Finite Group Databasebeta

← #7624013

Subgroups — #7624013

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61435C211yesmaximal, characteristic, complemented
2D8×#41C211yesmaximal, characteristic, complemented
3#33959C211yesmaximal, characteristic, complemented
4Q8×D16C211yesmaximal, characteristic, complemented
5#33926C211yesmaximal, characteristic, complemented
6#33925C211yesmaximal, characteristic, complemented
7#33920C213yesmaximal, complemented
8#33897C213yesmaximal, complemented
9#33891C213yesmaximal, complemented
10#33838C213yesmaximal, complemented
11#33751C213yesmaximal, complemented
12#33675C213yesmaximal, complemented
13#33670C213yesmaximal, complemented
14#33485C211yesmaximal, characteristic, complemented
15C4×#509C213yesmaximal, complemented
16#3482—21no—
17D4×#41—21no—
18C2×#509C2213yes—
19#3395C2211yescharacteristic, complemented
20#3391C2211yescharacteristic
21Q8×D8C2211yescharacteristic
22Q8×D8C2211yescharacteristic, complemented
23#3359—21no—
24#3358—21no—
25#3342C2213yescomplemented
26#3334C2213yes—
27#3302—23no—
28#3294—23no—
29#3283C2213yescomplemented
30D4×D8C2213yes—
31#3170C2213yescomplemented
32#3169C2213yes—
33#3107—23no—
34#3100—23no—
35C8×#41C2211yescharacteristic
36C4×#135C2213yescomplemented
37C2×C4×D8C2213yes—
38#2252C2213yescomplemented
39#2248C2213yescomplemented
40#2241C2213yescomplemented
41#2240C2213yescomplemented
42#2236C2216yescomplemented
43#2234C2216yescomplemented
44#2225C2213yescomplemented
45#2224C2213yes—
46#2224C2213yescomplemented
47#2224C2213yescomplemented
48#2220C2216yescomplemented
49#2220C2216yescomplemented
50#2208C2213yes—
51#2208C2213yescomplemented
52#2207C2213yescomplemented
53#2206C2213yescomplemented
54#2187C2216yes—
55#2186C2216yescomplemented
56Q8×C16C2211yescharacteristic, complemented
57Q8×C16C2211yescharacteristic, complemented
58#2177C2211yescharacteristic
59#2177C2213yescomplemented
60#2175C2213yes—
61#2175C2216yescomplemented
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—
77#557—21no—
78#556—23no—
79#552—43no—
80#550—21no—
81D4×Q8—21no—
82D4×Q8—21no—
83#548—41no—
84#548—41no—
85#547—23no—
86#547—23no—
87#546—43no—
88D42—23no—
89#533—23no—
90#532—23no—
91#529—43no—
92#525—43no—
93C4×#41—23no—
94C4×#41D411yescharacteristic
95C2×C4×Q8—41no—
96C2×C4×D4—23no—
97#509—212no—
98C2×SD32C2313yes—
99C2×SD32C2313yes—
100C2×SD32C2316yes—
101C2×D16C2313yes—
102C2×D16C2311yescharacteristic
103C2×M32C2313yes—
104C2×M32C2313yes—
105#500C2313yes—
106#500C2313yes—
107#494C2313yes—
108#493C2313yes—
109#487—23no—
110#486—23no—
111#485—26no—
112#480—26no—
113#475—23no—
114#474—23no—
115#474—23no—
116#474—23no—
117#467—23no—
118#467C2316yes—
119#466—23no—
120#466C2316yes—
121#462—23no—
122#461—23no—
123#459—23no—
124#459—23no—
125#452—26no—
126#447—26no—
127C8×Q8D411yescharacteristic
128C8×Q8C2311yescharacteristic
129C4×Q16—23no—
130C4×Q16C2313yes—
131C4×Q16C2313yes—
132C4×SD16—23no—
133C4×SD16—23no—
134C4×D8—23no—
135C4×D8—23no—
136C4×D8—23no—
137C4×D8C2313yes—
138C4×D8C2313yes—
139C4×D8C2313yes—
140C4×D8C2313yes—
141C4×D8C2311yescharacteristic
142C4×D8C2311yescharacteristic
143C8×D4D413yes—
144C8×D4C2313yes—
145#431D413yes—
146#427C2313yes—
147C2×#107C2313yes—
148#416—23no—
149C2×#102—23no—
150C2×C4×C8C2313yes—
151#367C2311yescharacteristic
