Finite Group Databasebeta

← #316870

Subgroups — #316870

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28204C212yesmaximal
2#22248C211yesmaximal, characteristic, complemented
3#22208C211yesmaximal, characteristic
4#10623C211yesmaximal, characteristic, complemented
5#10621C211yesmaximal, characteristic
6#10279C211yesmaximal, characteristic, complemented
7#10278C211yesmaximal, characteristic
8#9216C212yesmaximal, complemented
9#9215C212yesmaximal
10#8788C211yesmaximal, characteristic, complemented
11#8590C212yesmaximal, complemented
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×#476C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×C8×D4C2211yescharacteristic
17C4×C4⋊Q8C2212yes—
18C4×#124C2211yescharacteristic
19#2085—22no—
20#2042—22no—
21#1990C2212yes—
22#1984C2211yescharacteristic
23#1954C2211yescharacteristic
24#1927—22no—
25#1925—22no—
26#1923—22no—
27#1921—22no—
28#1921—22no—
29#1850—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
43C23×C8—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—
54C8×Q8—42no—
55C8×D4—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×C4×C8—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—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—21no—
106C22×C8—42no—
107C22×C8—82no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8#4114yes—
115#124—22no—
116#124—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—84no—
120C4×D4—22no—
121C4×D4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—22no—
134C2×C42—21no—
135C2×C42—42no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42#4112yes—
140C2×C42C2411yescharacteristic
141#107—28no—
142#107—44no—
143#106—44no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—42no—
151C4⋊C8—82no—
152#103—44no—
153#103—44no—
154#102—44no—
155#102—44no—
156C22⋊C8—42no—
157C22⋊C8—82no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—42no—
161#95—22no—
162#95—22no—
163#95—42no—
164#95—42no—
165C24—21no—
166C2×Q8—21no—
167C2×Q8—22no—
168C2×Q8—24no—
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8—82no—
173C2×Q8#13512yes—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—22no—
177C2×D4#13512yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12112yes—
188C22×C4#12112yes—
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C2×#4111yescharacteristic
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—
209C2×C8—84no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C22⋊C4—28no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—42no—
232C42—41no—
233C42—41no—
234C42—84no—
235C42C2×#4112yes—
236C42H(5,𝔽2)12yes—
237C23—22no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23C2×#12111yescharacteristic
243Q8—24no—
244Q8—42no—
245Q8—84no—
246Q8—84no—
247D4—24no—
248D4—42no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4#52511yescharacteristic
273C2×C4#53511yescharacteristic
274C2×C4#54612yes—
275C2×C4#55312yes—
276C2×C4C2×#13511yescharacteristic
277C2×C4C2×#13511yescharacteristic
278C2×C4#57712yes—
279C2×C4#57812yes—
280C2×C4#57812yes—
281C8—42no—
282C8—41no—
283C8—41no—
284C8—81no—
285C8—81no—
286C8—82no—
287C8—82no—
288C22—44no—
289C22—42no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22#253511yescharacteristic
294C22#304612yes—
295C22#305011yescharacteristic
296C22#305911yescharacteristic
297C22#306812yes—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—42no—
302C4—81no—
303C4—81no—
304C4—84no—
305C4—162no—
306C4#312112yes—
307C4#322211yescharacteristic
308C4#322311yescharacteristic
309C4#330012yes—
310C4#330012yes—
311C2—82no—
312C2#1941912yes—
313C2#1954712yes—
314C2#2814511yescharacteristic
315C2#2817311yescharacteristic
316C2#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. #10623C2semidirect
6. #10279C2semidirect
8. #9216C2semidirect
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
#222082
#106232
#106212
#102792
#102782
#92162
#92152
#87882
#85902