Finite Group Databasebeta

← #491238

Subgroups — #491238

Order 512.

454 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33776C211yesmaximal, characteristic, complemented
2#33488C211yesmaximal, characteristic, complemented
3#19256C211yesmaximal, characteristic, complemented
4#19101C211yesmaximal, characteristic
5#12473C211yesmaximal, characteristic, complemented
6#12472C211yesmaximal, characteristic, complemented
7#12469C211yesmaximal, characteristic, complemented
8#12468C211yesmaximal, characteristic, complemented
9#12435C211yesmaximal, characteristic, complemented
10#12434C211yesmaximal, characteristic, complemented
11#12432C211yesmaximal, characteristic, complemented
12#12431C211yesmaximal, characteristic, complemented
13#12036C211yesmaximal, characteristic, complemented
14#11966C211yesmaximal, characteristic, complemented
15#11953C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3147C2211yescharacteristic
18#3100—21no—
19#3077—21no—
20C2×C8×D4C2211yescharacteristic
21#2248C2211yescharacteristic, complemented
22#2247C2211yescharacteristic, complemented
23#2241C2211yescharacteristic, complemented
24#2240C2211yescharacteristic, complemented
25#2211—21no—
26#2211—21no—
27#2189—21no—
28#2189—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#1683—21no—
40#1683—21no—
41#1675C2211yescharacteristic
42#1675C2211yescharacteristic
43#1669C2211yescharacteristic
44#1669C2211yescharacteristic
45#1669C2211yescharacteristic
46#1669C2211yescharacteristic
47#1667C2211yescharacteristic
48#1667C2211yescharacteristic
49#1665—21no—
50#1665—21no—
51#1567C2211yescharacteristic
52#1564C2211yescharacteristic
53#1550—21no—
54#1540—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, complemented
61#1362C2211yescharacteristic, complemented
62#1362C2211yescharacteristic, complemented
63#1362C2211yescharacteristic, complemented
64#1362C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C2×#135—21no—
68C2×#135—21no—
69C23×C8—21no—
70#533—21no—
71#532—21no—
72C2×C4×D4D411yescharacteristic
73C2×Q32—21no—
74C2×Q32—21no—
75C2×SD32—21no—
76C2×SD32—21no—
77C2×M32—21no—
78C2×M32—21no—
79C22×C16—21no—
80C22×C16—21no—
81#500C2311yescharacteristic
82#495—21no—
83#495—21no—
84#494C2311yescharacteristic
85#494C2311yescharacteristic
86#493C2311yescharacteristic
87#492—21no—
88#487—21no—
89#480—41no—
90#478—41no—
91#478—41no—
92#476—21no—
93#476—21no—
94#474—21no—
95#474—21no—
96#474—21no—
97#474—21no—
98#467—21no—
99#467—21no—
100#459—21no—
101#459—21no—
102#447—41no—
103C4×Q16—21no—
104C4×Q16—21no—
105C4×D8—41no—
106C4×D8—41no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4D411yescharacteristic
110C8×D4D411yescharacteristic
111C8×D4C2311yescharacteristic
112C8×D4C2311yescharacteristic
113#431—21no—
114#428—41no—
115#423—21no—
116C2×C4⋊C8D411yescharacteristic
117#418—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#333—21no—
145#333—21no—
146#332—21no—
147#332—21no—
148#332—21no—
149#332—21no—
150#331—21no—
151#331—21no—
152C2×#41—41no—
153C22×D4D811yescharacteristic
154C23×C4—21no—
155#135—81no—
156#135—81no—
157#135—82no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×SD16—41no—
163C2×SD16—41no—
164C2×D8—21no—
165C2×D8—21no—
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—81no—
175C22×C8—81no—
176C22×C8C2×D411yescharacteristic
177C22×C8#4111yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181#127—21no—
182#125—41no—
183#124—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186#121—41no—
187#121—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4D811yescharacteristic
195C4×D4D811yescharacteristic
196C4×D4C2×D411yescharacteristic
197C4×D4C2×D411yescharacteristic
198#117—41no—
199C2×C4⋊C4D811yescharacteristic
