Finite Group Databasebeta

← #491217

Subgroups — #491217

Order 512.

445 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33704C211yesmaximal, characteristic, complemented
2#33483C211yesmaximal, characteristic, complemented
3#19361C211yesmaximal, characteristic, complemented
4#19353C211yesmaximal, characteristic
5#12480C211yesmaximal, characteristic, complemented
6#12479C211yesmaximal, characteristic, complemented
7#12476C211yesmaximal, characteristic, complemented
8#12475C211yesmaximal, characteristic, complemented
9#12472C211yesmaximal, characteristic, complemented
10#12471C211yesmaximal, characteristic, complemented
11#12468C211yesmaximal, characteristic, complemented
12#12467C211yesmaximal, characteristic, complemented
13#12025C211yesmaximal, characteristic, complemented
14#11973C211yesmaximal, characteristic, complemented
15#11961C211yesmaximal, characteristic, complemented
16#3252—21no—
17#3217—21no—
18#3160C2211yescharacteristic
19C2×#493C2211yescharacteristic
20#2966C2211yescharacteristic
21#2244—21no—
22#2244—21no—
23#2240C2211yescharacteristic
24#2240C2211yescharacteristic
25#2240C2211yescharacteristic
26#2240C2211yescharacteristic
27#2183C2211yescharacteristic, complemented
28Q8×C16C2211yescharacteristic, complemented
29#2169—21no—
30#2169C2211yescharacteristic, complemented
31#2168—21no—
32D4×C16C2211yescharacteristic, complemented
33#2164—21no—
34#2162C2211yescharacteristic
35#2140—21no—
36#2140—21no—
37#2108—21no—
38C4×M32C2211yescharacteristic
39#1692—21no—
40#1691—21no—
41#1688—21no—
42#1687—21no—
43#1676C2211yescharacteristic, complemented
44#1675C2211yescharacteristic, complemented
45#1674C2211yescharacteristic, complemented
46#1673C2211yescharacteristic, complemented
47#1668C2211yescharacteristic
48#1668C2211yescharacteristic
49#1667C2211yescharacteristic
50#1667C2211yescharacteristic
51#1568C2211yescharacteristic
52#1567C2211yescharacteristic
53#1559—21no—
54#1555—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
64#1361C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67C22×D8—21no—
68C22×D8—21no—
69#533—21no—
70C2×#127—21no—
71#529D411yescharacteristic
72C2×M32—21no—
73C2×M32D411yescharacteristic
74C2×M32D411yescharacteristic
75C22×C16—21no—
76#500C2311yescharacteristic
77#500C2311yescharacteristic
78#499—21no—
79#499—21no—
80#498—21no—
81#496—41no—
82#493—21no—
83#493—21no—
84#493—21no—
85#493C2311yescharacteristic
86#493C2311yescharacteristic
87#490—41no—
88#479—41no—
89#477—21no—
90#476—21no—
91#470—41no—
92#468—41no—
93#464—41no—
94#462—21no—
95#461—21no—
96#460—21no—
97#460—21no—
98#459—21no—
99#459—21no—
100#450—41no—
101#447—41no—
102#446D411yescharacteristic
103C8×Q8C2311yescharacteristic
104#436D411yescharacteristic
105#435—21no—
106C8×D4—21no—
107C8×D4C2311yescharacteristic
108#433—21no—
109#432—21no—
110#432D411yescharacteristic
111#427—21no—
112#427—21no—
113#423—21no—
114#405—21no—
115C2×C4×C8C2311yescharacteristic
116C4⋊C16—21no—
117C4⋊C16—21no—
118C4⋊C16C2311yescharacteristic
119C4⋊C16C2311yescharacteristic
120#357—21no—
121#357—21no—
122#357—21no—
123#357—21no—
124#357—21no—
125#357—21no—
126#357—21no—
127#357—21no—
128C22⋊C16—21no—
129C22⋊C16—21no—
130C22⋊C16—21no—
131#346—21no—
132#346D411yescharacteristic
133#346D411yescharacteristic
134C4×C16—21no—
135C4×C16—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#331—21no—
149#331—21no—
150#331—21no—
151#331—21no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—41no—
155C2×Q16—41no—
156C2×SD16—41no—
157C2×SD16—41no—
158C2×SD16—41no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—42no—
164C2×D8—42no—
165C2×D8—41no—
166C2×D8—41no—
167C2×D8—41no—
168C2×D8—41no—
169C2×D8—82no—
170C2×M16—21no—
171C2×M16—41no—
172C22×C8—21no—
173C22×C8C2×D411yescharacteristic
174C22×C8#4111yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177#127—21no—
178#127—21no—
179#127—41no—
180#127—41no—
181#127—41no—
182#126—21no—
183#125—41no—
184#125D811yescharacteristic
185#124D811yescharacteristic
186#123—21no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8D811yescharacteristic
