Finite Group Databasebeta

← #316960

Subgroups — #316960

Order 512.

579 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 142 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#28204C212yesmaximal
2#22248C211yesmaximal, characteristic, complemented
3#22214C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10622C211yesmaximal, characteristic
6#10279C211yesmaximal, characteristic, complemented
7#10278C211yesmaximal, characteristic
8#9224C212yesmaximal, complemented
9#9223C212yesmaximal
10#8788C211yesmaximal, characteristic, complemented
11#8590C212yesmaximal, complemented
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×#476C2212yes—
15C2×#446C2211yescharacteristic
16C2×#436C2211yescharacteristic
17C4×C4⋊Q8C2212yes—
18C4×#124C2211yescharacteristic
19#2085—22no—
20#2042—22no—
21#1990C2212yes—
22#1984C2211yescharacteristic
23#1954C2211yescharacteristic
24#1927—22no—
25#1925—22no—
26#1924—22no—
27#1921—22no—
28#1921—22no—
29#1851—22no—
30#1842C2211yescharacteristic
31#1834C2211yescharacteristic
32#1831C2212yes—
33#1803C2212yes—
34#1802C2212yes—
35#1744C2211yescharacteristic
36C2×#333C2212yes—
37C2×#333C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1277C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×M16—21no—
44C2×C4⋊Q8C2312yes—
45C2×#124D411yescharacteristic
46C2×C4×Q8—22no—
47C2×C4×Q8C2311yescharacteristic
48C2×C4×D4C2311yescharacteristic
49#500—24no—
50#477—24no—
51#477—42no—
52#476—24no—
53#476—42no—
54#446—42no—
55#436—42no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×#97—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#391D411yescharacteristic
76#390D411yescharacteristic
77#389—22no—
78#389—22no—
79#388—22no—
80#384C2312yes—
81#382D411yescharacteristic
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C22⋊C4—22no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#336D412yes—
95#336D412yes—
96#333—42no—
97#333—42no—
98C22×Q8—22no—
99C22×Q8C2×D411yescharacteristic
100C22×D4C2×D411yescharacteristic
101C23×C4—21no—
102C2×M16—42no—
103C2×M16—82no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—21no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8#4114yes—
115#124—42no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—84no—
119C4×D4—22no—
120C4×D4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C42—21no—
134C2×C42—42no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42#4112yes—
139C2×C42C2411yescharacteristic
140#107—28no—
141#107—44no—
142#106—44no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—42no—
150C4⋊C8—82no—
151#103—44no—
152#103—44no—
153#102—44no—
154#102—44no—
155C22⋊C8—42no—
156C22⋊C8—82no—
157#97—42no—
158C4×C8—22no—
159C4×C8—22no—
160#95—22no—
161#95—22no—
162#95—42no—
163#95—42no—
164C24—21no—
165C2×Q8—22no—
166C2×Q8—24no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8—82no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12112yes—
187C22×C4#12112yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191M16—84no—
192C2×C8—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C22⋊C4—44no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—21no—
229C42—21no—
230C42—42no—
231C42—41no—
232C42—41no—
233C42—84no—
234C42C2×#4112yes—
235C42H(5,𝔽2)12yes—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23C2×#12111yescharacteristic
242Q8—42no—
243Q8—42no—
244Q8—84no—
245Q8—84no—
246D4—42no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#52511yescharacteristic
272C2×C4#53511yescharacteristic
273C2×C4#54612yes—
274C2×C4#55312yes—
275C2×C4#57512yes—
276C2×C4#57712yes—
277C2×C4#57812yes—
278C2×C4#57811yescharacteristic
279C2×C4#57811yescharacteristic
280C8—42no—
281C8—41no—
282C8—41no—
283C8—81no—
284C8—81no—
285C8—82no—
286C8—82no—
287C22—81no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22—82no—
292C22#253511yescharacteristic
293C22#304612yes—
294C22#305411yescharacteristic
295C22#305911yescharacteristic
296C22#306812yes—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—81no—
301C4—81no—
302C4—84no—
303C4—81no—
304C4—162no—
305C4#312112yes—
306C4#322211yescharacteristic
307C4#322311yescharacteristic
308C4#332712yes—
309C4#332712yes—
310C2—161no—
311C2#1945512yes—
312C2#1954712yes—
313C2#2814811yescharacteristic
314C2#2817311yescharacteristic
315C2#2817711yescharacteristic
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
2. #22248C2semidirect
4. #10624C2semidirect
6. #10279C2semidirect
8. #9224C2semidirect
10. #8788C2semidirect
11. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#282042
#222482
#222142
#106242
#106222
#102792
#102782
#92242
#92232
#87882
#85902