Finite Group Databasebeta

← #324831

Subgroups — #324831

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30230C211yesmaximal, characteristic
2#29226C211yesmaximal, characteristic
3#29210C211yesmaximal, characteristic
4#22299C211yesmaximal, characteristic
5#10601C212yesmaximal
6#10593C212yesmaximal
7#9902C211yesmaximal, characteristic
8#9691C211yesmaximal, characteristic
9#9568C212yesmaximal
10#9567C212yesmaximal
11#8278C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×#125C2211yescharacteristic
15#2100—22no—
16#2100—22no—
17#2099—22no—
18#2059—22no—
19#1992—21no—
20#1991—21no—
21#1990C2212yes—
22#1990C2212yes—
23#1987C2212yes—
24#1987C2212yes—
25#1948C2212yes—
26#1947C2212yes—
27#1939—22no—
28#1939C2212yes—
29#1916—22no—
30#1846—21no—
31#1842C2211yescharacteristic
32#1836—21no—
33C8×C4⋊C4C2212yes—
34#1767C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36#1744C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#333C2212yes—
40C2×#333C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2211yescharacteristic
44#1277C2211yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2312yes—
47C2×#125—21no—
48C2×#125D411yescharacteristic
49C2×#107—22no—
50C2×#106—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2311yescharacteristic
55C2×C4⋊C8C2311yescharacteristic
56C2×C4⋊C8C2311yescharacteristic
57C2×C4⋊C8C2311yescharacteristic
58C2×#97—21no—
59C2×C4×C8—21no—
60C2×C4×C8—22no—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#389—22no—
64#389—22no—
65#389D412yes—
66#387—22no—
67#384—21no—
68#384—21no—
69#384—22no—
70#384C2312yes—
71#382—21no—
72#382D411yescharacteristic
73C4×C4⋊C4—22no—
74C4×C4⋊C4C2312yes—
75C43C2311yescharacteristic
76#340—22no—
77#340—24no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#336—22no—
86#336—22no—
87#336—21no—
88#336—21no—
89#336D412yes—
90#336D412yes—
91#335—22no—
92#335—22no—
93#333—24no—
94#333—24no—
95C22×Q8—22no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D412yes—
103C22×C8C2×D411yescharacteristic
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—84no—
109#125—22no—
110#125—81no—
111#125—81no—
112#125Q1614yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131#107—82no—
132#106—82no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—82no—
142C4⋊C8—82no—
143C4⋊C8#4112yes—
144C4⋊C8#4112yes—
145#97—42no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—42no—
151C4×C8—82no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—84no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4#12112yes—
164C22×C4#12112yes—
165C22×C4C22×D411yescharacteristic
166C22×C4C22×D411yescharacteristic
167C22×C4C2×#4111yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C4⋊C4—22no—
191C4⋊C4—28no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C4⋊C4C2×Q1614yes—
208C42—22no—
209C42—21no—
210C42—21no—
211C42—22no—
212C42—42no—
213C42—42no—
214C42—42no—
215C42C2×Q1612yes—
216C42#13512yes—
217C42C2×#4111yescharacteristic
218C42C2×#4111yescharacteristic
219C422−1+411yescharacteristic
220C422−1+411yescharacteristic
221C23C2×#12111yescharacteristic
222Q8—162no—
223Q8—162no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4#46714yes—
242C2×C4#53211yescharacteristic
243C2×C4#53611yescharacteristic
244C2×C4#54811yescharacteristic
245C2×C4#54811yescharacteristic
246C2×C4#55611yescharacteristic
247C2×C4#55611yescharacteristic
248C2×C4C22×Q1611yescharacteristic
249C2×C4C2×#13511yescharacteristic
250C2×C4#57711yescharacteristic
251C2×C4#57711yescharacteristic
252C2×C4#57712yes—
253C2×C4#57812yes—
254C8—42no—
255C8—42no—
256C8—41no—
257C8—41no—
258C8—42no—
259C8—81no—
260C8—81no—
261C8—82no—
262C22#254511yescharacteristic
263C22#304612yes—
264C22C2×#46711yescharacteristic
265C22#305711yescharacteristic
266C22#306812yes—
267C4—21no—
268C4—21no—
269C4—24no—
270C4—42no—
271C4—44no—
272C4—82no—
273C4—162no—
274C4—162no—
275C4#317411yescharacteristic
276C4#317711yescharacteristic
277C4#324212yes—
278C4#336011yescharacteristic
279C4#336011yescharacteristic
280C4#336311yescharacteristic
281C4#336311yescharacteristic
282C2#1941712yes—
283C2#1954412yes—
284C2#2863811yescharacteristic
285C2#2866411yescharacteristic
286C2#2867411yescharacteristic
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

Maximal subgroups

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

subgroupindex
#302302
#292262
#292102
#222992
#106012
#105932
#99022
#96912
#95682
#95672
#82782