Finite Group Databasebeta

← #329596

Subgroups — #329596

Order 512.

812 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 154 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#32956C212yesmaximal
2#29878C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic
4#10622C212yesmaximal, complemented
5#10481C214yesmaximal, complemented
6#10278C212yesmaximal, complemented
7#9128C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#500C2211yescharacteristic, complemented
11C2×#476C2214yes—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2087—22no—
16#2076—22no—
17#1990C2212yescomplemented
18#1954—22no—
19#1954C2212yes—
20#1954C2212yes—
21#1864—24no—
22#1853—22no—
23#1834C2211yescharacteristic, complemented
24#1802C2214yes—
25#1797C2212yes—
26#1786—22no—
27#1779—22no—
28C2×#335C2212yescomplemented
29C2×#333C2214yescomplemented
30#1296C2214yescomplemented
31#1277C2212yescomplemented
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C2×C4⋊Q8C2312yes—
35C2×C22⋊Q8—22no—
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4—42no—
39C2×C4×D4—42no—
40C2×C4×D4D412yes—
41C2×C4×D4C2312yes—
42C22×C4⋊C4D411yescharacteristic
43C22×C22⋊C4—21no—
44C22×C42C2311yescharacteristic
45#500—42no—
46#476—44no—
47#476—44no—
48#431—42no—
49#431—42no—
50C2×#107—22no—
51C2×#106—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2314yes—
55C2×#102—24no—
56C2×#102—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8D412yes—
62C2×C4×C8—21no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—22no—
66#389—22no—
67#389—22no—
68#384C2312yes—
69#380—22no—
70#380—22no—
71#340—22no—
72#340—24no—
73#340—24no—
74#336—22no—
75#335—42no—
76#333—44no—
77C25—21no—
78C22×Q8—22no—
79C22×D4—21no—
80C22×D4—22no—
81C22×D4—42no—
82C22×D4—42no—
83C22×D4Q1614yes—
84C22×D4C2×D412yes—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—84no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C22×C8—24no—
95C22×C8—22no—
96C22×C8C2×D412yes—
97C22×C8C2×D412yes—
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C22⋊Q8—84no—
101C4×D4—24no—
102C4×D4—24no—
103C4×D4—44no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—44no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—42no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42C2411yescharacteristic
126#107—44no—
127#106—48no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—42no—
132#102—48no—
133#102—48no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C22⋊C8—44no—
137C22⋊C8—84no—
138C22⋊C8—84no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—42no—
142#95—42no—
143#95—42no—
144C24—22no—
145C24—24no—
146C24—42no—
147C24—42no—
148C24—82no—
149C24—82no—
150C24#12111yescharacteristic
151C2×Q8—44no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—48no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—48no—
162C2×D4C2×Q1614yes—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—84no—
180C22×C4—82no—
181C22×C4—88no—
182C22×C4—88no—
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C2×Q1612yes—
187C22×C4#13512yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—44no—
196C2×C8—42no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C4⋊C4—24no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—84no—
212C42—21no—
213C42—22no—
214C42—21no—
215C42—22no—
216C42H(5,𝔽2)12yes—
217C42H(5,𝔽2)12yes—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—24no—
223C23—48no—
224C23—42no—
225C23—44no—
226C23—42no—
227C23—44no—
228C23—88no—
229C23—84no—
230C23—84no—
231C23—88no—
232C23#46712yes—
233C23C2×#12111yescharacteristic
234Q8—164no—
235D4—28no—
236D4—48no—
237D4—44no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—82no—
252C2×C4—88no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#46712yes—
264C2×C4#53412yes—
265C2×C4#53411yescharacteristic
266C2×C4#53511yescharacteristic
267C2×C4#53812yes—
268C2×C4C22×Q1611yescharacteristic
269C2×C4C2×#13511yescharacteristic
270C2×C4#57512yes—
271C2×C4#57714yes—
272C8—42no—
273C8—42no—
274C8—84no—
275C8—82no—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—44no—
282C22—48no—
283C22—816no—
284C22—82no—
285C22—82no—
286C22—84no—
287C22#261711yescharacteristic
288C22#304212yes—
289C22C2×#46711yescharacteristic
290C22#305411yescharacteristic
291C22#306812yes—
292C4—22no—
293C4—22no—
294C4—41no—
295C4—41no—
296C4—82no—
297C4—162no—
298C4—162no—
299C4#318912yes—
300C4#319112yes—
301C4#320012yes—
302C4#322212yes—
303C2—42no—
304C2—84no—
305C2#1939312yes—
306C2#1943012yes—
307C2#2972311yescharacteristic
308C2#2973111yescharacteristic
309C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #29878C2semidirect
4. #10622C2semidirect
5. #10481C2semidirect
6. #10278C2semidirect
7. #9128C2semidirect
8. #8599C2semidirect
10. C2×#500C22semidirect
17. #1990C22semidirect
23. #1834C22semidirect
28. C2×#335C22semidirect
29. C2×#333C22semidirect
30. #1296C22semidirect
31. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#329562
#298782
#217862
#106222
#104812
#102782
#91282
#85992