Finite Group Databasebeta

← #351015

Subgroups — #351015

Order 512.

622 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 170 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#33232C211yesmaximal, characteristic
2#33196C211yesmaximal, characteristic
3#33069C211yesmaximal, characteristic
4Q8×#97C211yesmaximal, characteristic
5#9902C211yesmaximal, characteristic
6#9705C211yesmaximal, characteristic
7#9569C214yesmaximal
8#9220C214yesmaximal
9#8786C211yesmaximal, characteristic
10C2×#491—22no—
11C2×#475C2212yes—
12C2×C8×Q8C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C42×Q8C2211yescharacteristic
16#1988C2212yes—
17#1960—21no—
18#1960—21no—
19#1956C2214yes—
20#1948—22no—
21#1947C2212yes—
22#1921—22no—
23#1863—22no—
24#1842C2211yescharacteristic
25#1831C2211yescharacteristic
26#1803—22no—
27#1803C2212yes—
28#1770—22no—
29#1770C2212yes—
30#1767—21no—
31#1767C2211yescharacteristic
32C4×#97—21no—
33C4×#97C2211yescharacteristic
34C2×#332C2214yes—
35#1296C2214yes—
36#1277C2214yes—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×C4×Q8D412yes—
42C2×C4×Q8C2311yescharacteristic
43#491—48no—
44#475—28no—
45C8×Q8—24no—
46C8×Q8—24no—
47C2×#107—22no—
48C2×#106—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—24no—
57C2×#103—24no—
58C2×#97—22no—
59C2×#97—21no—
60C2×#97D412yes—
61C2×#97D411yescharacteristic
62C2×#97D411yescharacteristic
63C2×C4×C8—22no—
64C2×C4×C8C2312yes—
65#389—22no—
66#389—22no—
67#384—22no—
68#384—22no—
69#384C2312yes—
70#382—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D412yes—
74C4×C4⋊C4C2311yescharacteristic
75C43—21no—
76C43C2311yescharacteristic
77#340—24no—
78#340—22no—
79#340—22no—
80#336—22no—
81#336—22no—
82#336—21no—
83#332—28no—
84C22×Q8—22no—
85C22×Q8C2×D411yescharacteristic
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8C2×D412yes—
90C22×C8C2×D412yes—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—44no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96#125—44no—
97C4×Q8—22no—
98C4×Q8—24no—
99C4×Q8—42no—
100C4×Q8#4114yes—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D411yescharacteristic
113C2×C42—21no—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42C2×D412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4112yes—
121C2×C42C2411yescharacteristic
122#107—44no—
123#106—44no—
124C4⋊C8—28no—
125C4⋊C8—22no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—44no—
129C4⋊C8#4114yes—
130#103—48no—
131#103—44no—
132#103—44no—
133#103—48no—
134#97—24no—
135#97—22no—
136#97—22no—
137#97—44no—
138#97—42no—
139C4×C8—28no—
140C4×C8—22no—
141C4×C8—22no—
142#95—22no—
143#95—21no—
144#95—42no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—44no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—42no—
151C2×Q8#12412yes—
152C2×Q8#13714yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4#12112yes—
159C22×C4#12112yes—
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D411yescharacteristic
162C22×C4C2×#4111yescharacteristic
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—24no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—44no—
175C2×C8—42no—
176C2×C8AGL(1,ℤ/8)14yes—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4#12412yes—
193C42—24no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—42no—
198C42—44no—
199C42—42no—
200C42—82no—
201C42#13512yes—
202C42#13712yes—
203C42C2×#4112yes—
204C42C2×#4112yes—
205C23C2×#12111yescharacteristic
206Q8—22no—
207Q8—24no—
208Q8—84no—
209Q8—84no—
210Q8#49114yescomplemented
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#46414yes—
231C2×C4#52512yes—
232C2×C4C2×#12411yescharacteristic
233C2×C4#52912yes—
234C2×C4#53211yescharacteristic
235C2×C4C2×#13511yescharacteristic
236C2×C4C2×AGL(1,ℤ/8)12yes—
237C2×C4C2×#13711yescharacteristic
238C2×C4C2×#13712yes—
239C2×C4#57512yes—
240C8—24no—
241C8—42no—
242C8—44no—
243C8—82no—
244C8—82no—
245C22#285011yescharacteristic
246C22C2×#46411yescharacteristic
247C22#303811yescharacteristic
248C22#305612yes—
249C22#306912yes—
250C4—24no—
251C4—22no—
252C4—24no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—162no—
257C4—162no—
258C4#310612yes—
259C4#311212yes—
260C4C2×#49111yescharacteristic
261C4#313511yescharacteristic
262C4#318112yes—
263C2#1917512yes—
264C2#1932112yes—
265C2#3303111yescharacteristic
266C2#3305311yescharacteristic
267C2#3306511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
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
210. Q8#491semidirect

Maximal subgroups

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

subgroupindex
#332322
#331962
#330692
Q8×#972
#99022
#97052
#95692
#92202
#87862