Finite Group Databasebeta

← #491109

Subgroups — #491109

Order 512.

491 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33748C211yesmaximal, characteristic, complemented
2D4×M32C211yesmaximal, characteristic, complemented
3#19258C211yesmaximal, characteristic, complemented
4#19255C211yesmaximal, characteristic, complemented
5#12472C211yesmaximal, characteristic, complemented
6#12471C211yesmaximal, characteristic, complemented
7#12468C211yesmaximal, characteristic, complemented
8#12467C211yesmaximal, characteristic, complemented
9#12456C211yesmaximal, characteristic, complemented
10#12453C211yesmaximal, characteristic, complemented
11#12433C211yesmaximal, characteristic, complemented
12#12430C211yesmaximal, characteristic, complemented
13#12025C211yesmaximal, characteristic, complemented
14#11986C211yesmaximal, characteristic, complemented
15#11985C211yesmaximal, characteristic, complemented
16C22×M32—21no—
17#3169C2211yescharacteristic
18#3169C2211yescharacteristic
19#3109—21no—
20#3036—21no—
21C2×C8×D4C2211yescharacteristic
22#2254C2211yescharacteristic, complemented
23#2246C2211yescharacteristic, complemented
24#2240C2211yescharacteristic, complemented
25#2240C2211yescharacteristic, complemented
26#2218—21no—
27#2218—21no—
28#2177—21no—
29#2177—21no—
30#2169C2211yescharacteristic
31#2169C2211yescharacteristic
32#2168—21no—
33#2168—21no—
34D4×C16C2211yescharacteristic
35D4×C16C2211yescharacteristic
36#2150C2211yescharacteristic
37#2112—21no—
38C4×M32C2211yescharacteristic
39#1681—21no—
40#1681—21no—
41#1677—21no—
42#1677—21no—
43#1675C2211yescharacteristic
44#1675C2211yescharacteristic
45#1673C2211yescharacteristic
46#1673C2211yescharacteristic
47#1667C2211yescharacteristic
48#1667C2211yescharacteristic
49#1667C2211yescharacteristic
50#1667C2211yescharacteristic
51#1568C2211yescharacteristic
52#1567C2211yescharacteristic
53#1551—21no—
54#1547—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, complemented
61#1364C2211yescharacteristic, complemented
62#1364C2211yescharacteristic, complemented
63#1361C2211yescharacteristic, complemented
64#1361C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C2×AGL(1,ℤ/8)—41no—
68C23×C8—21no—
69#532—21no—
70#532—21no—
71C2×C4×D4D411yescharacteristic
72C2×D16—21no—
73C2×D16—21no—
74C2×M32—21no—
75C2×M32—21no—
76C2×M32—21no—
77C2×M32—44no—
78C2×M32D411yescharacteristic
79C2×M32D411yescharacteristic
80C22×C16—21no—
81C22×C16—21no—
82#500C2311yescharacteristic
83#500C2311yescharacteristic
84#493C2311yescharacteristic
85#493C2311yescharacteristic
86#486—21no—
87#486—21no—
88#481—41no—
89#476—21no—
90#476—21no—
91#466—21no—
92#466—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—
104#442—41no—
105#442—41no—
106C4×D8—21no—
107C4×D8—21no—
108C4×D8—21no—
109C4×D8—21no—
110C8×D4—21no—
111C8×D4—21no—
112C8×D4D411yescharacteristic
113C8×D4D411yescharacteristic
114C8×D4C2311yescharacteristic
115C8×D4C2311yescharacteristic
116#432—21no—
117#423—21no—
118C2×C4⋊C8D411yescharacteristic
119#417—41no—
120C2×C22⋊C8—21no—
121C2×C4×C8C2311yescharacteristic
122#367—21no—
123#367—21no—
124C4⋊C16—21no—
125C4⋊C16C2311yescharacteristic
126C4⋊C16C2311yescharacteristic
127#357—21no—
128#357—21no—
129#357—21no—
130#357—21no—
131C22⋊C16—21no—
132C22⋊C16—21no—
133C22⋊C16—21no—
134C22⋊C16—21no—
135#346D411yescharacteristic
136#346D411yescharacteristic
137C4×C16C2311yescharacteristic
138C4×C16C2311yescharacteristic
139#335—21no—
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#335C2311yescharacteristic
144#334—21no—
145#333—21no—
146#333—21no—
147#332—21no—
148#332—21no—
149#331—21no—
150#331—21no—
151#331—21no—
152#331—21no—
153C2×#41—41no—
154C2×#41—41no—
155C22×D4D811yescharacteristic
156C23×C4—21no—
157AGL(1,ℤ/8)—84no—
158C2×SD16—41no—
159C2×SD16—41no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—41no—
165C2×D8—41no—
166C2×D8—41no—
167C2×D8—41no—
168C2×M16—41no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8C2×D411yescharacteristic
177C22×C8#4111yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180#127—21no—
181#127—21no—
182#124—41no—
183#124—41no—
184#121—41no—
185#121—41no—
186#121—41no—
