Finite Group Databasebeta

← #24919669

Subgroups — #24919669

Order 10624.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128401C211yesmaximal, characteristic, complemented
2#19128346C211yesmaximal, characteristic, complemented
3#19128056C211yesmaximal, characteristic, complemented
4#19128044C211yesmaximal, characteristic, complemented
5#19128016C211yesmaximal, characteristic, complemented
6#19128001C211yesmaximal, characteristic, complemented
7#19127953C211yesmaximal, characteristic, complemented
8#19127946C211yesmaximal, characteristic, complemented
9#19127916C211yesmaximal, characteristic, complemented
10#19127843C211yesmaximal, characteristic, complemented
11#19127833C211yesmaximal, characteristic, complemented
12#19127780C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14#19127699C211yesmaximal, characteristic, complemented
15C4×D8×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050578C2211yescharacteristic, complemented
19C2×#13050575C2211yescharacteristic
20C2×#13050573C2211yescharacteristic
21C2×#13050573C2211yescharacteristic, complemented
22#14581785—21no—
23#14581784C2211yescharacteristic, complemented
24#14581780C2211yescharacteristic
25#14581776—21no—
26#14581775—21no—
27#14581772C2211yescharacteristic
28#14581767C2211yescharacteristic
29#14581766—21no—
30#14581765—21no—
31#14581763C2211yescharacteristic
32#14581763C2211yescharacteristic, complemented
33#14581762—21no—
34C4×D332C2211yescharacteristic
35C4×C22⋊D83—21no—
36#14581750C2211yescharacteristic
37C4⋊C4×D83C2211yescharacteristic, complemented
38C2×(C2×D83)⋊C4—21no—
39#14581725C2211yescharacteristic, complemented
40#14581723C2211yescharacteristic, complemented
41C4⋊(C83⋊C8)C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581714C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic, complemented
52C2×D8×C83C2211yescharacteristic
53C4×D4×C83C2211yescharacteristic
54C4×D4×C83C2211yescharacteristic, complemented
55#107×C83C2211yescharacteristic, complemented
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic, complemented
58C4×C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050589—22no—
61#13050589—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—21no—
64C2×D332D411yescharacteristic
65C2×D332C2311yescharacteristic
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70D4×D83—22no—
71D4×D83—41no—
72C22×Dic83—21no—
73C2×C4×D83—21no—
74C2×C4×D83C2311yescharacteristic
75C2×C4×D83C2311yescharacteristic
76#13050578—41no—
77#13050575—41no—
78#13050573—41no—
79#13050573—41no—
80C2×C83⋊C8C2311yescharacteristic
81C2×C83⋊C8C2311yescharacteristic
82C4⋊Dic83—21no—
83C4⋊Dic83D411yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85C4⋊Dic83C2311yescharacteristic
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4D411yescharacteristic
97(C2×D83)⋊C4D411yescharacteristic
98C4×Dic83—21no—
99C2×D4×C83C2311yescharacteristic
100C2×D4×C83C2311yescharacteristic
101C22×C4×C83—21no—
102C22×C4×C83—21no—
103D8×C83—22no—
104C2×C8×C83C2311yescharacteristic
105C2×C8×C83C2311yescharacteristic
106C4⋊C4×C83C2311yescharacteristic
107C4⋊C4×C83C2311yescharacteristic
108C22⋊C4×C83—21no—
109C22⋊C4×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83—21no—
113C22×D83—21no—
114C22×D83—42no—
115C22×D83—42no—
116C22×D83—81no—
117C22×D83C2×D411yescharacteristic
118C22×D83C2×D411yescharacteristic
119D4×C83—21no—
120D4×C83—21no—
121D4×C83—21no—
122D4×C83#4112yes—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—81no—
126C2×C4×C83—81no—
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2411yescharacteristic
130C22⋊D83—42no—
131C22⋊D83—42no—
132C22⋊D83—42no—
133C22⋊D83—81no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—42no—
138C2×Dic83C2×D411yescharacteristic
139C2×Dic83C2×D411yescharacteristic
140D332—21no—
141D332—21no—
142D332—41no—
143D332—81no—
144C4×D83—21no—
145C4×D83—21no—
146C4×D83—21no—
147C4×D83—21no—
148C4×D83—81no—
149Dic166—21no—
150Dic166—21no—
151C8×C83—21no—
152C8×C83—21no—
153C83⋊C8—21no—
154C83⋊C8—21no—
155C22×C83—22no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83C22×D411yescharacteristic
162C2×D83—21no—
163C2×D83—21no—
164C2×D83—21no—
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—21no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—84no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83C2×#4111yescharacteristic
177C4×C83H(5,𝔽2)11yescharacteristic
178Dic83—21no—
179Dic83—21no—
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—42no—
184C2×C83—81no—
185C2×C83#54611yescharacteristic
186C2×C83#57511yescharacteristic
187C2×C83#57611yescharacteristic
188D83—41no—
189D83—41no—
190D83—81no—
191#3322—831nomaximal, Sylow
