Finite Group Databasebeta

← #507376

Subgroups — #507376

Order 512.

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

normal subgroups only

This group has 708 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
1#33216C211yesmaximal, characteristic, complemented
2#21233C211yesmaximal, characteristic
3#13480C212yesmaximal, complemented
4#13365C212yesmaximal, complemented
5#13149C212yesmaximal, complemented
6#13147C212yesmaximal, complemented
7#12868C212yesmaximal, complemented
8#12731C212yesmaximal
9#12600C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C2×#499C2211yescharacteristic, complemented
12C2×#427C2211yescharacteristic
13C2×#418—21no—
14C2×#405C2211yescharacteristic
15#2629—21no—
16#2309—21no—
17C2×#357C2212yes—
18C2×#357C2212yes—
19C2×#357C2212yes—
20C2×#357C2212yes—
21C2×C22⋊C16C2212yes—
22C2×#346C2211yescharacteristic, complemented
23C2×C4×C16C2211yescharacteristic, complemented
24#1992—21no—
25#1972—21no—
26#1972—21no—
27#1970C2212yes—
28#1970C2212yes—
29#1943—21no—
30#1942C2212yescomplemented
31#1845C2211yescharacteristic, complemented
32#1809—21no—
33#1808C2212yes—
34#1380C2212yescomplemented
35#1380C2212yescomplemented
36#1380C2212yescomplemented
37#1380C2212yescomplemented
38#1374C2212yescomplemented
39C23×D4—21no—
40C22×D8—24no—
41C22×D8—21no—
42C22×D8—24no—
43C22×D8—41no—
44C22×D8—41no—
45C22×D8C2312yes—
46C22×M16D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#125—21no—
49C2×#117—21no—
50C2×#117D411yescharacteristic
51C22×C16C2312yes—
52C22×C16C2312yes—
53#499—41no—
54#499—41no—
55#427—24no—
56C2×#107—21no—
57C2×#107C2312yes—
58C2×#107C2312yes—
59C2×#107C2312yes—
60C2×#106—21no—
61C2×#106—21no—
62#418—44no—
63C2×#102—22no—
64C2×#102—22no—
65#405—41no—
66#405—41no—
67C2×#97D411yescharacteristic
68C2×#97D411yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#398—42no—
72#396—42no—
73#390—22no—
74#390—22no—
75#390—42no—
76#389—22no—
77#384—21no—
78#380—22no—
79#379—42no—
80#357—24no—
81#357—24no—
82#357—42no—
83#357—42no—
84C22⋊C16—24no—
85#346—41no—
86C4×C16—41no—
87#340—22no—
88#340—22no—
89C25—41no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—41no—
97C22×D4—42no—
98C23×C4C2×D411yescharacteristic
99C2×D8—24no—
100C2×D8—22no—
101C2×D8—22no—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—48no—
106C2×D8—48no—
107C2×D8—42no—
108C2×D8—42no—
109C2×D8#4114yes—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16SD1614yes—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8#4114yes—
120C22×C8C2411yescharacteristic
121#125—42no—
122#125—42no—
123#117—41no—
124#117—41no—
125#117—42no—
126#117—42no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4D812yes—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4D812yes—
141C2×C42—41no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C16—22no—
145C2×C16—22no—
146C2×C16—22no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150#107—22no—
151#107—22no—
152#107—24no—
153#107—22no—
154#107—22no—
155#107—42no—
156#106—42no—
157#106—42no—
158#102—44no—
159#102—44no—
160#97—41no—
161#97—41no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166#95—42no—
167#95—42no—
168C24—44no—
169C24—42no—
170C24—82no—
171C24—82no—
172C24—162no—
173C24—162no—
174C24C2×D811yescharacteristic
175C2×D4—24no—
176C2×D4—48no—
177C2×D4—48no—
178C2×D4—44no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—82no—
185C2×D4—84no—
186C2×D4—84no—
187C2×D4—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4#12412yes—
196C22×C4C2×D812yes—
197C22×C4C2×D812yes—
198C22×C4C2×D811yescharacteristic
199C22×C4C2×SD1612yes—
200C22×C4C22×D411yescharacteristic
201D8—24no—
202D8—24no—
203D8—48no—
204D8—42no—
205D8—42no—
206M16—44no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8#12412yes—
216C2×C8C2×SD1612yes—
217C2×C8C2×SD1612yes—
218C2×C8C2×#4112yes—
219C2×C8C2×#4112yes—
220C2×C8H(5,𝔽2)11yescharacteristic
221C2×C8H(5,𝔽2)11yescharacteristic
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—84no—
230C4⋊C4—162no—
231C4⋊C4—162no—
232C22⋊C4—42no—
233C22⋊C4—82no—
234C22⋊C4—162no—
235C22⋊C4—162no—
236C22⋊C4—162no—
237C22⋊C4—162no—
238C22⋊C4—162no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—82no—
244C16—42no—
245C16—42no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—44no—
251C23—88no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—168no—
259C23—164no—
260C23—164no—
261C23—168no—
262C23#48612yes—
263C23C22×D811yescharacteristic
264D4—44no—
265D4—44no—
266D4—88no—
267D4—82no—
268D4—82no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#48612yes—
291C2×C4#52511yescharacteristic
292C2×C4C2×#12411yescharacteristic
293C2×C4#53411yescharacteristic
294C2×C4C22×SD1611yescharacteristic
295C2×C4#57511yescharacteristic
296C2×C4#57511yescharacteristic
297C8—22no—
298C8—22no—
299C8—22no—
300C8#55112yes—
301C8#55112yes—
302C22—24no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—42no—
307C22—84no—
308C22—84no—
309C22—84no—
310C22—1616no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22—162no—
315C22#310011yescharacteristic
316C22C2×#48611yescharacteristic
317C22#318611yescharacteristic
318C22#341512yes—
319C22#341612yes—
320C4—22no—
321C4—42no—
322C4—42no—
323C4—162no—
324C4—162no—
325C4—162no—
326C4#256111yescharacteristic
327C4#310111yescharacteristic
328C4#314011yescharacteristic
329C4#318711yescharacteristic
330C2—42no—
331C2—164no—
332C2#2910311yescharacteristic
333C2#3368211yescharacteristic
334C2#3368311yescharacteristic
335C2#3370212yes—
336C2#3376412yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33216C2semidirect
3. #13480C2semidirect
4. #13365C2semidirect
5. #13149C2semidirect
6. #13147C2semidirect
7. #12868C2semidirect
9. #12600C2semidirect
11. C2×#499C22semidirect
22. C2×#346C22semidirect
23. C2×C4×C16C22semidirect
30. #1942C22semidirect
31. #1845C22semidirect
34. #1380C22semidirect
35. #1380C22semidirect
36. #1380C22semidirect
37. #1380C22semidirect
38. #1374C22semidirect

Maximal subgroups

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

subgroupindex
#332162
#212332
#134802
#133652
#131492
#131472
#128682
#127312
#126002