152#367C2313yes—
153#366C2313yes—
154#366C2313yes—
155#366C2316yes—
156C4⋊C16C2313yes—
157C4⋊C16C2313yes—
158#358C2313yes—
159#358C2313yes—
160#358C2316yes—
161#357C2311yescharacteristic
162#357C2313yes—
163#357C2311yescharacteristic
164#357C2313yes—
165#357C2313yes—
166#357C2313yes—
167#357C2316yes—
168C22⋊C16C2316yes—
169#346C2311yescharacteristic
170#346C2313yes—
171C4×C16C2313yes—
172C4×C16C2313yes—
173C2×#41—23no—
174C2×#41—41no—
175C2×#41—43no—
176C2×#41—43no—
177C2×#41—88no—
178C2×#41D811yescharacteristic
179C22×Q8—41no—
180C22×Q8—41no—
181C22×D4—23no—
182C22×D4—43no—
183C23×C4—43no—
184#135—26no—
185#135—43no—
186C2×Q16—23no—
187C2×Q16C2413yes—
188C2×SD16—23no—
189C2×D8—26no—
190C2×D8—23no—
191C2×D8—23no—
192C2×D8—43no—
193C2×D8C2413yes—
194C2×D8C2411yescharacteristic
195C2×D8C2411yescharacteristic
196C22×C8C2×D413yes—
197C22×C8C2413yes—
198C4⋊Q8—23no—
199C4⋊Q8—23no—
200C4⋊Q8—23no—
201#127—23no—
202#126—46no—
203#125—43no—
204#125—43no—
205#124—43no—
206#124—43no—
207#123—46no—
208#123—46no—
209C22⋊Q8—26no—
210C22⋊Q8—43no—
211C22⋊Q8—43no—
212C22⋊Q8—43no—
213C22⋊Q8—43no—
214#121—26no—
215#121—43no—
216#121—43no—
217#121—43no—
218#121—43no—
219#120—46no—
220C4×Q8—23no—
221C4×Q8—23no—
222C4×Q8—41no—
223C4×Q8—43no—
224C4×Q8—43no—
225C4×Q8D811yescharacteristic
226C4×Q8C2×D411yescharacteristic
227C4×D4—21no—
228C4×D4—23no—
229C4×D4—23no—
230C4×D4—23no—
231C4×D4—21no—
232C4×D4—23no—
233C4×D4—43no—
234C4×D4—43no—
235C4×D4—43no—
236C4×D4—43no—
237C4×D4—43no—
238C4×D4D813yes—
239C4×D4C2×D413yes—
240#117—23no—
241#117—43no—
242#117—43no—
243#117D813yes—
244C2×C4⋊C4—23no—
245C2×C4⋊C4—43no—
246C2×C22⋊C4—43no—
247C2×C42—43no—
248C2×C42C2×D413yes—
249SD32—26no—
250SD32—26no—
251SD32—46no—
252D16—26no—
253D16#4114yescomplemented
254M32—26no—
255M32—43no—
256C2×C16C2411yescharacteristic
257C2×C16C2413yes—
258C2×C16C2413yes—
259C2×C16C2411yescharacteristic
260#107—23no—
261#107—23no—
262#107—23no—
263#107C2413yes—
264#107C2411yescharacteristic
265#107C2411yescharacteristic
266#107C2413yes—
267#106—23no—
268C4⋊C8C2×D413yes—
269C4⋊C8C2×D413yes—
270#103—23no—
271#103—23no—
272#103—26no—
273#102—21no—
274#102—23no—
275#102—23no—
276#102—23no—
277#102—21no—
278#102—23no—
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—
292C2×Q8—23no—
293C2×Q8—43no—
294C2×Q8—43no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8C2×D811yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—23no—
301C2×D4—46no—
302C2×D4—43no—
303C2×D4—46no—
304C2×D4—43no—
305C2×D4—43no—
306C2×D4—43no—
307C2×D4—83no—
308C2×D4—812no—
309C2×D4C2×D813yes—
310C22×C4—43no—
311C22×C4—41no—
312C22×C4—43no—
313C22×C4—43no—
314C22×C4—41no—
315C22×C4—43no—
316C22×C4—43no—
317C22×C4—43no—
318C22×C4—812no—
319C22×C4C2×D813yes—
320C22×C4C2×D813yes—
321C22×C4C22×D413yes—
322Q16—23no—
323Q16—23no—
324Q16—83no—
325SD16—46no—
326D8—23no—
327D8—21no—
328D8—21no—
329D8—23no—
330D8—46no—
331D8—83no—
332D8C2×#4112yes—
333D8C2×#4112yescomplemented
334C2×C8—23no—
335C2×C8—23no—
336C2×C8—43no—
337C2×C8C22×D411yescharacteristic
338C2×C8C22×D413yes—
339C2×C8C22×D413yes—
340C2×C8C2511yescharacteristic
341C4⋊C4—21no—
342C4⋊C4—23no—
343C4⋊C4—21no—
344C4⋊C4—23no—
345C4⋊C4—43no—
346C4⋊C4—43no—
347C4⋊C4—46no—
348C4⋊C4—43no—
349C4⋊C4—43no—
350C4⋊C4—43no—
351C4⋊C4—43no—
352C4⋊C4—43no—
