Finite Group Databasebeta

← #140929

Subgroups — #140929

Order 512.

455 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 102 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#10866C211yesmaximal, characteristic
4#8156C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#470—21no—
7C2×#467—21no—
8C2×#427—21no—
9C2×#405C2211yescharacteristic
10#2621C2212yes—
11#1971—21no—
12#1969—21no—
13#1960C412yescomplemented
14#1846C411yescharacteristic, complemented
15#1740C2212yes—
16C2×H(3,ℤ/4)—21no—
17#1458C2212yes—
18#1450—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—41no—
30#494—41no—
31#470—42no—
32#470—42no—
33#467—42no—
34#467—42no—
35#431—22no—
36#427—24no—
37#427—42no—
38C2×#107—21no—
39C2×#107—41no—
40C2×#106—21no—
41C2×#106—41no—
42C2×#103—22no—
43C2×#103—22no—
44#405—41no—
45#405—41no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#398—24no—
51#397—24no—
52#391—22no—
53#382C2×C412yes—
54#342—24no—
55H(3,ℤ/4)—42no—
56#324—22no—
57#323D414yes—
58#322—41no—
59C82—41no—
60C22×Q8—22no—
61C23×C4C2×D411yescharacteristic
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—41no—
65C2×Q16—41no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16—42no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16D812yes—
72C2×M16SD1612yes—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—22no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8D812yes—
81C22×C8SD1612yes—
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88#117—24no—
89#117—41no—
90#117—41no—
91#117—42no—
92#117—42no—
93#117—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4#4112yes—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4C2×D412yes—
102C2×C42—42no—
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D411yescharacteristic
105#107—44no—
106#107—42no—
107#106—44no—
108#106—42no—
109C4⋊C8—42no—
110#103—44no—
111#103—44no—
112C22⋊C8—24no—
113#97—41no—
114#97—41no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—42no—
120#95—24no—
121#95—44no—
122C24#12711yescharacteristic
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C4×D412yes—
138C22×C4#12112yes—
139C22×C4#12411yescharacteristic
140C22×C4C2×D811yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4C2×SD1611yescharacteristic
144Q16—84no—
145Q16—82no—
146Q16—82no—
147M16—42no—
148M16—42no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—44no—
160C2×C8#9911yescharacteristic
161C2×C8#9911yescharacteristic
162C2×C8#10212yes—
163C2×C8#10212yes—
164C2×C8#10212yes—
165C2×C8#10212yes—
166C2×C8#10412yes—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—84no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C22⋊C4—42no—
176C22⋊C4—44no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C22⋊C4—82no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—44no—
185C42—82no—
186C23—21no—
187C23—21no—
188C23—42no—
189C23#39011yescharacteristic
190C23#49211yescharacteristic
191C23#49311yescharacteristic
192Q8—82no—
193Q8—82no—
194Q8—82no—
195Q8—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—164no—
217C2×C4C2×#9911yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4C2×#10211yescharacteristic
220C2×C4C2×#10211yescharacteristic
221C2×C4C2×#10411yescharacteristic
222C2×C4#42111yescharacteristic
223C2×C4#48611yescharacteristic
224C2×C4#48611yescharacteristic
225C8—22no—
226C8—22no—
227C8—22no—
228C8—84no—
229C8#32712yes—
230C8#32712yes—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—81no—
238C22#195711yescharacteristic
239C22#196411yescharacteristic
240C22#198011yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—42no—
245C4—162no—
246C4—164no—
247C4—162no—
248C4C2×#32711yescharacteristic
249C4#150211yescharacteristic
250C4#151711yescharacteristic
251C4#151911yescharacteristic
252C2—21no—
253C2—21no—
254C2—41no—
255C2—41no—
256C2#968011yescharacteristic
257C2#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
13. #1960C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331802
#116942
#108662
#81562