Finite Group Databasebeta

← #281352

Subgroups — #281352

Order 512.

556 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#28274C211yesmaximal, characteristic
2#24994C212yesmaximal
3#22674C211yesmaximal, characteristic
4#10626C212yesmaximal
5#10438C212yesmaximal
6#9656C211yesmaximal, characteristic
7#9618C211yesmaximal, characteristic
8#8896C212yesmaximal
9#8815C211yesmaximal, characteristic
10#8651C211yesmaximal, characteristic
11#8207C211yesmaximal, characteristic
12C2×#446C2211yescharacteristic
13C2×#441—21no—
14C2×C4×Q16C2211yescharacteristic
15#2582C2212yes—
16Q8×C4⋊C4C2212yes—
17#2099—22no—
18#2072—22no—
19#2066—22no—
20#1984C2212yes—
21#1971—21no—
22#1948C2212yes—
23#1944—21no—
24#1943—21no—
25#1927—22no—
26#1920—22no—
27#1865—22no—
28#1844C2211yescharacteristic
29#1844C2211yescharacteristic
30#1834C2212yes—
31#1834C2212yes—
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
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×Q8—22no—
47C2×C4×Q8C2311yescharacteristic
48C2×C4×Q8C2312yes—
49#446—22no—
50#446—22no—
51#441—44no—
52C4×Q16—22no—
53C4×Q16—22no—
54C2×#107D411yescharacteristic
55C2×#106—21no—
56C2×#106—21no—
57C2×#106—22no—
58C2×#106—21no—
59C2×#106—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103D412yes—
68C2×#103D412yes—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#398—42no—
75#391—22no—
76#391—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—42no—
81#384—22no—
82#384C2312yes—
83#384C2312yes—
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#340—42no—
93#340D412yes—
94#336—22no—
95#336—22no—
96#336—21no—
97#335—22no—
98Q8⋊C8—22no—
99Q8⋊C8—24no—
100C82—22no—
101C22×Q8C2×D411yescharacteristic
102C22×Q8C2×D412yes—
103C2×Q16—22no—
104C2×Q16—22no—
105C2×Q16SD1614yes—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8C2×D412yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C4×Q8—22no—
114C4×Q8—21no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—22no—
120C4×Q8—21no—
121C4×Q8—44no—
122C4×Q8—82no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—42no—
141C2×C42C2411yescharacteristic
142#107—22no—
143#106—44no—
144#106—42no—
145#106—42no—
146#106—81no—
147#106—81no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—22no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—42no—
155#103—24no—
156#103—42no—
157#103—44no—
158#103—44no—
159#103—82no—
160#97—42no—
161#97—42no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8#4112yes—
166C4×C8#4112yes—
167C4×C8#4112yes—
168#95—21no—
169#95—42no—
170#95—42no—
171#95—42no—
172#95—42no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8C2×SD1614yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4H(5,𝔽2)11yescharacteristic
193Q16—28no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8#12312yes—
211C2×C8#12312yes—
212C2×C8C2×SD1612yes—
213C2×C8#13712yes—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—162no—
236C4⋊C4—162no—
237C42—41no—
238C42—42no—
239C42—42no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—84no—
244C42C2×#4111yescharacteristic
245C42C2×#4111yescharacteristic
246C42C2×#4111yescharacteristic
247C422−1+411yescharacteristic
248C23D4211yescharacteristic
249Q8—24no—
250Q8—24no—
251Q8—42no—
252Q8—42no—
253Q8—42no—
254Q8—41no—
255Q8—41no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4C2×#12311yescharacteristic
277C2×C4#53211yescharacteristic
278C2×C4#54211yescharacteristic
279C2×C4#54711yescharacteristic
280C2×C4#54711yescharacteristic
281C2×C4#55311yescharacteristic
282C2×C4C22×SD1611yescharacteristic
283C2×C4C2×#13711yescharacteristic
284C2×C4#57512yes—
285C2×C4#57711yescharacteristic
286C2×C4#57711yescharacteristic
287C2×C4#57812yes—
288C8—22no—
289C8—42no—
290C8—42no—
291C8—81no—
292C8—81no—
293C8—82no—
294C8#48314yes—
295C22#242211yescharacteristic
296C22#327712yes—
297C22#327811yescharacteristic
298C22D4×SD1611yescharacteristic
299C22#329012yes—
300C4—21no—
301C4—21no—
302C4—44no—
303C4—82no—
304C4—81no—
305C4—82no—
306C4—81no—
307C4—162no—
308C4—162no—
309C4C2×#48311yescharacteristic
310C4#309911yescharacteristic
311C4#317711yescharacteristic
312C4#318311yescharacteristic
313C4#333511yescharacteristic
314C4#334912yes—
315C4#335611yescharacteristic
316C2#1989612yes—
317C2#2032012yes—
318C2#2501411yescharacteristic
319C2#2504911yescharacteristic
320C2#2505011yescharacteristic
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
#249942
#226742
#106262
#104382
#96562
#96182
#88962
#88152
#86512
#82072