Finite Group Databasebeta

← #45663572

Subgroups — #45663572

Order 32896.

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

normal subgroups only

This group has 234 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#30657142C211yesmaximal, characteristic, complemented
3#30657017C211yesmaximal, characteristic
4#30656948C211yesmaximal, characteristic, complemented
5#30656484C211yesmaximal, characteristic, complemented
6#30656469C211yesmaximal, characteristic, complemented
7D4⋊C8×C257C211yesmaximal, characteristic
8C22×Dic514—21no—
9C2×Dic1028—21no—
10#22551592C2211yescharacteristic
11#22551592C2211yescharacteristic
12#22551591—21no—
13#22551571—21no—
14#22551569—22no—
15C4×Dic514C2211yescharacteristic
16D4×Dic257—21no—
17#22551526—22no—
18#22551525—21no—
19C4⋊#14523339C2211yescharacteristic
20#22551517—21no—
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—
32Dic1028—42no—
33C2×#14523339—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257—21no—
37C4⋊Dic257D411yescharacteristic
38C4⋊Dic257D412yes—
39#16861597—22no—
40#16861597—21no—
41C22⋊Dic257—22no—
42C4×Dic257—21no—
43C2×D4×C257D411yescharacteristic
44C22×C4×C257—21no—
45C2×C8×C257—21no—
46C2×C8×C257—21no—
47C4⋊C4×C257D411yescharacteristic
48C22⋊C4×C257—21no—
49C42×C257C2311yescharacteristic
50Dic514—21no—
51Dic514—44no—
52Dic514—41no—
53Dic514—41no—
54Dic514—41no—
55Dic514—41no—
56Dic514D812yes—
57C2×Dic257—21no—
58C2×Dic257—21no—
59C2×Dic257—22no—
60C2×Dic257—44no—
61C2×Dic257—41no—
62#14523339—41no—
63C23×C257—21no—
64D4×C257—21no—
65D4×C257Q1612yescomplemented
66C2×C4×C257—21no—
67C2×C4×C257—41no—
68C2×C4×C257—41no—
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C8×C257—22no—
73C8×C257—41no—
74C22×C257—22no—
75C22×C257—41no—
76C22×C257—41no—
77C22×C257#12011yescharacteristic
78C4×C257—21no—
79C4×C257—41no—
80C4×C257C2×D811yescharacteristic
81C4×C257C2×Q1611yescharacteristic
82C4×C257AGL(1,ℤ/8)11yescharacteristic
83C4×C257#13711yescharacteristic
84Dic257—22no—
85Dic257—42no—
86Dic257—41no—
87Dic257—81no—
88C2×C257—42no—
89C2×C257#44711yescharacteristic
90C2×C257#45111yescharacteristic
91C2×C257#45611yescharacteristic
92C257#164911yescharacteristic, complemented, Sylow
93#1649—2571nomaximal, Sylow
94D4×Q8—2571no—
95#500—2571no—
96#474—2571no—
97#462—2571no—
98#332—2571no—
99Q8⋊C8—2571no—
100D4⋊C8D25711yescharacteristic
101C22×Q8—5141no—
102C2×Q16—5141no—
103C4⋊Q8—2571no—
104C4⋊Q8—2571no—
105C4⋊Q8—5141no—
106C22⋊Q8—5142no—
107C22⋊Q8—5141no—
108C4×Q8—2571no—
109C4×D4—5141no—
110C4×D4C2×D25711yescharacteristic
111#107—5142no—
112#107—5141no—
113C4⋊C8—2571no—
114C4⋊C8C2×D25711yescharacteristic
115#103—5141no—
116#102—5141no—
117C4×C8C2×D25711yescharacteristic
118C2×Q8—2571no—
119C2×Q8—5142no—
120C2×Q8—5141no—
121C2×Q8—5141no—
122C2×Q8—10282no—
123C2×Q8—10282no—
124C2×D4D102811yescharacteristic
125C22×C4—21no—
126C22×C4—5142no—
127Q16—10282no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—5141no—
131C4⋊C4—2571no—
132C4⋊C4—2572no—
133C4⋊C4—5142no—
134C4⋊C4—5141no—
135C4⋊C4—5141no—
136C4⋊C4—5141no—
137C4⋊C4—5141no—
138C4⋊C4D102811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—5142no—
141C42—5141no—
142C42C22×D25711yescharacteristic
143C23—21no—
144Q8—2572no—
145Q8—5141no—
146Q8—10284no—
147Q8—10281no—
148Q8—10281no—
149Q8—10281no—
150Q8—10281no—
151D4—21no—
152D4Dic102812yescomplemented
153C2×C4—21no—
154C2×C4—5141no—
155C2×C4—5141no—
156C2×C4—5142no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—10284no—
160C2×C4—10281no—
161C2×C4D4×D25711yescharacteristic
162C2×C4D4×D25711yescharacteristic
163C2×C4C2×D102811yescharacteristic
164C8—22no—
165C8—41no—
166C8—10281no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2255156211yescharacteristic
171C4—21no—
172C4—5142no—
173C4—41no—
174C4—10282no—
175C4—10281no—
176C4—20561no—
177C4D8×D25711yescharacteristic
178C4C2×Dic102811yescharacteristic
179C4#2255161911yescharacteristic
180C4#2255162511yescharacteristic
181C2—42no—
182C2#3065687211yescharacteristic
183C2#3065689511yescharacteristic
184C2#3065692111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingD4⋊C8×C257C2
socleC22×C257#120
prime radicalC2×D4×C257D4
hypercenterD4⋊C8D257
hypocenterC257#1649
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. #30657142C2semidirect
4. #30656948C2semidirect
5. #30656484C2semidirect
6. #30656469C2semidirect
65. D4×C257Q16semidirect
92. C257#1649semidirect
152. D4Dic1028semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5142
#306571422
#306570172
#306569482
#306564842
#306564692
D4⋊C8×C2572
#1649257