Finite Group Databasebeta

← #351109

Subgroups — #351109

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33067C211yesmaximal, characteristic, complemented
2#33031C211yesmaximal, characteristic
3#30204C211yesmaximal, characteristic
4#20569C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9699C211yesmaximal, characteristic
7#9567C212yesmaximal, complemented
8#9564C212yesmaximal
9#9222C212yesmaximal
10#9216C212yesmaximal, complemented
11#8788C211yesmaximal, characteristic, complemented
12C2×#477C2211yescharacteristic
13C2×#460C2211yescharacteristic
14C2×C8×Q8C2212yes—
15#2859C2211yescharacteristic
16#2858C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#2073—22no—
19#2025—21no—
20#2025—21no—
21#1990C2212yes—
22#1987C2211yescharacteristic
23#1982C2211yescharacteristic
24#1957C2212yes—
25#1948C2212yes—
26#1925—22no—
27#1919—22no—
28#1880—22no—
29#1865—22no—
30#1831C2211yescharacteristic
31#1831C2211yescharacteristic
32#1803C2212yes—
33C8×C4⋊C4—22no—
34C8×C4⋊C4C2212yes—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2211yescharacteristic
37#1744—21no—
38#1744C2211yescharacteristic
39C2×#333C2212yes—
40C2×#331C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C22×SD16—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×#125—21no—
49C2×#124—22no—
50C2×C4×Q8D412yes—
51C2×C4×Q8C2311yescharacteristic
52#477—24no—
53#460—24no—
54C8×Q8—24no—
55C8×Q8—24no—
56C2×#107—21no—
57C2×#106—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×#103—21no—
64C2×#103—22no—
65C2×#102—22no—
66C2×#102—21no—
67C2×#102—22no—
68C2×#102—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#390—21no—
74#390—22no—
75#390—21no—
76#389—22no—
77#389—22no—
78#384C2312yes—
79#382—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4D412yes—
83C4×C4⋊C4C2311yescharacteristic
84C43—21no—
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#336—21no—
95#336D412yes—
96#336D411yescharacteristic
97#336D411yescharacteristic
98#333—24no—
99#331—24no—
100C22×Q8—21no—
101C22×Q8C2×D411yescharacteristic
102C22×D4—21no—
103C22×D4—21no—
104C22×D4—21no—
105C2×SD16—42no—
106C2×SD16—44no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115#127—22no—
116#127—22no—
117#125—82no—
118#124—84no—
119C4×Q8—24no—
120C4×Q8—24no—
121C4×Q8—22no—
122C4×Q8#4114yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4112yes—
143C2×C42C2411yescharacteristic
144#107—42no—
145#106—44no—
146C4⋊C8—28no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8#4114yes—
153#103—42no—
154#103—44no—
155#102—42no—
156#102—44no—
157#102—82no—
158#102—82no—
159C4×C8—28no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—81no—
163C4×C8—82no—
164#95—22no—
165#95—21no—
166#95—42no—
167C24—41no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8#12412yes—
173C2×Q8#13514yes—
174C2×D4—42no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—82no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—41no—
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—24no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C4⋊C4—28no—
212C4⋊C4—24no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C4⋊C4#12412yes—
225C4⋊C4#13514yes—
226C22⋊C4—84no—
227C22⋊C4—84no—
228C42—28no—
229C42—24no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—22no—
234C42—21no—
235C42—42no—
236C42—41no—
237C42—41no—
238C42—82no—
239C42C2×#4112yes—
240C42C2×#4112yes—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—82no—
245C23—82no—
246C23C2×#12111yescharacteristic
247Q8—22no—
248Q8—24no—
249Q8—162no—
250Q8#48814yes—
251D4—82no—
252D4—82no—
253D4—81no—
254D4—81no—
255D4—82no—
256D4—162no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#52512yes—
277C2×C4C2×#12411yescharacteristic
278C2×C4#52912yes—
279C2×C4#53211yescharacteristic
280C2×C4C2×#13512yes—
281C2×C4C2×#13512yes—
282C2×C4#57712yes—
283C2×C4#57711yescharacteristic
284C2×C4#57811yescharacteristic
285C8—42no—
286C8—42no—
287C8—44no—
288C8—81no—
289C8—81no—
290C8—82no—
291C22—161no—
292C22—161no—
293C22—161no—
294C22—162no—
295C22—162no—
296C22#285011yescharacteristic
297C22#304311yescharacteristic
298C22#304611yescharacteristic
299C22#305012yes—
300C22#306812yes—
301C4—24no—
302C4—22no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—44no—
307C4—161no—
308C4—162no—
309C4#312112yes—
310C4#312512yes—
311C4C2×#48811yescharacteristic
312C4#313111yescharacteristic
313C4#316812yes—
314C2—321no—
315C2#1940112yes—
316C2#1941912yes—
317C2#3303711yescharacteristic
318C2#3304411yescharacteristic
319C2#3306411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #33067C2semidirect
4. #20569C2semidirect
5. #9897C2semidirect
7. #9567C2semidirect
10. #9216C2semidirect
11. #8788C2semidirect

Maximal subgroups

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

subgroupindex
#330672
#330312
#302042
#205692
#98972
#96992
#95672
#95642
#92222
#92162
#87882