Finite Group Databasebeta

← #330437

Subgroups — #330437

Order 512.

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

normal subgroups only

This group has 579 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#33226C211yesmaximal, characteristic
2#29999C211yesmaximal, characteristic, complemented
3#29826C211yesmaximal, characteristic
4#21341C211yesmaximal, characteristic, complemented
5#10592C212yesmaximal, complemented
6#10591C212yesmaximal
7#10487C212yesmaximal, complemented
8#10474C212yesmaximal
9#9902C211yesmaximal, characteristic, complemented
10#9704C211yesmaximal, characteristic
11#8806C211yesmaximal, characteristic, complemented
12C2×#488—22no—
13C2×#475C2211yescharacteristic
14C2×#460C2211yescharacteristic
15#2859C2211yescharacteristic
16#2858C2211yescharacteristic
17#2295C2211yescharacteristic
18#2087—22no—
19#2068—22no—
20#2065—22no—
21#2055—22no—
22#1988C2211yescharacteristic
23#1984C2212yes—
24#1982C2211yescharacteristic
25#1958—21no—
26#1958—21no—
27#1957C2212yes—
28#1956C2212yes—
29#1947C2212yes—
30#1939C2212yes—
31#1916—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1803C2212yes—
35#1768—21no—
36#1767C2211yescharacteristic
37#1746—21no—
38C42×C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#335C2212yes—
41C2×#332C2212yes—
42C2×#331C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×SD16—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×#125—21no—
49C2×#124—21no—
50C2×#124—21no—
51C2×C4×Q8C2311yescharacteristic
52#488—48no—
53#475—42no—
54#460—42no—
55C2×#107—22no—
56C2×#106—21no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×#103—21no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—21no—
71C2×#102—22no—
72C2×#97—21no—
73C2×C4×C8D412yes—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2312yes—
77#391D412yes—
78#390—22no—
79#390—22no—
80#389—22no—
81#389—22no—
82#389D412yes—
83#387—22no—
84#384C2312yes—
85#382—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2311yescharacteristic
88#376—21no—
89C43C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—21no—
97#335—42no—
98#335—42no—
99#332—42no—
100#331—42no—
101C22×Q8—21no—
102C22×Q8C2×D411yescharacteristic
103C22×D4—21no—
104C22×D4—21no—
105C22×D4—21no—
106C2×SD16—42no—
107C2×SD16—82no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116#127—22no—
117#127—22no—
118#125—44no—
119#124—44no—
120#124—42no—
121#124—42no—
122C4×Q8—22no—
123C4×Q8—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#107—82no—
144#106—42no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—44no—
152C4⋊C8—82no—
153C4⋊C8—82no—
154#103—44no—
155#103—42no—
156#103—44no—
157#103—82no—
158#102—42no—
159#102—44no—
160#102—44no—
161#102—82no—
162#97—42no—
163C4×C8—24no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—21no—
172#95—42no—
173C24—41no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—41no—
180C2×Q8—42no—
181C2×Q8—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—82no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4#12112yes—
198C22×C4#12112yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202SD16—84no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8AGL(1,ℤ/8)14yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42#13512yes—
243C42#13712yes—
244C42H(5,𝔽2)12yes—
245C422−1+412yes—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—82no—
250C23—82no—
251C23C2×#12111yescharacteristic
252Q8—41no—
253Q8—41no—
254Q8—42no—
255Q8—82no—
256Q8—82no—
257D4—81no—
258D4—81no—
259D4—82no—
260D4—82no—
261D4—82no—
262D4—162no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#46414yes—
283C2×C4#53511yescharacteristic
284C2×C4#53611yescharacteristic
285C2×C4#53712yes—
286C2×C4#54212yes—
287C2×C4C2×#13511yescharacteristic
288C2×C4C2×AGL(1,ℤ/8)12yes—
289C2×C4C2×#13711yescharacteristic
290C2×C4#57612yes—
291C2×C4#57812yes—
292C8—24no—
293C8—42no—
294C8—81no—
295C8—82no—
296C8—81no—
297C8—82no—
298C22—162no—
299C22—161no—
300C22—161no—
301C22—162no—
302C22—161no—
303C22#262011yescharacteristic
304C22C2×#46411yescharacteristic
305C22#303811yescharacteristic
306C22#306112yes—
307C22#306712yes—
308C4—24no—
309C4—22no—
310C4—42no—
311C4—81no—
312C4—82no—
313C4—81no—
314C4—161no—
315C4—162no—
316C4#322011yescharacteristic
317C4#322311yescharacteristic
318C4#324312yes—
319C4#324712yes—
320C4#325412yes—
321C2—321no—
322C2#1921712yes—
323C2#1935612yes—
324C2#2982711yescharacteristic
325C2#2984911yescharacteristic
326C2#2986011yescharacteristic
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. #29999C2semidirect
4. #21341C2semidirect
5. #10592C2semidirect
7. #10487C2semidirect
9. #9902C2semidirect
11. #8806C2semidirect

Maximal subgroups

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

subgroupindex
#332262
#299992
#298262
#213412
#105922
#105912
#104872
#104742
#99022
#97042
#88062