Finite Group Databasebeta

← #323522

Subgroups — #323522

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33263C211yesmaximal, characteristic
2#29207C212yesmaximal
3#29177C212yesmaximal
4#21691C211yesmaximal, characteristic
5#10681C212yesmaximal
6#10680C212yesmaximal
7#9839C211yesmaximal, characteristic
8#9825C211yesmaximal, characteristic
9#8820C211yesmaximal, characteristic
10#8813C211yesmaximal, characteristic
11#8262C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#441—22no—
14#2859C2212yes—
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2088—24no—
18#2059—24no—
19#1990C2212yes—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1971—22no—
24#1956C2211yescharacteristic
25#1956C2211yescharacteristic
26#1949—24no—
27#1944—22no—
28#1943—22no—
29#1919—24no—
30#1845C2212yes—
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1844C2211yescharacteristic
34#1844C2211yescharacteristic
35#1844C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×Q16—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×C4⋊Q8C2311yescharacteristic
46C2×#125—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49#494—22no—
50#494Q814yes—
51#441—48no—
52C2×#107—22no—
53C2×#107—24no—
54C2×#106—24no—
55C2×#106—22no—
56C2×#106—22no—
57C2×#106—24no—
58C2×C4⋊C8D414yes—
59C2×#103—22no—
60C2×#103—22no—
61C2×#97D412yes—
62C2×#97D412yes—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2312yes—
65C2×C4×C8C2312yes—
66C2×C4×C8C2311yescharacteristic
67#391—21no—
68#391—21no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#384C2311yescharacteristic
76#384C2311yescharacteristic
77#384C2312yes—
78#384C2312yes—
79#382—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C43C2311yescharacteristic
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—22no—
91C82—22no—
92C22×Q8—21no—
93C22×Q8—21no—
94C2×Q16—28no—
95C2×Q16—44no—
96C2×Q16—44no—
97C22×C8C2×D414yes—
98C22×C8C2×D412yes—
99C22×C8C2×D414yes—
100C22×C8C2×D412yes—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8C2×Q812yes—
106C4⋊Q8C2×Q812yes—
107#125—84no—
108C4×Q8—44no—
109C4×Q8—44no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
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—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×D411yescharacteristic
132C2×C42C2411yescharacteristic
133#107—44no—
134#107—84no—
135#106—44no—
136#106—44no—
137#106—48no—
138#106—84no—
139C4⋊C8—28no—
140#103—44no—
141#103—44no—
142#97—24no—
143#97—24no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8C2×Q812yes—
150C4×C8#4112yes—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4#12712yes—
170C22×C4#12711yescharacteristic
171C22×C4#12711yescharacteristic
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D412yes—
174Q16—416no—
175C2×C8—24no—
176C2×C8—24no—
177C2×C8—22no—
178C2×C8—24no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—24no—
182C2×C8—24no—
183C2×C8C22⋊Q814yes—
184C2×C8#13514yes—
185C2×C8AGL(1,ℤ/8)14yes—
186C2×C8#13718yes—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C42—22no—
206C42—22no—
207C42—24no—
208C42—24no—
209C42—82no—
210C42—82no—
211C42C22×Q811yescharacteristic
212C42C2×#4111yescharacteristic
213C422−1+412yes—
214C23C2×#12711yescharacteristic
215Q8—44no—
216Q8—44no—
217Q8—82no—
218Q8—82no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4C2×C22⋊Q811yescharacteristic
243C2×C4#53611yescharacteristic
244C2×C4#54712yes—
245C2×C4D4×Q812yes—
246C2×C4C2×#13512yes—
247C2×C4C2×AGL(1,ℤ/8)12yes—
248C2×C4C2×#13714yes—
249C2×C4#57714yes—
250C8—28no—
251C8—44no—
252C8—42no—
253C8#47818yes—
254C22#254211yescharacteristic
255C22#314711yescharacteristic
256C22#315111yescharacteristic
257C22#315212yes—
258C22#315412yes—
259C4—41no—
260C4—41no—
261C4—42no—
262C4—82no—
263C4—82no—
264C4—161no—
265C4—162no—
266C4—161no—
267C4—162no—
268C4C2×#47812yes—
269C4#322812yes—
270C4#334414yes—
271C4#338714yes—
272C2#1964212yes—
273C2#1972111yescharacteristic
274C2#1974611yescharacteristic
275C2#2851411yescharacteristic
276C2#2855012yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#332632
#292072
#291772
#216912
#106812
#106802
#98392
#98252
#88202
#88132
#82622