Finite Group Databasebeta

← #281370

Subgroups — #281370

Order 512.

528 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#28274C211yesmaximal, characteristic
2#24997C212yesmaximal
3#22814C211yesmaximal, characteristic
4#10623C212yesmaximal
5#10439C212yesmaximal
6#9656C211yesmaximal, characteristic
7#9618C211yesmaximal, characteristic
8#8883C212yesmaximal
9#8814C211yesmaximal, characteristic
10#8651C211yesmaximal, characteristic
11#8207C211yesmaximal, characteristic
12C2×#446C2211yescharacteristic
13C2×C4×Q16C2211yescharacteristic
14#2582C2212yes—
15#2355C2212yes—
16#2101—22no—
17#2074—22no—
18#2066—22no—
19#1984C2212yes—
20#1971—21no—
21#1971—21no—
22#1948C2212yes—
23#1944—21no—
24#1927—22no—
25#1927—22no—
26#1919—22no—
27#1845C2212yes—
28#1844C2211yescharacteristic
29#1844C2211yescharacteristic
30#1834C2212yes—
31#1803C2212yes—
32#1803C2212yes—
33#1803C2212yes—
34#1776—21no—
35#1770C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×Q8⋊C8C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1277C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×Q8C2312yes—
48#446—22no—
49#446—22no—
50C4×Q16—22no—
51C4×Q16—22no—
52C2×#107—42no—
53C2×#107D411yescharacteristic
54C2×#106—21no—
55C2×#106—21no—
56C2×#106—22no—
57C2×#106—21no—
58C2×#106—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103D412yes—
67C2×#103D412yes—
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#398—42no—
74#391—22no—
75#391—22no—
76#391—22no—
77#389—22no—
78#389—22no—
79#387—42no—
80#384C2312yes—
81#384C2312yes—
82#382—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340D412yes—
93#336—22no—
94#336—22no—
95#336—21no—
96#335—22no—
97Q8⋊C8—22no—
98Q8⋊C8—24no—
99C82—22no—
100C22×Q8C2×D412yes—
101C22×Q8C2×D411yescharacteristic
102C2×Q16—22no—
103C2×Q16—22no—
104C2×Q16—22no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C4×Q8—22no—
113C4×Q8—21no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C42—22no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—42no—
136C2×C42C2411yescharacteristic
137#107—22no—
138#107—84no—
139#106—44no—
140#106—42no—
141#106—42no—
142#106—81no—
143#106—81no—
144C4⋊C8—22no—
145C4⋊C8—21no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—42no—
151#103—24no—
152#103—42no—
153#103—44no—
154#103—82no—
155#103—82no—
156#97—42no—
157#97—42no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8#4112yes—
162C4×C8#4112yes—
163C4×C8#4112yes—
164#95—21no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—22no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4H(5,𝔽2)11yescharacteristic
191Q16—44no—
192C2×C8—24no—
193C2×C8—22no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8#12312yes—
209C2×C8#12312yes—
210C2×C8#13512yes—
211C2×C8#13712yes—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—42no—
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—82no—
232C4⋊C4—162no—
233C42—41no—
234C42—42no—
235C42—41no—
236C42—42no—
237C42—41no—
238C42—41no—
239C42—162no—
240C42C2×#4111yescharacteristic
241C42C2×#4111yescharacteristic
242C42C2×#4111yescharacteristic
243C422−1+411yescharacteristic
244C23D4211yescharacteristic
245Q8—42no—
246Q8—42no—
247Q8—42no—
248Q8—42no—
249Q8—42no—
250Q8—41no—
251Q8—41no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
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×#12311yescharacteristic
273C2×C4#53211yescharacteristic
274C2×C4#54211yescharacteristic
275C2×C4#54711yescharacteristic
276C2×C4#54711yescharacteristic
277C2×C4#55311yescharacteristic
278C2×C4C2×#13511yescharacteristic
279C2×C4C2×#13711yescharacteristic
280C2×C4#57512yes—
281C2×C4#57611yescharacteristic
282C2×C4#57811yescharacteristic
283C2×C4#57812yes—
284C8—24no—
285C8—22no—
286C8—42no—
287C8—81no—
288C8—81no—
289C8—82no—
290C8#48514yes—
291C22#242211yescharacteristic
292C22#327611yescharacteristic
293C22#327712yes—
294C22#328711yescharacteristic
295C22#329112yes—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—82no—
300C4—81no—
301C4—82no—
302C4—81no—
303C4—162no—
304C4—162no—
305C4C2×#48511yescharacteristic
306C4#309811yescharacteristic
307C4#317711yescharacteristic
308C4#318311yescharacteristic
309C4#334211yescharacteristic
310C4#334912yes—
311C4#335711yescharacteristic
312C2#1996512yes—
313C2#2031812yes—
314C2#2501711yescharacteristic
315C2#2504911yescharacteristic
316C2#2505111yescharacteristic
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
#282742
#249972
#228142
#106232
#104392
#96562
#96182
#88832
#88142
#86512
#82072