Finite Group Databasebeta

← #139089

Subgroups — #139089

Order 512.

368 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 86 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#33774C211yesmaximal, characteristic
2#17977C211yesmaximal, characteristic, complemented
3#11961C211yesmaximal, characteristic
4#11857C211yesmaximal, characteristic, complemented
5#9832C211yesmaximal, characteristic
6#7217C211yesmaximal, characteristic, complemented
7#7145C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9C2×#493C2211yescharacteristic
10#2960—21no—
11#2248C412yescomplemented
12#2240C412yescomplemented
13#2207—22no—
14#2187—22no—
15#2162C2211yescharacteristic
16#2162C2211yescharacteristic
17#1957—21no—
18#1957—21no—
19#1834C2211yescharacteristic
20#1584—22no—
21#1584—22no—
22#1567C2211yescharacteristic
23#1493—21no—
24C2×#322C2211yescharacteristic
25#1361—22no—
26#1348—22no—
27C22×D8—21no—
28C22×D8—21no—
29#533—21no—
30C2×#127—21no—
31C2×SD32—24no—
32#500C2×C411yescharacteristic
33#494C2×C411yescharacteristic
34#493—21no—
35#493—21no—
36#493—21no—
37#493C2×C412yes—
38#487—21no—
39#467—21no—
40#467D412yes—
41#446—22no—
42#446—41no—
43C8×Q8—22no—
44C8×Q8—41no—
45C4×Q16—21no—
46C4×Q16D412yes—
47#424—41no—
48#423—21no—
49C2×#102—42no—
50C2×#102—41no—
51C2×#102—41no—
52#405—41no—
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#384—21no—
57C4⋊C16—21no—
58C4⋊C16—21no—
59#357—24no—
60C22⋊C16—21no—
61C22⋊C16—21no—
62C4×C16—21no—
63C4×C16C2×C412yes—
64#340—21no—
65#340—21no—
66#335—21no—
67#335D412yes—
68Q8⋊C8—24no—
69#322—21no—
70#322—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—41no—
74C2×Q16—21no—
75C2×Q16C22⋊C412yes—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—41no—
79C2×D8—41no—
80C2×D8—82no—
81C2×D8—82no—
82C2×M16—41no—
83C22×C8—21no—
84C22×C8D811yescharacteristic
85C22×C8D811yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8#4111yescharacteristic
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90#127—22no—
91#127—41no—
92#127—41no—
93#127—41no—
94#125—41no—
95C22⋊Q8—22no—
96C22⋊Q8—41no—
97C4×Q8—22no—
98C4×Q8—41no—
99#117—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C42C2×D411yescharacteristic
104SD32—84no—
105C2×C16—22no—
106C2×C16—41no—
107#107—21no—
108#107C22⋊C412yes—
109C4⋊C8—22no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C4⋊C8—41no—
113#103—22no—
114#103—22no—
115#102—82no—
116#102—84no—
117#102—82no—
118#97—42no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—42no—
124C4×C8C22×C411yescharacteristic
125C4×C8C2×D411yescharacteristic
126C4×C8C2×D411yescharacteristic
127C4×C8#4111yescharacteristic
128C24—81no—
129C2×Q8—22no—
130C2×Q8—41no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—82no—
135C2×D4—81no—
136C2×D4—81no—
137C2×D4—81no—
138C2×D4—81no—
139C2×D4—82no—
140C2×D4—81no—
141C2×D4—81no—
142C2×D4—81no—
143C2×D4—81no—
144C2×D4—81no—
145C2×D4—81no—
146C2×D4—81no—
147C22×C4—81no—
148C22×C4#12111yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4AGL(1,ℤ/8)11yescharacteristic
151Q16—42no—
152Q16—81no—
153D8—44no—
154D8—82no—
155D8—82no—
156D8—82no—
157M16—81no—
158M16—81no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8SD3212yescomplemented
170C2×C8C2×C22⋊C411yescharacteristic
171C2×C8C4×D411yescharacteristic
172C2×C8#12111yescharacteristic
173C2×C8#12311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C22⋊C4—42no—
186C42—42no—
187C42C4×D411yescharacteristic
188C42#12011yescharacteristic
189C42#13511yescharacteristic
190C42AGL(1,ℤ/8)11yescharacteristic
191C16—42no—
192C16—81no—
193C23—82no—
194C23—161no—
195C23—161no—
196C23—161no—
197C23—161no—
198C23—161no—
199C23—161no—
200C23#45911yescharacteristic
201Q8—42no—
202Q8—42no—
203Q8—82no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208D4—84no—
209D4—161no—
210D4—161no—
211D4—162no—
212D4—161no—
213D4—161no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—161no—
220C2×C4—161no—
221C2×C4—161no—
222C2×C4#38611yescharacteristic
223C2×C4C4×D811yescharacteristic
224C2×C4#44211yescharacteristic
225C2×C4#44711yescharacteristic
226C2×C4#44911yescharacteristic
227C2×C4#46311yescharacteristic
228C2×C4C2×SD3211yescharacteristic
229C2×C4#50811yescharacteristic
230C2×C4#50911yescharacteristic
231C2×C4#51011yescharacteristic
232C8—21no—
233C8—21no—
234C8—21no—
235C8—21no—
236C8—22no—
237C8—41no—
238C8—81no—
239C8—82no—
240C22—21no—
241C22—21no—
242C22—162no—
243C22—162no—
244C22—162no—
245C22—321no—
246C22—321no—
247C22—321no—
248C22—321no—
249C22#187911yescharacteristic
250C22#220711yescharacteristic
251C22#221111yescharacteristic
252C4—21no—
253C4—21no—
254C4—82no—
255C4—82no—
256C4—321no—
257C4#190311yescharacteristic
258C4#191011yescharacteristic
259C4C4×SD3211yescharacteristic
260C4#217511yescharacteristic
261C4AGL(1,ℤ/16)12yes—
262C4#218511yescharacteristic
263C4#218611yescharacteristic
264C2—21no—
265C2—21no—
266C2—322no—
267C2#929311yescharacteristic
268C2#1301311yescharacteristic
269C2#1304011yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17977C2semidirect
4. #11857C2semidirect
6. #7217C2semidirect
7. #7145C2semidirect
11. #2248C4semidirect
12. #2240C4semidirect
169. C2×C8SD32semidirect

Maximal subgroups

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

subgroupindex
#337742
#179772
#119612
#118572
#98322
#72172
#71452