Finite Group Databasebeta

← #330191

Subgroups — #330191

Order 512.

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

normal subgroups only

This group has 617 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#33225C211yesmaximal, characteristic, complemented
2#29868C211yesmaximal, characteristic
3#29864C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic
5#10541C214yesmaximal, complemented
6#10275C214yesmaximal
7#9705C211yesmaximal, characteristic, complemented
8#9702C211yesmaximal, characteristic, complemented
9#8785C211yesmaximal, characteristic, complemented
10C2×#487C2214yes—
11C2×#476C2212yes—
12C2×#474C2212yes—
13C2×#433—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2293C2211yescharacteristic
17#2084—22no—
18#2083—22no—
19#2071—22no—
20#2071—22no—
21#1988C2212yes—
22#1987C2212yes—
23#1956—21no—
24#1956—21no—
25#1956—22no—
26#1954C2212yes—
27#1939C2212yes—
28#1923—22no—
29#1831C2211yescharacteristic
30#1831C2211yescharacteristic
31#1767C2211yescharacteristic
32#1750—21no—
33C42×C8C2211yescharacteristic
34C2×#333C2214yes—
35C2×#332C2214yes—
36#1277C2214yes—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×C4⋊Q8C2312yes—
42C2×C4×D4C2311yescharacteristic
43C2×#117—21no—
44#487—44no—
45#487—44no—
46#476—44no—
47#474—44no—
48#433—44no—
49C2×#107—22no—
50C2×#106—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×#103—24no—
56C2×#103—24no—
57C2×#102—22no—
58C2×#102—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×#97—21no—
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D412yes—
65C2×C4×C8C2312yes—
66#390D412yes—
67#389D412yes—
68#388—22no—
69#384—22no—
70#384—22no—
71#384—21no—
72#384—21no—
73#384—22no—
74C4×C4⋊C4C2311yescharacteristic
75C4×C22⋊C4—21no—
76C43C2311yescharacteristic
77#340—24no—
78#340—24no—
79#336—22no—
80#336—22no—
81#336—22no—
82#336—21no—
83#333—44no—
84#332—44no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×D4C2×D411yescharacteristic
88C23×C4—21no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×D412yes—
93C22×C8C2×D412yes—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—44no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—84no—
102C4×D4—22no—
103C4×D4—22no—
104#117—42no—
105#117—42no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D411yescharacteristic
116C2×C4⋊C4C2×D412yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—21no—
120C2×C42—21no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42#4112yes—
125C2×C42C2411yescharacteristic
126#107—44no—
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—24no—
130C4⋊C8—22no—
131C4⋊C8—24no—
132C4⋊C8—44no—
133#103—48no—
134#103—48no—
135#102—44no—
136#102—44no—
137C22⋊C8—44no—
138C22⋊C8—44no—
139#97—42no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—42no—
145#95—22no—
146#95—21no—
147C24—21no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—44no—
153C2×Q8—84no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—24no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4#12112yes—
166C22×C4#12112yes—
167C22×C4C22×D411yescharacteristic
168C22×C4C22×D411yescharacteristic
169C22×C4C2×#4111yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—24no—
186C4⋊C4—44no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C22⋊C4—42no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C42—22no—
201C42—24no—
202C42—24no—
203C42—41no—
204C42—41no—
205C42#13512yes—
206C42AGL(1,ℤ/8)12yes—
207C42H(5,𝔽2)12yes—
208C42H(5,𝔽2)12yes—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×#12111yescharacteristic
215Q8—84no—
216Q8—84no—
217Q8—164no—
218D4—42no—
219D4—42no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4#46314yes—
239C2×C4#47114yes—
240C2×C4#53512yes—
241C2×C4#53511yescharacteristic
242C2×C4#53511yescharacteristic
243C2×C4#54012yes—
244C2×C4C2×#13511yescharacteristic
245C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
246C2×C4#57512yes—
247C2×C4#57814yes—
248C8—42no—
249C8—42no—
250C8—82no—
251C8—82no—
252C8—82no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—82no—
257C22—82no—
258C22#261911yescharacteristic
259C22C2×#46311yescharacteristic
260C22#304011yescharacteristic
261C22#305912yes—
262C22C2×#47112yes—
263C4—22no—
264C4—24no—
265C4—24no—
266C4—81no—
267C4—82no—
268C4—162no—
269C4—162no—
270C4#321212yes—
271C4#321412yes—
272C4#322312yes—
273C4#322312yes—
274C2—161no—
275C2#1919912yes—
276C2#1936512yes—
277C2#2979511yescharacteristic
278C2#2981211yescharacteristic
279C2#2981911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #33225C2semidirect
5. #10541C2semidirect
7. #9705C2semidirect
8. #9702C2semidirect
9. #8785C2semidirect

Maximal subgroups

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

subgroupindex
#332252
#298682
#298642
#212662
#105412
#102752
#97052
#97022
#87852