Finite Group Databasebeta

← #330363

Subgroups — #330363

Order 512.

584 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 142 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#30276C211yesmaximal, characteristic
2#29869C211yesmaximal, characteristic, complemented
3#29827C211yesmaximal, characteristic
4#22019C211yesmaximal, characteristic, complemented
5#10543C212yesmaximal, complemented
6#10539C212yesmaximal
7#10279C212yesmaximal, complemented
8#10269C212yesmaximal
9#9699C211yesmaximal, characteristic
10#9691C211yesmaximal, characteristic, complemented
11#8789C211yesmaximal, characteristic, complemented
12C2×#500C2212yes—
13C2×#486C2212yes—
14C2×#477C2212yes—
15C2×#477C2211yescharacteristic
16C2×#460C2211yescharacteristic
17#2868C2211yescharacteristic
18#2858C2211yescharacteristic
19#2295C2211yescharacteristic
20#2093—21no—
21#2093—21no—
22#2093—22no—
23#2083—22no—
24#2072—22no—
25#2067—22no—
26#2055—22no—
27#1987C2212yes—
28#1987C2211yescharacteristic
29#1984C2212yes—
30#1982C2211yescharacteristic
31#1939C2212yes—
32#1916—22no—
33#1842C2211yescharacteristic
34#1840—21no—
35#1832—21no—
36#1831C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#333C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#331C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C22×SD16—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×#127C2311yescharacteristic
50C2×#124—22no—
51C2×C4×Q8C2311yescharacteristic
52#500—42no—
53#500—42no—
54#486—42no—
55#486—42no—
56#477—42no—
57#477—42no—
58#477—42no—
59#460—42no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—21no—
63C2×#107—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#103—21no—
71C2×#103—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—21no—
75C2×#102—22no—
76C2×C4×C8—21no—
77C2×C4×C8C2312yes—
78#391D412yes—
79#390—22no—
80#390—21no—
81#390—21no—
82#389D412yes—
83#387—22no—
84#384—22no—
85#384—21no—
86#384—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2311yescharacteristic
89#376—21no—
90C43C2311yescharacteristic
91#340—22no—
92#340—24no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#336D411yescharacteristic
99#336D411yescharacteristic
100#336D412yes—
101#333—42no—
102#333—42no—
103#333—42no—
104#331—42no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8C2×D411yescharacteristic
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C2×SD16—42no—
112C2×SD16—82no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8C2×D412yes—
118C22×C8C2×D412yes—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—84no—
126#127—22no—
127#127—22no—
128#124—84no—
129C4×Q8—22no—
130C4×Q8—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D412yes—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42#4112yes—
149C2×C42C2411yescharacteristic
150#107—42no—
151#107—44no—
152#107—44no—
153#107—44no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—82no—
161C4⋊C8—82no—
162#103—44no—
163#103—42no—
164#102—42no—
165#102—44no—
166#102—44no—
167#102—82no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—81no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—21no—
175C24—41no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—82no—
184C2×Q8—84no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—82no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4#12112yes—
201C22×C4#12112yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4111yescharacteristic
205SD16—84no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C22⋊C4—84no—
238C22⋊C4—84no—
239C42—21no—
240C42—22no—
241C42—21no—
242C42—22no—
243C42—22no—
244C42—24no—
245C42—42no—
246C42—41no—
247C42—41no—
248C42H(5,𝔽2)12yes—
249C42H(5,𝔽2)12yes—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—82no—
254C23—82no—
255C23C2×#12111yescharacteristic
256Q8—41no—
257Q8—41no—
258Q8—42no—
259Q8—162no—
260Q8—162no—
261Q8—162no—
262D4—81no—
263D4—82no—
264D4—81no—
265D4—82no—
266D4—82no—
267D4—162no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4#47214yes—
286C2×C4#53511yescharacteristic
287C2×C4#53512yes—
288C2×C4#53511yescharacteristic
289C2×C4#54012yes—
290C2×C4#57712yes—
291C2×C4#57711yescharacteristic
292C2×C4#57811yescharacteristic
293C2×C4#57812yes—
294C2×C4#57812yes—
295C8—42no—
296C8—42no—
297C8—81no—
298C8—81no—
299C8—82no—
300C8—82no—
301C22—161no—
302C22—162no—
303C22—161no—
304C22—161no—
305C22—162no—
306C22#261911yescharacteristic
307C22#304411yescharacteristic
308C22#304611yescharacteristic
309C22#306312yes—
310C22C2×#47212yes—
311C4—22no—
312C4—24no—
313C4—42no—
314C4—81no—
315C4—81no—
316C4—82no—
317C4—161no—
318C4—162no—
319C4#322211yescharacteristic
320C4#322312yes—
321C4#322311yescharacteristic
322C4#322512yes—
323C4#322512yes—
324C2—321no—
325C2#1957312yes—
326C2#1959112yes—
327C2#2980511yescharacteristic
328C2#2981711yescharacteristic
329C2#2981811yescharacteristic
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. #29869C2semidirect
4. #22019C2semidirect
5. #10543C2semidirect
7. #10279C2semidirect
10. #9691C2semidirect
11. #8789C2semidirect

Maximal subgroups

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

subgroupindex
#302762
#298692
#298272
#220192
#105432
#105392
#102792
#102692
#96992
#96912
#87892