Finite Group Databasebeta

← #315017

Subgroups — #315017

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28813C211yesmaximal, characteristic, complemented
2#28485C211yesmaximal, characteristic
3#27767C212yesmaximal, complemented
4#10408C212yesmaximal, complemented
5#10348C212yesmaximal, complemented
6#9824C211yesmaximal, characteristic, complemented
7#9734C211yesmaximal, characteristic, complemented
8#9563C212yesmaximal, complemented
9#8644C211yesmaximal, characteristic, complemented
10C8×#107C211yesmaximal, characteristic, complemented
11C8×#102C211yesmaximal, characteristic, complemented
12C2×#500—21no—
13C2×#493C2211yescharacteristic
14C2×#486—22no—
15C2×#466—21no—
16C2×#466—21no—
17C2×C8×D4C2211yescharacteristic
18#2551C2212yes—
19#2541C2212yes—
20#2057—22no—
21#2041—22no—
22#2023—22no—
23#1999—22no—
24#1982C2212yes—
25#1982C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1947C2212yes—
29#1942—21no—
30#1934—22no—
31#1845C2211yescharacteristic
32#1802C2212yes—
33C4×#107C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#102C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#331C2212yes—
39C2×D4⋊C8C2211yescharacteristic
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C22×D8—21no—
46C22×D8—21no—
47C23×C8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×#127—22no—
51C2×#127C2312yes—
52C2×#121—22no—
53C2×#121—22no—
54C2×C4×D4C2311yescharacteristic
55#500—44no—
56#493—22no—
57#493—22no—
58#486—48no—
59#466—44no—
60#466—44no—
61C8×D4—22no—
62C8×D4—22no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107—21no—
66C2×#107D411yescharacteristic
67C2×#107D411yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—42no—
79C2×#102D411yescharacteristic
80C2×#102D411yescharacteristic
81C2×C22⋊C8—21no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#395—42no—
88#392—42no—
89#390—22no—
90#389—22no—
91#389—22no—
92#384C2312yes—
93C4×C4⋊C4C2312yes—
94C4×C4⋊C4C2311yescharacteristic
95#340—22no—
96#340—22no—
97#340D412yes—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—21no—
101#336—22no—
102#335—22no—
103#331—24no—
104D4⋊C8—22no—
105C82—22no—
106C22×Q8—42no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—42no—
111C22×D4—42no—
112C22×D4C2×D411yescharacteristic
113C23×C4—21no—
114C2×D8—42no—
115C2×D8—42no—
116C2×D8—42no—
117C2×D8—42no—
118C2×D8—81no—
119C2×D8—81no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—82no—
134C4⋊Q8—82no—
135#127—22no—
136#127—22no—
137#127—22no—
138#127—22no—
139#127—44no—
140#127—82no—
141#121—44no—
142#121—44no—
143#121—82no—
144#121—82no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42C2411yescharacteristic
165#107—42no—
166#107—44no—
167#107—81no—
168#107D814yes—
169#107SD1614yes—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8#4112yes—
176#102—22no—
177#102—44no—
178#102—44no—
179#102—44no—
180#102—44no—
181#102—44no—
182#102—44no—
183#102—82no—
184#102—84no—
185#102D814yescomplemented
186C22⋊C8—42no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—21no—
191C4×C8—44no—
192C4×C8—42no—
193C4×C8#4112yes—
194C4×C8#4112yes—
195#95—22no—
196#95—21no—
197#95—42no—
198C24—21no—
199C24—42no—
200C2×Q8—84no—
201C2×Q8—84no—
202C2×Q8—162no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—84no—
225C2×D4—84no—
226C2×D4—84no—
227C2×D4—84no—
228C2×D4—162no—
229C2×D4—162no—
230C2×D4C2×D812yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4H(5,𝔽2)11yescharacteristic
246D8—84no—
247D8—84no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—22no—
253C2×C8—21no—
254C2×C8—22no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—84no—
268C2×C8#12412yes—
269C2×C8#12412yes—
270C2×C8C2×D812yes—
271C2×C8C2×D812yes—
272C2×C8C2×SD1612yes—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4C2×D812yes—
287C4⋊C4C2×D814yes—
288C4⋊C4C2×SD1614yes—
289C4⋊C4#13512yes—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—84no—
293C22⋊C4—84no—
294C42—42no—
295C42—41no—
296C42—42no—
297C42—41no—
298C42C2×#4111yescharacteristic
299C42C2×#4111yescharacteristic
300C42C2×#4111yescharacteristic
301C422−1+411yescharacteristic
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23D4211yescharacteristic
317Q8—168no—
318D4—22no—
319D4—22no—
320D4—41no—
321D4—41no—
322D4—84no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—162no—
332D4—168no—
333D4—168no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4C2×#12411yescharacteristic
359C2×C4#53211yescharacteristic
360C2×C4#54211yescharacteristic
361C2×C4#54711yescharacteristic
362C2×C4#54811yescharacteristic
363C2×C4#55511yescharacteristic
364C2×C4C22×D811yescharacteristic
365C2×C4C22×D811yescharacteristic
366C2×C4C22×SD1611yescharacteristic
367C2×C4C2×#13511yescharacteristic
368C2×C4#57611yescharacteristic
369C2×C4#57611yescharacteristic
370C2×C4#57711yescharacteristic
371C2×C4#57811yescharacteristic
372C8—22no—
373C8—22no—
374C8—42no—
375C8—41no—
376C8—42no—
377C8—81no—
378C8—82no—
379C8#48614yescomplemented
380C22—42no—
381C22—42no—
382C22—42no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—162no—
388C22—162no—
389C22—162no—
390C22—162no—
391C22—162no—
392C22—162no—
393C22—162no—
394C22#249111yescharacteristic
395C22D4×D811yescharacteristic
396C22D4×D811yescharacteristic
397C22D4×SD1611yescharacteristic
398C22#328511yescharacteristic
399C22#329011yescharacteristic
400C22#329211yescharacteristic
401C4—21no—
402C4—21no—
403C4—44no—
404C4—42no—
405C4—44no—
406C4—42no—
407C4—81no—
408C4—162no—
409C4C2×#48611yescharacteristic
410C4#313611yescharacteristic
411C4#316911yescharacteristic
412C4#317511yescharacteristic
413C4#333411yescharacteristic
414C4#334111yescharacteristic
415C4#335811yescharacteristic
416C4#336211yescharacteristic
417C2—82no—
418C2—322no—
419C2D8211yescharacteristic
420C2D8×SD1611yescharacteristic
421C2#1997311yescharacteristic
422C2#1998611yescharacteristic
423C2#2776711yescharacteristic
424C2#2779111yescharacteristic
425C2#2780911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#28485C2
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
1. #28813C2semidirect
3. #27767C2semidirect
4. #10408C2semidirect
5. #10348C2semidirect
6. #9824C2semidirect
7. #9734C2semidirect
8. #9563C2semidirect
9. #8644C2semidirect
10. C8×#107C2semidirect
11. C8×#102C2semidirect
185. #102D8semidirect
379. C8#486semidirect

Maximal subgroups

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

subgroupindex
#288132
#284852
#277672
#104082
#103482
#98242
#97342
#95632
#86442
C8×#1072
C8×#1022