Finite Group Databasebeta

← #266727

Subgroups — #266727

Order 512.

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

normal subgroups only

This group has 962 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, complemented
2#28280C212yesmaximal, complemented
3C4×#493C211yesmaximal, characteristic
4#10576C212yesmaximal, complemented
5#10351C212yesmaximal, complemented
6#9870C212yesmaximal, complemented
7#9061C212yesmaximal
8#8924C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic, complemented
10C2×#493C2211yescharacteristic
11C2×#476—22no—
12C2×C4×D8—22no—
13#2804—22no—
14#2580—24no—
15#2571C2212yes—
16#2563—22no—
17#2487—22no—
18#2366C2212yes—
19D4×C4⋊C4—22no—
20C4×C4⋊Q8C2212yes—
21C4×#127C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2065—24no—
24#1982C2214yes—
25#1970—22no—
26#1957C2211yescharacteristic
27#1957C2211yescharacteristic
28#1948C2214yes—
29#1845C2211yescharacteristic
30#1842C2212yes—
31#1802—22no—
32C42×C8C2211yescharacteristic
33C2×#331C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C22×D8—22no—
38C2×C4⋊Q8D412yes—
39C2×#127C2311yescharacteristic
40C2×#127C2311yescharacteristic
41C2×C4×Q8—22no—
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—21no—
46C2×C4×D4D412yes—
47C22×C4⋊C4—22no—
48#493—22no—
49#493Q814yescomplemented
50#476—48no—
51C4×D8—48no—
52C2×#107—22no—
53C2×#106—24no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2314yes—
57C2×#102—22no—
58C2×#102—24no—
59C2×#102—22no—
60C2×#102—24no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#400—24no—
66#399—24no—
67#393—28no—
68#391—22no—
69#390—24no—
70#390—22no—
71#390—22no—
72#390—21no—
73#390D412yes—
74#389—24no—
75#389—22no—
76#389D414yes—
77#386—24no—
78#385—24no—
79#385—28no—
80#384D412yes—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#382—24no—
84#380—22no—
85#380—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2314yes—
88C43C2311yescharacteristic
89#340—24no—
90#331—24no—
91C22×Q8—22no—
92C22×D4—22no—
93C22×D4—21no—
94C22×D4—22no—
95C22×D4C2×D412yes—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—41no—
99C2×D8—28no—
100C2×D8—44no—
101C2×D8—44no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107#127—21no—
108#127—22no—
109#127—22no—
110#127C2×Q812yes—
111#127C2×Q812yes—
112#127#4112yes—
113C4×Q8—82no—
114C4×Q8—82no—
115C4×D4—44no—
116C4×D4—44no—
117C4×D4—44no—
118C4×D4—44no—
119C4×D4—44no—
120C4×D4—44no—
121C4×D4—84no—
122C4×D4—84no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—84no—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—42no—
141C2×C42—24no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2411yescharacteristic
147#107—44no—
148#106—48no—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—44no—
152C4⋊C8—82no—
153#102—48no—
154#102—44no—
155#102—44no—
156#102—84no—
157C4×C8—24no—
158C4×C8—21no—
159C4×C8—22no—
160C4×C8—42no—
161C4×C8C2×Q812yes—
162#95—24no—
163#95—28no—
164#95—24no—
165#95—24no—
166C24—22no—
167C24—41no—
168C2×Q8—44no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4C22⋊Q814yes—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—41no—
197C22×C4—84no—
198C22×C4—88no—
199C22×C4—84no—
200C22×C4—82no—
201C22×C4#12011yescharacteristic
202C22×C4#12012yes—
203C22×C4#12011yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D412yes—
206D8—416no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—44no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C22⋊C4—48no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—84no—
235C22⋊C4—88no—
236C22⋊C4—84no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—44no—
242C42—44no—
243C42#13514yes—
244C42AGL(1,ℤ/8)14yes—
245C42C22×Q811yescharacteristic
246C42C2×#4111yescharacteristic
247C422−1+412yes—
248C23—24no—
249C23—44no—
250C23—44no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—82no—
255C23—82no—
256C23—81no—
257C23—81no—
258C23C2×#12011yescharacteristic
259Q8—88no—
260D4—44no—
261D4—44no—
262D4—44no—
263D4—44no—
264D4—44no—
265D4—44no—
266D4—44no—
267D4—44no—
268D4—82no—
269D4—82no—
270D4—82no—
271D4—82no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—88no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—168no—
290C2×C4—164no—
291C2×C4#44914yes—
292C2×C4#44914yes—
293C2×C4#45312yes—
294C2×C4C2×C22⋊Q811yescharacteristic
295C2×C4#53611yescharacteristic
296C2×C4#54812yes—
297C2×C4D4×Q812yes—
298C2×C4C2×#13512yes—
299C2×C4C2×AGL(1,ℤ/8)12yes—
300C2×C4#57612yes—
301C2×C4#57712yes—
302C8—24no—
303C8—42no—
304C8—84no—
305C22—48no—
306C22—44no—
307C22—82no—
308C22—84no—
309C22—82no—
310C22—84no—
311C22—82no—
312C22—161no—
313C22—162no—
314C22—161no—
315C22#240911yescharacteristic
316C22C2×#44912yes—
317C22#301212yes—
318C22C2×#45311yescharacteristic
319C22#301811yescharacteristic
320C4—24no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—82no—
325C4—84no—
326C4—84no—
327C4—161no—
328C4#162014yes—
329C4#336012yes—
330C4#336612yes—
331C4#338012yes—
332C4#338412yes—
333C2—84no—
334C2—162no—
335C2C2×#162011yescharacteristic
336C2#1867812yes—
337C2#1868611yescharacteristic
338C2#2431211yescharacteristic
339C2#2432712yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #59864C2semidirect
2. #28280C2semidirect
4. #10576C2semidirect
5. #10351C2semidirect
6. #9870C2semidirect
8. #8924C2semidirect
9. #8585C2semidirect
49. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#282802
C4×#4932
#105762
#103512
#98702
#90612
#89242
#85852