Finite Group Databasebeta

← #24917641

Subgroups — #24917641

Order 10624.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128175C211yesmaximal, characteristic, complemented
2#19128039C211yesmaximal, characteristic, complemented
3#19128022C211yesmaximal, characteristic, complemented
4#19127988C211yesmaximal, characteristic, complemented
5#19127978C211yesmaximal, characteristic, complemented
6#19127913C211yesmaximal, characteristic, complemented
7#19127913C211yesmaximal, characteristic, complemented
8#19127851C211yesmaximal, characteristic, complemented
9#19127834C211yesmaximal, characteristic
10#19127811C211yesmaximal, characteristic, complemented
11#19127799C211yesmaximal, characteristic, complemented
12C4×#13050577C211yesmaximal, characteristic, complemented
13C4×#13050575C211yesmaximal, characteristic, complemented
14#19127762C211yesmaximal, characteristic, complemented
15#542×C83C211yesmaximal, characteristic, complemented
16C22×D332C2211yescharacteristic
17C2×#13050577C2211yescharacteristic
18C2×#13050577C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050575C2211yescharacteristic
21C4×D332C2211yescharacteristic
22C4×D332C2211yescharacteristic
23C4×D332C2211yescharacteristic
24C4×D332C2211yescharacteristic
25#14581754—21no—
26#14581754—21no—
27#14581751C2211yescharacteristic
28C22⋊C4×D83—21no—
29C22⋊C4×D83—21no—
30#14581725C2211yescharacteristic, complemented
31#14581725C2211yescharacteristic, complemented
32#14581723C2211yescharacteristic, complemented
33#14581723C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35C4⋊(C83⋊C8)C2211yescharacteristic, complemented
36#14581716C2211yescharacteristic, complemented
37#14581716C2211yescharacteristic, complemented
38#14581713C2211yescharacteristic, complemented
39#14581713C2211yescharacteristic, complemented
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44C22⋊(C83⋊C8)C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46#14581703C2211yescharacteristic, complemented
47C4×C83⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C83C2211yescharacteristic, complemented
49#126×C83—21no—
50#124×C83—21no—
51#124×C83C2211yescharacteristic, complemented
52#123×C83—21no—
53C22⋊Q8×C83—21no—
54C22⋊Q8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic, complemented
57C4×D4×C83C2211yescharacteristic, complemented
58#117×C83C2211yescharacteristic
59C23×D83—21no—
60C2×D332—21no—
61C2×D332—22no—
62C2×D332—22no—
63C2×D332C2311yescharacteristic
64C2×D332C2311yescharacteristic
65C2×C4×D83—21no—
66C2×C4×D83—21no—
67C2×C4×D83—21no—
68C2×C4×D83—21no—
69#13050577—22no—
70#13050577—22no—
71#13050575—22no—
72#13050575—22no—
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C2×C83⋊C8C2311yescharacteristic
76C2×C83⋊C8C2311yescharacteristic
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79C22⋊Dic83—21no—
80(C2×D83)⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84C4×Dic83—21no—
85C2×Q8×C83—21no—
86C2×Q8×C83C2311yescharacteristic
87C2×D4×C83—21no—
88C2×D4×C83C2311yescharacteristic
89C22×C4×C83—21no—
90C22×C4×C83C2311yescharacteristic
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83D411yescharacteristic
95C4⋊C4×C83D411yescharacteristic
96C4⋊C4×C83C2311yescharacteristic
97C4⋊C4×C83C2311yescharacteristic
98C22⋊C4×C83—21no—
99C22⋊C4×C83—21no—
100C22⋊C4×C83—21no—
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83D411yescharacteristic
104C22⋊C4×C83D411yescharacteristic
105C42×C83—21no—
106C42×C83C2311yescharacteristic
107C42×C83C2311yescharacteristic
108C23×C83—21no—
109C23×C83C2×D411yescharacteristic
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—24no—
113C22×D83—81no—
114C22×D83—81no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—81no—
118D4×C83—21no—
119D4×C83—21no—
120D4×C83—81no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—81no—
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C2×Dic83—21no—
134C2×Dic83—21no—
135D332—24no—
136D332—21no—
137D332—21no—
138D332#4112yes—
139D332#4112yes—
140C4×D83—81no—
141C4×D83—81no—
142C4×D83—81no—
143C4×D83—81no—
144C83⋊C8—21no—
145C83⋊C8—21no—
146C83⋊C8—21no—
147C83⋊C8—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83C22×D411yescharacteristic
155C2×D83—22no—
156C2×D83—22no—
157C2×D83—41no—
158C2×D83—41no—
159C2×D83—41no—
160C2×D83—41no—
161C2×D83—84no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—41no—
168C4×C83—41no—
169C4×C83—41no—
170C4×C83C2×#4111yescharacteristic
171C4×C83C2×#4111yescharacteristic
172Dic83—41no—
173Dic83—41no—
174C2×C83—41no—
175C2×C83—81no—
176C2×C83#52511yescharacteristic
177C2×C83#57611yescharacteristic
