Finite Group Databasebeta

← #491237

Subgroups — #491237

Order 512.

481 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33773C211yesmaximal, characteristic, complemented
2#33488C211yesmaximal, characteristic, complemented
3#19255C211yesmaximal, characteristic, complemented
4#19101C211yesmaximal, characteristic, complemented
5#12474C211yesmaximal, characteristic, complemented
6#12471C211yesmaximal, characteristic, complemented
7#12470C211yesmaximal, characteristic, complemented
8#12467C211yesmaximal, characteristic, complemented
9#12435C211yesmaximal, characteristic, complemented
10#12433C211yesmaximal, characteristic, complemented
11#12432C211yesmaximal, characteristic, complemented
12#12430C211yesmaximal, characteristic, complemented
13#12036C211yesmaximal, characteristic, complemented
14#11966C211yesmaximal, characteristic, complemented
15#11951C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3147C2211yescharacteristic
18#3100—21no—
19#3036—21no—
20C2×C8×D4C2211yescharacteristic
21#2248C2211yescharacteristic, complemented
22#2246C2211yescharacteristic, complemented
23#2241C2211yescharacteristic, complemented
24#2240C2211yescharacteristic, complemented
25#2210—21no—
26#2210—21no—
27#2186—21no—
28#2186—21no—
29#2169C2211yescharacteristic
30#2169C2211yescharacteristic
31#2168—21no—
32#2168—21no—
33D4×C16C2211yescharacteristic
34D4×C16C2211yescharacteristic
35#2163C2211yescharacteristic
36#2162C2211yescharacteristic
37#2112—21no—
38C2×C22⋊C16—21no—
39#1681—21no—
40#1681—21no—
41#1673C2211yescharacteristic
42#1673C2211yescharacteristic
43#1669C2211yescharacteristic
44#1669C2211yescharacteristic
45#1667C2211yescharacteristic
46#1667C2211yescharacteristic
47#1667C2211yescharacteristic
48#1667C2211yescharacteristic
49#1665—21no—
50#1665—21no—
51#1567C2211yescharacteristic
52#1564C2211yescharacteristic
53#1547—21no—
54#1540—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1364C2211yescharacteristic, complemented
60#1364C2211yescharacteristic, complemented
61#1362C2211yescharacteristic, complemented
62#1362C2211yescharacteristic, complemented
63#1361C2211yescharacteristic, complemented
64#1361C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C2×AGL(1,ℤ/8)—41no—
68C2×#135—21no—
69C2×#135—21no—
70C23×C8—21no—
71#532—21no—
72#532—21no—
73C2×C4×D4D411yescharacteristic
74C2×SD32—21no—
75C2×SD32—21no—
76C2×D16—21no—
77C2×D16—21no—
78C2×M32—21no—
79C2×M32—21no—
80C22×C16—21no—
81C22×C16—21no—
82#500C2311yescharacteristic
83#495—21no—
84#495—21no—
85#494C2311yescharacteristic
86#493C2311yescharacteristic
87#493C2311yescharacteristic
88#492—21no—
89#486—21no—
90#480—41no—
91#474—21no—
92#474—21no—
93#466—21no—
94#466—21no—
95#463—41no—
96#463—41no—
97#461—21no—
98#461—21no—
99#459—21no—
100#459—21no—
101#459—21no—
102#459—21no—
103#447—41no—
104C4×D8—21no—
105C4×D8—21no—
106C4×D8—41no—
107C4×D8—41no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4D411yescharacteristic
111C8×D4D411yescharacteristic
112C8×D4C2311yescharacteristic
113C8×D4C2311yescharacteristic
114#431—21no—
115#423—21no—
116C2×C4⋊C8D411yescharacteristic
117#417—41no—
118C2×C22⋊C8—21no—
119C2×C4×C8C2311yescharacteristic
120C4⋊C16D411yescharacteristic
121C4⋊C16D411yescharacteristic
122C4⋊C16C2311yescharacteristic
123C4⋊C16C2311yescharacteristic
124#358—21no—
125#358—21no—
126#357—21no—
127#357—21no—
128C22⋊C16—21no—
129C22⋊C16—21no—
130C22⋊C16—21no—
131C22⋊C16—21no—
132C22⋊C16—21no—
133C22⋊C16D411yescharacteristic
134C22⋊C16D411yescharacteristic
135#346—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335—21no—
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#334—21no—
144#332—21no—
145#332—21no—
146#332—21no—
147#332—21no—
148#331—21no—
149#331—21no—
150#331—21no—
151#331—21no—
152C2×#41—41no—
153C2×#41—41no—
154C22×D4D811yescharacteristic
155C23×C4—21no—
156AGL(1,ℤ/8)—84no—
157#135—81no—
158#135—81no—
159#135—82no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×SD16—41no—
163C2×SD16—41no—
164C2×SD16—41no—
165C2×SD16—41no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—41no—
171C2×D8—41no—
172C2×D8—41no—
173C2×D8—41no—
174C2×M16—41no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—81no—
181C22×C8—81no—
182C22×C8C2×D411yescharacteristic
