Finite Group Databasebeta

← #351166

Subgroups — #351166

Order 512.

611 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 150 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#33066C211yesmaximal, characteristic
2#33032C211yesmaximal, characteristic
3#30206C211yesmaximal, characteristic, complemented
4#20568C211yesmaximal, characteristic
5#9897C211yesmaximal, characteristic, complemented
6#9700C211yesmaximal, characteristic, complemented
7#9574C212yesmaximal, complemented
8#9569C212yesmaximal, complemented
9#9227C212yesmaximal
10#9225C212yesmaximal
11#8788C211yesmaximal, characteristic, complemented
12C2×#474C2211yescharacteristic
13C2×#461C2211yescharacteristic
14C2×#436C2212yes—
15#2862C2211yescharacteristic
16#2859C2211yescharacteristic
17C42×D4C2211yescharacteristic
18#2074—22no—
19#2026—21no—
20#2026—21no—
21#1990C2212yes—
22#1988C2211yescharacteristic
23#1981C2211yescharacteristic
24#1956C2212yes—
25#1947C2212yes—
26#1928—22no—
27#1918—22no—
28#1877—22no—
29#1862—22no—
30#1831C2211yescharacteristic
31#1831C2211yescharacteristic
32#1802C2212yes—
33#1770C2212yes—
34C4×C4⋊C8C2211yescharacteristic
35#1753—22no—
36#1749—21no—
37#1744C2211yescharacteristic
38#1728—21no—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C22×SD16—21no—
46C22×M16—22no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—21no—
50C2×C4×D4D412yes—
51C2×C4×D4C2311yescharacteristic
52C22×C42—21no—
53#474—24no—
54#461—24no—
55#436—24no—
56#436—24no—
57C2×#107—22no—
58C2×#107—21no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—21no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#102—22no—
68C2×#102—21no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#391—21no—
74#391—22no—
75#391—21no—
76#389—22no—
77#389—22no—
78#384C2312yes—
79#382—21no—
80#382—22no—
81C4×C4⋊C4D412yes—
82C4×C4⋊C4C2311yescharacteristic
83C4×C22⋊C4—21no—
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#336—21no—
95#336D412yes—
96#336D411yescharacteristic
97#336D411yescharacteristic
98#332—24no—
99#332—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×D4C2×D411yescharacteristic
103C23×C4—21no—
104C23×C4—22no—
105C2×SD16—44no—
106C2×SD16—42no—
107C2×M16—28no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8C2×D412yes—
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120#125—82no—
121C4×D4—22no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4#4114yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—82no—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4112yes—
144C2×C42C2411yescharacteristic
145#107—44no—
146#107—42no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8#4114yes—
153#103—44no—
154#103—42no—
155#103—82no—
156#103—82no—
157#102—44no—
158#102—42no—
159C22⋊C8—28no—
160#97—82no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—81no—
164#95—22no—
165#95—21no—
166#95—42no—
167#95—42no—
168#95—42no—
169C24—21no—
170C2×Q8—42no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4#12412yes—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C2×#4111yescharacteristic
199SD16—84no—
200M16—88no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C4⋊C4#12412yes—
232C22⋊C4—24no—
233C22⋊C4—44no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—21no—
238C42—21no—
239C42—42no—
240C42—42no—
241C42—82no—
242C42C2×#4112yes—
243C42C2×#4112yes—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23—42no—
249C23C2×#12111yescharacteristic
250Q8—82no—
251Q8—82no—
252Q8—81no—
253Q8—81no—
254Q8—82no—
255D4—22no—
256D4—22no—
257D4—42no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—88no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#52512yes—
283C2×C4C2×#12411yescharacteristic
284C2×C4#52912yes—
285C2×C4#53211yescharacteristic
286C2×C4#57512yes—
287C2×C4#57512yes—
288C2×C4#57612yes—
289C2×C4#57711yescharacteristic
290C2×C4#57811yescharacteristic
291C8—42no—
292C8—44no—
293C8—42no—
294C8—81no—
295C8—81no—
296C8—82no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—82no—
301C22—82no—
302C22#285011yescharacteristic
303C22#304111yescharacteristic
304C22#304611yescharacteristic
305C22#305412yes—
306C22#306912yes—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—82no—
312C4—161no—
313C4—161no—
314C4—162no—
315C4#311812yes—
316C4#312412yes—
317C4#314011yescharacteristic
318C4#314211yescharacteristic
319C4#318312yes—
320C2—82no—
321C2#1942412yes—
322C2#1945412yes—
323C2#3303611yescharacteristic
324C2#3305011yescharacteristic
325C2#3306511yescharacteristic
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
3. #30206C2semidirect
5. #9897C2semidirect
6. #9700C2semidirect
7. #9574C2semidirect
8. #9569C2semidirect
11. #8788C2semidirect

Maximal subgroups

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

subgroupindex
#330662
#330322
#302062
#205682
#98972
#97002
#95742
#95692
#92272
#92252
#87882