Finite Group Databasebeta

← #352666

Subgroups — #352666

Order 512.

761 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33294C211yesmaximal, characteristic, complemented
2#33277C211yesmaximal, characteristic
3#33230C211yesmaximal, characteristic
4#33225C211yesmaximal, characteristic, complemented
5#30305C211yesmaximal, characteristic, complemented
6#30277C211yesmaximal, characteristic
7#9828C212yesmaximal, complemented
8#9827C212yesmaximal
9#9822C212yesmaximal
10#9819C212yesmaximal, complemented
11C8×#97C211yesmaximal, characteristic, complemented
12C2×#498C2212yes—
13C2×#489—22no—
14C2×#487—22no—
15C2×#486C2212yes—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2858C2211yescharacteristic
20#2094—22no—
21#2094—22no—
22#2084—22no—
23#2058—22no—
24#1992—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1959—22no—
30#1957C2211yescharacteristic
31#1957C2212yes—
32#1957C2211yescharacteristic
33#1956—22no—
34#1956C2212yes—
35#1956C2212yes—
36#1956C2212yes—
37#1934—22no—
38#1934—22no—
39#1844C2212yes—
40#1836—22no—
41#1834C2212yes—
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×C82C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×#127C2311yescharacteristic
55C2×#125—21no—
56C2×#124—21no—
57C2×#124—21no—
58#498—24no—
59#498—24no—
60#489—48no—
61#487—48no—
62#486—24no—
63#486—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#97D412yes—
74C2×#97D412yes—
75C2×C4×C8D412yes—
76C2×C4×C8D412yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80C2×C4×C8C2312yes—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384C2311yescharacteristic
97#384C2312yes—
98#384C2311yescharacteristic
99#382—21no—
100C43C2311yescharacteristic
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#336D412yes—
107#336D412yes—
108#322—24no—
109C82—24no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—41no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×C8C2×D412yes—
117C22×C8C2×D414yes—
118C22×C8C2×D414yes—
119C22×C8C2×D412yes—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—82no—
131#127—22no—
132#127—22no—
133#125—44no—
134#124—44no—
135#124—44no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2411yescharacteristic
169#106—48no—
170#106—84no—
171#103—48no—
172#103—48no—
173#103—48no—
174#102—44no—
175#102—48no—
176#102—48no—
177#102—44no—
178#97—24no—
179#97SD1618yes—
180C4×C8—24no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—42no—
186C4×C8#4114yes—
187C4×C8#4114yes—
188#95—41no—
189#95—41no—
190C24—41no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4#12711yescharacteristic
217C22×C4#12711yescharacteristic
218C22×C4#12711yescharacteristic
219C22×C4#12711yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8#12418yes—
233C2×C8C2×SD1618yes—
234C2×C8AGL(1,ℤ/8)14yes—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42C2×SD1614yes—
269C42#13514yes—
270C42C2×#4112yes—
271C42C2×#4112yes—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23C2×#12711yescharacteristic
280Q8—84no—
281Q8—84no—
282Q8—84no—
283Q8—164no—
284D4—84no—
285D4—84no—
286D4—82no—
287D4—82no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4#49212yes—
312C2×C4#49212yes—
313C2×C4#49512yes—
314C2×C4#49512yes—
315C2×C4C2×#12412yes—
316C2×C4#53212yes—
317C2×C4#53211yescharacteristic
318C2×C4#53211yescharacteristic
319C2×C4C22×SD1612yes—
320C2×C4C2×#13512yes—
321C2×C4C2×AGL(1,ℤ/8)12yes—
322C2×C4#57512yes—
323C2×C4#57712yes—
324C2×C4#57812yes—
325C8—28no—
326C8—28no—
327C8—24no—
328C8—42no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22—161no—
334C22—161no—
335C22—161no—
336C22#285811yescharacteristic
337C22C2×#49211yescharacteristic
338C22C2×#49511yescharacteristic
339C22#315111yescharacteristic
340C22#315311yescharacteristic
341C22#315411yescharacteristic
342C22#315611yescharacteristic
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—162no—
348C4—161no—
349C4—161no—
350C4—161no—
351C4—162no—
352C4#313714yes—
353C4#316714yes—
354C4#317112yes—
355C4#318012yes—
356C2—321no—
357C2#1962111yescharacteristic
358C2#1962411yescharacteristic
359C2#1963911yescharacteristic
360C2#1964011yescharacteristic
361C2#3322911yescharacteristic
362C2#3323911yescharacteristic
363C2#3325111yescharacteristic
subgroupG/H
center, socleC23C2×#127
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. #33294C2semidirect
4. #33225C2semidirect
5. #30305C2semidirect
7. #9828C2semidirect
10. #9819C2semidirect
11. C8×#97C2semidirect

Maximal subgroups

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

subgroupindex
#332942
#332772
#332302
#332252
#303052
#302772
#98282
#98272
#98222
#98192
C8×#972