Finite Group Databasebeta

← #330189

Subgroups — #330189

Order 512.

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

normal subgroups only

This group has 743 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#33224C211yesmaximal, characteristic, complemented
2#29798C211yesmaximal, characteristic, complemented
3#29794C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic, complemented
5#10540C214yesmaximal, complemented
6#10261C214yesmaximal, complemented
7#9706C211yesmaximal, characteristic, complemented
8#9703C211yesmaximal, characteristic, complemented
9#8785C211yesmaximal, characteristic, complemented
10C2×#486C2214yes—
11C2×#461C2212yes—
12C2×#459C2212yes—
13C2×#433—21no—
14#2868C2211yescharacteristic
15#2867C2211yescharacteristic
16#2293C2211yescharacteristic
17#2043—22no—
18#2041—22no—
19#2013—22no—
20#2011—22no—
21#1982C2212yes—
22#1981C2212yes—
23#1957—21no—
24#1957—21no—
25#1957—22no—
26#1954C2212yes—
27#1939C2212yes—
28#1923—22no—
29#1831C2211yescharacteristic
30#1831C2211yescharacteristic
31#1767C2211yescharacteristic
32#1750—21no—
33C42×C8C2211yescharacteristic
34C2×#332C2214yes—
35C2×#331C2214yes—
36#1277C2214yes—
37C22×SD16—22no—
38C22×D8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×#127—21no—
42C2×#127—21no—
43C2×#127—22no—
44C2×#127—22no—
45C2×#127C2312yes—
46C2×C4×D4C2311yescharacteristic
47C2×#117—21no—
48#486—44no—
49#486—44no—
50#461—44no—
51#459—44no—
52#433—44no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#103—22no—
58C2×#103—24no—
59C2×#102—24no—
60C2×#102—24no—
61C2×#102—22no—
62C2×#102—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×#97—21no—
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D412yes—
69C2×C4×C8C2312yes—
70#390D412yes—
71#389D412yes—
72#388—22no—
73#384—21no—
74#384—21no—
75#384—22no—
76C4×C4⋊C4C2311yescharacteristic
77C4×C22⋊C4—21no—
78C43C2311yescharacteristic
79#336—22no—
80#336—22no—
81#336—22no—
82#336—21no—
83#332—44no—
84#331—44no—
85C22×Q8—22no—
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—42no—
90C22×D4—42no—
91C22×D4C2×D411yescharacteristic
92C23×C4—21no—
93C2×SD16—44no—
94C2×SD16—84no—
95C2×D8—44no—
96C2×D8—84no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—84no—
105#127—24no—
106#127—24no—
107#127—44no—
108#127—42no—
109#127—42no—
110#127—84no—
111#127—84no—
112C4×D4—22no—
113C4×D4—22no—
114#117—42no—
115#117—42no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4C2×D411yescharacteristic
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C42—21no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4112yes—
131C2×C42C2411yescharacteristic
132C4⋊C8—22no—
133C4⋊C8—24no—
134C4⋊C8—22no—
135C4⋊C8—24no—
136C4⋊C8—44no—
137#103—44no—
138#103—84no—
139#102—44no—
140#102—48no—
141#102—48no—
142#102—84no—
143C22⋊C8—44no—
144C22⋊C8—44no—
145#97—42no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—42no—
151#95—22no—
152#95—21no—
153C24—21no—
154C24—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—24no—
162C2×D4—44no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—84no—
172C2×D4—84no—
173C2×D4—84no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—84no—
177C2×D4—84no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4#12112yes—
186C22×C4#12112yes—
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C2×#4111yescharacteristic
190SD16—88no—
191D8—88no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—44no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—24no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—84no—
212C4⋊C4—84no—
213C22⋊C4—42no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C42—22no—
217C42—24no—
218C42—24no—
219C42—41no—
220C42—41no—
221C42#13512yes—
222C42AGL(1,ℤ/8)12yes—
223C42H(5,𝔽2)12yes—
224C42H(5,𝔽2)12yes—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23—42no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23—84no—
235C23C2×#12111yescharacteristic
236Q8—82no—
237Q8—82no—
238Q8—84no—
239D4—42no—
240D4—42no—
241D4—84no—
242D4—84no—
243D4—82no—
244D4—82no—
245D4—84no—
246D4—164no—
247D4—168no—
248D4—168no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4#46314yes—
265C2×C4#47114yes—
266C2×C4#53512yes—
267C2×C4#53511yescharacteristic
268C2×C4#53511yescharacteristic
269C2×C4#54012yes—
270C2×C4C2×#13511yescharacteristic
271C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
272C2×C4#57512yes—
273C2×C4#57614yes—
274C8—42no—
275C8—42no—
276C8—82no—
277C8—82no—
278C8—82no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—82no—
283C22—82no—
284C22—162no—
285C22—162no—
286C22—162no—
287C22—164no—
288C22—164no—
289C22#261911yescharacteristic
290C22C2×#46311yescharacteristic
291C22#304011yescharacteristic
292C22#305812yes—
293C22C2×#47112yes—
294C4—22no—
295C4—24no—
296C4—24no—
297C4—81no—
298C4—82no—
299C4—162no—
300C4#321212yes—
301C4#321412yes—
302C4#322012yes—
303C4#322012yes—
304C2—161no—
305C2—322no—
306C2#1919812yes—
307C2#1936412yes—
308C2#2979511yescharacteristic
309C2#2981011yescharacteristic
310C2#2981911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#29794C2
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
1. #33224C2semidirect
2. #29798C2semidirect
4. #21266C2semidirect
5. #10540C2semidirect
6. #10261C2semidirect
7. #9706C2semidirect
8. #9703C2semidirect
9. #8785C2semidirect

Maximal subgroups

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

subgroupindex
#332242
#297982
#297942
#212662
#105402
#102612
#97062
#97032
#87852