190#121D811yescharacteristic
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8C2×D411yescharacteristic
194C4×D4—21no—
195C4×D4C2×D411yescharacteristic
196#117—21no—
197#117—41no—
198C2×C42C2×D411yescharacteristic
199M32—41no—
200M32—41no—
201M32—42no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—42no—
205C2×C16C2×D411yescharacteristic
206C2×C16C2×D411yescharacteristic
207C2×C16C2×D411yescharacteristic
208C2×C16C2×D411yescharacteristic
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—41no—
214#106—41no—
215#106—41no—
216#106—41no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—41no—
224C4⋊C8C2×D411yescharacteristic
225C4⋊C8C2×D411yescharacteristic
226#103—41no—
227#103—41no—
228#103—41no—
229#102—41no—
230#102—41no—
231#102—41no—
232#102—41no—
233#102—41no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—41no—
238#97—21no—
239#97—21no—
240#97—41no—
241C4×C8—21no—
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246C24—81no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8C2×D811yescharacteristic
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—81no—
260C2×D4—81no—
261C2×D4—81no—
262C2×D4—161no—
263C2×D4—161no—
264C2×D4—162no—
265C2×D4C2×D811yescharacteristic
266C22×C4—21no—
267C22×C4#12111yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4#13511yescharacteristic
270Q16—161no—
271SD16—161no—
272SD16—161no—
273SD16—161no—
274D8—41no—
275D8—41no—
276D8—41no—
277D8—41no—
278D8—81no—
279D8—81no—
280D8—82no—
281D8—82no—
282D8—161no—
283D8—161no—
284M16—42no—
285M16—161no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8#12111yescharacteristic
298C2×C8#12111yescharacteristic
299C2×C8C22×D411yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4C2×D811yescharacteristic
312C4⋊C4C2×D811yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—81no—
319C42—21no—
320C42—81no—
321C42#12111yescharacteristic
322C42C2×D811yescharacteristic
323C42#13511yescharacteristic
324C42C22×D411yescharacteristic
325C16—21no—
326C16—21no—
327C16—21no—
328C16—21no—
329C16—41no—
330C16—41no—
331C23—21no—
332C23—81no—
333C23—81no—
334C23—162no—
335C23—162no—
336C23—162no—
337C23#46611yescharacteristic
338Q8—21no—
339Q8—21no—
340Q8—81no—
341Q8—81no—
342Q8—161no—
343D4—21no—
344D4—21no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—161no—
355D4—161no—
356D4—162no—
357D4—162no—
358D4—162no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4#46611yescharacteristic
372C2×C4#46611yescharacteristic
373C2×C4#46611yescharacteristic
374C2×C4C2×#12111yescharacteristic
375C2×C4C22×D811yescharacteristic
376C2×C4C2×#13511yescharacteristic
377C8—21no—
378C8—21no—
379C8—41no—
380C8—41no—
381C8—81no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8#47112yes—
386C8#57511yescharacteristic
387C8#57511yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—324no—
401C22C2×#46611yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—161no—
410C4—161no—
411C4#304011yescharacteristic
412C4C2×#47111yescharacteristic
413C4#341511yescharacteristic
414C4#341611yescharacteristic
415C2—41no—
416C2—81no—
417C2—321no—
418C2—321no—
419C2#1927611yescharacteristic
420C2#3361211yescharacteristic
421C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#19353C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33704C2semidirect
2. #33483C2semidirect
3. #19361C2semidirect
5. #12480C2semidirect
6. #12479C2semidirect
7. #12476C2semidirect
8. #12475C2semidirect
9. #12472C2semidirect
10. #12471C2semidirect
11. #12468C2semidirect
12. #12467C2semidirect
13. #12025C2semidirect
14. #11973C2semidirect
15. #11961C2semidirect
27. #2183C22semidirect
28. Q8×C16C22semidirect
30. #2169C22semidirect
32. D4×C16C22semidirect
43. #1676C22semidirect
44. #1675C22semidirect
45. #1674C22semidirect
46. #1673C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1364C22semidirect
62. #1364C22semidirect

Maximal subgroups

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

subgroupindex
#337042
#334832
#193612
#193532
#124802
#124792
#124762
#124752
#124722
#124712
#124682
#124672
#120252
#119732
#119612