Finite Group Databasebeta

← #324826

Subgroups — #324826

Order 512.

587 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30203C211yesmaximal, characteristic
2#28754C211yesmaximal, characteristic
3#28689C211yesmaximal, characteristic, complemented
4#22307C211yesmaximal, characteristic, complemented
5#10594C212yesmaximal, complemented
6#10591C212yesmaximal
7#9905C211yesmaximal, characteristic, complemented
8#9696C211yesmaximal, characteristic
9#9566C212yesmaximal, complemented
10#9563C212yesmaximal
11#8278C211yesmaximal, characteristic, complemented
12C2×#500—21no—
13C2×#489—21no—
14#2868C2211yescharacteristic
15#2858C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2083—22no—
18#2071—22no—
19#2042—22no—
20#2023—22no—
21#1990—21no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1984—22no—
25#1982C2212yes—
26#1981C2212yes—
27#1959—21no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1947C2212yes—
31#1947C2212yes—
32#1939C2212yes—
33#1916—22no—
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#332C2212yes—
41C2×#331C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8D411yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#124—21no—
52C2×#124—21no—
53C2×C4×Q8—21no—
54#500—44no—
55#489—44no—
56C2×#107—22no—
57C2×#107—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×#102—22no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8—22no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—21no—
74#390—21no—
75#390—21no—
76#390—21no—
77#390—21no—
78#389—21no—
79#389D412yes—
80#384—21no—
81#384—21no—
82#384—21no—
83#384D411yescharacteristic
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2312yes—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#335—24no—
102#332—24no—
103#331—24no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—41no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120C4⋊Q8—82no—
121C4⋊Q8—82no—
122C4⋊Q8D814yes—
123#127—22no—
124#127—22no—
125#124—42no—
126#124—42no—
127#124—82no—
128C4×Q8—82no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42C2411yescharacteristic
158#107—44no—
159#107—44no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—82no—
166C4⋊C8#4114yes—
167#103—44no—
168#103—82no—
169#102—44no—
170#102—82no—
171#102—82no—
172#102—82no—
173#97—42no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—42no—
177C4×C8—82no—
178#95—22no—
179#95—22no—
180C24—41no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—81no—
201C2×D4—81no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—82no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4C2×D814yes—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C42—42no—
261C42—42no—
262C42C2×D812yes—
263C42#13512yes—
264C42C2×#4112yes—
265C422−1+412yes—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23C2×#12111yescharacteristic
274Q8—84no—
275Q8—82no—
276Q8—82no—
277Q8—164no—
278D4—82no—
279D4—82no—
280D4—82no—
281D4—82no—
282D4—162no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#46612yes—
311C2×C4#46612yes—
312C2×C4#53211yescharacteristic
313C2×C4#53611yescharacteristic
314C2×C4#54812yes—
315C2×C4#55612yes—
316C2×C4C22×D811yescharacteristic
317C2×C4C2×#13511yescharacteristic
318C2×C4#57612yes—
319C2×C4#57712yes—
320C2×C4#57812yes—
321C8—44no—
322C8—42no—
323C8—42no—
324C8—82no—
325C8—82no—
326C22—161no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22#254511yescharacteristic
334C22#304111yescharacteristic
335C22#304411yescharacteristic
336C22C2×#46611yescharacteristic
337C22#305711yescharacteristic
338C22#306811yescharacteristic
339C22#306911yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—44no—
346C4—82no—
347C4—161no—
348C4—161no—
349C4—161no—
350C4#317512yes—
351C4#324312yes—
352C4#335812yes—
353C4#336512yes—
354C2—321no—
355C2#1938611yescharacteristic
356C2#1940411yescharacteristic
357C2#1950511yescharacteristic
358C2#1952911yescharacteristic
359C2#2863711yescharacteristic
360C2#2866311yescharacteristic
361C2#2867211yescharacteristic
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
3. #28689C2semidirect
4. #22307C2semidirect
5. #10594C2semidirect
7. #9905C2semidirect
9. #9566C2semidirect
11. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#302032
#287542
#286892
#223072
#105942
#105912
#99052
#96962
#95662
#95632
#82782