Finite Group Databasebeta

← #7624012

Subgroups — #7624012

Order 512.

1457 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61436C211yesmaximal, characteristic, complemented
2D8×#41C211yesmaximal, characteristic, complemented
3#33958C211yesmaximal, characteristic, complemented
4Q8×SD32C211yesmaximal, characteristic, complemented
5#33929C211yesmaximal, characteristic, complemented
6#33925C211yesmaximal, characteristic, complemented
7#33921C213yesmaximal, complemented
8#33899C213yesmaximal, complemented
9#33892C213yesmaximal, complemented
10#33840C213yesmaximal, complemented
11#33748C213yesmaximal, complemented
12#33675C213yesmaximal, complemented
13#33669C213yesmaximal, complemented
14#33485C211yesmaximal, characteristic, complemented
15C4×#509C213yesmaximal, complemented
16#3483—21no—
17D4×#41—21no—
18C2×#509C2213yes—
19#3391C2211yescharacteristic
20#3391C2211yescharacteristic, complemented
21Q8×Q16C2211yescharacteristic, complemented
22Q8×D8C2211yescharacteristic
23#3362—21no—
24#3358—21no—
25#3341C2213yescomplemented
26#3334C2213yes—
27#3305—23no—
28#3294—23no—
29#3287C2213yescomplemented
30D4×D8C2213yes—
31#3169C2213yes—
32#3169C2213yescomplemented
33#3106—23no—
34#3100—23no—
35C8×#41C2211yescharacteristic
36C4×#135C2213yescomplemented
37C2×C4×D8C2213yes—
38#2254C2213yescomplemented
39#2246C2213yescomplemented
40#2240C2213yescomplemented
41#2240C2213yescomplemented
42#2234C2216yescomplemented
43#2233C2216yescomplemented
44#2226C2213yes—
45#2226C2213yescomplemented
46#2225C2213yescomplemented
47#2224C2213yescomplemented
48#2218C2216yescomplemented
49#2218C2216yescomplemented
50#2207C2213yescomplemented
51#2206C2213yes—
52#2206C2213yescomplemented
53#2206C2213yescomplemented
54#2186C2216yescomplemented
55#2184C2216yes—
56Q8×C16C2211yescharacteristic, complemented
57Q8×C16C2211yescharacteristic, complemented
58#2177C2213yes—
59#2177C2216yescomplemented
60#2177C2213yescomplemented
61#2175C2211yescharacteristic
62#2175C2213yescomplemented
63C4×SD32C2213yes—
64C4×SD32C2213yescomplemented
65C4×D16C2213yes—
66C4×D16C2213yescomplemented
67#2169C2213yescomplemented
68#2169C2213yescomplemented
69#2163C2213yescomplemented
70#2159C2213yescomplemented
71#2140C2213yes—
72#2139C2213yescomplemented
73C4×M32C2213yescomplemented
74C22×#41—41no—
75C2×#135C2313yes—
76C22×D8C2313yes—
77Q82—21no—
78#555—23no—
79#555—43no—
80#550—21no—
81#550—21no—
82#550—23no—
83#550—41no—
84D4×Q8—21no—
85#548—41no—
86#547—23no—
87#546—43no—
88D42—23no—
89#532—23no—
90#532—23no—
91#532—43no—
92#525—43no—
93C4×#41—23no—
94C4×#41—41no—
95C4×#41D411yescharacteristic
96C2×C4×D4—23no—
97#509—212no—
98C2×SD32C2313yes—
99C2×SD32C2311yescharacteristic
100C2×D16C2313yes—
101C2×D16C2313yes—
102C2×D16C2316yes—
103C2×M32C2313yes—
104C2×M32C2313yes—
105#500C2313yes—
106#500C2313yes—
107#493C2313yes—
108#493C2313yes—
109#486—23no—
110#486—23no—
111#482—26no—
112#480—26no—
113#477—23no—
114#476—23no—
115#474—23no—
116#474—23no—
117#466—23no—
118#466—23no—
119#466C2316yes—
120#466C2316yes—
121#460—23no—
122#459—23no—
123#459—23no—
124#459—23no—
125#449—26no—
126#447—26no—
127C8×Q8D411yescharacteristic
128C8×Q8C2311yescharacteristic
129C4×Q16C2313yes—
130C4×Q16C2311yescharacteristic
131C4×SD16—23no—
132C4×SD16—23no—
133C4×D8—23no—
134C4×D8—23no—
135C4×D8—23no—
136C4×D8—23no—
137C4×D8C2313yes—
138C4×D8C2313yes—
139C4×D8C2313yes—
140C4×D8C2313yes—
141C4×D8C2313yes—
142C4×D8C2311yescharacteristic
143C8×D4D413yes—
