Finite Group Databasebeta

← #319411

Subgroups — #319411

Order 512.

950 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 182 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#33175C211yesmaximal, characteristic
2#27891C212yesmaximal, complemented
3#25903C212yesmaximal, complemented
4#21202C211yesmaximal, characteristic, complemented
5#10675C212yesmaximal, complemented
6#10307C212yesmaximal, complemented
7#9830C212yesmaximal, complemented
8#9822C212yesmaximal, complemented
9#8146C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#493C2211yescharacteristic
12C2×#486C2212yes—
13C2×#486C2212yes—
14C2×#469—22no—
15C2×#419—22no—
16C2×C4×M16C2211yescharacteristic
17#2856C2212yes—
18#2285C2211yescharacteristic
19#2033—24no—
20#2023—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1972—22no—
24#1960—22no—
25#1957C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1956C2212yes—
29#1956C2212yes—
30#1925—24no—
31#1874—24no—
32#1845C2212yes—
33#1844C2211yescharacteristic
34#1834C2212yes—
35C4×C22⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37C2×C82C2211yescharacteristic
38#1273C2212yes—
39C23×Q8—21no—
40C22×D8—22no—
41C22×M16D412yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2312yes—
48C2×#127C2312yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#117—21no—
53C22×C4⋊C4—22no—
54C22×C42C2311yescharacteristic
55#493—41no—
56#493—41no—
57#486—24no—
58#486—24no—
59#486—42no—
60#486—42no—
61#469—48no—
62C2×#107—24no—
63C2×#106—22no—
64#419—48no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×C22⋊C8D414yes—
73C4×M16—41no—
74C4×M16—41no—
75C2×#97D412yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80#390—22no—
81#390—22no—
82#390—22no—
83#384C2312yes—
84#384C2312yes—
85#382—21no—
86#380—22no—
87C4×C4⋊C4—22no—
88#340—24no—
89#340—24no—
90#340—24no—
91#336D414yes—
92#322—41no—
93C82—41no—
94C22×Q8—24no—
95C22×Q8—22no—
96C22×Q8—42no—
97C22×Q8—44no—
98C22×Q8—41no—
99C22×Q8—44no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—42no—
104C23×C4—41no—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C2×D8—44no—
108C2×D8—44no—
109C2×D8—44no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16D818yes—
113C22×C8C2×D414yes—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—24no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C4⋊Q8—42no—
124C4⋊Q8#4114yes—
125#127—22no—
126#127—22no—
127#127—22no—
128#127—22no—
129#121—48no—
130#121—48no—
131#121—84no—
132#117—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—42no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—41no—
152C2×C42C2×D412yes—
153C2×C42C2×D411yescharacteristic
154C2×C42C2411yescharacteristic
155#107—84no—
156#106—44no—
157#103—48no—
158#103—48no—
159#102—48no—
160#102—48no—
161#102—48no—
162#102—48no—
163#102—84no—
164C22⋊C8—44no—
165#97—42no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—21no—
169C4×C8—22no—
170C4×C8—21no—
171C4×C8—22no—
172C4×C8—21no—
173C4×C8—21no—
174#95—42no—
175C24—42no—
176C24#12711yescharacteristic
177C2×Q8—24no—
178C2×Q8—44no—
179C2×Q8—416no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—82no—
184C2×Q8—88no—
185C2×Q8—82no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—84no—
199C2×D4—84no—
200C2×D4—84no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4#12712yes—
215C22×C4#12711yescharacteristic
216C22×C4C2×D814yes—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D412yes—
219D8—88no—
220M16—48no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8C2×D814yes—
232C2×C8C2×D814yes—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—48no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C22⋊C4—82no—
247C22⋊C4—84no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—82no—
253C42C2×#4112yes—
254C42H(5,𝔽2)11yescharacteristic
255C42H(5,𝔽2)11yescharacteristic
256C23—21no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—82no—
266C23—82no—
267C23#49312yes—
268C23C2×#12711yescharacteristic
269Q8—48no—
270Q8—88no—
271Q8—84no—
272D4—84no—
273D4—84no—
274D4—84no—
275D4—84no—
276D4—84no—
277D4—84no—
278D4—168no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—88no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#49312yes—
299C2×C4#52511yescharacteristic
300C2×C4#53411yescharacteristic
301C2×C4#54612yes—
302C2×C4#54612yes—
303C2×C4C22×D812yes—
304C2×C4#57512yes—
305C2×C4#57514yes—
306C2×C4#57714yes—
307C8—24no—
308C8—24no—
309C8—44no—
310C8—44no—
311C22—22no—
312C22—24no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—162no—
317C22—162no—
318C22—162no—
319C22—162no—
320C22—162no—
321C22—162no—
322C22—162no—
323C22#253911yescharacteristic
324C22C2×#49311yescharacteristic
325C22#315311yescharacteristic
326C22#315312yes—
327C22#315412yes—
328C4—22no—
329C4—42no—
330C4—84no—
331C4—162no—
332C4—162no—
333C4#310012yes—
334C4#318612yes—
335C4#332114yes—
336C4#332614yes—
337C2—42no—
338C2—322no—
339C2#1963112yes—
340C2#1972411yescharacteristic
341C2#1975411yescharacteristic
342C2#2830111yescharacteristic
343C2#2836312yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33175C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #27891C2semidirect
3. #25903C2semidirect
4. #21202C2semidirect
5. #10675C2semidirect
6. #10307C2semidirect
7. #9830C2semidirect
8. #9822C2semidirect
9. #8146C2semidirect

Maximal subgroups

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

subgroupindex
#331752
#278912
#259032
#212022
#106752
#103072
#98302
#98222
#81462