200C2×C22⋊C4—21no—
201C2×C42C2×D411yescharacteristic
202Q32—81no—
203Q32—81no—
204SD32—81no—
205SD32—81no—
206M32—81no—
207M32—81no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—81no—
211C2×C16—81no—
212C2×C16C2×D411yescharacteristic
213C2×C16C2×D411yescharacteristic
214C2×C16C2×D411yescharacteristic
215C2×C16C2×D411yescharacteristic
216#107—21no—
217#107—21no—
218#107—41no—
219#107—41no—
220#107—41no—
221#107—41no—
222#106—41no—
223#106—41no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—41no—
230C4⋊C8C2×D411yescharacteristic
231C4⋊C8C2×D411yescharacteristic
232#103—41no—
233#103—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—
247C4×C8—41no—
248C4×C8C2×D411yescharacteristic
249C4×C8C2×D411yescharacteristic
250C4×C8#4111yescharacteristic
251C4×C8C2411yescharacteristic
252C24—21no—
253#41—82no—
254#41—82no—
255C2×Q8—41no—
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—81no—
265C2×D4C2×D811yescharacteristic
266C2×D4C2×D811yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4#12111yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4#13511yescharacteristic
278Q16—41no—
279Q16—41no—
280Q16—41no—
281Q16—41no—
282Q16—81no—
283Q16—81no—
284SD16—82no—
285SD16—82no—
286D8—41no—
287D8—41no—
288D8—81no—
289D8—82no—
290D8—82no—
291M16—82no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—81no—
306C2×C8—81no—
307C2×C8—81no—
308C2×C8#12111yescharacteristic
309C2×C8#12111yescharacteristic
310C2×C8C22×D411yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—81no—
320C4⋊C4—81no—
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—
330C42—81no—
331C42#12111yescharacteristic
332C42C2×D811yescharacteristic
333C42#13511yescharacteristic
334C42C22×D411yescharacteristic
335C16—21no—
336C16—21no—
337C16—21no—
338C16—21no—
339C16—41no—
340C16—41no—
341C23—21no—
342C23—21no—
343C23—42no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23#46611yescharacteristic
348Q8—81no—
349Q8—81no—
350Q8—81no—
351Q8—81no—
352Q8—161no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—161no—
362D4—162no—
363D4#51012yes—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—162no—
382C2×C4#46611yescharacteristic
383C2×C4#46611yescharacteristic
384C2×C4#46611yescharacteristic
385C2×C4C2×#12111yescharacteristic
386C2×C4C22×D811yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C8—21no—
389C8—21no—
390C8—21no—
391C8—41no—
392C8—41no—
393C8—81no—
394C8—81no—
395C8—81no—
396C8—81no—
397C8#57611yescharacteristic
398C8#57611yescharacteristic
399C22—21no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—84no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22C2×#46611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#304111yescharacteristic
423C4#306811yescharacteristic
424C4C2×#51011yescharacteristic
425C4#341411yescharacteristic
426C2—41no—
427C2—42no—
428C2—81no—
429C2—321no—
430C2#1938611yescharacteristic
431C2#3360611yescharacteristic
432C2#3361711yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#19101C2
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. #33776C2semidirect
2. #33488C2semidirect
3. #19256C2semidirect
5. #12473C2semidirect
6. #12472C2semidirect
7. #12469C2semidirect
8. #12468C2semidirect
9. #12435C2semidirect
10. #12434C2semidirect
11. #12432C2semidirect
12. #12431C2semidirect
13. #12036C2semidirect
14. #11966C2semidirect
15. #11953C2semidirect
21. #2248C22semidirect
22. #2247C22semidirect
23. #2241C22semidirect
24. #2240C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1362C22semidirect
62. #1362C22semidirect
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
#337762
#334882
#192562
#191012
#124732
#124722
#124692
#124682
#124352
#124342
#124322
#124312
#120362
#119662
#119532