Finite Group Databasebeta

← #324856

Subgroups — #324856

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30230C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29212C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10601C212yesmaximal
6#10593C212yesmaximal
7#9897C211yesmaximal, characteristic
8#9702C211yesmaximal, characteristic
9#9568C212yesmaximal
10#9567C212yesmaximal
11#8278C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2100—22no—
16#2100—22no—
17#2100—22no—
18#2059—22no—
19#1993—21no—
20#1990—21no—
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2212yes—
26#1987C2212yes—
27#1984—22no—
28#1948C2212yes—
29#1947C2212yes—
30#1939C2212yes—
31#1916—22no—
32#1847—21no—
33#1834—21no—
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C4×Q8—21no—
52C2×#107—22no—
53C2×#106—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#97—21no—
59C2×C4×C8—21no—
60C2×C4×C8—22no—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389D412yes—
70#384—21no—
71#384—21no—
72#384—21no—
73#384—21no—
74#384D411yescharacteristic
75#384C2311yescharacteristic
76#384C2311yescharacteristic
77#382—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4C2312yes—
80C43C2311yescharacteristic
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—21no—
94#336—21no—
95#336—21no—
96#336—21no—
97#336D412yes—
98#336D412yes—
99#335—24no—
100#333—24no—
101#333—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—82no—
116C4⋊Q8—82no—
117C4⋊Q8Q1614yes—
118#125—82no—
119C4×Q8—82no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C42—22no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42C2411yescharacteristic
150#107—82no—
151#106—82no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—82no—
158C4⋊C8#4114yes—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—42no—
163C4×C8—82no—
164#95—22no—
165#95—22no—
166#95—41no—
167#95—41no—
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C2×#4111yescharacteristic
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C4⋊C4—22no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4C2×Q1614yes—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—41no—
235C42—41no—
236C42—42no—
237C42—42no—
238C42C2×Q1612yes—
239C42#13512yes—
240C42C2×#4112yes—
241C422−1+412yes—
242C23C2×#12111yescharacteristic
243Q8—84no—
244Q8—162no—
245Q8—162no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4#46712yes—
277C2×C4#46712yes—
278C2×C4#53211yescharacteristic
279C2×C4#53611yescharacteristic
280C2×C4#54812yes—
281C2×C4#55612yes—
282C2×C4C22×Q1611yescharacteristic
283C2×C4C2×#13511yescharacteristic
284C2×C4#57712yes—
285C2×C4#57712yes—
286C2×C4#57812yes—
287C8—44no—
288C8—42no—
289C8—42no—
290C8—82no—
291C8—82no—
292C22#254511yescharacteristic
293C22#304211yescharacteristic
294C22#304411yescharacteristic
295C22C2×#46711yescharacteristic
296C22#305711yescharacteristic
297C22#306811yescharacteristic
298C22#306811yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—44no—
305C4—82no—
306C4—161no—
307C4—161no—
308C4—161no—
309C4—161no—
310C4#317812yes—
311C4#324212yes—
312C4#336012yes—
313C4#336312yes—
314C2#1939311yescharacteristic
315C2#1940511yescharacteristic
316C2#1951311yescharacteristic
317C2#1952911yescharacteristic
318C2#2863811yescharacteristic
319C2#2866611yescharacteristic
320C2#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
#292182
#292122
#223072
#106012
#105932
#98972
#97022
#95682
#95672
#82782