Finite Group Databasebeta

← #140980

Subgroups — #140980

Order 512.

427 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 82 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#33180C211yesmaximal, characteristic
2#11694C214yesmaximal
3#10814C211yesmaximal, characteristic
4#8157C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#470—21no—
7C2×#467—21no—
8C2×#405C2211yescharacteristic
9#2621C2212yes—
10#1971—21no—
11#1969—21no—
12#1960C412yescomplemented
13#1846C411yescharacteristic, complemented
14#1740C2212yes—
15#1735—21no—
16C2×H(3,ℤ/4)—21no—
17#1458C2212yes—
18#1452—21no—
19#1344—28no—
20#1281—21no—
21C22×Q16—21no—
22C22×Q16—21no—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4⋊Q8C2×C412yes—
26C2×C22⋊Q8—22no—
27C2×#117—22no—
28C2×#117C2311yescharacteristic
29#494—42no—
30#470—44no—
31#467—44no—
32#431—22no—
33C2×#107—21no—
34C2×#106—21no—
35C2×#103—22no—
36C2×#103—22no—
37#405—42no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#398—24no—
43#397—24no—
44#391—22no—
45#382C2×C412yes—
46#342—24no—
47#340—42no—
48H(3,ℤ/4)—24no—
49#324—22no—
50#323D414yes—
51#322—42no—
52C22×Q8—22no—
53C23×C4C2×D411yescharacteristic
54C2×Q16—42no—
55C2×Q16—42no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×Q16—42no—
59C2×Q16—42no—
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—41no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C4⋊Q8—42no—
71C4⋊Q8—42no—
72C22⋊Q8—44no—
73C22⋊Q8—44no—
74#117—24no—
75#117—41no—
76#117—41no—
77#117—42no—
78#117—42no—
79#117—42no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4#4112yes—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4C2×D412yes—
88C2×C42—42no—
89C2×C42C22×C411yescharacteristic
90C2×C42C2×D411yescharacteristic
91#107—42no—
92#106—42no—
93C4⋊C8—42no—
94#103—44no—
95#103—44no—
96C22⋊C8—24no—
97#97—41no—
98#97—41no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—42no—
102#95—24no—
103#95—44no—
104C24#12711yescharacteristic
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8—42no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—82no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C4×D412yes—
120C22×C4#12112yes—
121C22×C4#12411yescharacteristic
122C22×C4AGL(1,ℤ/8)12yes—
123C22×C4AGL(1,ℤ/8)12yes—
124Q16—84no—
125Q16—84no—
126M16—44no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—44no—
138C2×C8—44no—
139C2×C8#9912yes—
140C2×C8#10412yes—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—84no—
146C4⋊C4—82no—
147C4⋊C4—84no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—82no—
151C22⋊C4—84no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—44no—
157C42—82no—
158C23—21no—
159C23—21no—
160C23—42no—
161C23#39011yescharacteristic
162C23#49612yes—
163Q8—82no—
164Q8—82no—
165Q8—82no—
166Q8—82no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—84no—
178C2×C4—84no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—164no—
184C2×C4C2×#9911yescharacteristic
185C2×C4#41011yescharacteristic
186C2×C4#41811yescharacteristic
187C2×C4#41811yescharacteristic
188C2×C4C2×#10411yescharacteristic
189C2×C4#42111yescharacteristic
190C2×C4#49012yes—
191C8—22no—
192C8—22no—
193C8—41no—
194C8—41no—
195C8—84no—
196C22—22no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—81no—
201C22#196111yescharacteristic
202C22#196411yescharacteristic
203C22#198011yescharacteristic
204C4—22no—
205C4—42no—
206C4—42no—
207C4—162no—
208C4—164no—
209C4—162no—
210C4#150511yescharacteristic
211C4#150711yescharacteristic
212C4#151011yescharacteristic
213C4#151411yescharacteristic
214C2—21no—
215C2—21no—
216C2—42no—
217C2#968411yescharacteristic
218C2#1323012yes—
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
12. #1960C4semidirect
13. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331802
#116942
#108142
#81572