Finite Group Databasebeta

← #35302761

Subgroups — #35302761

Order 21248.

382 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24908425C211yesmaximal, characteristic
2#24908412C211yesmaximal, characteristic
3#24908227C211yesmaximal, characteristic
4#24907328C211yesmaximal, characteristic
5#24905347C211yesmaximal, characteristic
6#24905297C211yesmaximal, characteristic
7C2×#333×C83C211yesmaximal, characteristic
8C2×#14581724—21no—
9C2×C4⋊(C83⋊C8)C2211yescharacteristic
10C2×C4×C83⋊C8C2211yescharacteristic
11C2×C8×Dic83—21no—
12#19127571C2211yescharacteristic
13C4×C4⋊Dic83C2211yescharacteristic
14#19127464—21no—
15#19127463—21no—
16#19127463—21no—
17#19127463—21no—
18#19127463—21no—
19#19127455—21no—
20C2×C4⋊Q8×C83C2211yescharacteristic
21C2×C4⋊C8×C83C2211yescharacteristic
22C2×C4⋊C8×C83C2211yescharacteristic
23#333×C83C414yescomplemented
24C22×C83⋊C8—21no—
25C22×C83⋊C8D411yescharacteristic
26C22×C83⋊C8D411yescharacteristic
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83D411yescharacteristic
31C2×C4⋊Dic83D411yescharacteristic
32C2×C4×Dic83—21no—
33#14581724—42no—
34C4⋊(C83⋊C8)—21no—
35C4⋊(C83⋊C8)—21no—
36C4×C83⋊C8—21no—
37C4×C83⋊C8—21no—
38C8×Dic83—42no—
39C83⋊#95—21no—
40C22×Q8×C83—21no—
41C22×C8×C83—21no—
42C22×C8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83—21no—
45C4⋊Q8×C83—21no—
46C4⋊Q8×C83C2×C412yes—
47C2×C4⋊C4×C83—21no—
48C2×C4⋊C4×C83—21no—
49C2×C42×C83C2311yescharacteristic
50C4⋊C8×C83—21no—
51C4⋊C8×C83—21no—
52C4⋊C8×C83C2×C412yes—
53C4⋊C8×C83C2×C412yes—
54C22×Dic83—21no—
55C22×Dic83—21no—
56C22×Dic83—41no—
57C2×C83⋊C8—21no—
58C2×C83⋊C8—21no—
59C2×C83⋊C8—21no—
60C2×C83⋊C8—21no—
61C2×C83⋊C8—21no—
62C2×C83⋊C8—41no—
63C2×C83⋊C8—41no—
64C2×C83⋊C8—41no—
65C2×C83⋊C8SD1612yes—
66C4⋊Dic83—21no—
67C4⋊Dic83—21no—
68C4⋊Dic83—41no—
69C4⋊Dic83—41no—
70C4⋊Dic83—41no—
71C4⋊Dic83—41no—
72C4⋊Dic83—81no—
73C4⋊Dic83SD1612yes—
74C4⋊Dic83Q1612yes—
75C4×Dic83—41no—
76C4×Dic83—41no—
77C2×Q8×C83—22no—
78C2×Q8×C83—41no—
79C2×Q8×C83—81no—
80C2×Q8×C83—81no—
81C22×C4×C83—41no—
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83#4111yescharacteristic
85C2×C8×C83—22no—
86C2×C8×C83—22no—
87C2×C8×C83—41no—
88C2×C8×C83—41no—
89C2×C8×C83—41no—
90C2×C8×C83—41no—
91C4⋊C4×C83—22no—
92C4⋊C4×C83—22no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C42×C83C22×C411yescharacteristic
96C42×C83#4111yescharacteristic
97C42×C83#4111yescharacteristic
98C42×C83#4111yescharacteristic
99C23×C83#12111yescharacteristic
100Q8×C83—82no—
101Q8×C83—82no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—42no—
105C2×C4×C83—81no—
106C2×C4×C83—81no—
107C2×C4×C83#11711yescharacteristic
108C2×C4×C83C4×D411yescharacteristic
109C2×C4×C83C4×D411yescharacteristic
110C2×C4×C83#12311yescharacteristic
111C2×C4×C83#12411yescharacteristic
112C2×C4×C83#12611yescharacteristic
113C2×C4×C83C2×SD1611yescharacteristic
114C2×C4×C83C2×SD1611yescharacteristic
115C2×C4×C83C2×Q1611yescharacteristic
116C2×C4×C83#13511yescharacteristic
117C2×C4×C83#13511yescharacteristic
118C2×C4×C83#13511yescharacteristic
119C2×C4×C83#13711yescharacteristic
120C2×C4×C83#13711yescharacteristic
121C2×Dic83—22no—
122C2×Dic83—22no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—81no—
128C2×Dic83—81no—
129C2×Dic83—81no—
130C8×C83—42no—
131C8×C83—42no—
132C83⋊C8—22no—
133C83⋊C8—41no—
134C83⋊C8—81no—
135C22×C83#38811yescharacteristic
136C22×C83#46111yescharacteristic
137C22×C83#46411yescharacteristic
138C22×C83#46511yescharacteristic
139C22×C83#46711yescharacteristic
140C22×C83#47111yescharacteristic
141C22×C83#47211yescharacteristic
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—82no—
145C4×C83C4×SD1611yescharacteristic
146C4×C83C4×SD1611yescharacteristic
147C4×C83C4×Q1611yescharacteristic
148C4×C83#44111yescharacteristic
149C4×C83#48311yescharacteristic
150C4×C83#48511yescharacteristic
151C4×C83#48711yescharacteristic
152C4×C83#48811yescharacteristic
153Dic83—42no—
154Dic83—42no—
155Dic83—161no—
156#9567—831nomaximal, Sylow
157C2×C83#167111yescharacteristic
158C2×C83#167511yescharacteristic
159C2×C83#167811yescharacteristic
160C2×C83#168411yescharacteristic
161C2×C83#192711yescharacteristic
162C2×C83#193111yescharacteristic
