Finite Group Databasebeta

← #24919143

Subgroups — #24919143

Order 10624.

526 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581813C211yesmaximal, characteristic, complemented
2C2×D8×D83C211yesmaximal, characteristic, complemented
3#19128346C211yesmaximal, characteristic, complemented
4#19128344C211yesmaximal, characteristic, complemented
5#19128083C211yesmaximal, characteristic, complemented
6#19127941C211yesmaximal, characteristic, complemented
7#19127936C211yesmaximal, characteristic, complemented
8#19127917C211yesmaximal, characteristic, complemented
9#19127853C211yesmaximal, characteristic, complemented
10#19127850C211yesmaximal, characteristic, complemented
11#19127849C211yesmaximal, characteristic, complemented
12#19127843C211yesmaximal, characteristic, complemented
13C4×#13050575C211yesmaximal, characteristic, complemented
14#19127766C211yesmaximal, characteristic, complemented
15#463×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic, complemented
17C2×#13050589C2211yescharacteristic, complemented
18C2×D4×D83C2211yescharacteristic
19C2×D4×D83C2211yescharacteristic
20#14581813—24no—
21C2×#13050580C2211yescharacteristic, complemented
22C2×#13050579C2211yescharacteristic, complemented
23C2×#13050578C2211yescharacteristic, complemented
24C2×#13050577C2211yescharacteristic, complemented
25C2×D664C2211yescharacteristic
26C2×#13050575C2211yescharacteristic
27C2×#13050575C2211yescharacteristic
28D8×D83—24no—
29C2×C8×D83C2211yescharacteristic
30C2×C8×D83C2211yescharacteristic, complemented
31#14581780C2211yescharacteristic, complemented
32#14581776—21no—
33#14581774—21no—
34#14581772C2211yescharacteristic
35#14581768C2211yescharacteristic
36#14581767C2211yescharacteristic, complemented
37#14581766—21no—
38#14581765—21no—
39#14581763C2211yescharacteristic, complemented
40#14581762—21no—
41#14581762—21no—
42C4×D332C2211yescharacteristic
43C4×D332C2211yescharacteristic, complemented
44C2×(C2×D83)⋊C4—21no—
45C22⋊C4×D83—21no—
46#14581725C2211yescharacteristic, complemented
47#14581717C2211yescharacteristic, complemented
48#14581714C2211yescharacteristic, complemented
49#14581713C2211yescharacteristic, complemented
50#14581712C2211yescharacteristic, complemented
51(C2×D83)⋊C8C2211yescharacteristic
52(C2×D83)⋊C8C2211yescharacteristic, complemented
53C4×C83⋊C8C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic, complemented
55C2×D8×C83C2211yescharacteristic
56#124×C83C2211yescharacteristic, complemented
57C4×D4×C83C2211yescharacteristic, complemented
58C4⋊C8×C83C2211yescharacteristic, complemented
59#103×C83C2211yescharacteristic, complemented
60#102×C83C2211yescharacteristic, complemented
61C23×D83—21no—
62C23×D83—21no—
63#13050591—22no—
64#13050591—41no—
65#13050589—22no—
66#13050589—41no—
67C2×Dic166C2311yescharacteristic
68C2×D332—21no—
69C2×D332—21no—
70C2×D332D411yescharacteristic