183C22×C8#4111yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186#127—21no—
187#127—21no—
188#124—41no—
189#124—41no—
190#121—41no—
191#121—41no—
192#121—41no—
193#121—41no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4D811yescharacteristic
201C4×D4D811yescharacteristic
202C4×D4C2×D411yescharacteristic
203C4×D4C2×D411yescharacteristic
204C2×C4⋊C4D811yescharacteristic
205C2×C22⋊C4—21no—
206C2×C42C2×D411yescharacteristic
207SD32—81no—
208SD32—81no—
209D16—81no—
210D16—81no—
211M32—81no—
212M32—81no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—81no—
216C2×C16—81no—
217C2×C16C2×D411yescharacteristic
218C2×C16C2×D411yescharacteristic
219C2×C16C2×D411yescharacteristic
220C2×C16C2×D411yescharacteristic
221#107—21no—
222#107—21no—
223#107—41no—
224#107—41no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—41no—
231C4⋊C8C2×D411yescharacteristic
232C4⋊C8C2×D411yescharacteristic
233#103—41no—
234#103—41no—
235#102—41no—
236#102—41no—
237#102—41no—
238#102—41no—
239#102—41no—
240#102—41no—
241#102—41no—
242#102—41no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—41no—
248C4×C8—41no—
249C4×C8C2×D411yescharacteristic
250C4×C8C2×D411yescharacteristic
251C4×C8#4111yescharacteristic
252C4×C8C2411yescharacteristic
253C24—21no—
254#41—82no—
255#41—82no—
256#41—82no—
257#41—82no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—81no—
270C2×D4—81no—
271C2×D4C2×D811yescharacteristic
272C2×D4C2×D811yescharacteristic
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4#12111yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4#13511yescharacteristic
285Q16—41no—
286Q16—41no—
287Q16—81no—
288SD16—82no—
289SD16—82no—
290SD16—82no—
291SD16—82no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—81no—
297D8—81no—
298D8—82no—
299D8—82no—
300D8—82no—
301D8—82no—
302M16—82no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—81no—
317C2×C8—81no—
318C2×C8—81no—
319C2×C8#12111yescharacteristic
320C2×C8#12111yescharacteristic
321C2×C8C22×D411yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4C2×D811yescharacteristic
329C4⋊C4C2×D811yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C42#12111yescharacteristic
339C42C2×D811yescharacteristic
340C42#13511yescharacteristic
341C42C22×D411yescharacteristic
342C16—21no—
343C16—21no—
344C16—21no—
345C16—21no—
346C16—41no—
347C16—41no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23#46611yescharacteristic
356Q8—81no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—161no—
373D4—161no—
374D4—162no—
375D4—162no—
376D4#50912yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—162no—
394C2×C4—162no—
395C2×C4#46611yescharacteristic
396C2×C4#46611yescharacteristic
397C2×C4#46611yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C22×D811yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C8—21no—
402C8—21no—
403C8—21no—
404C8—41no—
405C8—41no—
406C8—81no—
407C8—81no—
408C8—81no—
409C8—81no—
410C8#57611yescharacteristic
411C8#57611yescharacteristic
412C22—21no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22C2×#46611yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—161no—
436C4—161no—
437C4#304111yescharacteristic
438C4#306811yescharacteristic
439C4C2×#50911yescharacteristic
440C4#341411yescharacteristic
441C2—41no—
442C2—42no—
443C2—81no—
444C2—321no—
445C2—321no—
446C2#1938611yescharacteristic
447C2#3360511yescharacteristic
448C2#3361711yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33773C2semidirect
2. #33488C2semidirect
3. #19255C2semidirect
4. #19101C2semidirect
5. #12474C2semidirect
6. #12471C2semidirect
7. #12470C2semidirect
8. #12467C2semidirect
9. #12435C2semidirect
10. #12433C2semidirect
11. #12432C2semidirect
12. #12430C2semidirect
13. #12036C2semidirect
14. #11966C2semidirect
15. #11951C2semidirect
21. #2248C22semidirect
22. #2246C22semidirect
23. #2241C22semidirect
24. #2240C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1364C22semidirect
60. #1364C22semidirect
61. #1362C22semidirect
62. #1362C22semidirect
63. #1361C22semidirect
64. #1361C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect

Maximal subgroups

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

subgroupindex
#337732
#334882
#192552
#191012
#124742
#124712
#124702
#124672
#124352
#124332
#124322
#124302
#120362
#119662
#119512