163C2×C83#193311yescharacteristic
164#1990—831no—
165#1987—831no—
166#1948—831no—
167C8×C4⋊C4—831no—
168C2×#333D8311yescharacteristic
169#1296—831no—
170#1277—831no—
171C83#956711yescharacteristic, complemented, Sylow
172C2×C4⋊Q8C2×D8311yescharacteristic
173C2×#106—1661no—
174C2×C4⋊C8—831no—
175C2×C4⋊C8C2×D8311yescharacteristic
176C2×C4⋊C8C2×D8311yescharacteristic
177C2×C4×C8—831no—
178C2×C4×C8—1661no—
179#384—831no—
180C4×C4⋊C4—831no—
181#340—1661no—
182#340—1661no—
183#340—1661no—
184#340—1661no—
185#340—1661no—
186#336—1661no—
187#333Dic8314yescomplemented
188C22×Q8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—831no—
192C22×C8—831no—
193C22×C8—1661no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C2×Dic8312yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—831no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—1661no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C42—1661no—
206C2×C42C22×D8311yescharacteristic
207#106—3322no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—1661no—
211C4⋊C8—1661no—
212C4⋊C8C2×Dic8312yes—
213C4⋊C8C2×Dic8312yes—
214C4×C8—3322no—
215C4×C8—1661no—
216C4×C8—1661no—
217#95—1661no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—81no—
221C2×Q8—81no—
222C22×C4—41no—
223C22×C4—3321no—
224C22×C4—1661no—
225C22×C4—1661no—
226C22×C4D4×D8311yescharacteristic
227C22×C4D4×D8311yescharacteristic
228C22×C4#1305059011yescharacteristic
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—3321no—
236C2×C8—3321no—
237C2×C8—3321no—
238C2×C8—832no—
239C2×C8—1661no—
240C2×C8—1661no—
241C2×C8—1661no—
242C2×C8—1661no—
243C2×C8—1661no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—3321no—
249C4⋊C4—3321no—
250C4⋊C4—3321no—
251C4⋊C4—3321no—
252C4⋊C4—832no—
253C4⋊C4—832no—
254C4⋊C4—6641no—
255C4⋊C4—1661no—
256C4⋊C4—1661no—
257C42—3321no—
258C42—3321no—
259C42C22×Dic8311yescharacteristic
260C42#1305058911yescharacteristic
261C42#1305058911yescharacteristic
262C42#1305059011yescharacteristic
263C23#1458176511yescharacteristic
264Q8—82no—
265Q8—82no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—3321no—
272C2×C4—3321no—
273C2×C4—3321no—
274C2×C4—3321no—
275C2×C4—6641no—
276C2×C4—6641no—
277C2×C4—6641no—
278C2×C4—1662no—
279C2×C4—1662no—
280C2×C4#1458175111yescharacteristic
281C2×C4D4×Dic8311yescharacteristic
282C2×C4D4×Dic8311yescharacteristic
283C2×C4#1458177411yescharacteristic
284C2×C4#1458177811yescharacteristic
285C2×C4#1458178511yescharacteristic
286C2×C4SD16×D8311yescharacteristic
287C2×C4SD16×D8311yescharacteristic
288C2×C4Q16×D8311yescharacteristic
289C2×C4#1458181311yescharacteristic
290C2×C4#1458181311yescharacteristic
291C2×C4#1458181511yescharacteristic
292C2×C4#1458182411yescharacteristic
293C2×C4#1458182411yescharacteristic
294C8—42no—
295C8—42no—
296C8—3321no—
297C8—6641no—
298C8—1662no—
299C22#1912759411yescharacteristic
300C22#1912790511yescharacteristic
301C22#1912792611yescharacteristic
302C22#1912793311yescharacteristic
303C22#1912794111yescharacteristic
304C22#1912796311yescharacteristic
305C22#1912797011yescharacteristic
306C4—21no—
307C4—21no—
308C4—82no—
309C4—13281no—
310C4—3322no—
311C4—3322no—
312C4SD16×Dic8311yescharacteristic
313C4SD16×Dic8311yescharacteristic
314C4Q16×Dic8311yescharacteristic
315C4#1912780811yescharacteristic
316C4#1912803011yescharacteristic
317C4#1912804011yescharacteristic
318C4#1912805011yescharacteristic
319C4#1912805711yescharacteristic
320C2#2490684111yescharacteristic
321C2#2490686111yescharacteristic
322C2#2490688211yescharacteristic
323C2#2490692411yescharacteristic
324C2#2490809911yescharacteristic
325C2#2490812711yescharacteristic
326C2#2490814111yescharacteristic
subgroupG/H
centerC23#14581765
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#333×C83C2
socle, prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenterC2×#333D83
hypocenterC83#9567
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
23. #333×C83C4semidirect
171. C83#9567semidirect
187. #333Dic83semidirect

Maximal subgroups

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

subgroupindex
#249084252
#249084122
#249082272
#249073282
#249053472
#249052972
C2×#333×C832
#956783