178C2×C83#57711yescharacteristic
179D83—42no—
180D83—42no—
181#3118—831nomaximal, Sylow
182C83#311811yescharacteristic, complemented, Sylow
183#542D8311yescharacteristic, complemented
184#518—831no—
185#485—831no—
186#482—831no—
187#476—831no—
188#474—831no—
189#463—831no—
190#463—831no—
191#450—831no—
192#447—831no—
193#442—831no—
194#440—831no—
195C4×SD16—831no—
196C4×D8—831no—
197#433—831no—
198C22×D4—831no—
199C2×SD16—831no—
200C2×SD16—831no—
201C2×D8—831no—
202C2×D8—831no—
203C4⋊Q8C2×D8311yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D8311yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D8311yescharacteristic
210#121C2×D8311yescharacteristic
211C4×Q8C2×D8311yescharacteristic, complemented
212C4×D4—831no—
213C4×D4—831no—
214C4×D4—831no—
215C4×D4—831no—
216C4×D4—1661no—
217C4×D4—1661no—
218C4×D4C2×D8311yescharacteristic, complemented
219#117—831no—
220#117C2×D8311yescharacteristic
221C2×C22⋊C4—1661no—
222C2×C22⋊C4—1661no—
223#107—831no—
224#107—831no—
225#106—831no—
226#106—831no—
227C4⋊C8—831no—
228C4⋊C8—831no—
229#103—831no—
230#103—831no—
231#102—831no—
232#102—831no—
233#102—831no—
234#102—831no—
235#102—831no—
236#102—831no—
237C22⋊C8—831no—
238C22⋊C8—831no—
239#97—831no—
240C4×C8—831no—
241C24—1661no—
242C2×Q8—21no—
243C2×Q8C22×D8311yescharacteristic
244C2×D4—21no—
245C2×D4—831no—
246C2×D4—831no—
247C2×D4—1662no—
248C2×D4—1662no—
249C2×D4—1661no—
250C2×D4C22×D8311yescharacteristic
251C22×C4—21no—
252C22×C4—1661no—
253C22×C4—1661no—
254C22×C4—1661no—
255C22×C4—1661no—
256C22×C4C22×D8311yescharacteristic
257SD16—1662no—
258SD16—1662no—
259D8—1662no—
260D8—1662no—
261C2×C8—831no—
262C2×C8—831no—
263C2×C8—831no—
264C2×C8—831no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—831no—
269C4⋊C4—831no—
270C4⋊C4C22⋊D8311yescharacteristic
271C4⋊C4C22⋊D8311yescharacteristic
272C4⋊C4C22×D8311yescharacteristic
273C4⋊C4C22×D8311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1661no—
280C22⋊C4—1661no—
281C22⋊C4—1661no—
282C22⋊C4—1661no—
283C22⋊C4—1661no—
284C22⋊C4C22⋊D8311yescharacteristic
285C22⋊C4C22⋊D8311yescharacteristic
286C42—21no—
287C42—1661no—
288C42C22×D8311yescharacteristic
289C42C22×D8311yescharacteristic
290C23—21no—
291C23—6641no—
292C23—6641no—
293C23—1664no—
294C23—1661no—
295C23—1661no—
296C23C2×C22⋊D8311yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—832no—
304D4—832no—
305D4—1664no—
306D4—1661no—
307D4—1661no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—6641no—
315C2×C4—6641no—
316C2×C4—6641no—
317C2×C4—6641no—
318C2×C4—1661no—
319C2×C4—1661no—
320C2×C4C2×C22⋊D8311yescharacteristic
321C2×C4C2×C22⋊D8311yescharacteristic
322C2×C4C2×C22⋊D8311yescharacteristic
323C2×C4C2×C22⋊D8311yescharacteristic
324C2×C4C2×C22⋊D8311yescharacteristic
325C2×C4C23×D8311yescharacteristic
326C8—1661no—
327C8—1661no—
328C8—1661no—
329C8—1661no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—3321no—
337C22—3321no—
338C22—3321no—
339C22—3321no—
340C22—6644no—
341C22—1662no—
342C22—1662no—
343C22C22×C22⋊D8311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—3321no—
353C4—3321no—
354C4#41×D8311yescharacteristic
355C4#41×D8311yescharacteristic
356C2—41no—
357C2—81no—
358C2—3322no—
359C2—3322no—
360C2#1912820811yescharacteristic
361C2#1912850411yescharacteristic
362C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#542×C83C2
socleC22×C83C22×D4
prime radical#19127834C2
hypercenter#542D83
hypocenterC83#3118
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #19128175C2semidirect
2. #19128039C2semidirect
3. #19128022C2semidirect
4. #19127988C2semidirect
5. #19127978C2semidirect
6. #19127913C2semidirect
7. #19127913C2semidirect
8. #19127851C2semidirect
10. #19127811C2semidirect
11. #19127799C2semidirect
12. C4×#13050577C2semidirect
13. C4×#13050575C2semidirect
14. #19127762C2semidirect
15. #542×C83C2semidirect
30. #14581725C22semidirect
31. #14581725C22semidirect
32. #14581723C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. C4⋊(C83⋊C8)C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
38. #14581713C22semidirect
39. #14581713C22semidirect
46. #14581703C22semidirect
47. C4×C83⋊C8C22semidirect
48. C4⋊Q8×C83C22semidirect
51. #124×C83C22semidirect
56. C4×Q8×C83C22semidirect
57. C4×D4×C83C22semidirect
182. C83#3118semidirect
183. #542D83semidirect
203. C4⋊Q8C2×D83semidirect
206. #124C2×D83semidirect
211. C4×Q8C2×D83semidirect
218. C4×D4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191281752
#191280392
#191280222
#191279882
#191279782
#191279132
#191278512
#191278342
#191278112
#191277992
C4×#130505772
C4×#130505752
#191277622
#542×C832
#311883