Finite Group Databasebeta

← #266739

Subgroups — #266739

Order 512.

902 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 200 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#59924C211yesmaximal, characteristic, complemented
2#28280C212yesmaximal
3C4×#492C211yesmaximal, characteristic
4#10574C212yesmaximal, complemented
5#10348C212yesmaximal
6#9870C212yesmaximal, complemented
7#9063C212yesmaximal
8#8928C212yesmaximal, complemented
9#8587C211yesmaximal, characteristic, complemented
10C2×#492C2211yescharacteristic
11C2×#476—22no—
12C2×C4×SD16—22no—
13#2811—22no—
14#2567—24no—
15#2567—22no—
16#2551—22no—
17#2551C2212yes—
18#2369—22no—
19#2369C2212yes—
20C4×C4⋊Q8C2212yes—
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2065—24no—
24#1982C2214yes—
25#1968—22no—
26#1957C2211yescharacteristic
27#1956C2211yescharacteristic
28#1948C2214yes—
29#1844C2211yescharacteristic
30#1842C2212yes—
31#1802—22no—
32C42×C8C2211yescharacteristic
33C2×#333C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C22×SD16—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—24no—
40C2×C4⋊Q8D412yes—
41C2×C4⋊Q8C2311yescharacteristic
42C2×#127C2311yescharacteristic
43C2×C4×Q8—22no—
44C2×C4×Q8D412yes—
45C2×C4×D4—22no—
46C2×C4×D4—21no—
47#492—22no—
48#492Q814yescomplemented
49#476—48no—
50C4×SD16—48no—
51C2×#106—24no—
52C2×#106—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2314yes—
56C2×#103—22no—
57C2×#102—24no—
58C2×#102—22no—
59C2×#102—24no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#395—24no—
65#395—28no—
66#395—24no—
67#391—28no—
68#391—24no—
69#391—22no—
70#391D412yes—
71#390—22no—
72#390—21no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—22no—
77#389—22no—
78#389D414yes—
79#384—24no—
80#384D412yes—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83C4×C4⋊C4—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2314yes—
88C43C2311yescharacteristic
89#340—24no—
90#333—24no—
91C22×Q8—24no—
92C22×Q8—22no—
93C22×Q8C2×D412yes—
94C22×D4—21no—
95C22×D4—22no—
96C23×C4—41no—
97C2×SD16—28no—
98C2×SD16—44no—
99C2×SD16—44no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—22no—
105C4⋊Q8—84no—
106C4⋊Q8—84no—
107C4⋊Q8—84no—
108C4⋊Q8SD1618yescomplemented
109C4⋊Q8C2×Q812yes—
110C4⋊Q8#4112yes—
111#127—21no—
112#127—22no—
113#127C2×Q812yes—
114C4×Q8—44no—
115C4×Q8—82no—
116C4×Q8—82no—
117C4×D4—44no—
118C4×D4—44no—
119C4×D4—44no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—42no—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2411yescharacteristic
144#106—44no—
145#106—48no—
146C4⋊C8—24no—
147C4⋊C8—24no—
148C4⋊C8—44no—
149C4⋊C8—82no—
150#103—44no—
151#102—44no—
152#102—48no—
153#102—84no—
154C4×C8—24no—
155C4×C8—21no—
156C4×C8—22no—
157C4×C8—42no—
158C4×C8C2×Q812yes—
159#95—28no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164C24—41no—
165C2×Q8—22no—
166C2×Q8—28no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×Q8—88no—
172C2×Q8—84no—
173C2×Q8C22⋊Q814yes—
174C2×D4—22no—
175C2×D4—24no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—41no—
189C22×C4—84no—
190C22×C4—82no—
191C22×C4#12011yescharacteristic
192C22×C4#12012yes—
193C22×C4#12011yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D412yes—
196SD16—416no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—44no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—28no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—88no—
223C4⋊C4—84no—
224C4⋊C4C2×SD1618yes—
225C22⋊C4—82no—
226C22⋊C4—84no—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—41no—
231C42—44no—
232C42—44no—
233C42—44no—
234C42—84no—
235C42C2×SD1614yes—
236C42#13714yes—
237C42C22×Q811yescharacteristic
238C42C2×#4111yescharacteristic
239C422−1+412yes—
240C23—42no—
241C23—82no—
242C23—82no—
243C23—81no—
244C23—81no—
245C23C2×#12011yescharacteristic
246Q8—44no—
247Q8—82no—
248Q8—88no—
249Q8—168no—
250D4—44no—
251D4—44no—
252D4—44no—
253D4—82no—
254D4—82no—
255D4—82no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—28no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—164no—
276C2×C4#44814yes—
277C2×C4#44814yes—
278C2×C4#45312yes—
279C2×C4C2×C22⋊Q811yescharacteristic
280C2×C4#53611yescharacteristic
281C2×C4#54812yes—
282C2×C4D4×Q812yes—
283C2×C4C22×SD1612yes—
284C2×C4C2×#13712yes—
285C2×C4#57612yes—
286C2×C4#57712yes—
287C8—24no—
288C8—42no—
289C8—84no—
290C22—84no—
291C22—82no—
292C22—161no—
293C22—162no—
294C22—161no—
295C22#240911yescharacteristic
296C22C2×#44812yes—
297C22#301212yes—
298C22C2×#45311yescharacteristic
299C22#301811yescharacteristic
300C4—24no—
301C4—21no—
302C4—21no—
303C4—48no—
304C4—44no—
305C4—41no—
306C4—84no—
307C4—82no—
308C4—161no—
309C4#162314yes—
310C4#336212yes—
311C4#336912yes—
312C4Q8×SD1612yes—
313C4#338512yes—
314C2—162no—
315C2C2×#162311yescharacteristic
316C2#1866412yes—
317C2#1868611yescharacteristic
318C2#2431211yescharacteristic
319C2#2432912yes—
subgroupG/H
center, socleC23C2×#120
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. #59924C2semidirect
4. #10574C2semidirect
6. #9870C2semidirect
8. #8928C2semidirect
9. #8587C2semidirect
48. #492Q8semidirect
108. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#599242
#282802
C4×#4922
#105742
#103482
#98702
#90632
#89282
#85872