Finite Group Databasebeta

← #241317

Subgroups — #241317

Order 512.

836 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 218 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#37643C211yesmaximal, characteristic
2#21964C212yesmaximal
3#20797C211yesmaximal, characteristic
4#10580C214yesmaximal, complemented
5#10579C214yesmaximal, complemented
6#8041C211yesmaximal, characteristic
7#7897C212yesmaximal
8C2×#427—22no—
9C2×#423C2212yes—
10C22×C4×C8C2211yescharacteristic
11#2845—22no—
12#2629—24no—
13#2629—22no—
14#2552—22no—
15#2552—22no—
16#2552C2214yes—
17#2285C2211yescharacteristic
18#2285C2212yes—
19#2285C2211yescharacteristic
20#1948—22no—
21#1948C2214yes—
22#1947—22no—
23#1947C2214yes—
24#1846—22no—
25#1845C2212yes—
26#1844C2212yes—
27#1809—24no—
28#1296C2214yes—
29#1279C2214yes—
30#1279C2214yes—
31C22×M16—22no—
32C23×C8—21no—
33C2×#125—22no—
34C2×#125—24no—
35C2×#117—22no—
36C2×#117—21no—
37C2×#117D412yes—
38C22×C42C2311yescharacteristic
39#427—48no—
40C2×#107—24no—
41C2×#107—24no—
42C2×#106—24no—
43C2×#106—24no—
44#423—24no—
45#423Q818yes—
46C2×C4⋊C8—22no—
47C2×C4⋊C8C2314yes—
48C2×C4×C8—21no—
49C2×C4×C8—22no—
50C2×C4×C8Q814yes—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#398—28no—
54#398—28no—
55#398—28no—
56#396—28no—
57#389—24no—
58#389—24no—
59#389—24no—
60#384—22no—
61#384—24no—
62#384D412yes—
63#384D412yes—
64#384C2312yes—
65#384C2312yes—
66#382—22no—
67#382—21no—
68#382D412yes—
69#380—22no—
70#380—22no—
71#380—22no—
72#380D414yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2314yes—
77#340—24no—
78#340—24no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D412yes—
81C2×M16—28no—
82C2×M16—82no—
83C2×M16—82no—
84C22×C8—24no—
85C22×C8—24no—
86C22×C8—24no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—24no—
90#125—84no—
91#125—84no—
92#125—84no—
93#117—44no—
94#117—44no—
95#117—82no—
96#117—82no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D414yes—
111C2×C4⋊C4C2×D414yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—41no—
115C2×C42—22no—
116C2×C42—24no—
117C2×C42—41no—
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×Q812yes—
121C2×C42C2×Q814yes—
122C2×C42#4112yes—
123C2×C42C2411yescharacteristic
124#107—48no—
125#107—84no—
126#106—48no—
127#106—84no—
128C4⋊C8—28no—
129C4⋊C8—24no—
130C4⋊C8C2×Q818yes—
131C4×C8—24no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8C2×Q812yes—
135C4×C8C2×Q812yes—
136#95—28no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—42no—
141C24#12011yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—42no—
152C22×C4#12012yes—
153C22×C4#12011yescharacteristic
154C22×C4C22⋊Q812yes—
155C22×C4C22⋊Q814yes—
156C22×C4C4⋊Q812yes—
157C22×C4C4⋊Q814yes—
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D412yes—
160M16—88no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—216no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—28no—
169C2×C8—44no—
170C2×C8—44no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—88no—
190C4⋊C4—88no—
191C4⋊C4—88no—
192C22⋊C4—82no—
193C22⋊C4—84no—
194C22⋊C4—82no—
195C42—44no—
196C42—44no—
197C42—44no—
198C42—82no—
199C42C22⋊Q812yes—
200C42C22⋊Q814yes—
201C42C4⋊Q812yes—
202C42C4⋊Q814yes—
203C42C22×Q811yescharacteristic
204C42C22×Q812yes—
205C42C2×#4111yescharacteristic
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23#39512yes—
211C23C2×#12011yescharacteristic
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4#39514yes—
235C2×C4#39512yes—
236C2×C4#45312yes—
237C2×C4#45612yes—
238C2×C4C2×C22⋊Q811yescharacteristic
239C2×C4C2×C22⋊Q812yes—
240C2×C4C2×C4⋊Q811yescharacteristic
241C2×C4C2×C4⋊Q812yes—
242C2×C4#57612yes—
243C2×C4#57712yes—
244C2×C4#57712yes—
245C2×C4#57812yes—
246C8—24no—
247C8—24no—
248C8—48no—
249C22—24no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22C2×#39511yescharacteristic
255C22#301212yes—
256C22#301312yes—
257C22C2×#45311yescharacteristic
258C22C2×#45611yescharacteristic
259C4—24no—
260C4—21no—
261C4—21no—
262C4—84no—
263C4—84no—
264C4—84no—
265C4—162no—
266C4#206014yes—
267C4#308112yes—
268C4#308212yes—
269C4#316312yes—
270C4#316412yes—
271C2—42no—
272C2#1868712yes—
273C2#1870111yescharacteristic
274C2#1870311yescharacteristic
275C2#2355712yes—
276C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
4. #10580C2semidirect
5. #10579C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#219642
#207972
#105802
#105792
#80412
#78972