Finite Group Databasebeta

← #305792

Subgroups — #305792

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28814C211yesmaximal, characteristic
2#26108C212yesmaximal
3Q8×#107C211yesmaximal, characteristic
4#10658C212yesmaximal
5#10580C212yesmaximal
6#9615C211yesmaximal, characteristic
7#9594C211yesmaximal, characteristic
8#8888C212yesmaximal
9#8814C211yesmaximal, characteristic
10#8645C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#499—21no—
13C2×C8×Q8C2211yescharacteristic
14#2552C2212yes—
15Q8×C4⋊C4C2212yes—
16#2099—22no—
17#2088—22no—
18#2059—22no—
19#1948C2212yes—
20#1948C2212yes—
21#1947—22no—
22#1947C2212yes—
23#1942—21no—
24#1939C2212yes—
25#1865—22no—
26#1845—21no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1844C2212yes—
30#1834C2212yes—
31#1803—22no—
32#1803C2212yes—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×C82C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×#125—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8C2311yescharacteristic
48#499—44no—
49C8×Q8—22no—
50C8×Q8Q814yes—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—22no—
56C2×#107D411yescharacteristic
57C2×#107D411yescharacteristic
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—42no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103D411yescharacteristic
68C2×#103D411yescharacteristic
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#398—42no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—42no—
79#384—22no—
80#384—22no—
81#384C2312yes—
82#384C2312yes—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2312yes—
85C4×C4⋊C4C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340D412yes—
93#340D411yescharacteristic
94#340D411yescharacteristic
95#336—21no—
96#336—22no—
97#335—22no—
98Q8⋊C8Q814yes—
99C82—22no—
100C22×Q8C2×D411yescharacteristic
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109#125—44no—
110#125—82no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8—44no—
114C4×Q8—82no—
115C4×Q8C2×Q812yes—
116C4×Q8C2×Q812yes—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D412yes—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—42no—
138C2×C42C2411yescharacteristic
139#107—22no—
140#107—42no—
141#107—42no—
142#107—82no—
143#107—81no—
144#107—81no—
145#107SD1614yes—
146#106—44no—
147#106—82no—
148#106—84no—
149C4⋊C8—24no—
150C4⋊C8—21no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—22no—
154C4⋊C8—82no—
155C4⋊C8C2×Q812yes—
156C4⋊C8C2×Q812yes—
157#103—22no—
158#103—22no—
159#103—44no—
160C4×C8—24no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—42no—
165C4×C8C2×Q812yes—
166C4×C8C2×Q812yes—
167C4×C8#4112yes—
168#95—22no—
169#95—21no—
170#95—42no—
171#95—42no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8C4⋊Q812yes—
176C2×Q8#13512yes—
177C2×Q8#13512yes—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4H(5,𝔽2)11yescharacteristic
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8C22⋊Q812yes—
210C2×C8C22⋊Q812yes—
211C2×C8C2×SD1612yes—
212C2×C8#13712yes—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—162no—
237C4⋊C4—162no—
238C4⋊C4C4⋊Q812yes—
239C4⋊C4C2×SD1614yes—
240C42—22no—
241C42—22no—
242C42—41no—
243C42—42no—
244C42—42no—
245C42—41no—
246C42—84no—
247C42C22×Q811yescharacteristic
248C42C22×Q811yescharacteristic
249C42C2×#4111yescharacteristic
250C422−1+411yescharacteristic
251C23D4211yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—22no—
255Q8#49914yescomplemented
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4C2×C22⋊Q811yescharacteristic
283C2×C4C2×C4⋊Q811yescharacteristic
284C2×C4#54411yescharacteristic
285C2×C4#54711yescharacteristic
286C2×C4D4×Q811yescharacteristic
287C2×C4#55411yescharacteristic
288C2×C4C22×SD1611yescharacteristic
289C2×C4C2×#13511yescharacteristic
290C2×C4C2×#13511yescharacteristic
291C2×C4C2×#13711yescharacteristic
292C2×C4#57611yescharacteristic
293C2×C4#57711yescharacteristic
294C2×C4#57711yescharacteristic
295C2×C4#57811yescharacteristic
296C8—24no—
297C8—22no—
298C8—41no—
299C8—41no—
300C8—42no—
301C8—42no—
302C8—82no—
303C8#47514yes—
304C22#244011yescharacteristic
305C22#327811yescharacteristic
306C22#328011yescharacteristic
307C22D4×SD1611yescharacteristic
308C22#328511yescharacteristic
309C22#329211yescharacteristic
310C22#329311yescharacteristic
311C4—24no—
312C4—21no—
313C4—21no—
314C4—44no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—82no—
319C4—162no—
320C4—162no—
321C4C2×#47511yescharacteristic
322C4#308411yescharacteristic
323C4C2×#49911yescharacteristic
324C4#316411yescharacteristic
325C4#333611yescharacteristic
326C4#333911yescharacteristic
327C4Q8×SD1611yescharacteristic
328C4#338711yescharacteristic
329C2#1990911yescharacteristic
330C2#1994411yescharacteristic
331C2#1997911yescharacteristic
332C2#1998611yescharacteristic
333C2#2604211yescharacteristic
334C2#2606711yescharacteristic
335C2#2608311yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
255. Q8#499semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#288142
#261082
Q8×#1072
#106582
#105802
#96152
#95942
#88882
#88142
#86452
C8×#1032