Finite Group Databasebeta

← #24918716

Subgroups — #24918716

Order 10624.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128306C211yesmaximal, characteristic
2#19128107C211yesmaximal, characteristic, complemented
3#19128100C211yesmaximal, characteristic, complemented
4#19128067C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19128025C211yesmaximal, characteristic, complemented
7#19128025C211yesmaximal, characteristic, complemented
8#19128019C211yesmaximal, characteristic, complemented
9#19128014C211yesmaximal, characteristic, complemented
10#19127998C211yesmaximal, characteristic, complemented
11#19127998C211yesmaximal, characteristic, complemented
12#19127988C211yesmaximal, characteristic, complemented
13#19127978C211yesmaximal, characteristic, complemented
14#19127762C211yesmaximal, characteristic, complemented
15#544×C83C211yesmaximal, characteristic, complemented
16#14581789C2211yescharacteristic
17#14581784C2211yescharacteristic
18#14581777—21no—
19#14581777—21no—
20#14581766C2211yescharacteristic
21#14581766C2211yescharacteristic
22#14581765—21no—
23#14581765—21no—
24C4×D332C2211yescharacteristic
25C4×D332C2211yescharacteristic
26#14581725C2211yescharacteristic
27#14581725C2211yescharacteristic
28#14581725C2211yescharacteristic
29#14581725C2211yescharacteristic
30#14581723C2211yescharacteristic
31#14581723C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35C4⋊(C83⋊C8)C2211yescharacteristic
36#14581712C2211yescharacteristic
37#14581712C2211yescharacteristic
38#14581712C2211yescharacteristic
39#14581712C2211yescharacteristic
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44C22⋊(C83⋊C8)C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46#14581703C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83C2211yescharacteristic
50C4⋊Q8×C83C2211yescharacteristic
51#125×C83—21no—
52#125×C83C2211yescharacteristic
53#125×C83C2211yescharacteristic
54C22⋊Q8×C83—21no—
55C22⋊Q8×C83—21no—
56#117×C83C2211yescharacteristic
57#117×C83C2211yescharacteristic
58C2×C4⋊C4×C83C2211yescharacteristic
59C2×D332—21no—
60C2×D332—21no—
61C2×D332C2311yescharacteristic
62C2×D332C2311yescharacteristic
63C2×C22⋊D83—21no—
64C2×C22⋊D83—21no—
65C2×C4×D83—21no—
66C2×C4×D83—21no—
67C2×C83⋊C8C2311yescharacteristic
68C2×C83⋊C8C2311yescharacteristic
69C2×C83⋊C8C2311yescharacteristic
70C2×C83⋊C8C2311yescharacteristic
71C4⋊Dic83C2311yescharacteristic
72C4⋊Dic83C2311yescharacteristic
73Dic83⋊C4—21no—
74Dic83⋊C4—21no—
75(C2×D83)⋊C4—21no—
76(C2×D83)⋊C4—21no—
77(C2×D83)⋊C4—21no—
78(C2×D83)⋊C4—21no—
79C2×Q8×C83—21no—
80C2×Q8×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83C2311yescharacteristic
83C4⋊C4×C83—21no—
84C4⋊C4×C83—21no—
85C4⋊C4×C83—21no—
86C4⋊C4×C83—21no—
87C4⋊C4×C83—21no—
88C4⋊C4×C83—21no—
89C4⋊C4×C83—21no—
90C4⋊C4×C83D411yescharacteristic
91C4⋊C4×C83D411yescharacteristic
92C4⋊C4×C83C2311yescharacteristic
93C4⋊C4×C83C2311yescharacteristic
94C4⋊C4×C83C2311yescharacteristic
95C4⋊C4×C83C2311yescharacteristic
96C22⋊C4×C83—21no—
97C22⋊C4×C83D411yescharacteristic
98C22⋊C4×C83D411yescharacteristic
99C42×C83—21no—
100C42×C83C2311yescharacteristic
101C42×C83C2311yescharacteristic
102C23×C83C2×D411yescharacteristic
103C22×D83—21no—
104C22×D83—21no—
105Q8×C83—81no—
106Q8×C83—81no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—81no—
114C2×C4×C83C2×D411yescharacteristic
115C2×C4×C83C2×D411yescharacteristic
116C2×C4×C83C2×D411yescharacteristic
117C2×C4×C83C2×D411yescharacteristic
118C2×C4×C83C2×D411yescharacteristic
119C2×C4×C83C2411yescharacteristic
120C22⋊D83—81no—
121C22⋊D83—81no—
122C2×Dic83—21no—
123C2×Dic83—21no—
124D332—21no—
125D332—21no—
126D332—21no—
127D332—21no—
128D332—81no—
129D332—81no—
130C4×D83—81no—
131C4×D83—81no—
132C83⋊C8—21no—
133C83⋊C8—21no—
134C83⋊C8—21no—
135C83⋊C8—21no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83C22×D411yescharacteristic
140C2×D83—41no—
141C2×D83—41no—
142C2×D83—41no—
143C2×D83—41no—
144C2×D83—41no—
145C2×D83—41no—
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—21no—
150C4×C83—21no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—41no—
