Finite Group Databasebeta

← #141567

Subgroups — #141567

Order 512.

322 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 72 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#28677C211yesmaximal, characteristic
2#11795C211yesmaximal, characteristic
3#11782C211yesmaximal, characteristic
4#11574C211yesmaximal, characteristic
5#11568C211yesmaximal, characteristic
6#8621C211yesmaximal, characteristic
7C8×#101C211yesmaximal, characteristic
8C2×#498C411yescharacteristic
9C2×C8×Q8C2211yescharacteristic
10#2567C2211yescharacteristic
11#2546C2211yescharacteristic
12#2088—21no—
13#2088—21no—
14#1988C411yescharacteristic
15#1987C411yescharacteristic
16#1941—21no—
17#1941—21no—
18C4×#106C411yescharacteristic
19C4×#101C2211yescharacteristic
20#1568—21no—
21#1467C2211yescharacteristic
22#1467C2211yescharacteristic
23#1461C2211yescharacteristic
24C8×M16—21no—
25#1354—22no—
26#1346—22no—
27#1325—22no—
28#1319—22no—
29#1290—21no—
30#1283—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8C2×C411yescharacteristic
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4×Q8C2311yescharacteristic
35#498—41no—
36#498—41no—
37C8×Q8—41no—
38C8×Q8—41no—
39#432—21no—
40#432—21no—
41#431—21no—
42C2×#106—21no—
43C2×#106—21no—
44C2×#106—21no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×#101—21no—
49C4×M16—21no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#398—21no—
54#398—21no—
55#395—21no—
56#395—21no—
57#391—21no—
58#391—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#384—21no—
64C4×C4⋊C4C2×C411yescharacteristic
65C4×C4⋊C4C2×C411yescharacteristic
66#343D412yes—
67#340—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#335—41no—
73#334—41no—
74#324—21no—
75#324—21no—
76#324—21no—
77#324—21no—
78#324—21no—
79#324D412yes—
80#322—41no—
81C82—41no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8#4111yescharacteristic
85C2×M16—22no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—82no—
95C4×Q8—41no—
96C4×Q8—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C2×D411yescharacteristic
108C2×C4⋊C4C2×D411yescharacteristic
109C2×C4⋊C4#4111yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115#106—24no—
116#106—42no—
117#106—41no—
118#106—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125#101—24no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129#97—41no—
130#97—41no—
131#97—41no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—41no—
139C4×C8—41no—
140#95—21no—
141#95—21no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—161no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4#12411yescharacteristic
164C22×C4#12711yescharacteristic
165M16—24no—
166M16—42no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—42no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8#9911yescharacteristic
179C2×C8#9911yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—161no—
196C4⋊C4—161no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42—81no—
205C23#39011yescharacteristic
206Q8—44no—
207Q8—162no—
208Q8—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
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—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
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×C4C2×#9911yescharacteristic
237C2×C4#41011yescharacteristic
238C2×C4#44312yes—
239C2×C4#47112yes—
240C8—21no—
241C8—44no—
242C8—41no—
243C8—41no—
244C8—41no—
245C8—82no—
246C8—161no—
247C8—161no—
248C8#35612yes—
249C22—21no—
250C22—21no—
251C22#196411yescharacteristic
252C22#197411yescharacteristic
253C22#197711yescharacteristic
254C4—22no—
255C4—22no—
256C4—21no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—82no—
262C4—82no—
263C4—161no—
264C4—161no—
265C4C2×#35611yescharacteristic
266C4#213511yescharacteristic
267C2—21no—
268C2—21no—
269C2#981311yescharacteristic
270C2#1322711yescharacteristic
271C2#1322911yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#390
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#286772
#117952
#117822
#115742
#115682
#86212
C8×#1012