Finite Group Databasebeta

← #24919623

Subgroups — #24919623

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#19128415C211yesmaximal, characteristic, complemented
2#19128339C211yesmaximal, characteristic, complemented
3#19128062C211yesmaximal, characteristic, complemented
4#19128040C211yesmaximal, characteristic, complemented
5#19128026C211yesmaximal, characteristic, complemented
6#19127979C211yesmaximal, characteristic, complemented
7#19127944C211yesmaximal, characteristic, complemented
8#19127935C211yesmaximal, characteristic, complemented
9#19127918C211yesmaximal, characteristic, complemented
10#19127833C211yesmaximal, characteristic, complemented
11#19127813C211yesmaximal, characteristic, complemented
12#19127769C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14#19127697C211yesmaximal, characteristic, complemented
15#449×C83C211yesmaximal, characteristic, complemented
16C2×D4×D83C2211yescharacteristic
17C2×#13050575C2211yescharacteristic
18C2×#13050573C2211yescharacteristic
19C22×C83⋊C8C2211yescharacteristic, complemented
20#14581791C2211yescharacteristic, complemented
21#14581781C2211yescharacteristic
22#14581779—21no—
23#14581778—21no—
24#14581775—21no—
25#14581768C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581764—21no—
28#14581764—21no—
29#14581763C2211yescharacteristic
30#14581760—21no—
31D4×Dic83C2211yescharacteristic
32D4×Dic83C2211yescharacteristic, complemented
33C4×C22⋊D83C2211yescharacteristic
34#14581751C2211yescharacteristic, complemented
35#14581750C2211yescharacteristic
36#14581748—21no—
37C2×(C2×D83)⋊C4—21no—
38#14581726C2211yescharacteristic, complemented
39#14581725C2211yescharacteristic, complemented
40#14581724C2211yescharacteristic, complemented
41Dic83⋊C8C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic, complemented
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51#14581702C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic, complemented
54C2×D8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C2×Dic166—21no—
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C2×C22⋊D83—21no—
65C2×C22⋊D83D411yescharacteristic
66D4×D83—22no—
67D4×D83—41no—
68C22×Dic83—21no—
69C22×Dic83—21no—
70C2×C4×D83C2311yescharacteristic
71#13050575—41no—
72#13050573—41no—
73C2×C83⋊C8—41no—
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C4⋊Dic83C2311yescharacteristic
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81Dic83⋊C4D411yescharacteristic
82C22⋊Dic83—21no—
83C22⋊Dic83—21no—
84C22⋊Dic83—21no—
85C22⋊Dic83—21no—
86C22⋊Dic83D411yescharacteristic
87(C2×D83)⋊C4—21no—
88(C2×D83)⋊C4—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4D411yescharacteristic
93C4×Dic83—21no—
94C4×Dic83—21no—
95C4×Dic83C2311yescharacteristic
96#41×C83—22no—
97#41×C83—41no—
98C2×Q8×C83C2311yescharacteristic
99C2×D4×C83—21no—
100C2×D4×C83—21no—
101C2×D4×C83C2311yescharacteristic
102C2×D4×C83C2311yescharacteristic
103C22×C4×C83—21no—
104C22×C4×C83C2311yescharacteristic
105SD16×C83—41no—
106D8×C83—22no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83C2×D411yescharacteristic
114C22×D83—21no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—81no—
118C22×D83C2×D411yescharacteristic
119Q8×C83—21no—
120Q8×C83—21no—
121D4×C83—21no—
122D4×C83—21no—
123D4×C83—21no—
124D4×C83—42no—
125D4×C83—81no—
126D4×C83#4112yes—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—42no—
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2411yescharacteristic
134C22⋊D83—41no—
135C22⋊D83—42no—
136C22⋊D83—81no—
137C22⋊D83—81no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—42no—
142C2×Dic83—81no—
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145C2×Dic83C2×D411yescharacteristic
146D332—21no—
147D332—21no—
148C4×D83—21no—
149C4×D83—21no—
150Dic166—81no—
151C8×C83—21no—
152C8×C83—21no—
153C83⋊C8—21no—
154C83⋊C8—21no—
155C22×C83—22no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83C22×D411yescharacteristic
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—41no—
174C4×C83—41no—
175C4×C83C2×#4111yescharacteristic
176C4×C83H(5,𝔽2)11yescharacteristic
177Dic83—21no—
178Dic83—21no—
179Dic83—21no—
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—42no—
184C2×C83—41no—
185C2×C83—81no—
186C2×C83#54611yescharacteristic
187C2×C83#57511yescharacteristic
188C2×C83#57611yescharacteristic
189D83—41no—
190D83—81no—
191#3319—831nomaximal, Sylow
192C83#331911yescharacteristic, complemented, Sylow
193#555—831no—
194#546—831no—
195#489—831no—