155C4×C832−1+411yescharacteristic
156C4×C832−1+411yescharacteristic
157Dic83—41no—
158Dic83—41no—
159C2×C83—41no—
160C2×C83#53611yescharacteristic
161C2×C83#57611yescharacteristic
162C2×C83#57711yescharacteristic
163D83—81no—
164D83—81no—
165#3241—831nomaximal, Sylow
166C83#324111yescharacteristic, complemented, Sylow
167#544D8311yescharacteristic, complemented
168#540—831no—
169#501—831no—
170#499—831no—
171#490—831no—
172#486—831no—
173#482—831no—
174#482—831no—
175#481—831no—
176#480—831no—
177#478—831no—
178#478—831no—
179#476—831no—
180#474—831no—
181#433—831no—
182C4⋊Q8—21no—
183C4⋊Q8C2×D8311yescharacteristic
184C4⋊Q8C2×D8311yescharacteristic
185#127—831no—
186#125—21no—
187#125—831no—
188#125C2×D8311yescharacteristic
189#125C2×D8311yescharacteristic
190#123—1661no—
191#123—1661no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—831no—
195#121—831no—
196#121—1661no—
197#121—1661no—
198C4×D4—831no—
199C4×D4—831no—
200#117C2×D8311yescharacteristic
201#117C2×D8311yescharacteristic
202C2×C4⋊C4C2×D8311yescharacteristic
203#107—831no—
204#107—831no—
205#107—831no—
206#107—831no—
207#106—831no—
208#106—831no—
209#106—831no—
210#106—831no—
211C4⋊C8—831no—
212C4⋊C8—831no—
213#102—831no—
214#102—831no—
215#102—831no—
216#102—831no—
217#102—831no—
218#102—831no—
219#102—831no—
220#102—831no—
221C22⋊C8—831no—
222C22⋊C8—831no—
223#97—831no—
224C4×C8—831no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×D4—831no—
228C2×D4—831no—
229C2×D4—1661no—
230C2×D4—1661no—
231C2×D4—1661no—
232C2×D4—1661no—
233C22×C4—21no—
234C22×C4—1661no—
235C22×C4—1661no—
236C22×C4C22×D8311yescharacteristic
237C2×C8—831no—
238C2×C8—831no—
239C2×C8—831no—
240C2×C8—831no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—831no—
249C4⋊C4—831no—
250C4⋊C4—1661no—
251C4⋊C4—1661no—
252C4⋊C4C22⋊D8311yescharacteristic
253C4⋊C4C22⋊D8311yescharacteristic
254C4⋊C4C22×D8311yescharacteristic
255C4⋊C4C22×D8311yescharacteristic
256C4⋊C4C22×D8311yescharacteristic
257C4⋊C4C22×D8311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—1661no—
260C22⋊C4—1661no—
261C22⋊C4—1661no—
262C22⋊C4—1661no—
263C22⋊C4C22⋊D8311yescharacteristic
264C22⋊C4C22⋊D8311yescharacteristic
265C42—21no—
266C42C22×D8311yescharacteristic
267C42C22×D8311yescharacteristic
268C23—1661no—
269C23—1661no—
270C23C2×C22⋊D8311yescharacteristic
271Q8—81no—
272Q8—81no—
273D4—6641no—
274D4—6641no—
275D4—6641no—
276D4—6641no—
277D4—1661no—
278D4—1661no—
279D4—1661no—
280D4—1661no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—81no—
288C2×C4—6641no—
289C2×C4—6641no—
290C2×C4—1661no—
291C2×C4—1661no—
292C2×C4C2×C22⋊D8311yescharacteristic
293C2×C4C2×C22⋊D8311yescharacteristic
294C2×C4C2×C22⋊D8311yescharacteristic
295C2×C4C2×C22⋊D8311yescharacteristic
296C2×C4C2×C22⋊D8311yescharacteristic
297C2×C4C23×D8311yescharacteristic
298C8—1661no—
299C8—1661no—
300C8—1661no—
301C8—1661no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—3321no—
306C22—3321no—
307C22—3321no—
308C22—3321no—
309C22—3321no—
310C22—3321no—
311C22C22×C22⋊D8311yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3321no—
322C4—3321no—
323C4#1458184011yescharacteristic
324C4#1458184011yescharacteristic
325C2—41no—
326C2—6641no—
327C2—6641no—
328C2#1912827511yescharacteristic
329C2#1912850411yescharacteristic
330C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#544×C83C2
socleC22×C83C22×D4
prime radical#19128306C2
hypercenter#544D83
hypocenterC83#3241
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #19128107C2semidirect
3. #19128100C2semidirect
4. #19128067C2semidirect
5. #19128049C2semidirect
6. #19128025C2semidirect
7. #19128025C2semidirect
8. #19128019C2semidirect
9. #19128014C2semidirect
10. #19127998C2semidirect
11. #19127998C2semidirect
12. #19127988C2semidirect
13. #19127978C2semidirect
14. #19127762C2semidirect
15. #544×C83C2semidirect
166. C83#3241semidirect
167. #544D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191281072
#191281002
#191280672
#191280492
#191280252
#191280192
#191280142
#191279982
#191279882
#191279782
#191277622
#544×C832
#324183