Finite Group Databasebeta

← #491305

Subgroups — #491305

Order 512.

438 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 128 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#33774C211yesmaximal, characteristic
2#33500C211yesmaximal, characteristic, complemented
3#19345C211yesmaximal, characteristic, complemented
4#19203C211yesmaximal, characteristic
5#12479C212yesmaximal, complemented
6#12475C212yesmaximal, complemented
7#12444C212yesmaximal, complemented
8#12441C212yesmaximal, complemented
9#12036C211yesmaximal, characteristic, complemented
10#11961C211yesmaximal, characteristic
11#11950C211yesmaximal, characteristic, complemented
12#3255—21no—
13#3208—21no—
14#3170C2211yescharacteristic
15C2×#493C2211yescharacteristic
16#2960C2211yescharacteristic
17#2248C2212yes—
18#2240C2212yes—
19#2207—21no—
20#2207—21no—
21#2187—22no—
22#2183—21no—
23#2183C2212yescomplemented
24Q8×C16—21no—
25Q8×C16C2212yescomplemented
26#2164—22no—
27#2162C2211yescharacteristic
28#2162C2211yescharacteristic
29#1700—22no—
30#1688—22no—
31#1674C2212yescomplemented
32#1670C2212yescomplemented
33#1668C2212yes—
34#1668C2212yes—
35#1567C2211yescharacteristic
36#1564C2211yescharacteristic
37#1561—21no—
38#1545—21no—
39#1372C2212yescomplemented
40#1371C2212yescomplemented
41#1364C2211yescharacteristic, complemented
42#1364C2211yescharacteristic, complemented
43#1362C2211yescharacteristic, complemented
44#1362C2211yescharacteristic, complemented
45#1361C2212yes—
46#1361C2212yes—
47C22×D8—21no—
48C22×D8—21no—
49#533—21no—
50#533D411yescharacteristic
51C2×#127—21no—
52C2×SD32—22no—
53C2×SD32—22no—
54#500C2311yescharacteristic
55#494C2311yescharacteristic
56#493—21no—
57#493—21no—
58#493—21no—
59#493C2312yes—
60#492—41no—
61#487—21no—
62#486—41no—
63#475—22no—
64#475—41no—
65#470—42no—
66#467—21no—
67#467—21no—
68#462—22no—
69#462—41no—
70#460—22no—
71#460—22no—
72#450—42no—
73#446—21no—
74#446D412yes—
75C8×Q8—21no—
76C8×Q8C2312yes—
77C4×Q16—21no—
78C4×Q16—21no—
79#431—21no—
80#424—21no—
81#423—21no—
82#423D411yescharacteristic
83#405—21no—
84C2×C4×C8C2311yescharacteristic
85C4⋊C16—22no—
86C4⋊C16—22no—
87C4⋊C16D411yescharacteristic
88C4⋊C16D411yescharacteristic
89C4⋊C16C2311yescharacteristic
90C4⋊C16C2311yescharacteristic
91#357—22no—
92#357—22no—
93C22⋊C16—21no—
94C22⋊C16D412yes—
95#346—22no—
96C4×C16—22no—
97C4×C16—21no—
98C4×C16C2312yes—
99#335—21no—
100#335C2311yescharacteristic
101#335C2311yescharacteristic
102#335C2312yes—
103#334—21no—
104#332—22no—
105#332—22no—
106#331—22no—
107#331—22no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—41no—
111C2×Q16—21no—
112C2×Q16—21no—
113C2×Q16—42no—
114C2×SD16—42no—
115C2×SD16—42no—
116C2×D8—22no—
117C2×D8—22no—
118C2×D8—41no—
119C2×D8—41no—
120C2×D8—81no—
121C2×D8—81no—
122C2×D8—82no—
123C2×M16—21no—
124C2×M16—41no—
125C22×C8C2×D411yescharacteristic
126C22×C8#4111yescharacteristic
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8D811yescharacteristic
130C4⋊Q8D811yescharacteristic
131#127—22no—
132#127—41no—
133#127—41no—
134#127—41no—
135#125—21no—
136#125—41no—
137C22⋊Q8—21no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8D812yes—
141C4×Q8—21no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8C2×D412yes—
145#117—21no—
146#117—41no—
147C2×C42C2×D411yescharacteristic
