Finite Group Databasebeta

← #45663488

Subgroups — #45663488

Order 32896.

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

normal subgroups only

This group has 228 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
1D4×Dic514C211yesmaximal, characteristic
2#30657088C211yesmaximal, characteristic, complemented
3#30657033C211yesmaximal, characteristic, complemented
4#30657027C211yesmaximal, characteristic
5#30656484C211yesmaximal, characteristic, complemented
6#30656469C211yesmaximal, characteristic, complemented
7D4⋊C8×C257C211yesmaximal, characteristic
8C22×Dic514—21no—
9#22551592C2211yescharacteristic
10#22551592C2211yescharacteristic
11#22551591—21no—
12#22551571—21no—
13#22551569—22no—
14C4×Dic514C2211yescharacteristic
15D4×Dic257—21no—
16#22551526—21no—
17#22551523—21no—
18C4⋊#14523339C2211yescharacteristic
19#22551517—21no—
20#22551517—22no—
21#22551513—21no—
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—22no—
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514—42no—
29C2×Dic514—42no—
30C2×Dic514D411yescharacteristic
31C22×Dic257—22no—
32C2×#14523339—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257D411yescharacteristic
37C4⋊Dic257D412yes—
38#16861597—22no—
39#16861597—21no—
40C22⋊Dic257—22no—
41C4×Dic257—21no—
42C2×D4×C257D411yescharacteristic
43C22×C4×C257—21no—
44C2×C8×C257—21no—
45C2×C8×C257—21no—
46C4⋊C4×C257D411yescharacteristic
47C22⋊C4×C257—21no—
48C42×C257C2311yescharacteristic
49Dic514—21no—
50Dic514—44no—
51Dic514—41no—
52Dic514—41no—
53Dic514—42no—
54Dic514SD1612yes—
55C2×Dic257—21no—
56C2×Dic257—21no—
57C2×Dic257—22no—
58C2×Dic257—44no—
59C2×Dic257—41no—
60#14523339—41no—
61C23×C257—21no—
62D4×C257—21no—
63D4×C257Q1612yes—
64C2×C4×C257—21no—
65C2×C4×C257—41no—
66C2×C4×C257—41no—
67C2×C4×C257C2×D411yescharacteristic
68C2×C4×C257C2×D411yescharacteristic
69C2×C4×C257C2×D411yescharacteristic
70C8×C257—41no—
71C8×C257—41no—
72C22×C257—22no—
73C22×C257—41no—
74C22×C257—41no—
75C22×C257#12011yescharacteristic
76C4×C257—21no—
77C4×C257—41no—
78C4×C257C2×SD1611yescharacteristic
79C4×C257C2×Q1611yescharacteristic
80C4×C257AGL(1,ℤ/8)11yescharacteristic
81C4×C257#13711yescharacteristic
82Dic257—22no—
83Dic257—42no—
84Dic257—41no—
85Dic257—81no—
86C2×C257—42no—
87C2×C257#45011yescharacteristic
88C2×C257#45111yescharacteristic
89C2×C257#45411yescharacteristic
90C257#163711yescharacteristic, complemented, Sylow
91#1637—2571nomaximal, Sylow
92D4×Q8—2571no—
93#487—2571no—
94#477—2571no—
95#476—2571no—
96#332—2571no—
97Q8⋊C8—2571no—
98D4⋊C8D25711yescharacteristic
99C22×Q8—5141no—
100C4⋊Q8—2571no—
101C4⋊Q8—2571no—
102C4⋊Q8—5141no—
103C22⋊Q8—5142no—
104C22⋊Q8—5141no—
105C4×Q8—2571no—
106C4×D4—5141no—
107C4×D4C2×D25711yescharacteristic
108#107—5141no—
109#106—5141no—
110C4⋊C8—2571no—
111C4⋊C8C2×D25711yescharacteristic
112#103—5142no—
113#103—5141no—
114#102—5141no—
115C4×C8C2×D25711yescharacteristic
116C2×Q8—2571no—
117C2×Q8—5142no—
118C2×Q8—5141no—
119C2×Q8—5141no—
120C2×Q8—10282no—
121C2×Q8—10282no—
122C2×D4D102811yescharacteristic
123C22×C4—21no—
124C22×C4—5142no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—5141no—
128C4⋊C4—2572no—
129C4⋊C4—2571no—
130C4⋊C4—5142no—
131C4⋊C4—5141no—
132C4⋊C4—5141no—
133C4⋊C4—5141no—
134C4⋊C4—5141no—
135C4⋊C4D102811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—5142no—
138C42—5141no—
139C42C22×D25711yescharacteristic
140C23—21no—
141Q8—2572no—
142Q8—5141no—
143Q8—10282no—
144Q8—10284no—
145Q8—10281no—
146Q8—10281no—
147D4—21no—
148D4Dic102812yes—
149C2×C4—21no—
150C2×C4—5141no—
151C2×C4—5141no—
152C2×C4—5142no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—10284no—
156C2×C4—10281no—
157C2×C4D4×D25711yescharacteristic
158C2×C4D4×D25711yescharacteristic
159C2×C4C2×D102811yescharacteristic
160C8—41no—
161C8—41no—
162C8—10281no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#2255156211yescharacteristic
167C4—21no—
168C4—5142no—
169C4—41no—
170C4—10282no—
171C4—10281no—
172C4—20561no—
173C4SD16×D25711yescharacteristic
174C4C2×Dic102811yescharacteristic
175C4#2255161611yescharacteristic
176C4#2255162511yescharacteristic
177C2—42no—
178C2#3065688911yescharacteristic
179C2#3065689511yescharacteristic
180C2#3065691111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingD4⋊C8×C257C2
socleC22×C257#120
prime radicalC2×D4×C257D4
hypercenterD4⋊C8D257
hypocenterC257#1637
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30657088C2semidirect
3. #30657033C2semidirect
5. #30656484C2semidirect
6. #30656469C2semidirect
90. C257#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5142
#306570882
#306570332
#306570272
#306564842
#306564692
D4⋊C8×C2572
#1637257