Finite Group Databasebeta

← #138017

Subgroups — #138017

Order 512.

392 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 88 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#33419C211yesmaximal, characteristic
2#33133C211yesmaximal, characteristic
3#11706C212yesmaximal
4#10814C211yesmaximal, characteristic
5#7262C212yesmaximal
6C2×#467C2211yescharacteristic
7C2×#405C2211yescharacteristic
8#2625—21no—
9#2164—24no—
10#2150—21no—
11#2112C2212yes—
12C4×M32—21no—
13#1971C411yescharacteristic, complemented
14#1971C411yescharacteristic, complemented
15#1944C411yescharacteristic, complemented
16#1944C411yescharacteristic, complemented
17#1933—21no—
18#1933—21no—
19#1894—21no—
20#1807C2211yescharacteristic
21#1740—21no—
22#1735C2212yes—
23C2×H(3,ℤ/4)—21no—
24#1348—24no—
25C22×Q16C2×C411yescharacteristic
26C22×M16D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C2×#117—22no—
30C2×#117D411yescharacteristic
31C2×M32—22no—
32#467—22no—
33#467—22no—
34C2×#107C2×C411yescharacteristic
35C2×#106C2×C411yescharacteristic
36C2×#106C2×C411yescharacteristic
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40#405—22no—
41#405—22no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#400—41no—
47#398—41no—
48#397—41no—
49#397—41no—
50#391—21no—
51#391—21no—
52#389—21no—
53#389—21no—
54#380—21no—
55C4⋊C16—24no—
56C22⋊C16—22no—
57C22⋊C16D414yes—
58#346—22no—
59C4×C16—22no—
60#342—41no—
61#342—41no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340—22no—
67H(3,ℤ/4)—24no—
68C22×Q8—21no—
69C23×C4C2×D411yescharacteristic
70C2×Q16—22no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D412yes—
83C22×C8#4112yes—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86#117—22no—
87#117—22no—
88#117—24no—
89#117—42no—
90#117—42no—
91#117—42no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4SD1611yescharacteristic
100C2×C4⋊C4SD1611yescharacteristic
101C2×C22⋊C4—42no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4D811yescharacteristic
105C2×C22⋊C4D811yescharacteristic
106C2×C42—42no—
107C2×C42C22⋊C411yescharacteristic
108C2×C42C22⋊C411yescharacteristic
109M32—42no—
110M32—42no—
111C2×C16—24no—
112#107—22no—
113#106—41no—
114#106—41no—
115#103—42no—
116#103—42no—
117#103—41no—
118#103—41no—
119#97—22no—
120#97—22no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—41no—
132C24C2×D811yescharacteristic
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—41no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—82no—
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4#12112yes—
154C22×C4C2×SD1611yescharacteristic
155C22×C4#13512yes—
156Q16—82no—
157M16—44no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8#10412yes—
170C2×C8C4×D412yes—
171C2×C8#12011yescharacteristic
172C2×C8#12311yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—44no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4—161no—
185C4⋊C4—161no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—44no—
189C22⋊C4—82no—
190C22⋊C4—82no—
191C22⋊C4—161no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42—82no—
198C16—44no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23C2×#10211yescharacteristic
204C23#46612yes—
205Q8—81no—
206Q8—81no—
207Q8—81no—
208Q8—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—84no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—162no—
237C2×C4—162no—
238C2×C4#38611yescharacteristic
239C2×C4#41011yescharacteristic
240C2×C4#41011yescharacteristic
241C2×C4#41611yescharacteristic
242C2×C4C2×#10411yescharacteristic
243C2×C4#42111yescharacteristic
244C2×C4#46512yes—
245C8—22no—
246C8—21no—
247C8—22no—
248C8—21no—
249C8—41no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—81no—
256C22—81no—
257C22#150211yescharacteristic
258C22#151711yescharacteristic
259C22#189311yescharacteristic
260C4—22no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—161no—
266C4—161no—
267C4—164no—
268C4—161no—
269C4—161no—
270C4#150011yescharacteristic
271C4#152111yescharacteristic
272C4#187111yescharacteristic
273C4#189911yescharacteristic
274C2—21no—
275C2—21no—
276C2—42no—
277C2#911411yescharacteristic
278C2#1307511yescharacteristic
279C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1971C4semidirect
14. #1971C4semidirect
15. #1944C4semidirect
16. #1944C4semidirect

Maximal subgroups

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

subgroupindex
#334192
#331332
#117062
#108142
#72622