Finite Group Databasebeta

← #120535

Subgroups — #120535

Order 512.

463 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 136 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#21972C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic
3#8066C212yesmaximal
4#7988C211yesmaximal, characteristic
5#7856C212yesmaximal
6#2920C2212yes—
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1950—24no—
12#1939—22no—
13#1850—22no—
14#1842C412yes—
15#1831C412yes—
16#1810—22no—
17#1299—24no—
18#1290—22no—
19#1274—24no—
20C23×C8D412yes—
21C2×#117Q811yescharacteristic
22C2×#117Q811yescharacteristic
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#431—22no—
33#431—22no—
34C2×#108—24no—
35#423—22no—
36#423—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—24no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C412yes—
43C2×C4⋊C8D412yes—
44#407—22no—
45#407—22no—
46C2×C22⋊C8—21no—
47C4×M16D414yes—
48C4×M16Q814yes—
49C2×C4×C8—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54#376D412yes—
55C43C2×C412yes—
56#341—24no—
57#336—24no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×Q811yescharacteristic
60C23×C4#4111yescharacteristic
61C2×M16—24no—
62C2×M16—24no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—82no—
68C22×C8C22⋊C414yes—
69#117—24no—
70#117—42no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42C4⋊C412yes—
78C2×C42C4⋊C412yes—
79C2×C42C22×C411yescharacteristic
80C2×C42C2×D412yes—
81C2×C42C2×D411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×Q812yes—
84C2×C42#4112yes—
85C2×C42#4112yes—
86C2×C42#4111yescharacteristic
87#108—84no—
88C4⋊C8—24no—
89C4⋊C8—22no—
90C4⋊C8—22no—
91C4⋊C8—24no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—84no—
97C22⋊C8—41no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—41no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103#97—24no—
104#97—24no—
105C4×C8—24no—
106C4×C8—24no—
107C4×C8—42no—
108#95—22no—
109#95—22no—
110C24C22⋊Q811yescharacteristic
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—81no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C2×C4⋊C411yescharacteristic
120C22×C4#11711yescharacteristic
121C22×C4#12011yescharacteristic
122C22×C4#12111yescharacteristic
123C22×C4#12111yescharacteristic
124C22×C4C22⋊Q811yescharacteristic
125C22×C4C22⋊Q812yes—
126C22×C4#12311yescharacteristic
127C22×C4#12411yescharacteristic
128C22×C4#12512yes—
129C22×C4#12612yes—
130C22×C4C4⋊Q812yes—
131C22×C4C4⋊Q811yescharacteristic
132M16—44no—
133M16—44no—
134M16—84no—
135C2×C8—24no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—82no—
145C4⋊C4—24no—
146C4⋊C4—42no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C42—22no—
150C42—24no—
151C42—42no—
152C42—42no—
153C42—82no—
154C42#10814yescomplemented
155C42C4×D412yes—
156C42C4×Q812yes—
157C42#12112yes—
158C42#12112yes—
159C42C22⋊Q812yes—
160C42C22⋊Q812yes—
161C42#12312yes—
162C42#12512yes—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#38011yescharacteristic
168C23#39511yescharacteristic
169C23#40011yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4#38611yescharacteristic
194C2×C4#38712yes—
195C2×C4#38912yes—
196C2×C4#39111yescharacteristic
197C2×C4#39411yescharacteristic
198C2×C4#39612yes—
199C2×C4#39812yes—
200C2×C4#39811yescharacteristic
201C2×C4C2×#10812yes—
202C2×C4#43012yes—
203C8—44no—
204C8—84no—
205C8—84no—
206C8—81no—
207C8—81no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22#144111yescharacteristic
219C22#181011yescharacteristic
220C22#181111yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—82no—
229C4#191111yescharacteristic
230C4#191211yescharacteristic
231C4#195014yes—
232C2—21no—
233C2—21no—
234C2—41no—
235C2—41no—
236C2#1122411yescharacteristic
237C2#1123311yescharacteristic
238C2#1124611yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
154. C42#108semidirect

Maximal subgroups

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

subgroupindex
#219722
#219692
#80662
#79882
#78562