Finite Group Databasebeta

← #491112

Subgroups — #491112

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33751C211yesmaximal, characteristic, complemented
2D4×M32C211yesmaximal, characteristic, complemented
3#19258C211yesmaximal, characteristic
4#19256C211yesmaximal, characteristic, complemented
5#12473C211yesmaximal, characteristic, complemented
6#12472C211yesmaximal, characteristic, complemented
7#12469C211yesmaximal, characteristic, complemented
8#12468C211yesmaximal, characteristic, complemented
9#12455C211yesmaximal, characteristic, complemented
10#12452C211yesmaximal, characteristic, complemented
11#12435C211yesmaximal, characteristic, complemented
12#12432C211yesmaximal, characteristic, complemented
13#12025C211yesmaximal, characteristic, complemented
14#11987C211yesmaximal, characteristic, complemented
15#11986C211yesmaximal, characteristic, complemented
16C22×M32—21no—
17#3170C2211yescharacteristic
18#3169C2211yescharacteristic
19#3109—21no—
20#3077—21no—
21C2×C8×D4C2211yescharacteristic
22#2252C2211yescharacteristic, complemented
23#2248C2211yescharacteristic, complemented
24#2241C2211yescharacteristic, complemented
25#2240C2211yescharacteristic, complemented
26#2220—21no—
27#2220—21no—
28#2175—21no—
29#2175—21no—
30#2169C2211yescharacteristic
31#2169C2211yescharacteristic
32#2168—21no—
33#2168—21no—
34D4×C16C2211yescharacteristic
35D4×C16C2211yescharacteristic
36#2150C2211yescharacteristic
37#2112—21no—
38C4×M32C2211yescharacteristic
39#1683—21no—
40#1683—21no—
41#1677—21no—
42#1677—21no—
43#1675C2211yescharacteristic
44#1675C2211yescharacteristic
45#1675C2211yescharacteristic
46#1675C2211yescharacteristic
47#1669C2211yescharacteristic
48#1669C2211yescharacteristic
49#1667C2211yescharacteristic
50#1667C2211yescharacteristic
51#1568C2211yescharacteristic
52#1567C2211yescharacteristic
53#1551—21no—
54#1550—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, complemented
61#1365C2211yescharacteristic, complemented
62#1365C2211yescharacteristic, complemented
63#1362C2211yescharacteristic, complemented
64#1362C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C23×C8—21no—
68#533—21no—
69#532—21no—
70C2×C4×D4D411yescharacteristic
71C2×SD32—21no—
72C2×SD32—21no—
73C2×M32—21no—
74C2×M32—21no—
75C2×M32—21no—
76C2×M32—44no—
77C2×M32D411yescharacteristic
78C2×M32D411yescharacteristic
79C22×C16—21no—
80C22×C16—21no—
81#500C2311yescharacteristic
82#500C2311yescharacteristic
83#494C2311yescharacteristic
84#493C2311yescharacteristic
85#487—21no—
86#486—21no—
87#481—41no—
88#478—41no—
89#478—41no—
90#476—21no—
91#476—21no—
92#476—21no—
93#476—21no—
94#474—21no—
95#474—21no—
96#467—21no—
97#467—21no—
98#466—21no—
99#466—21no—
100#459—21no—
101#459—21no—
102#447—41no—
103#442—41no—
104#442—41no—
105C4×Q16—21no—
106C4×Q16—21no—
107C4×D8—21no—
108C4×D8—21no—
109C8×D4—21no—
110C8×D4—21no—
111C8×D4D411yescharacteristic
112C8×D4D411yescharacteristic
113C8×D4C2311yescharacteristic
114C8×D4C2311yescharacteristic
115#432—21no—
116#428—41no—
117#423—21no—
118C2×C4⋊C8D411yescharacteristic
119#418—41no—
120C2×C22⋊C8—21no—
121C2×C4×C8C2311yescharacteristic
122#366—21no—
123#366—21no—
124C4⋊C16—21no—
125C4⋊C16C2311yescharacteristic
126C4⋊C16C2311yescharacteristic
127#358—21no—
128#358—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#333—21no—
148#333—21no—
149#332—21no—
150#332—21no—
151#331—21no—
152#331—21no—
153C2×#41—41no—
154C22×D4D811yescharacteristic
155C23×C4—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—41no—
161C2×D8—41no—
162C2×M16—41no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8C2×D411yescharacteristic
171C22×C8#4111yescharacteristic
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175#127—21no—
176#125—41no—
177#124—41no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180#121—41no—
181#121—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4D811yescharacteristic
189C4×D4D811yescharacteristic
