25711yescharacteristic
143C23—5141no—
144Q8—2572no—
145Q8—5141no—
146Q8—41no—
147Q8—41no—
148Q8—41no—
149Q8—41no—
150Q8—10284no—
151D4—2572no—
152D4—5141no—
153C2×C4—22no—
154C2×C4—5141no—
155C2×C4—5141no—
156C2×C4—5141no—
157C2×C4—41no—
158C2×C4—10284no—
159C2×C4—10281no—
160C2×C4—10281no—
161C2×C4D4×D25711yescharacteristic
162C2×C4D4×D25711yescharacteristic
163C2×C4C2×C22⋊D25711yescharacteristic
164C8—22no—
165C8—10281no—
166C8—10281no—
167C22—5142no—
168C22—10281no—
169C22—10281no—
170C22#2255156011yescharacteristic
171C4—21no—
172C4—5142no—
173C4—42no—
174C4—10281no—
175C4—10281no—
176C4—81no—
177C4D8×D25711yescharacteristic
178C4Q16×D25711yescharacteristic
179C4#2255161611yescharacteristic
180C4#2255162411yescharacteristic
181C2—10282no—
182C2#3065686911yescharacteristic
183C2#3065689111yescharacteristic
184C2#3065691911yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#500×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#500D257
hypocenterC257#1649
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #30656944C2semidirect
4. #30656483C2semidirect
5. #30656469C2semidirect
7. #500×C257C2semidirect
55. D1028Q16semidirect
92. C257#1649semidirect
95. #500D257semidirect

Maximal subgroups

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

subgroupindex
#306574022
#306570142
#306569442
#306564832
#306564692
#306564662
#500×C2572
#1649257