71C2×D332C2311yescharacteristic
72C2×D332C2311yescharacteristic
73C2×C22⋊D83—21no—
74C2×C22⋊D83—21no—
75C2×C22⋊D83—21no—
76C2×C22⋊D83—21no—
77D4×D83—22no—
78D4×D83—22no—
79D4×D83—41no—
80D4×D83—41no—
81C22×Dic83—21no—
82C2×C4×D83—21no—
83C2×C4×D83—21no—
84C2×C4×D83C2311yescharacteristic
85C2×C4×D83C2311yescharacteristic
86#13050580—22no—
87#13050579—22no—
88#13050578—41no—
89#13050577—41no—
90D664—22no—
91#13050575—22no—
92#13050575D414yescomplemented
93C2×C83⋊C8C2311yescharacteristic
94C2×C83⋊C8C2311yescharacteristic
95C8×D83—22no—
96C8×D83—22no—
97C4⋊Dic83—21no—
98C4⋊Dic83D411yescharacteristic
99C4⋊Dic83C2311yescharacteristic
100Dic83⋊C4—21no—
101C22⋊Dic83—21no—
102(C2×D83)⋊C4—21no—
103(C2×D83)⋊C4—21no—
104(C2×D83)⋊C4—21no—
105(C2×D83)⋊C4—21no—
106(C2×D83)⋊C4—21no—
107(C2×D83)⋊C4—21no—
108(C2×D83)⋊C4D411yescharacteristic
109(C2×D83)⋊C4D411yescharacteristic
110C2×Q8×C83C2311yescharacteristic
111C2×D4×C83C2311yescharacteristic
112C2×D4×C83C2311yescharacteristic
113C22×C4×C83—21no—
114SD16×C83—22no—
115D8×C83—22no—
116C2×C8×C83C2311yescharacteristic
117C2×C8×C83C2311yescharacteristic
118C4⋊C4×C83C2311yescharacteristic
119C22⋊C4×C83—21no—
120C22⋊C4×C83—21no—
121C22⋊C4×C83—21no—
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C23×C83—21no—
125C22×D83—21no—
126C22×D83—21no—
127C22×D83—42no—
128C22×D83—42no—
129C22×D83—42no—
130C22×D83—42no—
131C22×D83—81no—
132C22×D83—81no—
133C22×D83C2×D411yescharacteristic
134C22×D83C2×D411yescharacteristic
135Q8×C83—21no—
136Q8×C83—21no—
137D4×C83—21no—
138D4×C83—21no—
139D4×C83—21no—
140D4×C83C2×D412yes—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—81no—
144C2×C4×C83C2×D411yescharacteristic
145C2×C4×C83C2×D411yescharacteristic
146C2×C4×C83C2411yescharacteristic
147C22⋊D83—42no—
148C22⋊D83—42no—
149C22⋊D83—42no—
150C22⋊D83—81no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83—42no—
154C2×Dic83C2×D411yescharacteristic
155C2×Dic83C2×D411yescharacteristic
156D332—22no—
157D332—21no—
158D332—21no—
159D332—21no—
160D332—42no—
161D332—81no—
162D332C2×D412yes—
163C4×D83—21no—
164C4×D83—21no—
165C4×D83—21no—
166C4×D83—21no—
167C4×D83—42no—
168C4×D83—81no—
169Dic166—21no—
170Dic166—21no—
171C8×C83—21no—
172C8×C83—21no—
173C83⋊C8—21no—
174C83⋊C8C2×D412yes—
175C22×C83—22no—
176C22×C83—41no—
177C22×C83—41no—
178C22×C83—41no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83C22×D411yescharacteristic
182C2×D83—21no—
183C2×D83—21no—
184C2×D83—21no—
185C2×D83—22no—
186C2×D83—21no—
187C2×D83—21no—
188C2×D83—41no—
189C2×D83—41no—
190C2×D83—41no—
191C2×D83—41no—
192C2×D83—41no—
193C2×D83—84no—
194C2×D83—84no—
195C2×D83#13512yescomplemented
196C4×C83—21no—
197C4×C83—21no—
198C4×C83—41no—
199C4×C83—41no—
200C4×C83C22×D411yescharacteristic
201C4×C83H(5,𝔽2)11yescharacteristic
202Dic83—21no—
203Dic83—21no—
204Dic83—41no—
205Dic83—41no—
206C2×C83—42no—
