Finite Group Databasebeta

← #319326

Subgroups — #319326

Order 512.

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

normal subgroups only

This group has 898 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#33171C211yesmaximal, characteristic
2#27890C212yesmaximal, complemented
3#25598C212yesmaximal, complemented
4#21318C211yesmaximal, characteristic, complemented
5#10675C212yesmaximal, complemented
6#10305C212yesmaximal, complemented
7#9829C212yesmaximal, complemented
8#9827C212yesmaximal, complemented
9#8144C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×#493C2211yescharacteristic
12C2×#492C2212yes—
13C2×#486C2212yes—
14C2×#466—22no—
15C22×C4×C8C2211yescharacteristic
16#2856C2212yes—
17#2285C2211yescharacteristic
18#2034—24no—
19#2000—24no—
20#1991—22no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1970—22no—
25#1957C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1956C2212yes—
29#1926—24no—
30#1880—24no—
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1834C2212yes—
34C4×C22⋊C8C2212yes—
35#1735—22no—
36C2×#322C2211yescharacteristic
37C2×#322C2211yescharacteristic
38#1273C2212yes—
39C22×SD16—24no—
40C22×D8—22no—
41C23×C8D412yes—
42#533—24no—
43#533—41no—
44#533—41no—
45C2×C4⋊Q8C2312yes—
46C2×#127C2312yes—
47C2×#125—21no—
48C2×C22⋊Q8—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×C4×Q8—22no—
53C2×#117—21no—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#493—41no—
57#493—41no—
58#492—24no—
59#492—42no—
60#486—24no—
61#486—42no—
62#466—48no—
63C2×#107—22no—
64C2×#106—24no—
65C2×#103—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×C22⋊C8D414yes—
71C2×C4×C8—41no—
72C2×C4×C8—41no—
73C2×C4×C8D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#390—22no—
79#390—22no—
80#390—22no—
81#384C2312yes—
82#384C2312yes—
83#382—21no—
84#380—22no—
85C4×C4⋊C4—22no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#336D414yes—
91#322—41no—
92#322—41no—
93C22×Q8—22no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—42no—
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C2×SD16—48no—
101C2×SD16—48no—
102C2×SD16—84no—
103C2×D8—44no—
104C2×D8—44no—
105C2×D8—44no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8SD1618yescomplemented
109C22×C8C2×D414yes—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8#4114yes—
117#127—22no—
118#127—22no—
119#127—22no—
120#127—22no—
121#125—24no—
122#125—42no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125#121—48no—
126#121—84no—
127#121—84no—
128C4×Q8—44no—
129C4×Q8—44no—
130#117—24no—
131#117—42no—
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—42no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42C2×D412yes—
153C2×C42C2×D411yescharacteristic
154C2×C42C2411yescharacteristic
155#107—44no—
156#106—84no—
157#103—84no—
158#102—48no—
159#102—48no—
160#102—48no—
161#102—84no—
162C22⋊C8—44no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—21no—
170C4×C8—21no—
171C4×C8—42no—
172#95—42no—
173C24—42no—
174C24#12711yescharacteristic
175C2×Q8—24no—
176C2×Q8—42no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—84no—
192C2×D4—84no—
193C2×D4—84no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#12712yes—
205C22×C4#12711yescharacteristic
206C22×C4C2×SD1614yes—
207C22×C4C22×D411yescharacteristic
208C22×C4C22×D412yes—
209SD16—816no—
210D8—88no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—48no—
222C2×C8C2×SD1614yes—
223C2×C8C2×SD1614yes—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—48no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—84no—
237C22⋊C4—84no—
238C22⋊C4—82no—
239C22⋊C4—84no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—22no—
244C42—44no—
245C42—82no—
246C42C2×#4112yes—
247C42H(5,𝔽2)11yescharacteristic
248C42H(5,𝔽2)11yescharacteristic
249C23—22no—
250C23—21no—
251C23—22no—
252C23—21no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23#49212yes—
261C23C2×#12711yescharacteristic
262Q8—48no—
263Q8—84no—
264D4—84no—
265D4—84no—
266D4—84no—
267D4—84no—
268D4—84no—
269D4—84no—
270D4—168no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4#49212yes—
290C2×C4#52511yescharacteristic
291C2×C4#53411yescharacteristic
292C2×C4#54612yes—
293C2×C4#54612yes—
294C2×C4C22×SD1612yes—
295C2×C4#57512yes—
296C2×C4#57514yes—
297C2×C4#57614yes—
298C8—24no—
299C8—24no—
300C8—44no—
301C8—44no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—42no—
306C22—42no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22#253911yescharacteristic
315C22C2×#49211yescharacteristic
316C22#315012yes—
317C22#315311yescharacteristic
318C22#315512yes—
319C4—22no—
320C4—42no—
321C4—84no—
322C4—162no—
323C4—162no—
324C4#310312yes—
325C4#318712yes—
326C4#330914yes—
327C4#331914yes—
328C2—42no—
329C2—322no—
330C2#1960811yescharacteristic
331C2#1965211yescharacteristic
332C2#1969512yes—
333C2#2830511yescharacteristic
334C2#2834912yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
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. #27890C2semidirect
3. #25598C2semidirect
4. #21318C2semidirect
5. #10675C2semidirect
6. #10305C2semidirect
7. #9829C2semidirect
8. #9827C2semidirect
9. #8144C2semidirect
108. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#331712
#278902
#255982
#213182
#106752
#103052
#98292
#98272
#81442