148SD32—82no—
149SD32—82no—
150C2×C16—21no—
151C2×C16—21no—
152C2×C16C2×D412yes—
153C2×C16C2×D412yes—
154#107—21no—
155#107—21no—
156#106—42no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—22no—
163C4⋊C8—41no—
164C4⋊C8C2×D412yes—
165#103—42no—
166#103—41no—
167#103—41no—
168#103—42no—
169#102—42no—
170#102—42no—
171C22⋊C8—41no—
172#97—22no—
173C4×C8—22no—
174C4×C8—41no—
175C4×C8C2×D411yescharacteristic
176C4×C8C2×D411yescharacteristic
177C4×C8#4111yescharacteristic
178C4×C8C2411yescharacteristic
179C24—81no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C2×D812yes—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—161no—
193C2×D4—161no—
194C2×D4—162no—
195C22×C4#12111yescharacteristic
196C22×C4C2×D811yescharacteristic
197C22×C4#13511yescharacteristic
198Q16—41no—
199Q16—41no—
200Q16—81no—
201Q16—162no—
202SD16—162no—
203SD16—162no—
204D8—42no—
205D8—42no—
206D8—82no—
207D8—82no—
208D8—82no—
209M16—81no—
210M16—161no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—22no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8#12111yescharacteristic
219C2×C8#12111yescharacteristic
220C2×C8C22×D411yescharacteristic
221C2×C8C2×#4111yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4C2×D812yes—
234C22⋊C4—22no—
235C22⋊C4—81no—
236C42—22no—
237C42—81no—
238C42#12111yescharacteristic
239C42C2×D811yescharacteristic
240C42#13511yescharacteristic
241C42C22×D411yescharacteristic
242C16—22no—
243C16—22no—
244C16—41no—
245C16—41no—
246C23—82no—
247C23—162no—
248C23—162no—
249C23—162no—
250C23#46611yescharacteristic
251Q8—22no—
252Q8—22no—
253Q8—42no—
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—81no—
258D4—82no—
259D4—82no—
260D4—82no—
261D4—82no—
262D4—162no—
263D4—162no—
264D4—162no—
265D4—162no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4#46611yescharacteristic
276C2×C4#46611yescharacteristic
277C2×C4#46611yescharacteristic
278C2×C4C2×#12111yescharacteristic
279C2×C4C22×D811yescharacteristic
280C2×C4C2×#13511yescharacteristic
281C8—21no—
282C8—21no—
283C8—21no—
284C8—42no—
285C8—82no—
286C8—81no—
287C8—81no—
288C8#57711yescharacteristic
289C8#57711yescharacteristic
290C22—21no—
291C22—21no—
292C22—21no—
293C22—162no—
294C22—162no—
295C22—162no—
296C22—324no—
297C22C2×#46611yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—161no—
304C4—161no—
305C4#222312yes—
306C4#304311yescharacteristic
307C4#306811yescharacteristic
308C4#341511yescharacteristic
309C4#341611yescharacteristic
310C2—41no—
311C2—322no—
312C2#1939811yescharacteristic
313C2#3361211yescharacteristic
314C2C2×#222311yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×#493C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33500C2semidirect
3. #19345C2semidirect
5. #12479C2semidirect
6. #12475C2semidirect
7. #12444C2semidirect
8. #12441C2semidirect
9. #12036C2semidirect
11. #11950C2semidirect
23. #2183C22semidirect
25. Q8×C16C22semidirect
31. #1674C22semidirect
32. #1670C22semidirect
39. #1372C22semidirect
40. #1371C22semidirect
41. #1364C22semidirect
42. #1364C22semidirect
43. #1362C22semidirect
44. #1362C22semidirect

Maximal subgroups

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

subgroupindex
#337742
#335002
#193452
#192032
#124792
#124752
#124442
#124412
#120362
#119612
#119502