Finite Group Databasebeta

← #508305

Subgroups — #508305

Order 512.

772 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#2219C211yesmaximal, characteristic, complemented
2C2×#2112C211yesmaximal, characteristic
3#32931C212yesmaximal
4#13494C212yesmaximal, complemented
5#13383C212yesmaximal, complemented
6#13382C212yesmaximal, complemented
7#13306C212yesmaximal, complemented
8#13062C212yesmaximal, complemented
9#12836C211yesmaximal, characteristic, complemented
10C24×C8C2211yescharacteristic
11C22×Q32C2211yescharacteristic, complemented
12C22×M32C2211yescharacteristic
13C22×M32C2211yescharacteristic
14C2×#467C2212yes—
15C2×#467C2212yes—
16C2×#467C2212yes—
17#2848—22no—
18#2219—24no—
19C2×#367C2211yescharacteristic, complemented
20C2×#367C2211yescharacteristic, complemented
21C2×#366C2211yescharacteristic, complemented
22C2×#358C2212yescomplemented
23C2×#358C2212yescomplemented
24C2×#358C2212yescomplemented
25#2112—24no—
26C2×C22⋊C16C2212yes—
27C2×C22⋊C16C2212yes—
28#1892—22no—
29#1892—22no—
30#1892—22no—
31#1808C2212yes—
32#1808C2212yes—
33#1808C2212yes—
34#1380C2212yescomplemented
35#1380C2212yescomplemented
36#1380C2212yescomplemented
37C24×C4D411yescharacteristic
38C22×Q16C2312yes—
39C23×C8—22no—
40C23×C8—22no—
41C23×C8—41no—
42C23×C8—41no—
43C23×C8—41no—
44C23×C8—41no—
45C23×C8D411yescharacteristic
46C23×C8D412yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×Q32—22no—
53C2×Q32—41no—
54C2×Q32—41no—
55C2×M32—22no—
56C2×M32—41no—
57C2×M32—41no—
58C2×M32—41no—
59C2×M32—41no—
60C2×M32D414yes—
61C22×C16C2312yes—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64#467—24no—
65#467—24no—
66#467—42no—
67#467—42no—
68#467—42no—
69#467—42no—
70C2×#107C2312yes—
71C2×#107C2312yes—
72C2×#107C2312yes—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76#385—42no—
77#385—42no—
78#385—42no—
79#380—22no—
80#380—22no—
81#380—22no—
82#379—42no—
83#367—22no—
84#367—41no—
85#366—41no—
86#358—24no—
87#358—42no—
88#358—42no—
89C22⋊C16—24no—
90C22⋊C16—24no—
91#340—22no—
92#340—22no—
93#340—22no—
94C25D811yescharacteristic
95C22×Q8—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4D811yescharacteristic
103C23×C4D812yes—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16—22no—
109C2×Q16—22no—
110C2×Q16—22no—
111C2×Q16—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—24no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—44no—
129C22×C8—44no—
130C22×C8—44no—
131C22×C8—44no—
132C22×C8C2×D412yes—
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D412yes—
135C22×C8#4112yes—
136C22×C8C2411yescharacteristic
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155Q32—44no—
156M32—44no—
157M32—44no—
158C2×C16—21no—
159C2×C16—21no—
160C2×C16—21no—
161C2×C16—21no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16C2×D412yes—
166C2×C16C2×D412yes—
167#107—22no—
168#107—22no—
169#107—22no—
170#107—22no—
171#107—22no—
172#107—22no—
173#103—44no—
174#103—44no—
175#103—44no—
176#95—42no—
177#95—42no—
178#95—42no—
179C24—22no—
180C24—24no—
181C24—41no—
182C24—41no—
183C24—42no—
184C24C2×D811yescharacteristic
185C24C2×D812yes—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4#12112yes—
217C22×C4C2×D812yes—
218C22×C4C2×D811yescharacteristic
219C22×C4#13512yes—
220C22×C4C22×D411yescharacteristic
221Q16—42no—
222Q16—42no—
223Q16—42no—
224Q16—42no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—48no—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8#12112yes—
241C2×C8C22×D411yescharacteristic
242C2×C8C2×#4111yescharacteristic
243C2×C8H(5,𝔽2)11yescharacteristic
244C2×C8H(5,𝔽2)11yescharacteristic
245C2×C8H(5,𝔽2)11yescharacteristic
246C2×C8H(5,𝔽2)11yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—84no—
259C22⋊C4—84no—
260C22⋊C4—84no—
261C22⋊C4—84no—
262C16—22no—
263C16—22no—
264C16—42no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—48no—
276C23—44no—
277C23—44no—
278C23—48no—
279C23—44no—
280C23#46612yes—
281C23#51014yes—
282C23C22×D811yescharacteristic
283Q8—82no—
284Q8—82no—
285Q8—82no—
286Q8—82no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#46612yes—
315C2×C4C2×#12111yescharacteristic
316C2×C4#53411yescharacteristic
317C2×C4#53411yescharacteristic
318C2×C4C2×#13511yescharacteristic
319C2×C4#57511yescharacteristic
320C2×C4#57511yescharacteristic
321C8—22no—
322C8—42no—
323C8#53411yescharacteristic
324C8#53511yescharacteristic
325C8#53811yescharacteristic
326C8#53811yescharacteristic
327C22—22no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—24no—
332C22—416no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—42no—
340C22C2×#46611yescharacteristic
341C22#318611yescharacteristic
342C22#318611yescharacteristic
343C22C2×#51012yes—
344C22#341412yes—
345C4—22no—
346C4—42no—
347C4—162no—
348C4—162no—
349C4—162no—
350C4—162no—
351C4#261711yescharacteristic
352C4#305411yescharacteristic
353C4#319111yescharacteristic
354C4#319111yescharacteristic
355C2—42no—
356C2—44no—
357C2#2973011yescharacteristic
358C2#3360612yes—
359C2#3376111yescharacteristic
360C2#3376212yes—
361C2#3376311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2219C2semidirect
4. #13494C2semidirect
5. #13383C2semidirect
6. #13382C2semidirect
7. #13306C2semidirect
8. #13062C2semidirect
9. #12836C2semidirect
11. C22×Q32C22semidirect
19. C2×#367C22semidirect
20. C2×#367C22semidirect
21. C2×#366C22semidirect
22. C2×#358C22semidirect
23. C2×#358C22semidirect
24. C2×#358C22semidirect
34. #1380C22semidirect
35. #1380C22semidirect
36. #1380C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22192
C2×#21122
#329312
#134942
#133832
#133822
#133062
#130622
#128362