Finite Group Databasebeta

← #241261

Subgroups — #241261

Order 512.

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

normal subgroups only

This group has 754 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#37609C211yesmaximal, characteristic
2#21987C212yesmaximal
3C2×#1836C211yesmaximal, characteristic
4#10586C212yesmaximal, complemented
5#10585C212yesmaximal, complemented
6#10584C214yesmaximal, complemented
7#8036C212yesmaximal
8#8030C211yesmaximal, characteristic
9C2×#432C2212yes—
10C22×#97C2211yescharacteristic
11#2845C2211yescharacteristic
12#2845C2211yescharacteristic
13#2844C2212yes—
14#2666—22no—
15#2666—22no—
16#2627—22no—
17#2612—22no—
18#2569—22no—
19#2549—22no—
20C2×#384C2211yescharacteristic
21#2283C2212yes—
22C2×#382C2211yescharacteristic
23#1948—22no—
24#1947—22no—
25#1853—22no—
26#1852—22no—
27#1847C2214yes—
28#1836—22no—
29#1836C2212yes—
30#1836C2212yes—
31#1836C2214yes—
32C2×#340—22no—
33#1295C2212yes—
34#1295C2214yes—
35#1295C2214yes—
36#1295C2212yes—
37C23×C8—21no—
38C2×C22⋊Q8—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4D412yes—
42C22×C42C2311yescharacteristic
43#432—24no—
44#432Q818yescomplemented
45C2×#107—24no—
46C2×#106—24no—
47C2×C4⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×#97—21no—
50C2×#97—22no—
51C2×#97Q814yescomplemented
52C2×#97C2312yes—
53C2×#97C2314yes—
54#400—22no—
55#400—22no—
56#400—22no—
57#400—22no—
58#398—22no—
59#398—22no—
60#398—22no—
61#398—22no—
62#398—22no—
63#398—22no—
64#398—22no—
65#398—22no—
66#395—22no—
67#395—22no—
68#395—22no—
69#395—22no—
70#391—22no—
71#391—22no—
72#389—22no—
73#389—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#385—22no—
79#385—22no—
80#384—21no—
81#384—22no—
82#384C2312yes—
83#383—22no—
84#382—22no—
85#382D412yes—
86#382C2311yescharacteristic
87#382C2312yes—
88#382C2312yes—
89#382C2311yescharacteristic
90#381—22no—
91#380—22no—
92#380D412yes—
93#380D412yes—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4—22no—
97#376—22no—
98C2×#95—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103C22×Q8—22no—
104C23×C4—22no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C22×C8—24no—
109C22×C8—22no—
110C22×C8—24no—
111C22×C8—42no—
112C22×C8—42no—
113C22⋊Q8—84no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2×Q812yes—
144C2×C42C2×Q814yes—
145C2×C42#4112yes—
146C2×C42C2411yescharacteristic
147#107—84no—
148#106—84no—
149C4⋊C8—28no—
150C22⋊C8—28no—
151#97—24no—
152#97—22no—
153#97—24no—
154#97C2×Q814yes—
155#97C2×Q818yes—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C24#12011yescharacteristic
175C2×Q8—82no—
176C2×Q8—82no—
177C2×Q8—82no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4—84no—
193C22×C4—82no—
194C22×C4#12012yes—
195C22×C4#12011yescharacteristic
196C22×C4C22⋊Q812yes—
197C22×C4C22⋊Q814yes—
198C22×C4C4⋊Q812yes—
199C22×C4C4⋊Q814yes—
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D412yes—
202C2×C8—24no—
203C2×C8—28no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—48no—
207C2×C8—42no—
208C2×C8—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C4⋊C4—84no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C42—82no—
233C42—82no—
234C42C22⋊Q812yes—
235C42C22⋊Q814yes—
236C42C4⋊Q812yes—
237C42C4⋊Q814yes—
238C42C22×Q811yescharacteristic
239C42C22×Q812yes—
240C42C2×#4111yescharacteristic
241C23—22no—
242C23—22no—
243C23#39512yes—
244C23#45412yes—
245C23#45512yes—
246C23C2×#12011yescharacteristic
247Q8—164no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—84no—
280C2×C4—164no—
281C2×C4—164no—
282C2×C4#39512yes—
283C2×C4#39514yes—
284C2×C4C2×C22⋊Q811yescharacteristic
285C2×C4C2×C22⋊Q812yes—
286C2×C4C2×C4⋊Q811yescharacteristic
287C2×C4C2×C4⋊Q812yes—
288C8—48no—
289C8—44no—
290C22—24no—
291C22—24no—
292C22—22no—
293C22—22no—
294C22#206114yes—
295C22C2×#39511yescharacteristic
296C22C2×#45511yescharacteristic
297C22C2×#45411yescharacteristic
298C22#302112yes—
299C22#302212yes—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—24no—
306C4—82no—
307C4—82no—
308C4—82no—
309C4—82no—
310C4—82no—
311C4—82no—
312C4—162no—
313C2—24no—
314C2#1905512yes—
315C2#1905911yescharacteristic
316C2#1906011yescharacteristic
317C2C2×#206111yescharacteristic
318C2#2356212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10586C2semidirect
5. #10585C2semidirect
6. #10584C2semidirect
44. #432Q8semidirect
51. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#376092
#219872
C2×#18362
#105862
#105852
#105842
#80362
#80302