Finite Group Databasebeta

← #506470

Subgroups — #506470

Order 512.

930 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 190 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×#2235C212yesmaximal, complemented
2C2×#2187C212yesmaximal
3#33172C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic
5#13527C212yesmaximal, complemented
6#13327C212yesmaximal, complemented
7#13237C214yesmaximal, complemented
8C4×C22⋊C16C211yesmaximal, characteristic, complemented
9C23×D8C2211yescharacteristic
10C22×SD32C2214yes—
11C2×#494C2211yescharacteristic, complemented
12C2×#467—21no—
13C2×#467C2212yes—
14C22×#107C2211yescharacteristic
15C22×#102—21no—
16C22×C4×C8C2211yescharacteristic
17#2857—21no—
18C2×#390—21no—
19#2235—28no—
20#2187—28no—
21C2×#367C2214yescomplemented
22C2×#358C2214yescomplemented
23C2×#357C2214yes—
24C2×C22⋊C16C2212yes—
25C2×C4×C16C2212yescomplemented
26#1970—21no—
27#1970—21no—
28#1970C2212yes—
29#1970C2212yes—
30#1969—21no—
31#1969C2212yescomplemented
32#1956—21no—
33#1845C2211yescharacteristic, complemented
34#1374C2212yescomplemented
35C23×D4—21no—
36C22×Q16—21no—
37C22×Q16C2312yes—
38C22×D8—24no—
39C22×D8—21no—
40C22×D8—24no—
41C22×D8—42no—
42C22×D8C2312yes—
43C23×C8D411yescharacteristic
44C23×C8C2311yescharacteristic
45C2×C4⋊Q8—21no—
46C2×C22⋊Q8—22no—
47C22×C4⋊C4—21no—
48C22×C22⋊C4—42no—
49C22×C42D411yescharacteristic
50C2×SD32—28no—
51C2×SD32—28no—
52C2×SD32—44no—
53C22×C16C2314yes—
54#494—42no—
55#467—24no—
56#467—24no—
57#467—44no—
58C2×#107—24no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—44no—
68C2×C4×C8—42no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—22no—
74#390—22no—
75#390—22no—
76#390—42no—
77#389—42no—
78#384—21no—
79#380—42no—
80#367—28no—
81#358—28no—
82#357—28no—
83C22⋊C16D418yes—
84C4×C16—42no—
85C25—41no—
86C22×Q8—22no—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—42no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4—41no—
95C23×C4—41no—
96C23×C4D812yes—
97C23×C4C2×D411yescharacteristic
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—24no—
101C2×Q16—24no—
102C2×Q16—42no—
103C2×Q16—42no—
104C2×Q16—42no—
105C2×D8—24no—
106C2×D8—24no—
107C2×D8—22no—
108C2×D8—24no—
109C2×D8—416no—
110C2×D8—42no—
111C2×D8—42no—
112C2×D8#4114yes—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8D814yes—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D414yes—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2411yescharacteristic
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—88no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42D812yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144SD32—416no—
145C2×C16—24no—
146C2×C16—24no—
147C2×C16C2×D418yes—
148#107—22no—
149#107—24no—
150#107—22no—
151#107—42no—
152#107—42no—
153#103—44no—
154#103—44no—
155#102—44no—
156#102—44no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—41no—
161#95—44no—
162C24—44no—
163C24—42no—
164C24—82no—
165C24—84no—
166C24—82no—
167C24—84no—
168C24C2×D811yescharacteristic
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
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—42no—
196C22×C4—42no—
197C22×C4—82no—
198C22×C4—84no—
199C22×C4SD3218yescomplemented
200C22×C4#12712yes—
201C22×C4C2×D812yes—
202C22×C4C2×D811yescharacteristic
203C22×C4C2×D812yes—
204C22×C4C2×D812yes—
205C22×C4C22×D411yescharacteristic
206Q16—44no—
207Q16—44no—
208Q16—84no—
209D8—28no—
210D8—48no—
211D8—44no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—28no—
219C2×C8—44no—
220C2×C8#12712yes—
221C2×C8C2×D812yes—
222C2×C8C2×D812yes—
223C2×C8C22×D412yes—
224C2×C8C2×#4112yes—
225C2×C8H(5,𝔽2)12yes—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—84no—
235C22⋊C4—88no—
236C22⋊C4—88no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C16—28no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—44no—
247C23—88no—
248C23—816no—
249C23—82no—
250C23—84no—
251C23—84no—
252C23—82no—
253C23—164no—
254C23—168no—
255C23—164no—
256C23#49312yes—
257C23C2×SD3214yes—
258C23C22×D811yescharacteristic
259Q8—82no—
260Q8—82no—
261Q8—82no—
262Q8—82no—
263D4—44no—
264D4—44no—
265D4—88no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—48no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—82no—
286C2×C4—164no—
287C2×C4#49312yes—
288C2×C4C2×SD3214yes—
289C2×C4C2×SD3214yes—
290C2×C4#52511yescharacteristic
291C2×C4C2×#12711yescharacteristic
292C2×C4#53411yescharacteristic
293C2×C4C22×D811yescharacteristic
294C2×C4#57511yescharacteristic
295C2×C4#57511yescharacteristic
296C8—22no—
297C8—22no—
298C8—22no—
299C8#54614yes—
300C22—24no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—88no—
305C22—84no—
306C22—1616no—
307C22—162no—
308C22—162no—
309C22—164no—
310C22#224814yes—
311C22#310011yescharacteristic
312C22C2×#49311yescharacteristic
313C22#318611yescharacteristic
314C22C22×SD3212yes—
315C22#341412yes—
316C4—22no—
317C4—24no—
318C4—24no—
319C4—162no—
320C4—162no—
321C4—162no—
322C4#253911yescharacteristic
323C4#310011yescharacteristic
324C4#315311yescharacteristic
325C4#318611yescharacteristic
326C2—24no—
327C2—164no—
328C2#2830111yescharacteristic
329C2#3366112yes—
330C2C2×#224811yescharacteristic
331C2#3371911yescharacteristic
332C2#3375312yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2235C2semidirect
3. #33172C2semidirect
5. #13527C2semidirect
6. #13327C2semidirect
7. #13237C2semidirect
8. C4×C22⋊C16C2semidirect
11. C2×#494C22semidirect
21. C2×#367C22semidirect
22. C2×#358C22semidirect
25. C2×C4×C16C22semidirect
31. #1969C22semidirect
33. #1845C22semidirect
34. #1374C22semidirect
199. C22×C4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
C2×#21872
#331722
C2×#19702
#135272
#133272
#132372
C4×C22⋊C162