196#485—831no—
197#482—831no—
198#474—831no—
199#468—831no—
200#466—831no—
201#463—831no—
202#449D8311yescharacteristic, complemented
203#447—831no—
204#442—831no—
205#435—831no—
206#418—831no—
207#416—831no—
208C2×#41C2×D8311yescharacteristic
209C22×D4—831no—
210C2×SD16C2×D8311yescharacteristic, complemented
211C2×D8—831no—
212C2×D8C2×D8311yescharacteristic
213C2×M16—831no—
214C22×C8—831no—
215C4⋊Q8—831no—
216#124—831no—
217#124—1661no—
218#124—1661no—
219#123—1661no—
220C22⋊Q8—831no—
221#121—831no—
222#121—831no—
223#121—1661no—
224#121—1661no—
225#121C2×D8311yescharacteristic, complemented
226#120—1661no—
227C4×D4—831no—
228C4×D4—831no—
229C4×D4—831no—
230#117—831no—
231#117—831no—
232#117—1661no—
233C2×C22⋊C4—1661no—
234#107—831no—
235#107—831no—
236#106—831no—
237C4⋊C8—831no—
238#103—831no—
239#103—831no—
240#103C2×D8311yescharacteristic, complemented
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102C2×D8311yescharacteristic
248C22⋊C8—831no—
249C22⋊C8C2×D8311yescharacteristic, complemented
250#97—831no—
251C24—1661no—
252#41—22no—
253#41—41no—
254C2×Q8—1661no—
255C2×Q8C22×D8311yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3321no—
259C2×D4—831no—
260C2×D4—831no—
261C2×D4—1661no—
262C2×D4—1662no—
263C2×D4—1661no—
264C2×D4—1661no—
265C2×D4C22×D8311yescharacteristic
266C2×D4C22×D8311yescharacteristic
267C22×C4—21no—
268C22×C4—831no—
269C22×C4—1661no—
270C22×C4—1661no—
271C22×C4C22×D8311yescharacteristic
272SD16—41no—
273D8—22no—
274D8—3321no—
275M16—3321no—
276C2×C8—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8C22×D8311yescharacteristic
280C2×C8C22×D8311yescharacteristic
281C4⋊C4—831no—
282C4⋊C4—831no—
283C4⋊C4—831no—
284C4⋊C4—831no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4C22×D8311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—831no—
290C22⋊C4—831no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C22⋊C4—1661no—
298C22⋊C4—1661no—
299C22⋊C4—1661no—
300C42—831no—
301C42—1661no—
302C42—1661no—
303C23—21no—
304C23—21no—
305C23—3322no—
306C23—3322no—
307C23—831no—
308C23—6641no—
309C23—1661no—
310C23D4×D8311yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—6641no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—81no—
319D4—3322no—
320D4—3321no—
321D4—6641no—
322D4—6641no—
323D4—1661no—
324D4—1661no—
325D4#1305058912yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—3322no—
332C2×C4—831no—
333C2×C4—831no—
334C2×C4—831no—
335C2×C4—6641no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4D4×D8311yescharacteristic
342C2×C4C23×D8311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1661no—
346C8—1661no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3321no—
357C22—3321no—
358C22—3321no—
359C22—6644no—
360C22—1661no—
361C22—1661no—
362C22—1661no—
363C22C2×D4×D8311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3321no—
368C4—3321no—
369C4—3321no—
370C4—1661no—
371C4—1661no—
372C4—1661no—
373C4C2×#1305058911yescharacteristic
374C4#1458183711yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—3321no—
379C2—6641no—
380C2#1912833811yescharacteristic
381C2#1912849811yescharacteristic
382C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#449×C83C2
socleC22×C83C22×D4
hypercenter#449D83
hypocenterC83#3319
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #19128415C2semidirect
2. #19128339C2semidirect
3. #19128062C2semidirect
4. #19128040C2semidirect
5. #19128026C2semidirect
6. #19127979C2semidirect
7. #19127944C2semidirect
8. #19127935C2semidirect
9. #19127918C2semidirect
10. #19127833C2semidirect
11. #19127813C2semidirect
12. #19127769C2semidirect
13. #19127704C2semidirect
14. #19127697C2semidirect
15. #449×C83C2semidirect
19. C22×C83⋊C8C22semidirect
20. #14581791C22semidirect
32. D4×Dic83C22semidirect
34. #14581751C22semidirect
38. #14581726C22semidirect
39. #14581725C22semidirect
40. #14581724C22semidirect
41. Dic83⋊C8C22semidirect
42. #14581716C22semidirect
43. #14581716C22semidirect
48. #14581713C22semidirect
51. #14581702C22semidirect
53. C2×SD16×C83C22semidirect
55. #121×C83C22semidirect
56. #103×C83C22semidirect
58. C22⋊C8×C83C22semidirect
192. C83#3319semidirect
202. #449D83semidirect
210. C2×SD16C2×D83semidirect
225. #121C2×D83semidirect
240. #103C2×D83semidirect
249. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284152
#191283392
#191280622
#191280402
#191280262
#191279792
#191279442
#191279352
#191279182
#191278332
#191278132
#191277692
#191277042
#191276972
#449×C832
#331983