Finite Group Databasebeta

← #241175

Subgroups — #241175

Order 512.

860 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 206 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#35120C211yesmaximal, characteristic
2#21361C212yesmaximal
3#14164C211yesmaximal, characteristic
4#10572C214yesmaximal
5#10569C212yesmaximal
6#10568C212yesmaximal
7#8025C212yesmaximal
8#7906C211yesmaximal, characteristic
9C2×#431C2212yes—
10#2855—22no—
11#2659C2212yes—
12#2646C2212yes—
13#2612—24no—
14#2606—22no—
15#2605—22no—
16#2329—22no—
17#2283C2212yes—
18C2×#383C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1949—24no—
22#1853C2214yes—
23#1852C2214yes—
24C4×#107—22no—
25C4×#106—22no—
26#1444C2212yes—
27#1442C2212yes—
28#1436—22no—
29#1436—22no—
30#1294C2214yes—
31#1294C2214yes—
32#1280C2211yescharacteristic
33#1280C2211yescharacteristic
34#1280C2212yes—
35C2×C22⋊Q8—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2312yes—
38C22×C42—21no—
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41#431—24no—
42#431Q818yescomplemented
43C2×#107—24no—
44C2×#106—24no—
45C2×C4⋊C8—22no—
46C2×C22⋊C8C2312yes—
47C2×C22⋊C8C2312yes—
48C2×C4×C8—22no—
49#400—24no—
50#398—24no—
51#398—24no—
52#395—24no—
53#391—24no—
54#391—22no—
55#391—22no—
56#389—24no—
57#389—22no—
58#389—22no—
59#387—24no—
60#387—24no—
61#387—24no—
62#387—24no—
63#385—24no—
64#383—22no—
65#383—22no—
66#383Q814yescomplemented
67#382D414yes—
68#381—22no—
69#380—22no—
70#380C2314yes—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D412yes—
74C4×C4⋊C4D412yes—
75C4×C22⋊C4—22no—
76#376—22no—
77#376—22no—
78#376—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95C2311yescharacteristic
87C2×#95C2311yescharacteristic
88C2×#95C2312yes—
89#340—24no—
90#340—24no—
91C22×Q8—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2411yescharacteristic
99C22×C8—22no—
100C22×C8—22no—
101C22⋊Q8—44no—
102C22⋊Q8—44no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D414yes—
116C2×C22⋊C4—24no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—81no—
125C2×C42—81no—
126C2×C42C2×D412yes—
127C2×C42C2×D412yes—
128C2×C42C2×Q814yes—
129C2×C42C2×Q812yes—
130C2×C42#4112yes—
131#107—48no—
132#106—48no—
133C4⋊C8—28no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8C2×Q814yes—
137C22⋊C8C2×Q814yes—
138C4×C8—28no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—22no—
146#95—22no—
147#95—24no—
148#95—24no—
149#95—22no—
150#95—22no—
151#95—24no—
152#95—22no—
153#95—44no—
154#95—41no—
155#95—41no—
156#95—44no—
157#95—82no—
158#95C2×Q814yes—
159C24C22×D411yescharacteristic
160C2×Q8—44no—
161C2×Q8—82no—
162C2×Q8—82no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
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—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12012yes—
196C22×C4C22⋊Q812yes—
197C22×C4C22⋊Q812yes—
198C22×C4C4⋊Q812yes—
199C22×C4C4⋊Q812yes—
200C22×C4#13514yes—
201C22×C4C22×D412yes—
202C22×C4C22×Q812yes—
203C22×C4C22×Q811yescharacteristic
204C22×C4C2×#4111yescharacteristic
205C2×C8—24no—
206C2×C8—24no—
207C2×C8—44no—
208C2×C8—44no—
209C4⋊C4—28no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—48no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C4⋊C4—88no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C42—44no—
223C42—44no—
224C42—82no—
225C42—82no—
226C42—84no—
227C42C22⋊Q814yes—
228C42C22⋊Q814yes—
229C42C4⋊Q814yes—
230C42C4⋊Q814yes—
231C23—21no—
232C23—21no—
233C23—22no—
234C23C2×#12011yescharacteristic
235C23C2×C22⋊Q811yescharacteristic
236C23C2×C4⋊Q811yescharacteristic
237C23C2×#13512yes—
238C23#57512yes—
239Q8—164no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
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—82no—
283C2×C4#39514yes—
284C2×C4#39512yes—
285C2×C4#39512yes—
286C2×C4C2×C22⋊Q812yes—
287C2×C4C2×C4⋊Q812yes—
288C8—44no—
289C8—44no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22C2×#39511yescharacteristic
299C22#300212yes—
300C22#302111yescharacteristic
301C22#302211yescharacteristic
302C22#302512yes—
303C22#307612yes—
304C22#316011yescharacteristic
305C22#316211yescharacteristic
306C4—24no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—22no—
311C4—44no—
312C4—84no—
313C4—84no—
314C4—84no—
315C4—82no—
316C4—84no—
317C2—22no—
318C2—21no—
319C2—21no—
320C2#1820912yes—
321C2#1821612yes—
322C2#2355111yescharacteristic
323C2#2356211yescharacteristic
324C2#2356411yescharacteristic
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

Complements

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

NH ≅ G/Ntype
42. #431Q8semidirect
66. #383Q8semidirect

Maximal subgroups

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

subgroupindex
#351202
#213612
#141642
#105722
#105692
#105682
#80252
#79062