Finite Group Databasebeta

← #324980

Subgroups — #324980

Order 512.

506 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30230C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#28693C211yesmaximal, characteristic
4#22311C211yesmaximal, characteristic
5#10601C212yesmaximal
6#10593C212yesmaximal
7#9899C211yesmaximal, characteristic
8#9702C211yesmaximal, characteristic
9#9576C212yesmaximal
10#9575C212yesmaximal
11#8278C211yesmaximal, characteristic
12C2×#501—21no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2100—22no—
17#2100—22no—
18#2100—22no—
19#2085—22no—
20#1992—21no—
21#1990—21no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2212yes—
27#1987C2212yes—
28#1985—22no—
29#1948C2212yes—
30#1947C2212yes—
31#1939C2212yes—
32#1917—22no—
33#1834—21no—
34#1831C2211yescharacteristic
35#1770C2212yes—
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—
52#501—44no—
53C2×#107—22no—
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#97—21no—
60C2×#97—22no—
61C2×C4×C8—21no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389D412yes—
71#384—21no—
72#384—21no—
73#384—21no—
74#384—21no—
75#384D411yescharacteristic
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#382—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4C2312yes—
81C43C2311yescharacteristic
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#340—22no—
92#336—22no—
93#336—22no—
94#336—21no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336D412yes—
99#336D412yes—
100#335—24no—
101#333—24no—
102#333—24no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—82no—
118C4⋊Q8—82no—
119#125—42no—
120#125—42no—
121C4×Q8—82no—
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—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C42—22no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42C2411yescharacteristic
152#107—44no—
153#106—44no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—82no—
160C4⋊C8#4114yes—
161#97—42no—
162#97—82no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—42no—
166#95—22no—
167#95—22no—
168#95—41no—
169#95—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8—81no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C2×#4111yescharacteristic
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—24no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—41no—
238C42—41no—
239C42—42no—
240C42—42no—
241C42AGL(1,ℤ/8)12yes—
242C42#13712yes—
243C42C2×#4112yes—
244C422−1+412yes—
245C23C2×#12111yescharacteristic
246Q8—84no—
247Q8—162no—
248Q8—162no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
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—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4#46812yes—
280C2×C4#46812yes—
281C2×C4#53211yescharacteristic
282C2×C4#53611yescharacteristic
283C2×C4#54812yes—
284C2×C4#55612yes—
285C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
286C2×C4C2×#13711yescharacteristic
287C2×C4#57712yes—
288C2×C4#57812yes—
289C2×C4#57812yes—
290C8—44no—
291C8—42no—
292C8—42no—
293C8—82no—
294C8—82no—
295C22#254511yescharacteristic
296C22#304211yescharacteristic
297C22#304411yescharacteristic
298C22C2×#46811yescharacteristic
299C22#305911yescharacteristic
300C22#306811yescharacteristic
301C22#306811yescharacteristic
302C4—22no—
303C4—22no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4—161no—
310C4—161no—
311C4—161no—
312C4—161no—
313C4#317812yes—
314C4#324212yes—
315C4#336712yes—
316C4#337412yes—
317C2#1942711yescharacteristic
318C2#1943911yescharacteristic
319C2#1951511yescharacteristic
320C2#1953111yescharacteristic
321C2#2864611yescharacteristic
322C2#2866611yescharacteristic
323C2#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
#286932
#223112
#106012
#105932
#98992
#97022
#95762
#95752
#82782