Finite Group Databasebeta

← #45663471

Subgroups — #45663471

Order 32896.

234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 61 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#30657089C211yesmaximal, characteristic, complemented
2#30657027C211yesmaximal, characteristic, complemented
3#30657027C211yesmaximal, characteristic, complemented
4#30656483C211yesmaximal, characteristic, complemented
5D4⋊#14523339C211yesmaximal, characteristic, complemented
6D4⋊#14523339C211yesmaximal, characteristic, complemented
7#547×C257C211yesmaximal, characteristic
8#22551592C2211yescharacteristic
9#22551523—21no—
10#22551523—21no—
11C4⋊#14523339C2211yescharacteristic
12C4⋊#14523339C2211yescharacteristic
13#22551515—22no—
14#22551513—21no—
15#22551513—21no—
16C4×#14523339C2211yescharacteristic
17C2×#41×C257—21no—
18C4⋊Q8×C257C2211yescharacteristic
19#123×C257—22no—
20C22⋊Q8×C257—21no—
21#121×C257—21no—
22C4×D4×C257C2211yescharacteristic
23C4×D4×C257C2211yescharacteristic
24C2×C4⋊C4×C257—21no—
25C2×Dic514—21no—
26C2×#14523339—21no—
27C2×#14523339—21no—
28C2×#14523339—21no—
29C4⋊Dic257—21no—
30C4⋊Dic257—21no—
31#41×C257—42no—
32#41×C257—42no—
33C2×Q8×C257—21no—
34C2×D4×C257—22no—
35C2×D4×C257D411yescharacteristic
36C2×D4×C257D411yescharacteristic
37C22×C4×C257—22no—
38C22×C4×C257—21no—
39C22×C4×C257—21no—
40C4⋊C4×C257—22no—
41C4⋊C4×C257—21no—
42C4⋊C4×C257D411yescharacteristic
43C4⋊C4×C257D411yescharacteristic
44C4⋊C4×C257D412yes—
45C22⋊C4×C257—22no—
46C22⋊C4×C257—21no—
47C22⋊C4×C257—21no—
48C42×C257C2311yescharacteristic
49Dic514—42no—
50C2×Dic257—41no—
51#14523339—41no—
52#14523339—41no—
53#14523339—41no—
54C23×C257—21no—
55C23×C257—21no—
56Q8×C257—41no—
57Q8×C257—41no—
58D4×C257—21no—
59D4×C257—21no—
60D4×C257—21no—
61D4×C257—44no—
62D4×C257SD1612yes—
63C2×C4×C257—21no—
64C2×C4×C257—21no—
65C2×C4×C257—22no—
66C2×C4×C257—44no—
67C2×C4×C257—41no—
68C2×C4×C257—41no—
69C2×C4×C257—41no—
70C2×C4×C257—41no—
71C2×C4×C257C2×D411yescharacteristic
72C2×C4×C257C2×D411yescharacteristic
73C2×C4×C257C2×D411yescharacteristic
74C22×C257—22no—
75C22×C257—41no—
76C22×C257—41no—
77C22×C257—41no—
78C22×C257—41no—
79C22×C257—41no—
80C22×C257#12011yescharacteristic
81C4×C257—21no—
82C4×C257—41no—
83C4×C257—42no—
84C4×C257—41no—
85C4×C257C2×SD1611yescharacteristic
86C4×C257#13511yescharacteristic
87C4×C257AGL(1,ℤ/8)11yescharacteristic
88C4×C257#13711yescharacteristic
89Dic257—81no—
90C2×C257—42no—
91C2×C257—81no—
92C2×C257#45011yescharacteristic
93C2×C257#45211yescharacteristic
94C2×C257#45411yescharacteristic
95C257#163511yescharacteristic, complemented, Sylow
96#1635—2571nomaximal, Sylow
97#547D25711yescharacteristic
98#487—2571no—
99#476—2571no—
100#476—2571no—
101#332—2571no—
102D4⋊C8—2571no—
103D4⋊C8—2571no—
104C2×#41—21no—
105C4⋊Q8—2571no—
106C4⋊Q8C2×D25711yescharacteristic
107#123—22no—
108C22⋊Q8—21no—
109#121—21no—
110C4×D4C2×D25711yescharacteristic
111C4×D4C2×D25711yescharacteristic
112C2×C4⋊C4—21no—
113#106—5141no—
114#106—5141no—
115C4⋊C8—2571no—
116C4⋊C8—2571no—
117#103—5142no—
118#102—5141no—
119#102—5141no—
120C4×C8—2571no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×Q8—5141no—
125C2×D4—22no—
126C2×D4C22⋊D25711yescharacteristic
127C2×D4C22⋊D25711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C2×C8—5141no—
132C2×C8—5141no—
133C2×C8—5141no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—5141no—
137C4⋊C4—5141no—
138C4⋊C4C22⋊D25712yes—
139C4⋊C4C22⋊D25711yescharacteristic
140C4⋊C4C22⋊D25711yescharacteristic
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C42C22×D25711yescharacteristic
145C23—21no—
146C23—21no—
147Q8—41no—
148Q8—41no—
149Q8—10282no—
150D4—21no—
151D4—21no—
152D4—21no—
153D4—44no—
154D4#1686160612yes—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—10281no—
164C2×C4C2×C22⋊D25711yescharacteristic
165C2×C4C2×C22⋊D25711yescharacteristic
166C2×C4C2×C22⋊D25711yescharacteristic
167C8—10281no—
168C8—10281no—
169C8—10281no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#2255156111yescharacteristic
177C4—21no—
178C4—41no—
179C4—42no—
180C4—41no—
181C4—20561no—
182C4C2×#1686160611yescharacteristic
183C4#2255161211yescharacteristic
184C4#2255161811yescharacteristic
185C4#2255162211yescharacteristic
186C2—42no—
187C2—81no—
188C2#3065688811yescharacteristic
189C2#3065689911yescharacteristic
190C2#3065691011yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting, prime radical#547×C257C2
socleC22×C257#120
hypercenter#547D257
hypocenterC257#1635
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #30657089C2semidirect
2. #30657027C2semidirect
3. #30657027C2semidirect
4. #30656483C2semidirect
5. D4⋊#14523339C2semidirect
6. D4⋊#14523339C2semidirect
95. C257#1635semidirect

Maximal subgroups

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

subgroupindex
#306570892
#306570272
#306564832
D4⋊#145233392
#547×C2572
#1635257