Finite Group Databasebeta

← #330720

Subgroups — #330720

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#29994C211yesmaximal, characteristic
3#29865C211yesmaximal, characteristic
4#22009C211yesmaximal, characteristic
5#10592C212yesmaximal, complemented
6#10589C212yesmaximal, complemented
7#10484C212yesmaximal
8#10481C212yesmaximal
9#9897C211yesmaximal, characteristic, complemented
10#9697C211yesmaximal, characteristic, complemented
11#8812C211yesmaximal, characteristic, complemented
12C2×#476C2211yescharacteristic
13C2×#474C2211yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2293C2211yescharacteristic
17#2086—22no—
18#2085—22no—
19#2074—22no—
20#2073—22no—
21#2067—22no—
22#2026—21no—
23#2026—21no—
24#1990C2212yes—
25#1988C2211yescharacteristic
26#1987C2211yescharacteristic
27#1956C2212yes—
28#1954C2212yes—
29#1947C2212yes—
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×#107—22no—
54C2×#106—21no—
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—82no—
115C4×D4—22no—
116C4×D4—22no—
117#117—82no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D411yescharacteristic
128C2×C4⋊C4C2×D412yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C42—21no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D412yes—
135C2×C42C2×D411yescharacteristic
136C2×C42#4112yes—
137C2×C42C2411yescharacteristic
138#107—42no—
139#107—82no—
140#106—42no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—82no—
148C4⋊C8—82no—
149#103—82no—
150#103—82no—
151#102—42no—
152#102—42no—
153#102—82no—
154C22⋊C8—82no—
155C22⋊C8—82no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—81no—
159#95—22no—
160#95—21no—
161#95—42no—
162#95—42no—
163#95—42no—
164C24—21no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—82no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4#12112yes—
185C22×C4#12112yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C2×#4111yescharacteristic
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C22⋊C4—44no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C42—22no—
226C42—21no—
227C42—22no—
228C42—21no—
229C42—22no—
230C42—42no—
231C42—42no—
232C42H(5,𝔽2)12yes—
233C422−1+412yes—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23C2×#12111yescharacteristic
240Q8—82no—
241Q8—82no—
242Q8—162no—
243D4—41no—
244D4—41no—
245D4—42no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4#53511yescharacteristic
270C2×C4#53611yescharacteristic
271C2×C4#53712yes—
272C2×C4#54212yes—
273C2×C4#57612yes—
274C2×C4#57712yes—
275C2×C4#57711yescharacteristic
276C2×C4#57812yes—
277C2×C4#57811yescharacteristic
278C8—42no—
279C8—42no—
280C8—81no—
281C8—82no—
282C8—81no—
283C8—82no—
284C22—82no—
285C22—82no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22#262011yescharacteristic
290C22#304111yescharacteristic
291C22#304611yescharacteristic
292C22#306312yes—
293C22#306712yes—
294C4—22no—
295C4—42no—
296C4—42no—
297C4—82no—
298C4—81no—
299C4—162no—
300C4—161no—
301C4—161no—
302C4#322111yescharacteristic
303C4#322311yescharacteristic
304C4#324212yes—
305C4#325712yes—
306C4#326212yes—
307C2—161no—
308C2#1955612yes—
309C2#1958712yes—
310C2#2983911yescharacteristic
311C2#2985611yescharacteristic
312C2#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. #10592C2semidirect
6. #10589C2semidirect
9. #9897C2semidirect
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
#299942
#298652
#220092
#105922
#105892
#104842
#104812
#98972
#96972
#88122