Finite Group Databasebeta

← #315293

Subgroups — #315293

Order 512.

523 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28830C211yesmaximal, characteristic
2#28683C211yesmaximal, characteristic
3#27757C212yesmaximal
4#10419C212yesmaximal
5#10353C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9569C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8286C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#446C2211yescharacteristic
14#2571C2212yes—
15#2546C2212yes—
16#2087—22no—
17#2085—22no—
18#2074—22no—
19#2043—22no—
20#1990C2212yes—
21#1988C2212yes—
22#1981C2212yes—
23#1960—22no—
24#1956C2212yes—
25#1947C2212yes—
26#1944—21no—
27#1944—21no—
28#1941—21no—
29#1935—22no—
30#1846—21no—
31#1844C2211yescharacteristic
32#1803C2212yes—
33C4×#106C2211yescharacteristic
34#1770C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8C2311yescharacteristic
48#500—22no—
49#500—22no—
50#446—22no—
51#446—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—22no—
55C2×#106—21no—
56C2×#106—21no—
57C2×#106D411yescharacteristic
58C2×#106D411yescharacteristic
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103D412yes—
69C2×#97—21no—
70C2×#97—22no—
71C2×C4×C8—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#400—42no—
76#398—42no—
77#391—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384C2312yes—
83#382—22no—
84#382—22no—
85C4×C4⋊C4C2311yescharacteristic
86C4×C4⋊C4C2312yes—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—42no—
92#340D412yes—
93#340D412yes—
94#336—22no—
95#336—21no—
96#335—22no—
97#332—24no—
98Q8⋊C8—22no—
99#322—22no—
100C22×Q8—22no—
101C22×Q8C2×D411yescharacteristic
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8C2×D412yes—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—44no—
114C4⋊Q8—82no—
115C4×Q8—21no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—21no—
135C2×C42C2411yescharacteristic
136#107—42no—
137#107—42no—
138#107—82no—
139#106—22no—
140#106—22no—
141#106—42no—
142#106—81no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—42no—
149C4⋊C8#4112yes—
150#103—42no—
151#103—44no—
152#103—82no—
153#103—82no—
154#103—82no—
155#103—82no—
156#97—42no—
157#97—82no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—42no—
163C4×C8#4112yes—
164C4×C8#4112yes—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—42no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4H(5,𝔽2)11yescharacteristic
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8#12412yes—
209C2×C8#12412yes—
210C2×C8#13712yes—
211C2×C8#13712yes—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—162no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—42no—
238C42—42no—
239C42—41no—
240C42C2×#4111yescharacteristic
241C42C2×#4111yescharacteristic
242C42C2×#4111yescharacteristic
243C422−1+411yescharacteristic
244C23D4211yescharacteristic
245Q8—42no—
246Q8—41no—
247Q8—41no—
248Q8—82no—
249Q8—82no—
250Q8—82no—
251Q8—82no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4C2×#12411yescharacteristic
273C2×C4#53211yescharacteristic
274C2×C4#54211yescharacteristic
275C2×C4#54711yescharacteristic
276C2×C4#54811yescharacteristic
277C2×C4#55511yescharacteristic
278C2×C4C2×#13711yescharacteristic
279C2×C4C2×#13711yescharacteristic
280C2×C4#57512yes—
281C2×C4#57612yes—
282C2×C4#57711yescharacteristic
283C2×C4#57811yescharacteristic
284C8—22no—
285C8—42no—
286C8—42no—
287C8—81no—
288C8—81no—
289C8—82no—
290C8#49114yes—
291C22#249111yescharacteristic
292C22#326812yes—
293C22#327611yescharacteristic
294C22#327811yescharacteristic
295C22#328912yes—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—81no—
302C4—81no—
303C4—162no—
304C4—162no—
305C4C2×#49111yescharacteristic
306C4#313711yescharacteristic
307C4#317611yescharacteristic
308C4#318311yescharacteristic
309C4#335512yes—
310C4#337211yescharacteristic
311C4#337411yescharacteristic
312C2#2026112yes—
313C2#2026812yes—
314C2#2777711yescharacteristic
315C2#2780511yescharacteristic
316C2#2781011yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#288302
#286832
#277572
#104192
#103532
#98362
#97312
#95692
#86512
#82862
#82112