Finite Group Databasebeta

← #329588

Subgroups — #329588

Order 512.

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

normal subgroups only

This group has 836 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#32953C212yesmaximal
2#29808C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic
4#10622C212yesmaximal, complemented
5#10485C214yesmaximal, complemented
6#10274C212yesmaximal, complemented
7#9132C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#494C2211yescharacteristic, complemented
11C2×#474C2214yes—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2056—22no—
16#2001—22no—
17#1956C2212yescomplemented
18#1954—22no—
19#1954C2212yes—
20#1954C2212yes—
21#1862—24no—
22#1852—22no—
23#1834C2211yescharacteristic, complemented
24#1802C2214yes—
25#1797C2212yes—
26#1786—22no—
27#1779—22no—
28C2×#335C2212yescomplemented
29C2×#332C2214yescomplemented
30#1296C2214yescomplemented
31#1277C2212yescomplemented
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C22×Q16—22no—
35C2×C4⋊Q8C2312yes—
36C2×C22⋊Q8—22no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—42no—
40C2×C4×D4—42no—
41C2×C4×D4D412yes—
42C2×C4×D4C2312yes—
43C22×C4⋊C4D411yescharacteristic
44C22×C22⋊C4—21no—
45C22×C42C2311yescharacteristic
46#494—42no—
47#474—44no—
48#474—44no—
49#431—42no—
50#431—42no—
51C2×#107—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2314yes—
55C2×#103—24no—
56C2×#102—24no—
57C2×#102—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8D412yes—
63C2×C4×C8—21no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#391—22no—
67#389—22no—
68#389—22no—
69#384C2312yes—
70#380—22no—
71#380—22no—
72#340—22no—
73#340—24no—
74#336—22no—
75#335—42no—
76#332—44no—
77C25—21no—
78C22×Q8—22no—
79C22×D4—21no—
80C22×D4—22no—
81C22×D4—42no—
82C22×D4—42no—
83C22×D4Q1614yescomplemented
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
94C2×Q16—44no—
95C2×Q16—84no—
96C22×C8—24no—
97C22×C8—22no—
98C22×C8C2×D412yes—
99C22×C8C2×D412yes—
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C22⋊Q8—84no—
103C4×D4—24no—
104C4×D4—24no—
105C4×D4—44no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—44no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×D412yes—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4—42no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42C2411yescharacteristic
128#107—48no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—24no—
132C4⋊C8—42no—
133#103—84no—
134#102—48no—
135#102—48no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—44no—
139C22⋊C8—84no—
140C22⋊C8—84no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—42no—
144#95—42no—
145#95—42no—
146C24—22no—
147C24—24no—
148C24—42no—
149C24—42no—
150C24—82no—
151C24—82no—
152C24#12111yescharacteristic
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—48no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—48no—
165C2×D4C2×Q1614yes—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—84no—
183C22×C4—82no—
184C22×C4—88no—
185C22×C4—88no—
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C2×Q1612yes—
190C22×C4#13512yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C2×#4111yescharacteristic
194Q16—88no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—44no—
200C2×C8—42no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—44no—
204C4⋊C4—24no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C22⋊C4—84no—
215C42—21no—
216C42—22no—
217C42—21no—
218C42—22no—
219C42H(5,𝔽2)12yes—
220C42H(5,𝔽2)12yes—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—24no—
226C23—48no—
227C23—42no—
228C23—44no—
229C23—42no—
230C23—44no—
231C23—88no—
232C23—84no—
233C23—84no—
234C23—88no—
235C23#46712yes—
236C23C2×#12111yescharacteristic
237Q8—84no—
238Q8—84no—
239D4—28no—
240D4—48no—
241D4—44no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—82no—
256C2×C4—88no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4#46712yes—
268C2×C4#53412yes—
269C2×C4#53411yescharacteristic
270C2×C4#53511yescharacteristic
271C2×C4#53812yes—
272C2×C4C22×Q1611yescharacteristic
273C2×C4C2×#13511yescharacteristic
274C2×C4#57512yes—
275C2×C4#57614yes—
276C8—42no—
277C8—42no—
278C8—84no—
279C8—82no—
280C22—22no—
281C22—24no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—44no—
286C22—48no—
287C22—816no—
288C22—82no—
289C22—82no—
290C22—84no—
291C22#261711yescharacteristic
292C22#304112yes—
293C22C2×#46711yescharacteristic
294C22#305411yescharacteristic
295C22#306912yes—
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—162no—
302C4—162no—
303C4#318912yes—
304C4#319112yes—
305C4#319912yes—
306C4#322012yes—
307C2—42no—
308C2—84no—
309C2#1938712yes—
310C2#1942412yes—
311C2#2972111yescharacteristic
312C2#2973111yescharacteristic
313C2#2975211yescharacteristic
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. #29808C2semidirect
4. #10622C2semidirect
5. #10485C2semidirect
6. #10274C2semidirect
7. #9132C2semidirect
8. #8599C2semidirect
10. C2×#494C22semidirect
17. #1956C22semidirect
23. #1834C22semidirect
28. C2×#335C22semidirect
29. C2×#332C22semidirect
30. #1296C22semidirect
31. #1277C22semidirect
83. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#329532
#298082
#217862
#106222
#104852
#102742
#91322
#85992