190C4×D4C2×D411yescharacteristic
191C4×D4C2×D411yescharacteristic
192#117—41no—
193C2×C4⋊C4D811yescharacteristic
194C2×C22⋊C4—21no—
195C2×C42C2×D411yescharacteristic
196SD32—42no—
197SD32—42no—
198M32—22no—
199M32—41no—
200M32—42no—
201M32—42no—
202M32—42no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—42no—
206C2×C16—42no—
207C2×C16C2×D411yescharacteristic
208C2×C16C2×D411yescharacteristic
209C2×C16C2×D411yescharacteristic
210C2×C16C2×D411yescharacteristic
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—41no—
216#107—41no—
217#106—41no—
218#106—41no—
219#106—41no—
220#106—41no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—41no—
227C4⋊C8C2×D411yescharacteristic
228C4⋊C8C2×D411yescharacteristic
229#103—41no—
230#103—41no—
231#102—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—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—41no—
246#97—41no—
247C4×C8C2×D411yescharacteristic
248C4×C8C2×D411yescharacteristic
249C4×C8#4111yescharacteristic
250C4×C8C2411yescharacteristic
251C24—21no—
252#41—162no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—81no—
263C2×D4C2×D811yescharacteristic
264C2×D4C2×D811yescharacteristic
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—81no—
273C22×C4#12111yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4#13511yescharacteristic
276Q16—41no—
277Q16—41no—
278Q16—81no—
279D8—41no—
280D8—41no—
281D8—81no—
282D8—82no—
283D8—82no—
284M16—82no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8#12111yescharacteristic
302C2×C8#12111yescharacteristic
303C2×C8C22×D411yescharacteristic
304C2×C8C2×#4111yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4C2×D811yescharacteristic
315C4⋊C4C2×D811yescharacteristic
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C42—81no—
324C42#12111yescharacteristic
325C42C2×D811yescharacteristic
326C42#13511yescharacteristic
327C42C22×D411yescharacteristic
328C16—21no—
329C16—21no—
330C16—21no—
331C16—41no—
332C16—41no—
333C16AGL(1,ℤ/8)12yes—
334C23—21no—
335C23—21no—
336C23—42no—
337C23—42no—
338C23—81no—
339C23—81no—
340C23#46611yescharacteristic
341Q8—81no—
342Q8—81no—
343Q8—161no—
344Q8—161no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—161no—
354D4—162no—
355D4#51012yes—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—162no—
374C2×C4#46611yescharacteristic
375C2×C4#46611yescharacteristic
376C2×C4#46611yescharacteristic
377C2×C4C2×#12111yescharacteristic
378C2×C4C22×D811yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C8—21no—
381C8—21no—
382C8—21no—
383C8—41no—
384C8—41no—
385C8—81no—
386C8—81no—
387C8—81no—
388C8—81no—
389C8C2×AGL(1,ℤ/8)11yescharacteristic
390C8#57511yescharacteristic
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22C2×#46611yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4#303611yescharacteristic
415C4#306711yescharacteristic
416C4C2×#51011yescharacteristic
417C4#341411yescharacteristic
418C2—41no—
419C2—42no—
420C2—81no—
421C2—321no—
422C2#1925511yescharacteristic
423C2#3360611yescharacteristic
424C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#19258C2
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. #33751C2semidirect
2. D4×M32C2semidirect
4. #19256C2semidirect
5. #12473C2semidirect
6. #12472C2semidirect
7. #12469C2semidirect
8. #12468C2semidirect
9. #12455C2semidirect
10. #12452C2semidirect
11. #12435C2semidirect
12. #12432C2semidirect
13. #12025C2semidirect
14. #11987C2semidirect
15. #11986C2semidirect
22. #2252C22semidirect
23. #2248C22semidirect
24. #2241C22semidirect
25. #2240C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1365C22semidirect
62. #1365C22semidirect
63. #1362C22semidirect
64. #1362C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect

Maximal subgroups

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

subgroupindex
#337512
D4×M322
#192582
#192562
#124732
#124722
#124692
#124682
#124552
#124522
#124352
#124322
#120252
#119872
#119862