187#121—41no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4D811yescharacteristic
195C4×D4D811yescharacteristic
196C4×D4C2×D411yescharacteristic
197C4×D4C2×D411yescharacteristic
198C2×C4⋊C4D811yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C2×D411yescharacteristic
201D16—42no—
202D16—42no—
203M32—22no—
204M32—41no—
205M32—42no—
206M32—42no—
207M32—42no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—42no—
211C2×C16—42no—
212C2×C16C2×D411yescharacteristic
213C2×C16C2×D411yescharacteristic
214C2×C16C2×D411yescharacteristic
215C2×C16C2×D411yescharacteristic
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#106—41no—
221#106—41no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—41no—
228C4⋊C8C2×D411yescharacteristic
229C4⋊C8C2×D411yescharacteristic
230#103—41no—
231#103—41no—
232#102—41no—
233#102—41no—
234#102—41no—
235#102—41no—
236#102—41no—
237#102—41no—
238#102—41no—
239#102—41no—
240#102—41no—
241#102—41no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—41no—
247#97—41no—
248C4×C8C2×D411yescharacteristic
249C4×C8C2×D411yescharacteristic
250C4×C8#4111yescharacteristic
251C4×C8C2411yescharacteristic
252C24—21no—
253#41—82no—
254#41—82no—
255#41—162no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4C2×D811yescharacteristic
270C2×D4C2×D811yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4#12111yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4#13511yescharacteristic
283SD16—82no—
284SD16—82no—
285D8—41no—
286D8—41no—
287D8—41no—
288D8—41no—
289D8—81no—
290D8—81no—
291D8—82no—
292D8—82no—
293D8—82no—
294D8—82no—
295M16—82no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8#12111yescharacteristic
313C2×C8#12111yescharacteristic
314C2×C8C22×D411yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4C2×D811yescharacteristic
322C4⋊C4C2×D811yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C42#12111yescharacteristic
332C42C2×D811yescharacteristic
333C42#13511yescharacteristic
334C42C22×D411yescharacteristic
335C16—21no—
336C16—21no—
337C16—21no—
338C16—41no—
339C16—41no—
340C16AGL(1,ℤ/8)12yescomplemented
341C23—21no—
342C23—21no—
343C23—42no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23#46611yescharacteristic
349Q8—81no—
350Q8—81no—
351Q8—161no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—161no—
365D4—161no—
366D4—162no—
367D4—162no—
368D4#50912yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—162no—
386C2×C4—162no—
387C2×C4#46611yescharacteristic
388C2×C4#46611yescharacteristic
389C2×C4#46611yescharacteristic
390C2×C4C2×#12111yescharacteristic
391C2×C4C22×D811yescharacteristic
392C2×C4C2×#13511yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—41no—
397C8—41no—
398C8—81no—
399C8—81no—
400C8—81no—
401C8—81no—
402C8C2×AGL(1,ℤ/8)11yescharacteristic
403C8#57511yescharacteristic
404C22—21no—
405C22—21no—
406C22—22no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—84no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22C2×#46611yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—161no—
428C4—161no—
429C4#303611yescharacteristic
430C4#306711yescharacteristic
431C4C2×#50911yescharacteristic
432C4#341411yescharacteristic
433C2—41no—
434C2—42no—
435C2—81no—
436C2—321no—
437C2—321no—
438C2#1925511yescharacteristic
439C2#3360511yescharacteristic
440C2#3361811yescharacteristic
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. #33748C2semidirect
2. D4×M32C2semidirect
3. #19258C2semidirect
4. #19255C2semidirect
5. #12472C2semidirect
6. #12471C2semidirect
7. #12468C2semidirect
8. #12467C2semidirect
9. #12456C2semidirect
10. #12453C2semidirect
11. #12433C2semidirect
12. #12430C2semidirect
13. #12025C2semidirect
14. #11986C2semidirect
15. #11985C2semidirect
22. #2254C22semidirect
23. #2246C22semidirect
24. #2240C22semidirect
25. #2240C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1364C22semidirect
62. #1364C22semidirect
63. #1361C22semidirect
64. #1361C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect
340. C16AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#337482
D4×M322
#192582
#192552
#124722
#124712
#124682
#124672
#124562
#124532
#124332
#124302
#120252
#119862
#119852