Finite Group Databasebeta

← #45663433

Subgroups — #45663433

Order 32896.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#30657460C211yesmaximal, characteristic, complemented
2#30657084C211yesmaximal, characteristic, complemented
3#30656937C211yesmaximal, characteristic, complemented
4#30656933C211yesmaximal, characteristic
5#30656484C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic, complemented
7Q8⋊C8×C257C211yesmaximal, characteristic, complemented
8C2×#16861605—21no—
9C2×D2056—21no—
10#22551592C2211yescharacteristic
11#22551589C2211yescharacteristic
12#22551587—22no—
13#22551580—21no—
14#22551568—21no—
15C4×Dic514—21no—
16C4×D1028C2211yescharacteristic
17#22551550—21no—
18C4⋊#14523339C2211yescharacteristic
19#22551514—22no—
20#22551514—21no—
21#22551512—21no—
22C4×Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×D1028—21no—
27C2×D1028—21no—
28C2×D1028—21no—
29C2×D1028D411yescharacteristic
30C2×C4×D257—21no—
31#16861605—42no—
32D2056—42no—
33C2×#14523339—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257D411yescharacteristic
36C4⋊Dic257D412yes—
37#16861597—22no—
38#16861595—22no—
39#16861595—22no—
40#16861595—21no—
41C4×Dic257—22no—
42C2×Q8×C257D411yescharacteristic
43C2×C8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257D411yescharacteristic
47C42×C257—21no—
48C42×C257C2311yescharacteristic
49C22×D257—21no—
50C22×D257—41no—
51Dic514—41no—
52Dic514—41no—
53D1028—21no—
54D1028—21no—
55D1028—42no—
56D1028—41no—
57D1028—41no—
58D1028—41no—
59D1028—41no—
60C2×Dic257—21no—
61C2×Dic257—22no—
62C4×D257—41no—
63C4×D257—41no—
64#14523339—41no—
65Q8×C257—21no—
66Q8×C257—21no—
67C2×C4×C257—21no—
68C2×C4×C257—21no—
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C8×C257—41no—
73C8×C257—41no—
74C22×C257#12011yescharacteristic
75C2×D257—41no—
76C2×D257—41no—
77C2×D257—41no—
78C2×D257—81no—
79C2×D257—81no—
80C2×D257—81no—
81C4×C257—21no—
82C4×C257—41no—
83C4×C257—41no—
84C4×C257#13511yescharacteristic
85C4×C257#13511yescharacteristic
86C4×C257AGL(1,ℤ/8)11yescharacteristic
87C4×C257AGL(1,ℤ/8)11yescharacteristic
88Dic257—41no—
89Dic257—42no—
90C2×C257#44911yescharacteristic
91C2×C257#44911yescharacteristic
92C2×C257#45411yescharacteristic
93D257—81no—
94D257—161no—
95C257#162811yescharacteristic, complemented, Sylow
96#1628—2571nomaximal, Sylow
97#556—2571no—
98#486—2571no—
99#460—2571no—
100#459—2571no—
101#332—2571no—
102Q8⋊C8D25711yescharacteristic, complemented
103D4⋊C8—2571no—
104C2×SD16—5141no—
105C2×D8—5141no—
106C4⋊Q8—2571no—
107#127—2571no—
108#126—5142no—
109#124—5141no—
110C22⋊Q8—5141no—
111C4×Q8—5141no—
112C4×Q8C2×D25711yescharacteristic
113C4×D4—2571no—
114#117—5141no—
115C4⋊C8—2571no—
116C4⋊C8C2×D25711yescharacteristic
117#102—5141no—
118#102—5142no—
119#102—5141no—
120C4×C8C2×D25711yescharacteristic
121C2×Q8—5141no—
122C2×Q8D102811yescharacteristic
123C2×D4—2571no—
124C2×D4—5141no—
125C2×D4—5141no—
126C2×D4—5141no—
127C22×C4—5141no—
128SD16—10282no—
129D8—10282no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—5141no—
133C4⋊C4—2571no—
134C4⋊C4—2572no—
135C4⋊C4—21no—
136C4⋊C4—5142no—
137C4⋊C4—5141no—
138C4⋊C4D102811yescharacteristic
139C22⋊C4—5142no—
140C22⋊C4—5142no—
141C22⋊C4—5141no—
142C42—21no—
143C42—5142no—
144C42C22×D25711yescharacteristic
145C23—5141no—
146C23—10281no—
147Q8—21no—
148Q8—21no—
149Q8—10281no—
150Q8—10281no—
151D4—5141no—
152D4—5141no—
153D4—10282no—
154D4—10281no—
155D4—10281no—
156D4—10281no—
157D4—10281no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—5141no—
161C2×C4—5142no—
162C2×C4—10281no—
163C2×C4—10281no—
164C2×C4D4×D25711yescharacteristic
165C2×C4D4×D25711yescharacteristic
166C2×C4C2×D102811yescharacteristic
167C8—41no—
168C8—41no—
169C8—10281no—
170C22—10281no—
171C22—10281no—
172C22—10281no—
173C22—20561no—
174C22—20561no—
175C22—20561no—
176C22#2255156211yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—10281no—
181C4—10282no—
182C4#2255161411yescharacteristic
183C4#2255161511yescharacteristic
184C4#2255161911yescharacteristic
185C4#2255162011yescharacteristic
186C2—20561no—
187C2—41121no—
188C2#3065688411yescharacteristic
189C2#3065688511yescharacteristic
190C2#3065691111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingQ8⋊C8×C257C2
socleC22×C257#120
prime radical#30656933C2
hypercenterQ8⋊C8D257
hypocenterC257#1628
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #30657460C2semidirect
2. #30657084C2semidirect
3. #30656937C2semidirect
5. #30656484C2semidirect
6. #30656466C2semidirect
7. Q8⋊C8×C257C2semidirect
95. C257#1628semidirect
102. Q8⋊C8D257semidirect

Maximal subgroups

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

subgroupindex
#306574602
#306570842
#306569372
#306569332
#306564842
#306564662
Q8⋊C8×C2572
#1628257