Finite Group Databasebeta

← #322556

Subgroups — #322556

Order 512.

858 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 174 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#33254C211yesmaximal, characteristic
2#28750C212yesmaximal, complemented
3#28654C212yesmaximal, complemented
4#21691C211yesmaximal, characteristic, complemented
5C2×#1720C212yesmaximal, complemented
6#10317C211yesmaximal, characteristic, complemented
7#10316C211yesmaximal, characteristic, complemented
8#10307C211yesmaximal, characteristic
9#10307C211yesmaximal, characteristic
10#9827C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#498C2212yes—
13C2×#498C2211yescharacteristic
14C2×#494C2211yescharacteristic
15C2×#493C2211yescharacteristic
16C2×#487C2212yes—
17C2×#486C2212yes—
18C2×#486C2212yes—
19C2×#486C2212yes—
20C2×#441—22no—
21#2858C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2065—24no—
25#2042—24no—
26#1990C2212yes—
27#1983—24no—
28#1972—22no—
29#1972—22no—
30#1971—22no—
31#1957C2212yes—
32#1957C2212yes—
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2212yes—
36#1926—24no—
37#1844C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40#1720—28no—
41C2×#322C2212yes—
42#1273C2211yescharacteristic
43#1273C2211yescharacteristic
44C22×Q16—22no—
45C22×D8—22no—
46C2×C4⋊Q8C2312yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#127C2312yes—
50C2×#124—22no—
51C2×#124—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54#498—24no—
55#498—42no—
56#498—42no—
57#494—22no—
58#494D414yes—
59#493—42no—
60#487—24no—
61#487—42no—
62#486—24no—
63#486—24no—
64#486—24no—
65#486—24no—
66#486—42no—
67#486—42no—
68#441—48no—
69C2×#106—24no—
70C2×#106—22no—
71C2×#106—22no—
72C2×C4⋊C8D414yes—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#97D412yes—
83C2×#97D412yes—
84C2×C4×C8C2312yes—
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#391—21no—
89#391—21no—
90#390—22no—
91#390—22no—
92#390—22no—
93#390—22no—
94#389—22no—
95#389—22no—
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C43C2311yescharacteristic
101#340—24no—
102#336D414yes—
103#322—24no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C2×Q16—28no—
111C2×Q16—44no—
112C2×Q16—44no—
113C2×D8—44no—
114C2×D8—44no—
115C2×D8—44no—
116C22×C8C2×D414yes—
117C22×C8C2×D414yes—
118C22×C8C2×D412yes—
119C22×C8C2×D412yes—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8C2×D412yes—
129C4⋊Q8C2×D412yes—
130#127—22no—
131#127—22no—
132#127—22no—
133#127—22no—
134#124—48no—
135#124—84no—
136C4×Q8—44no—
137C4×Q8—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2411yescharacteristic
159#106—48no—
160#106—44no—
161#106—44no—
162C4⋊C8—44no—
163#103—48no—
164#103—44no—
165#103—44no—
166#103—84no—
167#102—48no—
168#102—48no—
169#102—48no—
170#102—48no—
171#102—84no—
172#97—24no—
173#97—42no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8C2×D412yes—
180C4×C8#4112yes—
181#95—42no—
182#95—42no—
183C24—42no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4#12712yes—
214C22×C4#12711yescharacteristic
215C22×C4#12711yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D412yes—
218Q16—416no—
219D8—88no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8#12114yes—
231C2×C8AGL(1,ℤ/8)14yes—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C22⋊C4—84no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C22⋊C4—84no—
250C42—22no—
251C42—22no—
252C42—24no—
253C42—24no—
254C42—82no—
255C42—82no—
256C42C22×D411yescharacteristic
257C42C2×#4111yescharacteristic
258C42H(5,𝔽2)12yes—
259C23—82no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—82no—
266C23C2×#12711yescharacteristic
267Q8—44no—
268Q8—44no—
269Q8—82no—
270Q8—82no—
271Q8—84no—
272Q8—84no—
273D4—84no—
274D4—84no—
275D4—84no—
276D4—84no—
277D4—84no—
278D4—84no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4C2×#12111yescharacteristic
300C2×C4#53511yescharacteristic
301C2×C4D4212yes—
302C2×C4#55012yes—
303C2×C4C2×AGL(1,ℤ/8)12yes—
304C2×C4#57514yes—
305C2×C4#57512yes—
306C2×C4#57714yes—
307C8—24no—
308C8—24no—
309C8—44no—
310C8—44no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22—162no—
315C22—162no—
316C22—162no—
317C22—162no—
318C22#254111yescharacteristic
319C22#315011yescharacteristic
320C22#315111yescharacteristic
321C22#315312yes—
322C22#315412yes—
323C4—41no—
324C4—41no—
325C4—42no—
326C4—82no—
327C4—82no—
328C4—161no—
329C4—161no—
330C4—162no—
331C4#303612yes—
332C4#321312yes—
333C4#327114yes—
334C4#340114yes—
335C2—322no—
336C2#1969212yes—
337C2#1972111yescharacteristic
338C2#1975011yescharacteristic
339C2#2844511yescharacteristic
340C2#2848112yes—
subgroupG/H
center, socleC23C2×#127
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
2. #28750C2semidirect
3. #28654C2semidirect
4. #21691C2semidirect
5. C2×#1720C2semidirect
6. #10317C2semidirect
7. #10316C2semidirect
10. #9827C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332542
#287502
#286542
#216912
C2×#17202
#103172
#103162
#103072
#98272
#82632