207C2×C83—81no—
208C2×C83D4211yescharacteristic
209C2×C83C2×#13511yescharacteristic
210C2×C83#57611yescharacteristic
211D83—22no—
212D83—41no—
213D83—42no—
214D83—81no—
215#3280—831nomaximal, Sylow
216C83#328011yescharacteristic, complemented, Sylow
217C2×#135—831no—
218C22×D8—831no—
219#546—831no—
220#546—831no—
221#495—831no—
222#467—831no—
223#466—831no—
224#463—831no—
225#463D8311yescharacteristic, complemented
226#450—831no—
227#450—831no—
228#449—831no—
229#449—831no—
230C4×D8—831no—
231C8×D4—831no—
232C2×#41—831no—
233C2×#41—831no—
234C22×D4—831no—
235C22×D4—831no—
236#135—1664no—
237C2×Q16—831no—
238C2×SD16—831no—
239C2×SD16—831no—
240C2×SD16—831no—
241C2×SD16C2×D8311yescharacteristic, complemented
242C2×D8—831no—
243C2×D8—831no—
244C2×D8—831no—
245C2×D8—1664no—
246C2×D8C2×D8311yescharacteristic
247C22×C8—831no—
248C22×C8—831no—
249#124—831no—
250#124C2×D8311yescharacteristic, complemented
251#123—1661no—
252#123—1661no—
253C22⋊Q8—831no—
254C22⋊Q8—831no—
255#121—831no—
256#121—831no—
257#121—1661no—
258#121—1661no—
259#120—1661no—
260#120—1661no—
261C4×D4—831no—
262C4×D4—831no—
263C4×D4C2×D8311yescharacteristic, complemented
264C2×C22⋊C4—1661no—
265C2×C22⋊C4—1661no—
266#107—831no—
267C4⋊C8C2×D8311yescharacteristic, complemented
268#103—831no—
269#103C2×D8311yescharacteristic, complemented
270#102—831no—
271#102—831no—
272#102—831no—
273#102C2×D8311yescharacteristic, complemented
274C22⋊C8—831no—
275C22⋊C8—831no—
276C4×C8—831no—
277C24—1661no—
278C24—1661no—
279#41—3321no—
280#41—3321no—
281#41—1662no—
282#41—1662no—
283C2×Q8—831no—
284C2×Q8C22×D8311yescharacteristic
285C2×D4—3321no—
286C2×D4—3321no—
287C2×D4—831no—
288C2×D4—831no—
289C2×D4—831no—
290C2×D4—1661no—
291C2×D4—1661no—
292C2×D4—1661no—
293C2×D4—1661no—
294C2×D4—1662no—
295C2×D4—1661no—
296C2×D4—1662no—
297C2×D4—1661no—
298C2×D4C22×D8311yescharacteristic
299C2×D4C22×D8311yescharacteristic
300C22×C4—21no—
301C22×C4—831no—
302C22×C4—831no—
303C22×C4—1661no—
304C22×C4—1661no—
305C22×C4—1661no—
306Q16—1662no—
307SD16—22no—
308SD16—3321no—
309SD16—3321no—
310SD16—1662no—
311D8—22no—
312D8—834no—
313D8—1662no—
314D8—1662no—
315C2×C8—831no—
316C2×C8—831no—
317C2×C8—1662no—
318C2×C8—1662no—
319C2×C8C22×D8311yescharacteristic
320C2×C8C22×D8311yescharacteristic
321C4⋊C4—831no—
322C4⋊C4—831no—
323C4⋊C4—1661no—
324C4⋊C4—1661no—
325C4⋊C4C22×D8311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—831no—
330C22⋊C4—831no—
331C22⋊C4—1661no—
332C22⋊C4—1661no—
333C22⋊C4—1661no—
334C22⋊C4—1661no—
335C22⋊C4—1661no—
336C22⋊C4—1661no—
337C22⋊C4—1661no—
338C42C22×D8311yescharacteristic
339C23—21no—
340C23—21no—
341C23—3322no—
342C23—3322no—
343C23—3322no—
344C23—3322no—
345C23—831no—
346C23—831no—
347C23—6641no—
348C23—6641no—
349C23—1661no—
350C23—1661no—
351Q8—21no—
352Q8—21no—
353Q8—1661no—
354Q8—1661no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—3322no—
