Finite Group Databasebeta

← #138528

Subgroups — #138528

Order 512.

374 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 84 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#29773C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic
3#11892C212yesmaximal
4C2×#1322C211yesmaximal, characteristic
5#7232C212yesmaximal
6C2×#467C2211yescharacteristic
7#2920—21no—
8C22×C4⋊C8C2211yescharacteristic
9#2855—21no—
10C2×C22⋊C16C2212yes—
11#2086C411yescharacteristic
12#2086C411yescharacteristic
13#2066—21no—
14#2066—21no—
15#2056C411yescharacteristic
16#2056C411yescharacteristic
17#1950—24no—
18#1892—21no—
19#1810—21no—
20#1810C2212yes—
21#1807C2211yescharacteristic
22#1350—24no—
23#1322—21no—
24#1322—21no—
25C22×Q16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#467—22no—
32#467—22no—
33#432—41no—
34#432—41no—
35#431—41no—
36#431—41no—
37C2×#108—22no—
38C2×#108—22no—
39C2×#107C2×C411yescharacteristic
40C2×#106—21no—
41#423—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51#407—22no—
52C4×M16—22no—
53#391—21no—
54#391—21no—
55#389—21no—
56#389—21no—
57#387—41no—
58#387—41no—
59#385—41no—
60#381—41no—
61#380—21no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64#341—22no—
65#341D414yes—
66#340—21no—
67#340—21no—
68#340C2×C411yescharacteristic
69#340C2×C411yescharacteristic
70C22×Q8—21no—
71C23×C4SD1611yescharacteristic
72C23×C4SD1611yescharacteristic
73C23×C4C2×D411yescharacteristic
74C2×Q16—22no—
75C2×Q16—81no—
76C2×M16—24no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—81no—
80C22×C8—81no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8C2×D411yescharacteristic
86C22×C8#4112yes—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22⋊C411yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42D811yescharacteristic
107C2×C16—42no—
108C2×C16—82no—
109#108—82no—
110#108—82no—
111#107—22no—
112#106—81no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119#103—42no—
120#103—81no—
121#103—81no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#97—42no—
125C4×C8—42no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133C24C2×SD1611yescharacteristic
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—41no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4#10211yescharacteristic
156C22×C4#10211yescharacteristic
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4#12011yescharacteristic
159C22×C4#12311yescharacteristic
160C22×C4C2×D811yescharacteristic
161C22×C4AGL(1,ℤ/8)11yescharacteristic
162C22×C4#13711yescharacteristic
163Q16—82no—
164M16—44no—
165M16—44no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—44no—
176C2×C8—81no—
177C2×C8C4×D412yes—
178C2×C8#12112yes—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C22⋊C4—82no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C16—82no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—41no—
205C23C2×#10211yescharacteristic
206C23#44811yescharacteristic
207C23#48111yescharacteristic
208Q8—81no—
209Q8—81no—
210Q8—81no—
211Q8—81no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#35912yes—
247C2×C4#38611yescharacteristic
248C2×C4#41711yescharacteristic
249C2×C4#44711yescharacteristic
250C2×C4#48011yescharacteristic
251C8—22no—
252C8—22no—
253C8—41no—
254C8—41no—
255C8—84no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—42no—
264C22—42no—
265C22#187411yescharacteristic
266C22C2×#35911yescharacteristic
267C22#214311yescharacteristic
268C22#223912yes—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—161no—
277C4—161no—
278C4—161no—
279C4—161no—
280C4#191211yescharacteristic
281C4#191311yescharacteristic
282C2—21no—
283C2—21no—
284C2—22no—
285C2—41no—
286C2#920411yescharacteristic
287C2#1299011yescharacteristic
288C2#1300911yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#297732
#219692
#118922
C2×#13222
#72322