Finite Group Databasebeta

← #325644

Subgroups — #325644

Order 512.

569 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 150 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#30278C211yesmaximal, characteristic
2#29357C211yesmaximal, characteristic, complemented
3#29348C211yesmaximal, characteristic
4#22299C211yesmaximal, characteristic, complemented
5#10544C212yesmaximal, complemented
6#10539C212yesmaximal
7#9738C212yesmaximal
8#9730C212yesmaximal, complemented
9#9707C211yesmaximal, characteristic
10#9697C211yesmaximal, characteristic, complemented
11#8278C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#498C2211yescharacteristic
14C2×#486C2211yescharacteristic
15C2×#486C2211yescharacteristic
16#2862C2211yescharacteristic
17#2858C2211yescharacteristic
18C4×#125C2211yescharacteristic
19#2092—22no—
20#2092—22no—
21#2074—21no—
22#2073—21no—
23#2026—21no—
24#2025—21no—
25#1987C2212yes—
26#1987C2212yes—
27#1982C2212yes—
28#1981C2212yes—
29#1939—22no—
30#1939C2212yes—
31#1935—22no—
32#1934—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×#125D411yescharacteristic
50C2×#124—22no—
51#500—24no—
52#498—42no—
53#486—24no—
54#486—42no—
55C2×#107—22no—
56C2×#106—21no—
57C2×#106—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#102—21no—
66C2×#102—21no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#97—21no—
71C2×C4×C8—22no—
72C2×C4×C8—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#391—21no—
77#391—22no—
78#390—21no—
79#390—22no—
80#390—21no—
81#389D412yes—
82#387—22no—
83#382—22no—
84#382D411yescharacteristic
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2312yes—
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#336—22no—
93#336—22no—
94#336—21no—
95#336—21no—
96#336D412yes—
97#336D412yes—
98#335—24no—
99#333—42no—
100#333—42no—
101#332—42no—
102#331—42no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121#127—22no—
122#127—22no—
123#125—24no—
124#124—84no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42—22no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#107—44no—
144#106—42no—
145#106—42no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—82no—
153C4⋊C8—82no—
154#103—44no—
155#103—82no—
156#102—42no—
157#102—42no—
158#102—44no—
159#102—44no—
160#102—82no—
161#97—81no—
162C4×C8—22no—
163C4×C8—81no—
164C4×C8—82no—
165C4×C8#4114yes—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—42no—
170#95—42no—
171C24—41no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—82no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C2×#4111yescharacteristic
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8#12414yes—
216C4⋊C4—28no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C4⋊C4#13514yes—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—22no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42C2×#4112yes—
242C42H(5,𝔽2)12yes—
243C23—82no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—82no—
248C23C2×#12111yescharacteristic
249Q8—82no—
250Q8—82no—
251Q8—162no—
252Q8—162no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—81no—
257D4—81no—
258D4—162no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4C2×#12411yescharacteristic
280C2×C4#53511yescharacteristic
281C2×C4#55012yes—
282C2×C4#55912yes—
283C2×C4C2×#13512yes—
284C2×C4#57611yescharacteristic
285C2×C4#57712yes—
286C2×C4#57812yes—
287C2×C4#57811yescharacteristic
288C8—24no—
289C8—41no—
290C8—41no—
291C8—82no—
292C8—82no—
293C8—82no—
294C22—161no—
295C22—161no—
296C22—162no—
297C22—161no—
298C22—162no—
299C22#254811yescharacteristic
300C22#304612yes—
301C22#305011yescharacteristic
302C22#306111yescharacteristic
303C22#306812yes—
304C4—22no—
305C4—42no—
306C4—42no—
307C4—44no—
308C4—82no—
309C4—161no—
310C4—161no—
311C4—161no—
312C4#313712yes—
313C4#322111yescharacteristic
314C4#322311yescharacteristic
315C4#339312yes—
316C4#339612yes—
317C2—321no—
318C2#1941912yes—
319C2#1955112yes—
320C2#2874011yescharacteristic
321C2#2876011yescharacteristic
322C2#2876811yescharacteristic
subgroupG/H
center, socleC23C2×#121
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
2. #29357C2semidirect
4. #22299C2semidirect
5. #10544C2semidirect
8. #9730C2semidirect
10. #9697C2semidirect
11. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#302782
#293572
#293482
#222992
#105442
#105392
#97382
#97302
#97072
#96972
#82782