359D4—3322no—
360D4—3322no—
361D4—3322no—
362D4—832no—
363D4—6641no—
364D4—6641no—
365D4—1662no—
366D4—1661no—
367D4—1661no—
368D4—1661no—
369D4D4×D8312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—3322no—
374C2×C4—3322no—
375C2×C4—831no—
376C2×C4—831no—
377C2×C4—6641no—
378C2×C4—1661no—
379C2×C4—1661no—
380C2×C4—1661no—
381C2×C4—1661no—
382C2×C4—1661no—
383C2×C4—1661no—
384C2×C4D4×D8311yescharacteristic
385C2×C4D4×D8311yescharacteristic
386C2×C4C23×D8311yescharacteristic
387C8—21no—
388C8—21no—
389C8—832no—
390C8—1661no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3321no—
398C22—3321no—
399C22—3321no—
400C22—3321no—
401C22—3321no—
402C22—832no—
403C22—6644no—
404C22—6644no—
405C22—1662no—
406C22—1661no—
407C22—1661no—
408C22—1661no—
409C22—1661no—
410C22—1661no—
411C22C2×D4×D8311yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3321no—
417C4—3321no—
418C4—1661no—
419C4—1661no—
420C4C2×D4×D8311yescharacteristic
421C4#1458183811yescharacteristic
422C2—42no—
423C2—81no—
424C2—3322no—
425C2—3321no—
426C2—6641no—
427C2—1662no—
428C2#1912833511yescharacteristic
429C2#135×D8311yescharacteristic
430C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#463×C83C2
socleC22×C83C22×D4
hypercenter#463D83
hypocenterC83#3280
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#14581813C2semidirect
2. C2×D8×D83C2semidirect
3. #19128346C2semidirect
4. #19128344C2semidirect
5. #19128083C2semidirect
6. #19127941C2semidirect
7. #19127936C2semidirect
8. #19127917C2semidirect
9. #19127853C2semidirect
10. #19127850C2semidirect
11. #19127849C2semidirect
12. #19127843C2semidirect
13. C4×#13050575C2semidirect
14. #19127766C2semidirect
15. #463×C83C2semidirect
16. C2×#13050591C22semidirect
17. C2×#13050589C22semidirect
21. C2×#13050580C22semidirect
22. C2×#13050579C22semidirect
23. C2×#13050578C22semidirect
24. C2×#13050577C22semidirect
30. C2×C8×D83C22semidirect
31. #14581780C22semidirect
36. #14581767C22semidirect
39. #14581763C22semidirect
43. C4×D332C22semidirect
46. #14581725C22semidirect
47. #14581717C22semidirect
48. #14581714C22semidirect
49. #14581713C22semidirect
50. #14581712C22semidirect
52. (C2×D83)⋊C8C22semidirect
53. C4×C83⋊C8C22semidirect
54. C2×SD16×C83C22semidirect
56. #124×C83C22semidirect
57. C4×D4×C83C22semidirect
58. C4⋊C8×C83C22semidirect
59. #103×C83C22semidirect
60. #102×C83C22semidirect
92. #13050575D4semidirect
195. C2×D83#135semidirect
216. C83#3280semidirect
225. #463D83semidirect
241. C2×SD16C2×D83semidirect
250. #124C2×D83semidirect
263. C4×D4C2×D83semidirect
267. C4⋊C8C2×D83semidirect
269. #103C2×D83semidirect
273. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818132
C2×D8×D832
#191283462
#191283442
#191280832
#191279412
#191279362
#191279172
#191278532
#191278502
#191278492
#191278432
C4×#130505752
#191277662
#463×C832
#328083