Finite Group Databasebeta

← #325388

Subgroups — #325388

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30304C211yesmaximal, characteristic, complemented
2#29149C211yesmaximal, characteristic
3#29139C211yesmaximal, characteristic
4#22260C211yesmaximal, characteristic
5#10622C212yesmaximal, complemented
6#10616C212yesmaximal, complemented
7#9903C211yesmaximal, characteristic, complemented
8#9897C211yesmaximal, characteristic, complemented
9#8648C212yesmaximal, complemented
10#8644C212yesmaximal, complemented
11#8278C211yesmaximal, characteristic, complemented
12C2×C8×D4C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×#127C2211yescharacteristic
16#2083—22no—
17#2057—22no—
18#2025—22no—
19#2023—22no—
20#1961—21no—
21#1957—21no—
22#1954C2212yes—
23#1869—22no—
24#1869—22no—
25#1845C2211yescharacteristic
26#1845C2211yescharacteristic
27#1836—21no—
28#1834—21no—
29#1834C2211yescharacteristic
30#1834C2211yescharacteristic
31#1831C2211yescharacteristic
32#1802C2212yes—
33#1802C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C23×C8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×#127D411yescharacteristic
50C2×#125—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4C2312yes—
54C8×D4—24no—
55C8×D4—24no—
56C2×#107—22no—
57C2×#107—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×#102—22no—
62C2×#102—22no—
63C2×#102—22no—
64C2×#102—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#390—21no—
73#390—21no—
74#390D412yes—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384D411yescharacteristic
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#382—21no—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#335—24no—
102C22×Q8—41no—
103C22×D4—21no—
104C22×D4—21no—
105C22×D4C2×D412yes—
106C23×C4—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—44no—
111C22×C8—82no—
112C22×C8—82no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—82no—
116C4⋊Q8—82no—
117C4⋊Q8—82no—
118#127—22no—
119#127Q1614yes—
120#125—82no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—82no—
126C4×D4—82no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
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—21no—
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⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C2×D412yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—82no—
158#107—82no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165#102—82no—
166#102—82no—
167#102—82no—
168#102—82no—
169C22⋊C8—44no—
170C22⋊C8—82no—
171C22⋊C8—82no—
172#97—42no—
173C4×C8—22no—
174C4×C8—42no—
175C4×C8#4114yes—
176#95—41no—
177#95—41no—
178C24—22no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8—82no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4C2×Q1614yes—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C2×#4111yescharacteristic
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—88no—
227C2×C8#12414yes—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—41no—
257C42—41no—
258C42C2×Q1612yes—
259C42#13512yes—
260C42C2×#4112yes—
261C422−1+412yes—
262C23—24no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23C2×#12111yescharacteristic
270Q8—164no—
271D4—24no—
272D4—24no—
273D4—42no—
274D4—42no—
275D4—84no—
276D4—84no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#46712yes—
309C2×C4#46712yes—
310C2×C4C2×#12411yescharacteristic
311C2×C4#53611yescharacteristic
312C2×C4#54712yes—
313C2×C4#55612yes—
314C2×C4C22×Q1611yescharacteristic
315C2×C4C2×#13511yescharacteristic
316C2×C4#57612yes—
317C2×C4#57612yes—
318C2×C4#57712yes—
319C8—24no—
320C8—42no—
321C8—82no—
322C8—82no—
323C8—82no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22#254711yescharacteristic
332C22#304111yescharacteristic
333C22#304211yescharacteristic
334C22C2×#46711yescharacteristic
335C22#305811yescharacteristic
336C22#306811yescharacteristic
337C22#306911yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—82no—
344C4—161no—
345C4—161no—
346C4—161no—
347C4—161no—
348C4#313612yes—
349C4#324112yes—
350C4#333812yes—
351C4#334112yes—
352C2—84no—
353C2#1938711yescharacteristic
354C2#1939311yescharacteristic
355C2#1950611yescharacteristic
356C2#1951411yescharacteristic
357C2#2870511yescharacteristic
358C2#2872811yescharacteristic
359C2#2873511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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
1. #30304C2semidirect
5. #10622C2semidirect
6. #10616C2semidirect
7. #9903C2semidirect
8. #9897C2semidirect
9. #8648C2semidirect
10. #8644C2semidirect
11. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#303042
#291492
#291392
#222602
#106222
#106162
#99032
#98972
#86482
#86442
#82782