Finite Group Databasebeta

← #330633

Subgroups — #330633

Order 512.

524 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 138 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#30206C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic
3#29829C211yesmaximal, characteristic
4#22013C211yesmaximal, characteristic
5#10594C212yesmaximal, complemented
6#10592C212yesmaximal, complemented
7#10485C212yesmaximal
8#10476C212yesmaximal
9#9902C211yesmaximal, characteristic, complemented
10#9708C211yesmaximal, characteristic, complemented
11#8810C211yesmaximal, characteristic, complemented
12C2×#474C2211yescharacteristic
13C2×#461C2211yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2293C2211yescharacteristic
17#2086—22no—
18#2074—22no—
19#2068—22no—
20#2067—22no—
21#2055—22no—
22#2026—21no—
23#2026—21no—
24#1990C2212yes—
25#1988C2211yescharacteristic
26#1981C2211yescharacteristic
27#1956C2212yes—
28#1954C2212yes—
29#1947C2212yes—
30#1939C2212yes—
31#1924—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1818—21no—
35#1802C2212yes—
36#1798—21no—
37#1767C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#335C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×SD16—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×D4C2311yescharacteristic
50C2×#117—21no—
51#474—42no—
52#461—42no—
53C2×#107—21no—
54C2×#107—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—21no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#102—21no—
67C2×#102—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—21no—
71C2×C4×C8C2312yes—
72#391—22no—
73#391—21no—
74#391—21no—
75#390D412yes—
76#389—22no—
77#389—22no—
78#389D412yes—
79#388—22no—
80#384C2312yes—
81#382—21no—
82#382—22no—
83C4×C4⋊C4C2311yescharacteristic
84C4×C22⋊C4—21no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—21no—
93#336D411yescharacteristic
94#336D411yescharacteristic
95#336D412yes—
96#335—42no—
97#335—42no—
98#332—42no—
99#332—42no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×D4C2×D411yescharacteristic
103C23×C4—21no—
104C2×SD16—42no—
105C2×SD16—82no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116#125—82no—
117C4×D4—22no—
118C4×D4—22no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D412yes—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C42—21no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42#4112yes—
139C2×C42C2411yescharacteristic
140#107—42no—
141#107—82no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—82no—
149C4⋊C8—82no—
150C4⋊C8—82no—
151#103—42no—
152#103—82no—
153#103—82no—
154#103—82no—
155#102—42no—
156#102—82no—
157C22⋊C8—82no—
158C22⋊C8—82no—
159#97—81no—
160C4×C8—22no—
161C4×C8—22no—
162#95—22no—
163#95—21no—
164#95—42no—
165#95—42no—
166#95—42no—
167C24—21no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—22no—
179C2×D4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4#12112yes—
190C22×C4#12112yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C2×#4111yescharacteristic
194SD16—84no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—84no—
227C22⋊C4—42no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—42no—
236C42—42no—
237C42H(5,𝔽2)12yes—
238C422−1+412yes—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23C2×#12111yescharacteristic
245Q8—81no—
246Q8—81no—
247Q8—82no—
248Q8—82no—
249Q8—82no—
250D4—41no—
251D4—41no—
252D4—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#53511yescharacteristic
277C2×C4#53611yescharacteristic
278C2×C4#53712yes—
279C2×C4#54212yes—
280C2×C4#57612yes—
281C2×C4#57612yes—
282C2×C4#57711yescharacteristic
283C2×C4#57812yes—
284C2×C4#57811yescharacteristic
285C8—42no—
286C8—42no—
287C8—81no—
288C8—82no—
289C8—81no—
290C8—82no—
291C22—82no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#262011yescharacteristic
297C22#304111yescharacteristic
298C22#304611yescharacteristic
299C22#306112yes—
300C22#306912yes—
301C4—22no—
302C4—42no—
303C4—42no—
304C4—82no—
305C4—81no—
306C4—162no—
307C4—161no—
308C4—161no—
309C4#322011yescharacteristic
310C4#322311yescharacteristic
311C4#324312yes—
312C4#325712yes—
313C4#326212yes—
314C2—161no—
315C2#1951112yes—
316C2#1955012yes—
317C2#2983911yescharacteristic
318C2#2984911yescharacteristic
319C2#2986211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30206C2semidirect
5. #10594C2semidirect
6. #10592C2semidirect
9. #9902C2semidirect
10. #9708C2semidirect
11. #8810C2semidirect

Maximal subgroups

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

subgroupindex
#302062
#299942
#298292
#220132
#105942
#105922
#104852
#104762
#99022
#97082
#88102