Finite Group Databasebeta

← #242832

Subgroups — #242832

Order 512.

761 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 194 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#21867C212yesmaximal
2#21375C211yesmaximal, characteristic
3C2×#1844C211yesmaximal, characteristic, complemented
4#10620C214yesmaximal, complemented
5#10619C214yesmaximal, complemented
6#8040C212yesmaximal, complemented
7#7875C211yesmaximal, characteristic, complemented
8C2×#529C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2212yes—
11C22×#106—22no—
12C22×C4×C8C2211yescharacteristic
13C2×#384C2212yes—
14#1992—22no—
15#1984C2212yes—
16#1959—22no—
17#1954C2212yes—
18#1874—22no—
19#1872—22no—
20#1869—22no—
21#1844—21no—
22#1844—21no—
23#1844C2214yes—
24#1803C2214yes—
25#1802C2214yes—
26#1750—21no—
27#1750—21no—
28#1296C2218yes—
29#1277C2214yes—
30C23×C8D412yes—
31#529—24no—
32#529—24no—
33C2×#126—21no—
34C2×#125D411yescharacteristic
35C2×#124D411yescharacteristic
36C2×#123—21no—
37C2×C22⋊Q8D411yescharacteristic
38C2×#121D411yescharacteristic
39C2×C4×Q8C2311yescharacteristic
40C2×C4×D4—21no—
41C2×C4×D4C2311yescharacteristic
42C2×#117—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C42C2311yescharacteristic
47#432—24no—
48#432—24no—
49#432—24no—
50C2×#106—24no—
51C2×#106—24no—
52C2×#106—48no—
53C2×C4⋊C8C2314yes—
54C2×C4⋊C8C2312yes—
55C2×#103—24no—
56C2×#102—24no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×#97—22no—
62C2×#97—21no—
63C2×C4×C8—24no—
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8C2312yes—
67#384—22no—
68#384—22no—
69#384—22no—
70#384C2314yes—
71#340—24no—
72#340—24no—
73#336—22no—
74#336—22no—
75C22×Q8C2×D411yescharacteristic
76C22×D4—21no—
77C22×D4C2×D411yescharacteristic
78C23×C4—21no—
79C23×C4—42no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4#4111yescharacteristic
83C22×C8—24no—
84C22×C8—24no—
85C22×C8—24no—
86C22×C8—22no—
87C22×C8—24no—
88C22×C8C2×D412yes—
89C22×C8C2×D412yes—
90#126—44no—
91#125—24no—
92#124—24no—
93#123—44no—
94C22⋊Q8—24no—
95#121—24no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—42no—
101C4×D4—42no—
102#117—42no—
103#117—42no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—48no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—84no—
116C2×C4⋊C4—84no—
117C2×C4⋊C4C2×D411yescharacteristic
118C2×C4⋊C4C2×D411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42#4114yes—
128C2×C42#4114yes—
129C2×C42#4111yescharacteristic
130C2×C42C2411yescharacteristic
131#106—48no—
132#106—48no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8—24no—
136#103—84no—
137#102—84no—
138C22⋊C8—44no—
139C22⋊C8—42no—
140C22⋊C8—44no—
141C22⋊C8—44no—
142#97—44no—
143#97—42no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—24no—
147C24—21no—
148C24#12111yescharacteristic
149C2×Q8—22no—
150C2×Q8—24no—
151C2×D4—22no—
152C2×D4—24no—
153C2×D4—42no—
154C2×D4—44no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4—84no—
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12314yes—
174C22×C4#12312yes—
175C22×C4#12412yes—
176C22×C4#12614yes—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C2×#4111yescharacteristic
180C2×C8—216no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—44no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8AGL(1,ℤ/8)14yes—
192C2×C8#13714yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—48no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—84no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42—41no—
212C42—41no—
213C42#12312yes—
214C42#12314yes—
215C42#12412yes—
216C42#12614yes—
217C42C2×#4112yes—
218C42C2×#4112yes—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23#39712yes—
228C23#47114yes—
229C23C2×#12111yescharacteristic
230Q8—44no—
231D4—44no—
232D4—84no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—28no—
241C2×C4—24no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—168no—
255C2×C4#39712yes—
256C2×C4#39712yes—
257C2×C4#39712yes—
258C2×C4C2×#12311yescharacteristic
259C2×C4C2×#12312yes—
260C2×C4C2×#12411yescharacteristic
261C2×C4C2×#12612yes—
262C2×C4C2×AGL(1,ℤ/8)12yes—
263C2×C4C2×#13712yes—
264C2×C4#57714yes—
265C8—24no—
266C8—24no—
267C8—84no—
268C8—82no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—82no—
277C22—82no—
278C22#207914yes—
279C22C2×#39711yescharacteristic
280C22#303811yescharacteristic
281C22#303911yescharacteristic
282C22#305712yes—
283C22C2×#47112yes—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—164no—
292C4#309212yes—
293C4#309914yes—
294C4#314112yes—
295C2—24no—
296C2—161no—
297C2#1935212yes—
298C2#1935712yes—
299C2#2363011yescharacteristic
300C2#2364011yescharacteristic
301C2C2×#207911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
3. C2×#1844C2semidirect
4. #10620C2semidirect
5. #10619C2semidirect
6. #8040C2semidirect
7. #7875C2semidirect

Maximal subgroups

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

subgroupindex
#218672
#213752
C2×#18442
#106202
#106192
#80402
#78752