Finite Group Databasebeta

← #45663474

Subgroups — #45663474

Order 32896.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#30657392C211yesmaximal, characteristic
2#30657088C211yesmaximal, characteristic, complemented
3#30657028C211yesmaximal, characteristic, complemented
4#30657027C211yesmaximal, characteristic, complemented
5#30656484C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic, complemented
7D4⋊C8×C257C211yesmaximal, characteristic, complemented
8C2×#16861618—21no—
9#22551592C2211yescharacteristic
10#22551592C2211yescharacteristic
11#22551574—22no—
12#22551571—21no—
13#22551563—21no—
14C4×D1028C2211yescharacteristic
15C4⋊C4×D257—21no—
16#22551524—21no—
17#22551523—21no—
18C4⋊#14523339C2211yescharacteristic
19#22551517—22no—
20#22551514—21no—
21#22551513—21no—
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25#16861618—42no—
26#16861618—42no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×D1028D411yescharacteristic
30C2×C22⋊D257—22no—
31C2×C4×D257—22no—
32C2×C4×D257—21no—
33C2×#14523339—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257—21no—
37C4⋊Dic257D411yescharacteristic
38C4⋊Dic257D412yes—
39#16861597—22no—
40C22⋊Dic257—22no—
41#16861595—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
49C22×D257—21no—
50Dic514—41no—
51Dic514—41no—
52Dic514—42no—
53D1028—21no—
54D1028SD1612yes—
55C22⋊D257—44no—
56C2×Dic257—21no—
57C2×Dic257—22no—
58C2×Dic257—41no—
59C4×D257—44no—
60C4×D257—41no—
61C4×D257—41no—
62#14523339—41no—
63C23×C257—21no—
64D4×C257—21no—
65D4×C257—21no—
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—41no—
73C8×C257—41no—
74C22×C257—41no—
75C22×C257—41no—
76C22×C257—41no—
77C22×C257#12011yescharacteristic
78C2×D257—22no—
79C2×D257—41no—
80C2×D257—41no—
81C4×C257—21no—
82C4×C257—41no—
83C4×C257C2×SD1611yescharacteristic
84C4×C257#13511yescharacteristic
85C4×C257AGL(1,ℤ/8)11yescharacteristic
86C4×C257#13711yescharacteristic
87Dic257—42no—
88Dic257—41no—
89Dic257—81no—
90C2×C257—81no—
91C2×C257#45011yescharacteristic
92C2×C257#45211yescharacteristic
93C2×C257#45411yescharacteristic
94D257—42no—
95C257#163511yescharacteristic, complemented, Sylow
96#1635—2571nomaximal, Sylow
97#547—2571no—
98#487—2571no—
99#476—2571no—
100#476—2571no—
101#332—2571no—
102D4⋊C8—2571no—
103D4⋊C8D25711yescharacteristic, complemented
104C2×#41—5141no—
105C4⋊Q8—2571no—
106C4⋊Q8—2571no—
107#123—5142no—
108C22⋊Q8—5141no—
109#121—5141no—
110C4×D4—2571no—
111C4×D4C2×D25711yescharacteristic
112C2×C4⋊C4—5141no—
113#106—5141no—
114#106—5141no—
115C4⋊C8—2571no—
116C4⋊C8C2×D25711yescharacteristic
117#103—5142no—
118#102—5141no—
119#102—5141no—
120C4×C8C2×D25711yescharacteristic
121#41—10282no—
122#41—10282no—
123C2×Q8—5141no—
124C2×Q8—5141no—
125C2×D4—2571no—
126C2×D4—5142no—
127C2×D4D102811yescharacteristic
128C22×C4—21no—
129C22×C4—5142no—
130C22×C4—5141no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—5141no—
134C4⋊C4—2572no—
135C4⋊C4—2571no—
136C4⋊C4—5142no—
137C4⋊C4—5141no—
138C4⋊C4—5141no—
139C4⋊C4—5141no—
140C4⋊C4D102811yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—5142no—
143C22⋊C4—5141no—
144C42C22×D25711yescharacteristic
145C23—21no—
146C23—5141no—
147Q8—10282no—
148Q8—10281no—
149Q8—10281no—
150D4—2572no—
151D4—21no—
152D4—21no—
153D4—5141no—
154D4—10284no—
155C2×C4—21no—
156C2×C4—5141no—
157C2×C4—5142no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—10284no—
161C2×C4—10281no—
162C2×C4—10281no—
163C2×C4—10281no—
164C2×C4D4×D25711yescharacteristic
165C2×C4D4×D25711yescharacteristic
166C2×C4C2×D102811yescharacteristic
167C8—41no—
168C8—41no—
169C8—10281no—
170C22—5142no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—10281no—
175C22—10281no—
176C22#2255156211yescharacteristic
177C4—21no—
178C4—41no—
179C4—10282no—
180C4—10281no—
181C4—20561no—
182C4SD16×D25711yescharacteristic
183C4#2255161411yescharacteristic
184C4#2255161611yescharacteristic
185C4#2255162511yescharacteristic
186C2—10282no—
187C2—81no—
188C2#3065688911yescharacteristic
189C2#3065690111yescharacteristic
190C2#3065691111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingD4⋊C8×C257C2
socleC22×C257#120
prime radical#30657392C2
hypercenterD4⋊C8D257
hypocenterC257#1635
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. #30657028C2semidirect
4. #30657027C2semidirect
5. #30656484C2semidirect
6. #30656466C2semidirect
7. D4⋊C8×C257C2semidirect
95. C257#1635semidirect
103. D4⋊C8D257semidirect

Maximal subgroups

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

subgroupindex
#306573922
#306570882
#306570282
#306570272
#306564842
#306564662
D4⋊C8×C2572
#1635257