Finite Group Databasebeta

← #266765

Subgroups — #266765

Order 512.

884 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 184 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#59864C211yesmaximal, characteristic
2#29290C212yesmaximal, complemented
3C4×#492C211yesmaximal, characteristic
4#10577C212yesmaximal, complemented
5#10352C212yesmaximal, complemented
6#9870C212yesmaximal, complemented
7#9064C212yesmaximal
8#8920C212yesmaximal
9#8585C211yesmaximal, characteristic
10C2×#492C2211yescharacteristic
11C2×C4×SD16—22no—
12#2804—22no—
13#2580—24no—
14#2571C2212yes—
15#2563—22no—
16#2487—22no—
17#2366C2212yes—
18D4×C4⋊C4—22no—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2087—24no—
23#1987—22no—
24#1987C2214yes—
25#1968—22no—
26#1957C2211yescharacteristic
27#1956C2211yescharacteristic
28#1947—22no—
29#1947C2214yes—
30#1844C2211yescharacteristic
31#1842C2212yes—
32C42×C8C2211yescharacteristic
33C2×#331C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C22×SD16—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8C2311yescharacteristic
40C2×#127C2311yescharacteristic
41C2×C4×Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×D4—24no—
44C2×C4×D4—22no—
45C2×C4×D4D412yes—
46C22×C4⋊C4—22no—
47#492—22no—
48#492Q814yescomplemented
49C4×SD16—48no—
50C2×#107—24no—
51C2×#106—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2314yes—
55C2×#103—22no—
56C2×#102—22no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#400—24no—
62#399—24no—
63#393—28no—
64#391—22no—
65#391—21no—
66#390—24no—
67#390—22no—
68#390D412yes—
69#389—24no—
70#389—22no—
71#389—22no—
72#389D414yes—
73#386—24no—
74#385—24no—
75#385—28no—
76#384D412yes—
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#382—24no—
80#380—22no—
81#380—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2314yes—
85C43C2311yescharacteristic
86#340—24no—
87#340—24no—
88#340—24no—
89#331—24no—
90C22×Q8—22no—
91C22×Q8—21no—
92C22×D4—22no—
93C22×D4C2×D412yes—
94C23×C4—24no—
95C23×C4—22no—
96C2×SD16—28no—
97C2×SD16—44no—
98C2×SD16—44no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—21no—
105C4⋊Q8—22no—
106C4⋊Q8C2×Q812yes—
107#127—22no—
108#127C2×Q812yes—
109#127#4112yes—
110C4×Q8—44no—
111C4×Q8—82no—
112C4×Q8—82no—
113C4×D4—44no—
114C4×D4—44no—
115C4×D4—44no—
116C4×D4—84no—
117C4×D4—84no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C2×D414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—22no—
136C2×C42—24no—
137C2×C42—41no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D412yes—
142C2×C42C2411yescharacteristic
143#107—84no—
144#106—44no—
145C4⋊C8—24no—
146C4⋊C8—24no—
147C4⋊C8—44no—
148C4⋊C8—82no—
149#103—44no—
150#102—44no—
151C4×C8—24no—
152C4×C8—21no—
153C4×C8—22no—
154C4×C8—42no—
155C4×C8C2×Q812yes—
156#95—24no—
157#95—28no—
158#95—24no—
159#95—24no—
160#95—42no—
161C24—22no—
162C2×Q8—22no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—44no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×D4—22no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4C22⋊Q814yes—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—41no—
183C22×C4—48no—
184C22×C4—44no—
185C22×C4—41no—
186C22×C4—84no—
187C22×C4—88no—
188C22×C4#12011yescharacteristic
189C22×C4#12012yes—
190C22×C4#12011yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D412yes—
193SD16—416no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—44no—
201C2×C8—44no—
202C2×C8—44no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—84no—
219C4⋊C4—82no—
220C4⋊C4—84no—
221C22⋊C4—48no—
222C22⋊C4—82no—
223C22⋊C4—84no—
224C22⋊C4—88no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—41no—
229C42—44no—
230C42—44no—
231C42—82no—
232C42#13514yes—
233C42#13714yes—
234C42C22×Q811yescharacteristic
235C42C2×#4111yescharacteristic
236C422−1+412yes—
237C23—24no—
238C23—44no—
239C23—44no—
240C23—42no—
241C23—42no—
242C23C2×#12011yescharacteristic
243Q8—44no—
244Q8—82no—
245Q8—88no—
246D4—44no—
247D4—44no—
248D4—44no—
249D4—44no—
250D4—44no—
251D4—82no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—88no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—168no—
273C2×C4#45214yes—
274C2×C4#45214yes—
275C2×C4#45312yes—
276C2×C4C2×C22⋊Q811yescharacteristic
277C2×C4#53611yescharacteristic
278C2×C4#54812yes—
279C2×C4D4×Q812yes—
280C2×C4C2×#13512yes—
281C2×C4C2×#13712yes—
282C2×C4#57612yes—
283C2×C4#57712yes—
284C8—24no—
285C8—42no—
286C8—84no—
287C22—48no—
288C22—44no—
289C22—82no—
290C22—84no—
291C22—82no—
292C22#240911yescharacteristic
293C22C2×#45212yes—
294C22#301212yes—
295C22C2×#45311yescharacteristic
296C22#301811yescharacteristic
297C4—24no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—82no—
302C4—84no—
303C4—82no—
304C4—84no—
305C4—161no—
306C4#163014yes—
307C4#336112yes—
308C4#337212yes—
309C4#338112yes—
310C4#338512yes—
311C2—84no—
312C2C2×#163011yescharacteristic
313C2#1867912yes—
314C2#1868611yescharacteristic
315C2#2431211yescharacteristic
316C2#2433112yes—
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
2. #29290C2semidirect
4. #10577C2semidirect
5. #10352C2semidirect
6. #9870C2semidirect
48. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#292902
C4×#4922
#105772
#103522
#98702
#90642
#89202
#85852