353C4⋊C4C2×D813yes—
354C4⋊C4C2×D813yes—
355C22⋊C4—43no—
356C22⋊C4—46no—
357C22⋊C4—41no—
358C22⋊C4—43no—
359C22⋊C4—43no—
360C22⋊C4—43no—
361C22⋊C4—41no—
362C22⋊C4—43no—
363C22⋊C4—46no—
364C22⋊C4C2×D816yes—
365C42—43no—
366C42—43no—
367C42—43no—
368C42C2×D813yes—
369C42C2×D813yes—
370C42C22×D411yescharacteristic
371C42C22×D413yes—
372C16—23no—
373C16—21no—
374C16—23no—
375C16C2×#4112yescomplemented
376C23—41no—
377C23—41no—
378C23—43no—
379C23—86no—
380C23—86no—
381C23—163no—
382C23C22×D813yes—
383Q8—43no—
384Q8—43no—
385Q8—86no—
386Q8—86no—
387Q8#50914yescomplemented
388D4—26no—
389D4—22no—
390D4—22no—
391D4—41no—
392D4—41no—
393D4—43no—
394D4—43no—
395D4—86no—
396D4—86no—
397D4—812no—
398D4—86no—
399D4—163no—
400C2×C4—23no—
401C2×C4—26no—
402C2×C4—23no—
403C2×C4—43no—
404C2×C4—41no—
405C2×C4—43no—
406C2×C4—43no—
407C2×C4—41no—
408C2×C4—43no—
409C2×C4—86no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—86no—
413C2×C4—86no—
414C2×C4—86no—
415C2×C4—86no—
416C2×C4—86no—
417C2×C4C22×D811yescharacteristic
418C2×C4C22×D813yes—
419C2×C4C22×D811yescharacteristic
420C2×C4C22×D813yes—
421C2×C4C22×D813yes—
422C2×C4C23×D411yescharacteristic
423C8—21no—
424C8—23no—
425C8—23no—
426C8C22×#4111yescharacteristic
427C8#58511yescharacteristic
428C22—23no—
429C22—23no—
430C22—23no—
431C22—42no—
432C22—42no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—83no—
438C22—83no—
439C22—83no—
440C22—1612no—
441C22C23×D811yescharacteristic
442C4—21no—
443C4—21no—
444C4—23no—
445C4—23no—
446C4—81no—
447C4—83no—
448C4—83no—
449C4—81no—
450C4—83no—
451C4C2×#50916yes—
452C4D4×#4111yescharacteristic
453C4#358411yescharacteristic
454C2—43no—
455C2—82no—
456C2—82no—
457C2—163no—
458C2D8×#4111yescharacteristic
459C2C22×#50911yescharacteristic
460C2#6263711yescharacteristic
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. #61435C2semidirect
2. D8×#41C2semidirect
3. #33959C2semidirect
4. Q8×D16C2semidirect
5. #33926C2semidirect
6. #33925C2semidirect
7. #33920C2semidirect
8. #33897C2semidirect
9. #33891C2semidirect
10. #33838C2semidirect
11. #33751C2semidirect
12. #33675C2semidirect
13. #33670C2semidirect
14. #33485C2semidirect
15. C4×#509C2semidirect
19. #3395C22semidirect
22. Q8×D8C22semidirect
25. #3342C22semidirect
29. #3283C22semidirect
31. #3170C22semidirect
36. C4×#135C22semidirect
38. #2252C22semidirect
39. #2248C22semidirect
40. #2241C22semidirect
41. #2240C22semidirect
42. #2236C22semidirect
43. #2234C22semidirect
44. #2225C22semidirect
46. #2224C22semidirect
47. #2224C22semidirect
48. #2220C22semidirect
49. #2220C22semidirect
51. #2208C22semidirect
52. #2207C22semidirect
53. #2206C22semidirect
55. #2186C22semidirect
56. Q8×C16C22semidirect
57. Q8×C16C22semidirect
59. #2177C22semidirect
61. #2175C22semidirect
62. #2175C22semidirect
64. C4×SD32C22semidirect
66. C4×D16C22semidirect
67. #2169C22semidirect
68. #2169C22semidirect
69. #2163C22semidirect
70. #2159C22semidirect
72. #2139C22semidirect
73. C4×M32C22semidirect
253. D16#41semidirect
333. D8C2×#41semidirect
375. C16C2×#41semidirect
387. Q8#509semidirect

Maximal subgroups

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

subgroupindex
#614352
D8×#412
#339592
Q8×D162
#339262
#339252
#339202
#338972
#338912
#338382
#337512
#336752
#336702
#334852
C4×#5092