Finite Group Databasebeta

← #330713

Subgroups — #330713

Order 512.

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

normal subgroups only

This group has 512 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#29998C211yesmaximal, characteristic
3#29865C211yesmaximal, characteristic
4#22009C211yesmaximal, characteristic
5#10601C212yesmaximal, complemented
6#10596C212yesmaximal, complemented
7#10485C212yesmaximal
8#10480C212yesmaximal
9#9899C211yesmaximal, characteristic, complemented
10#9697C211yesmaximal, characteristic, complemented
11#8812C211yesmaximal, characteristic, complemented
12C2×#476C2211yescharacteristic
13C2×#474C2211yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2293C2211yescharacteristic
17#2088—22no—
18#2085—22no—
19#2074—21no—
20#2074—21no—
21#2073—22no—
22#2065—22no—
23#2026—22no—
24#1990C2212yes—
25#1988C2211yescharacteristic
26#1987C2211yescharacteristic
27#1956C2212yes—
28#1954C2212yes—
29#1948C2212yes—
30#1939C2212yes—
31#1923—22no—
32#1842C2211yescharacteristic
33#1831C2211yescharacteristic
34#1818—21no—
35#1802C2212yes—
36#1798—21no—
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×C4×D4C2311yescharacteristic
49C2×#117—21no—
50#476—42no—
51#474—42no—
52C2×#107—21no—
53C2×#106—21no—
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×#103—22no—
62C2×#103—22no—
63C2×#102—21no—
64C2×#102—21no—
65C2×#102—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C4×C8—21no—
69C2×C4×C8C2312yes—
70#391—21no—
71#391—21no—
72#391—22no—
73#390D412yes—
74#389—22no—
75#389—22no—
76#389D412yes—
77#388—22no—
78#384C2312yes—
79#382—22no—
80#382—21no—
81C4×C4⋊C4C2311yescharacteristic
82C4×C22⋊C4—21no—
83C43C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#336—22no—
91#336—21no—
92#336—22no—
93#336D411yescharacteristic
94#336D411yescharacteristic
95#336D412yes—
96#335—42no—
97#334—42no—
98#333—42no—
99#332—42no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×D4C2×D411yescharacteristic
103C23×C4—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114#125—81no—
115#125—81no—
116C4×D4—22no—
117C4×D4—22no—
118#117—82no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D412yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42#4112yes—
138C2×C42C2411yescharacteristic
139#107—42no—
140#106—42no—
141#106—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—
150#103—82no—
151#103—82no—
152#102—42no—
153#102—42no—
154#102—82no—
155C22⋊C8—82no—
156C22⋊C8—82no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—81no—
160#95—22no—
161#95—21no—
162#95—42no—
163#95—42no—
164#95—42no—
165C24—21no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4#12112yes—
186C22×C4#12112yes—
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C2×#4111yescharacteristic
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C42—21no—
227C42—22no—
228C42—22no—
229C42—21no—
230C42—22no—
231C42—42no—
232C42—42no—
233C42H(5,𝔽2)12yes—
234C422−1+412yes—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23C2×#12111yescharacteristic
241Q8—82no—
242Q8—82no—
243Q8—162no—
244D4—41no—
245D4—41no—
246D4—42no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#53511yescharacteristic
271C2×C4#53611yescharacteristic
272C2×C4#53712yes—
273C2×C4#54212yes—
274C2×C4#57611yescharacteristic
275C2×C4#57712yes—
276C2×C4#57712yes—
277C2×C4#57711yescharacteristic
278C2×C4#57812yes—
279C8—42no—
280C8—42no—
281C8—81no—
282C8—82no—
283C8—81no—
284C8—82no—
285C22—82no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22#262011yescharacteristic
291C22#304211yescharacteristic
292C22#304511yescharacteristic
293C22#306312yes—
294C22#306712yes—
295C4—22no—
296C4—42no—
297C4—42no—
298C4—82no—
299C4—81no—
300C4—162no—
301C4—161no—
302C4—161no—
303C4#322111yescharacteristic
304C4#322311yescharacteristic
305C4#324212yes—
306C4#325712yes—
307C4#326312yes—
308C2—161no—
309C2#1956212yes—
310C2#1957912yes—
311C2#2983811yescharacteristic
312C2#2985611yescharacteristic
313C2#2986011yescharacteristic
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. #10601C2semidirect
6. #10596C2semidirect
9. #9899C2semidirect
10. #9697C2semidirect
11. #8812C2semidirect

Maximal subgroups

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

subgroupindex
#302062
#299982
#298652
#220092
#106012
#105962
#104852
#104802
#98992
#96972
#88122