Finite Group Databasebeta

← #241158

Subgroups — #241158

Order 512.

840 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 202 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#34638C211yesmaximal, characteristic
2#21986C212yesmaximal
3#13963C211yesmaximal, characteristic
4#10572C214yesmaximal
5#10571C214yesmaximal
6#8022C212yesmaximal
7#7910C211yesmaximal, characteristic
8C2×#432C2212yes—
9#2647—22no—
10#2647—22no—
11#2646—21no—
12#2646—21no—
13#2646C2214yes—
14#2402—24no—
15#2296—22no—
16#2285C2212yes—
17C2×#382C2211yescharacteristic
18C2×#376C2211yescharacteristic
19C2×#376C2211yescharacteristic
20#1949—24no—
21#1853C2214yes—
22#1852C2214yes—
23#1847—22no—
24#1836—22no—
25#1444C2212yes—
26#1442C2212yes—
27#1436—22no—
28C4⋊#95—21no—
29#1434—21no—
30#1294C2214yes—
31#1294C2214yes—
32#1280C2214yes—
33C2×#117—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2312yes—
36C22×C42—21no—
37C22×C42D412yes—
38C22×C42C2311yescharacteristic
39#432—24no—
40#432Q818yes—
41C2×#107—24no—
42C2×#106—24no—
43C2×C4⋊C8—22no—
44C2×C22⋊C8C2314yes—
45C2×#97—22no—
46#400—24no—
47#400—24no—
48#400—24no—
49#400—24no—
50#398—24no—
51#398—24no—
52#398—24no—
53#398—24no—
54#389—22no—
55#389—22no—
56#389—22no—
57#389—22no—
58#385—24no—
59#385—24no—
60#384D412yes—
61#384D412yes—
62#382—22no—
63#382—22no—
64#382—22no—
65#380—22no—
66#380C2314yes—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D414yes—
71#376—22no—
72#376—22no—
73#376—22no—
74#376Q814yes—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95D411yescharacteristic
82C2×#95D411yescharacteristic
83C2×#95C2312yes—
84C2×#95C2311yescharacteristic
85C2×#95C2311yescharacteristic
86#340—24no—
87#340—24no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—24no—
97#117—44no—
98#117—44no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C22⋊C4—24no—
116C2×C42—24no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—44no—
123C2×C42C2×D412yes—
124C2×C42C2×D412yes—
125C2×C42C2×Q814yes—
126C2×C42C2×Q812yes—
127C2×C42#4112yes—
128#107—84no—
129#106—84no—
130C4⋊C8—28no—
131C22⋊C8—24no—
132C22⋊C8C2×Q818yes—
133#97—28no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—22no—
139#95—22no—
140#95—24no—
141#95—24no—
142#95—22no—
143#95—22no—
144#95—24no—
145#95—21no—
146#95—21no—
147#95—44no—
148#95—44no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95—82no—
153#95C2×Q812yes—
154#95C2×Q812yes—
155C24C22×D411yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4#12012yes—
191C22×C4#12011yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4C22⋊Q812yes—
194C22×C4C22⋊Q812yes—
195C22×C4C4⋊Q812yes—
196C22×C4C4⋊Q812yes—
197C22×C4C22×D412yes—
198C22×C4C22×Q812yes—
199C22×C4C22×Q811yescharacteristic
200C22×C4C2×#4111yescharacteristic
201C2×C8—28no—
202C2×C8—44no—
203C2×C8—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—84no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—88no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C42—44no—
220C42—44no—
221C42—48no—
222C42—48no—
223C42C22⋊Q814yes—
224C42C22⋊Q814yes—
225C42C4⋊Q814yes—
226C42C4⋊Q814yes—
227C23—22no—
228C23—21no—
229C23—21no—
230C23C2×#12011yescharacteristic
231C23C2×C22⋊Q811yescharacteristic
232C23C2×C4⋊Q811yescharacteristic
233C23#57512yes—
234C23#57512yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#39512yes—
283C2×C4#39512yes—
284C2×C4#39514yes—
285C2×C4C2×C22⋊Q812yes—
286C2×C4C2×C4⋊Q812yes—
287C8—48no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22C2×#39511yescharacteristic
297C22#300512yes—
298C22#301812yes—
299C22#302711yescharacteristic
300C22#302811yescharacteristic
301C22#308011yescharacteristic
302C22#308011yescharacteristic
303C22#316612yes—
304C4—22no—
305C4—22no—
306C4—24no—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—82no—
311C4—84no—
312C4—82no—
313C4—84no—
314C4—82no—
315C4—162no—
316C2—21no—
317C2—21no—
318C2—22no—
319C2#1833412yes—
320C2#1834112yes—
321C2#2355311yescharacteristic
322C2#2356011yescharacteristic
323C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#346382
#219862
#139632
#105722
#105712
#80222
#79102