144C8×D4C2313yes—
145#431D413yes—
146#427C2313yes—
147C2×#107C2313yes—
148#416—23no—
149C2×#102—23no—
150C2×C4×C8C2313yes—
151#367C2313yes—
152#367C2313yes—
153#367C2316yes—
154#366C2311yescharacteristic
155#366C2313yes—
156C4⋊C16C2313yes—
157C4⋊C16C2313yes—
158#358C2311yescharacteristic
159#358C2313yes—
160#357C2313yes—
161#357C2313yes—
162#357C2311yescharacteristic
163#357C2313yes—
164#357C2313yes—
165#357C2313yes—
166#357C2316yes—
167#357C2316yes—
168C22⋊C16C2316yes—
169#346C2311yescharacteristic
170#346C2313yes—
171C4×C16C2313yes—
172C4×C16C2313yes—
173C2×#41—23no—
174C2×#41—41no—
175C2×#41—41no—
176C2×#41—43no—
177C2×#41—43no—
178C2×#41—43no—
179C2×#41—88no—
180C2×#41D811yescharacteristic
181C22×Q8—41no—
182C22×D4—23no—
183C22×D4—43no—
184C23×C4—43no—
185#135—26no—
186#135—43no—
187C2×Q16C2411yescharacteristic
188C2×SD16—23no—
189C2×D8—26no—
190C2×D8—23no—
191C2×D8—23no—
192C2×D8—23no—
193C2×D8—43no—
194C2×D8C2413yes—
195C2×D8C2413yes—
196C2×D8C2411yescharacteristic
197C22×C8C2×D413yes—
198C22×C8C2413yes—
199C4⋊Q8—23no—
200C4⋊Q8—23no—
201C4⋊Q8—43no—
202#127—23no—
203#127—23no—
204#127—43no—
205#124—46no—
206#124—43no—
207#124—43no—
208#123—46no—
209C22⋊Q8—43no—
210C22⋊Q8—43no—
211#121—26no—
212#121—26no—
213#121—43no—
214#121—43no—
215#121—43no—
216#121—43no—
217#121—43no—
218#121—43no—
219#121—46no—
220#120—46no—
221C4×Q8—21no—
222C4×Q8—23no—
223C4×Q8—41no—
224C4×Q8D811yescharacteristic
225C4×Q8C2×D411yescharacteristic
226C4×D4—21no—
227C4×D4—23no—
228C4×D4—23no—
229C4×D4—23no—
230C4×D4—23no—
231C4×D4—23no—
232C4×D4—43no—
233C4×D4—43no—
234C4×D4—43no—
235C4×D4—43no—
236C4×D4—43no—
237C4×D4—43no—
238C4×D4—43no—
239C4×D4D813yes—
240C4×D4C2×D413yes—
241#117—23no—
242#117—43no—
243#117—43no—
244#117D813yes—
245C2×C4⋊C4—23no—
246C2×C22⋊C4—43no—
247C2×C42—43no—
248C2×C42C2×D413yes—
249SD32—26no—
250SD32#4114yescomplemented
251D16—26no—
252D16—26no—
253D16—46no—
254M32—26no—
255M32—43no—
256C2×C16C2411yescharacteristic
257C2×C16C2413yes—
258C2×C16C2413yes—
259C2×C16C2411yescharacteristic
260#107—23no—
261#107—23no—
262#107—23no—
263#107C2411yescharacteristic
264#107C2413yes—
265#107C2411yescharacteristic
266#107C2413yes—
267#106—23no—
268C4⋊C8C2×D413yes—
269C4⋊C8C2×D413yes—
270#103—21no—
271#103—23no—
272#102—21no—
273#102—23no—
274#102—23no—
275#102—23no—
276#102—23no—
277#102—23no—
278#102—26no—
279#102—26no—
280C22⋊C8C2×D416yes—
281C4×C8C2×D413yes—
282C4×C8C2×D413yes—
283C4×C8C2413yes—
284C4×C8C2411yescharacteristic
285C24—43no—
286#41—24no—
287#41—46no—
288#41—83no—
289#41—812no—
290#41—812no—
291#41—84no—
292#41—84no—
293#41—166no—
294C2×Q8—21no—
295C2×Q8—43no—
296C2×Q8—43no—
297C2×Q8—84no—
298C2×Q8C2×D811yescharacteristic
299C2×D4—21no—
300C2×D4—23no—
301C2×D4—23no—
302C2×D4—46no—
303C2×D4—43no—
304C2×D4—46no—
305C2×D4—46no—
306C2×D4—43no—
307C2×D4—43no—
308C2×D4—43no—
309C2×D4—43no—
310C2×D4—43no—
311C2×D4—83no—
312C2×D4—812no—
313C2×D4C2×D813yes—
314C22×C4—43no—
315C22×C4—43no—
316C22×C4—43no—
317C22×C4—43no—
318C22×C4—41no—
319C22×C4—43no—
320C22×C4—43no—
321C22×C4—43no—
322C22×C4—812no—
323C22×C4C2×D813yes—
324C22×C4C2×D813yes—
325C22×C4C22×D413yes—
326Q16—21no—
327Q16C2×#4112yescomplemented
328SD16—46no—
329D8—23no—
330D8—23no—
331D8—21no—
332D8—23no—
333D8—23no—
334D8—46no—
335D8—83no—
336D8—83no—
337D8C2×#4112yes—
338C2×C8—23no—
339C2×C8—23no—
340C2×C8—43no—
341C2×C8C22×D411yescharacteristic
342C2×C8C22×D413yes—
343C2×C8C22×D413yes—
344C2×C8C2511yescharacteristic
