Finite Group Databasebeta

← #325615

Subgroups — #325615

Order 512.

623 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30276C211yesmaximal, characteristic
2#28755C211yesmaximal, characteristic, complemented
3#28752C211yesmaximal, characteristic
4#22299C211yesmaximal, characteristic, complemented
5#10545C212yesmaximal, complemented
6#10540C212yesmaximal
7#9738C212yesmaximal
8#9731C212yesmaximal, complemented
9#9704C211yesmaximal, characteristic
10#9694C211yesmaximal, characteristic, complemented
11#8278C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#491—21no—
15C2×#488—21no—
16C2×#486C2211yescharacteristic
17C2×#486C2211yescharacteristic
18#2868C2211yescharacteristic
19#2858C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2093—22no—
22#2093—22no—
23#2043—22no—
24#2042—22no—
25#1982C2212yes—
26#1981C2212yes—
27#1981C2212yes—
28#1981C2212yes—
29#1960—21no—
30#1958—21no—
31#1939—22no—
32#1939C2212yes—
33#1916—22no—
34#1842C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#332C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#125D411yescharacteristic
52C2×#124—21no—
53C2×#124—21no—
54#500—24no—
55#500—42no—
56#491—44no—
57#488—44no—
58#486—24no—
59#486—42no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#102—21no—
75C2×#102—21no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—22no—
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#390—22no—
85#390—22no—
86#389D412yes—
87#387—22no—
88#384—22no—
89#384—22no—
90#382D411yescharacteristic
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2312yes—
93C43C2311yescharacteristic
94#336—22no—
95#336—22no—
96#336—21no—
97#336—21no—
98#336D412yes—
99#336D412yes—
100#335—24no—
101#332—42no—
102#332—42no—
103#332—42no—
104#331—42no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—82no—
126#127—22no—
127#127—22no—
128#125—22no—
129#125D814yes—
130#124—42no—
131#124—42no—
132#124—82no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D412yes—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C42—22no—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D412yes—
150C2×C42C2×D411yescharacteristic
151C2×C42#4112yes—
152C2×C42C2411yescharacteristic
153#107—42no—
154#107—42no—
155#107—44no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—82no—
163C4⋊C8—82no—
164#103—44no—
165#103—44no—
166#103—44no—
167#103—82no—
168#103—82no—
169#103—82no—
170#102—42no—
171#102—42no—
172#102—44no—
173#102—44no—
174#102—82no—
175#97—42no—
176C4×C8—22no—
177C4×C8—42no—
178C4×C8—82no—
179C4×C8#4114yes—
180#95—22no—
181#95—22no—
182#95—22no—
183C24—41no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—82no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C2×#4111yescharacteristic
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8#12414yes—
232C4⋊C4—22no—
233C4⋊C4—28no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4C2×D814yes—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42C2×D812yes—
258C42#13512yes—
259C42C2×#4112yes—
260C42H(5,𝔽2)12yes—
261C23—82no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—82no—
266C23C2×#12111yescharacteristic
267Q8—82no—
268Q8—82no—
269Q8—82no—
270Q8—82no—
271Q8—82no—
272Q8—82no—
273D4—82no—
274D4—82no—
275D4—82no—
276D4—81no—
277D4—81no—
278D4—162no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4#46614yes—
301C2×C4C2×#12411yescharacteristic
302C2×C4#53511yescharacteristic
303C2×C4#55012yes—
304C2×C4#55912yes—
305C2×C4C22×D811yescharacteristic
306C2×C4C2×#13511yescharacteristic
307C2×C4#57612yes—
308C2×C4#57712yes—
309C2×C4#57812yes—
310C8—24no—
311C8—41no—
312C8—41no—
313C8—82no—
314C8—82no—
315C8—82no—
316C22—161no—
317C22—161no—
318C22—162no—
319C22—161no—
320C22—162no—
321C22#254811yescharacteristic
322C22#304612yes—
323C22C2×#46611yescharacteristic
324C22#305811yescharacteristic
325C22#306912yes—
326C4—22no—
327C4—24no—
328C4—42no—
329C4—44no—
330C4—82no—
331C4—161no—
332C4—161no—
333C4—161no—
334C4#313712yes—
335C4#322111yescharacteristic
336C4#322311yescharacteristic
337C4#339112yes—
338C4#339612yes—
339C2—321no—
340C2#1941612yes—
341C2#1954612yes—
342C2#2873611yescharacteristic
343C2#2876011yescharacteristic
344C2#2876711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28755C2semidirect
4. #22299C2semidirect
5. #10545C2semidirect
8. #9731C2semidirect
10. #9694C2semidirect
11. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#302762
#287552
#287522
#222992
#105452
#105402
#97382
#97312
#97042
#96942
#82782