Finite Group Databasebeta

← #314614

Subgroups — #314614

Order 512.

523 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 136 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#21330C211yesmaximal, characteristic
2#20811C211yesmaximal, characteristic
3#10668C212yesmaximal, complemented
4#10628C212yesmaximal, complemented
5#9584C214yesmaximal, complemented
6#8861C212yesmaximal
7#8730C211yesmaximal, characteristic
8#8410C212yesmaximal
9#3458C2211yescharacteristic
10C2×#432—21no—
11#2905C2211yescharacteristic
12C4×#117C2211yescharacteristic
13#1995—21no—
14#1995C2212yes—
15#1994—21no—
16#1994C2212yes—
17#1843C2211yescharacteristic
18#1843C2211yescharacteristic
19#1832—21no—
20#1798—21no—
21#1798C2212yes—
22#1767—21no—
23#1767C2212yes—
24#1733—21no—
25C2×#339—21no—
26#1571C2214yes—
27#1542C2214yes—
28#1481C2214yes—
29#1458C2212yes—
30#1301C2214yes—
31#1276C2212yes—
32#1274C2212yes—
33C22×M16—21no—
34#544—24no—
35#544—22no—
36#543—22no—
37#533—21no—
38#533—42no—
39#533C2312yes—
40C2×#117—21no—
41C2×#117D411yescharacteristic
42C2×#117C2311yescharacteristic
43C22×C42D411yescharacteristic
44#432—44no—
45#431C2314yes—
46#424—42no—
47#423—21no—
48#423C2312yes—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C22⋊C8—22no—
56C2×#97—22no—
57C4×C4⋊C4—22no—
58C4×C22⋊C4—22no—
59#376D411yescharacteristic
60#376C2311yescharacteristic
61C43D411yescharacteristic
62C43C2311yescharacteristic
63#339—22no—
64#339—22no—
65#339—22no—
66#339—22no—
67#336—22no—
68#336—22no—
69#335—44no—
70#333—24no—
71#332—24no—
72Q8⋊C8—44no—
73#324—22no—
74#323—22no—
75C23×C4—21no—
76C23×C4C2×D411yescharacteristic
77C2×M16—22no—
78C2×M16—41no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C4⋊Q8—22no—
85C4⋊Q8—44no—
86C4⋊Q8#4114yes—
87#125—22no—
88#125—22no—
89#125—44no—
90C22⋊Q8—24no—
91C22⋊Q8—44no—
92C22⋊Q8—44no—
93C22⋊Q8—42no—
94C4×Q8—24no—
95C4×Q8—42no—
96#117—22no—
97#117—22no—
98#117—22no—
99#117—22no—
100#117—22no—
101#117—41no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4#4112yes—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4#4112yes—
110C2×C42—21no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—41no—
115C2×C42Q1614yes—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2411yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—24no—
124C4⋊C8—24no—
125C4⋊C8—24no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131C22⋊C8—24no—
132C22⋊C8—44no—
133#97—44no—
134C4×C8—24no—
135#95—22no—
136#95—22no—
137C24#12311yescharacteristic
138C2×Q8—24no—
139C2×Q8—42no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—82no—
151C22×C4#12311yescharacteristic
152C22×C4#12312yes—
153C22×C4C2×Q1612yes—
154C22×C4AGL(1,ℤ/8)12yes—
155C22×C4C22×D411yescharacteristic
156C22×C4C2×#4112yes—
157M16—42no—
158M16—42no—
159M16—82no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—82no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C22⋊C4—24no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—24no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42—82no—
192C42—82no—
193C42C2×Q1612yes—
194C42C2×Q1612yes—
195C42#13712yes—
196C42C2×#4112yes—
197C42H(5,𝔽2)12yes—
198C23—21no—
199C23—21no—
200C23—42no—
201C23#48412yes—
202C23C2×#12311yescharacteristic
203Q8—84no—
204Q8—162no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—84no—
224C2×C4—162no—
225C2×C4#48412yes—
226C2×C4#52511yescharacteristic
227C2×C4#53511yescharacteristic
228C2×C4#55114yes—
229C2×C4C22×Q1611yescharacteristic
230C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C4C2×#13711yescharacteristic
232C2×C4#57511yescharacteristic
233C8—84no—
234C8—82no—
235C8—82no—
236C8—82no—
237C22—22no—
238C22—22no—
239C22—42no—
240C22—42no—
241C22—81no—
242C22#248211yescharacteristic
243C22C2×#48411yescharacteristic
244C22#309511yescharacteristic
245C4—22no—
246C4—24no—
247C4—42no—
248C4—42no—
249C4—42no—
250C4—42no—
251C4—81no—
252C4—84no—
253C4#310411yescharacteristic
254C4#311211yescharacteristic
255C4#320911yescharacteristic
256C4#321311yescharacteristic
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2758611yescharacteristic
261C2#2759912yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2482
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #10668C2semidirect
4. #10628C2semidirect
5. #9584C2semidirect

Maximal subgroups

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

subgroupindex
#213302
#208112
#106682
#106282
#95842
#88612
#87302
#84102