345C4⋊C4—21no—
346C4⋊C4—23no—
347C4⋊C4—21no—
348C4⋊C4—23no—
349C4⋊C4—43no—
350C4⋊C4—41no—
351C4⋊C4—43no—
352C4⋊C4—43no—
353C4⋊C4—43no—
354C4⋊C4C2×D813yes—
355C4⋊C4C2×D813yes—
356C22⋊C4—43no—
357C22⋊C4—41no—
358C22⋊C4—43no—
359C22⋊C4—43no—
360C22⋊C4—43no—
361C22⋊C4—43no—
362C22⋊C4—43no—
363C22⋊C4—46no—
364C22⋊C4—46no—
365C22⋊C4C2×D816yes—
366C42—41no—
367C42—43no—
368C42—43no—
369C42C2×D813yes—
370C42C2×D813yes—
371C42C22×D411yescharacteristic
372C42C22×D413yes—
373C16—23no—
374C16—21no—
375C16—23no—
376C16C2×#4112yescomplemented
377C23—41no—
378C23—43no—
379C23—43no—
380C23—86no—
381C23—86no—
382C23—163no—
383C23C22×D813yes—
384Q8—22no—
385Q8—41no—
386Q8—86no—
387Q8—163no—
388Q8#50914yescomplemented
389D4—26no—
390D4—22no—
391D4—41no—
392D4—43no—
393D4—43no—
394D4—43no—
395D4—43no—
396D4—86no—
397D4—86no—
398D4—812no—
399D4—86no—
400D4—86no—
401D4—163no—
402D4—163no—
403D4—166no—
404C2×C4—23no—
405C2×C4—26no—
406C2×C4—23no—
407C2×C4—43no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—43no—
411C2×C4—41no—
412C2×C4—43no—
413C2×C4—86no—
414C2×C4—82no—
415C2×C4—86no—
416C2×C4—86no—
417C2×C4—86no—
418C2×C4—86no—
419C2×C4—86no—
420C2×C4—86no—
421C2×C4C22×D811yescharacteristic
422C2×C4C22×D813yes—
423C2×C4C22×D811yescharacteristic
424C2×C4C22×D813yes—
425C2×C4C22×D813yes—
426C2×C4C23×D411yescharacteristic
427C8—21no—
428C8—23no—
429C8—23no—
430C8C22×#4111yescharacteristic
431C8#58511yescharacteristic
432C22—23no—
433C22—23no—
434C22—23no—
435C22—42no—
436C22—81no—
437C22—81no—
438C22—83no—
439C22—83no—
440C22—83no—
441C22—83no—
442C22—83no—
443C22—83no—
444C22—1612no—
445C22C23×D811yescharacteristic
446C4—21no—
447C4—21no—
448C4—23no—
449C4—23no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—83no—
454C4—83no—
455C4C2×#50916yes—
456C4D4×#4111yescharacteristic
457C4#358411yescharacteristic
458C2—43no—
459C2—82no—
460C2—163no—
461C2—163no—
462C2D8×#4111yescharacteristic
463C2C22×#50911yescharacteristic
464C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61436C2semidirect
2. D8×#41C2semidirect
3. #33958C2semidirect
4. Q8×SD32C2semidirect
5. #33929C2semidirect
6. #33925C2semidirect
7. #33921C2semidirect
8. #33899C2semidirect
9. #33892C2semidirect
10. #33840C2semidirect
11. #33748C2semidirect
12. #33675C2semidirect
13. #33669C2semidirect
14. #33485C2semidirect
15. C4×#509C2semidirect
20. #3391C22semidirect
21. Q8×Q16C22semidirect
25. #3341C22semidirect
29. #3287C22semidirect
32. #3169C22semidirect
36. C4×#135C22semidirect
38. #2254C22semidirect
39. #2246C22semidirect
40. #2240C22semidirect
41. #2240C22semidirect
42. #2234C22semidirect
43. #2233C22semidirect
45. #2226C22semidirect
46. #2225C22semidirect
47. #2224C22semidirect
48. #2218C22semidirect
49. #2218C22semidirect
50. #2207C22semidirect
52. #2206C22semidirect
53. #2206C22semidirect
54. #2186C22semidirect
56. Q8×C16C22semidirect
57. Q8×C16C22semidirect
59. #2177C22semidirect
60. #2177C22semidirect
62. #2175C22semidirect
64. C4×SD32C22semidirect
66. C4×D16C22semidirect
67. #2169C22semidirect
68. #2169C22semidirect
69. #2163C22semidirect
70. #2159C22semidirect
72. #2139C22semidirect
73. C4×M32C22semidirect
250. SD32#41semidirect
327. Q16C2×#41semidirect
376. C16C2×#41semidirect
388. Q8#509semidirect

Maximal subgroups

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

subgroupindex
#614362
D8×#412
#339582
Q8×SD322
#339292
#339252
#339212
#338992
#338922
#338402
#337482
#336752
#336692
#334852
C4×#5092