Finite Group Databasebeta

← #141166

Subgroups — #141166

Order 512.

494 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 114 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#33172C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C2×#1348C214yesmaximal
4#10820C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#467—21no—
7C2×#467C2212yes—
8C22×C4×C8C2211yescharacteristic
9#2857—21no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12#1969—21no—
13#1969C412yescomplemented
14#1956—21no—
15#1845C411yescharacteristic, complemented
16C2×#340—21no—
17C2×#340C2212yes—
18C4⋊#95—21no—
19#1374—21no—
20#1348—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C22×C4⋊C4—22no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#494—42no—
31#467—24no—
32#467—24no—
33#467—44no—
34C2×#107—21no—
35C2×#107C2×C412yes—
36C2×#103—22no—
37C2×#103—22no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#391—22no—
44#389—41no—
45#389—41no—
46#384—21no—
47#380—41no—
48#380—41no—
49C2×#95—41no—
50C2×#95—41no—
51C4⋊C16—42no—
52C22⋊C16—24no—
53#340—22no—
54#340—42no—
55#340D414yes—
56C22×Q8—22no—
57C23×C4—42no—
58C23×C4D811yescharacteristic
59C23×C4D811yescharacteristic
60C23×C4C2×D411yescharacteristic
61C2×Q16—24no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D412yes—
77C22×C8#4112yes—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C22⋊Q8—44no—
81C22⋊Q8—44no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C42SD1611yescharacteristic
101C2×C16—42no—
102C2×C16—42no—
103C2×C16—42no—
104#107—24no—
105#107—42no—
106#103—44no—
107#103—42no—
108#103—42no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—41no—
113#95—42no—
114#95—42no—
115#95—82no—
116#95—82no—
117C24C2×D811yescharacteristic
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4#10211yescharacteristic
143C22×C4#10211yescharacteristic
144C22×C4#10211yescharacteristic
145C22×C4#10211yescharacteristic
146C22×C4SD3212yes—
147C22×C4Q3212yes—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4#12112yes—
150C22×C4C2×SD1611yescharacteristic
151C22×C4AGL(1,ℤ/8)12yes—
152Q16—84no—
153Q16—84no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—44no—
166C2×C8#9912yes—
167C2×C8C4×D412yes—
168C2×C8#12411yescharacteristic
169C2×C8#12711yescharacteristic
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—84no—
181C22⋊C4—84no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—41no—
187C42—41no—
188C16—82no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23C2×#10211yescharacteristic
194C23#45912yes—
195C23C2×SD3211yescharacteristic
196C23C2×Q3211yescharacteristic
197C23#51012yes—
198Q8—82no—
199Q8—82no—
200Q8—82no—
201Q8—82no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—162no—
230C2×C4—162no—
231C2×C4—162no—
232C2×C4—162no—
233C2×C4#35812yes—
234C2×C4#35812yes—
235C2×C4#36012yes—
236C2×C4#39011yescharacteristic
237C2×C4C2×#9911yescharacteristic
238C2×C4#41011yescharacteristic
239C2×C4#41811yescharacteristic
240C2×C4#42111yescharacteristic
241C2×C4#42111yescharacteristic
242C2×C4#46012yes—
243C8—21no—
244C8—22no—
245C8—21no—
246C8—22no—
247C8—41no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22#151711yescharacteristic
257C22#151911yescharacteristic
258C22#198211yescharacteristic
259C22C2×#35811yescharacteristic
260C22#214111yescharacteristic
261C22C2×#36011yescharacteristic
262C22#214511yescharacteristic
263C22#220812yes—
264C22#220912yes—
265C4—22no—
266C4—22no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—164no—
271C4—162no—
272C4—162no—
273C4#151011yescharacteristic
274C4#151411yescharacteristic
275C4#196311yescharacteristic
276C4#196411yescharacteristic
277C2—22no—
278C2—22no—
279C2#971511yescharacteristic
280C2#1185711yescharacteristic
281C2#1185911yescharacteristic
282C2#1188711yescharacteristic
283C2#1188911yescharacteristic
284C2#1325311yescharacteristic
285C2#1325511yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
13. #1969C4semidirect
15. #1845C4semidirect

Maximal subgroups

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

subgroupindex
#331722
#127832
C2×#13482
#108202