192C83#332211yescharacteristic, complemented, Sylow
193#553—831no—
194#546—831no—
195#488—831no—
196#485—831no—
197#480—831no—
198#478—831no—
199#469—831no—
200#468—831no—
201#463—831no—
202#449—831no—
203#447—831no—
204C4×D8D8311yescharacteristic, complemented
205#436—831no—
206#418—831no—
207#417—831no—
208C2×#41—831no—
209C22×D4—831no—
210C2×SD16—831no—
211C2×D8—831no—
212C2×D8C2×D8311yescharacteristic
213C2×M16—831no—
214C2×M16—831no—
215#126—1661no—
216#125—831no—
217#124—831no—
218#123—1661no—
219#123—1661no—
220C22⋊Q8—831no—
221#121—831no—
222#121—831no—
223#121—831no—
224#121—1661no—
225#121—1661no—
226#120—1661no—
227C4×D4—831no—
228C4×D4—1661no—
229C4×D4C2×D8311yescharacteristic
230C4×D4C2×D8311yescharacteristic, complemented
231#117—831no—
232C2×C4⋊C4—831no—
233C2×C22⋊C4—1661no—
234#107—831no—
235#107C2×D8311yescharacteristic, complemented
236#106—831no—
237C4⋊C8—831no—
238#103—831no—
239#103—831no—
240#103—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102C2×D8311yescharacteristic
247#102C2×D8311yescharacteristic, complemented
248C22⋊C8—831no—
249C22⋊C8—831no—
250C4×C8C2×D8311yescharacteristic, complemented
251C24—1661no—
252#41—3321no—
253#41—1662no—
254C2×Q8—831no—
255C2×D4—3321no—
256C2×D4—831no—
257C2×D4—831no—
258C2×D4—1661no—
259C2×D4—1661no—
260C2×D4—1661no—
261C2×D4—1661no—
262C2×D4—1662no—
263C2×D4—1661no—
264C2×D4C22×D8311yescharacteristic
265C2×D4C22×D8311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—831no—
269C22×C4—831no—
270C22×C4—1661no—
271C22×C4—1661no—
272SD16—3321no—
273D8—22no—
274D8—3321no—
275M16—3321no—
276M16—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8C22×D8311yescharacteristic
280C2×C8C22×D8311yescharacteristic
281C4⋊C4—831no—
282C4⋊C4—831no—
283C4⋊C4—831no—
284C4⋊C4—1661no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4—1661no—
288C4⋊C4C22×D8311yescharacteristic
289C4⋊C4C22×D8311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—831no—
293C22⋊C4—831no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C22⋊C4—1661no—
298C22⋊C4—1661no—
299C22⋊C4—1661no—
300C22⋊C4—1661no—
301C42—1661no—
302C42C22×D8311yescharacteristic
303C23—21no—
304C23—21no—
305C23—3322no—
306C23—3322no—
307C23—831no—
308C23—831no—
309C23—6641no—
310C23—1661no—
311Q8—1661no—
312Q8—1661no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—3321no—
317D4—3322no—
318D4—3322no—
319D4—3322no—
320D4—6641no—
321D4—6641no—
322D4—1661no—
323D4—1661no—
324D4#1305059012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—3322no—
330C2×C4—831no—
331C2×C4—831no—
332C2×C4—6641no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4D4×D8311yescharacteristic
341C2×C4D4×D8311yescharacteristic
342C2×C4C23×D8311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1661no—
346C8—1661no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—3321no—
354C22—3321no—
355C22—3321no—
356C22—6644no—
357C22—1661no—
358C22—1661no—
359C22—1661no—
360C22—1661no—
361C22—1661no—
362C22—1661no—
363C22C2×D4×D8311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3321no—
369C4—3321no—
370C4—3321no—
371C4—1661no—
372C4—1661no—
373C4C2×#1305059011yescharacteristic
374C4#1458183811yescharacteristic
375C2—42no—
376C2—81no—
377C2—3321no—
378C2—3321no—
379C2—6641no—
380C2#1912834711yescharacteristic
381C2#1912849811yescharacteristic
382C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
hypercenterC4×D8D83
hypocenterC83#3322
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #19128401C2semidirect
2. #19128346C2semidirect
3. #19128056C2semidirect
4. #19128044C2semidirect
5. #19128016C2semidirect
6. #19128001C2semidirect
7. #19127953C2semidirect
8. #19127946C2semidirect
9. #19127916C2semidirect
10. #19127843C2semidirect
11. #19127833C2semidirect
12. #19127780C2semidirect
13. #19127704C2semidirect
14. #19127699C2semidirect
15. C4×D8×C83C2semidirect
18. C2×#13050578C22semidirect
21. C2×#13050573C22semidirect
23. #14581784C22semidirect
32. #14581763C22semidirect
37. C4⋊C4×D83C22semidirect
39. #14581725C22semidirect
40. #14581723C22semidirect
41. C4⋊(C83⋊C8)C22semidirect
42. #14581717C22semidirect
43. #14581717C22semidirect
44. #14581715C22semidirect
51. (C2×D83)⋊C8C22semidirect
54. C4×D4×C83C22semidirect
55. #107×C83C22semidirect
57. #102×C83C22semidirect
58. C4×C8×C83C22semidirect
192. C83#3322semidirect
204. C4×D8D83semidirect
230. C4×D4C2×D83semidirect
235. #107C2×D83semidirect
247. #102C2×D83semidirect
250. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284012
#191283462
#191280562
#191280442
#191280162
#191280012
#191279532
#191279462
#191279162
#191278432
#191278332
#191277802
#191277042
#191276992
C4×D8×C832
#332283