Finite Group Databasebeta

← #35323466

Subgroups — #35323466

Order 21248.

450 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128105C211yesmaximal, characteristic, complemented
2C2×#19127502C211yesmaximal, characteristic
3#24908330C211yesmaximal, characteristic
4#24905555C211yesmaximal, characteristic, complemented
5#24905554C211yesmaximal, characteristic, complemented
6#24905540C211yesmaximal, characteristic, complemented
7C2×#357×C83C211yesmaximal, characteristic
8C2×#14581791—21no—
9C22×C83⋊C16C2211yescharacteristic
10#19128105—22no—
11#19128105—22no—
12C2×#14581726C2211yescharacteristic
13C2×#14581726C2211yescharacteristic
14C2×#14581725—21no—
15C2×#14581713—21no—
16C2×C8×Dic83C2211yescharacteristic
17#19127604—21no—
18#19127502—22no—
19C22×D8×C83C2211yescharacteristic
20C22×C16×C83C2211yescharacteristic
21C2×#107×C83C2211yescharacteristic
22#357×C83—22no—
23C22×Dic166—41no—
24C22×C83⋊C8D411yescharacteristic
25#14581791—24no—
26#14581791—42no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×Dic83⋊C4—41no—
30C2×C22⋊Dic83—41no—
31C2×C22⋊Dic83—41no—
32C2×C4×Dic83D411yescharacteristic
33C2×C83⋊C16—21no—
34C2×C83⋊C16—21no—
35C2×C83⋊C16—21no—
36#14581726—21no—
37#14581726—21no—
38#14581726—21no—
39#14581726D412yes—
40#14581725—24no—
41#14581713—42no—
42C8×Dic83—21no—
43C8×Dic83—21no—
44C22×D4×C83—21no—
45C2×D8×C83—21no—
46C2×D8×C83—21no—
47C2×D8×C83—21no—
48C2×D8×C83—21no—
49C2×D8×C83—21no—
50C2×D8×C83Q812yescomplemented
51C22×C8×C83C2311yescharacteristic
52C2×C4⋊C4×C83—21no—
53C2×C16×C83—21no—
54C2×C16×C83—21no—
55C2×C16×C83—21no—
56#107×C83—21no—
57#107×C83D412yes—
58C2×Dic166—44no—
59C2×Dic166—82no—
60C2×Dic166—82no—
61C22×Dic83—41no—
62C22×Dic83—41no—
63C22×Dic83D811yescharacteristic
64C22×Dic83D811yescharacteristic
65C2×C83⋊C8—21no—
66C2×C83⋊C8—21no—
67C2×C83⋊C8SD1612yes—
68C4⋊Dic83—22no—
69C4⋊Dic83—41no—
70C4⋊Dic83—41no—
71C4⋊Dic83—41no—
72Dic83⋊C4—44no—
73C22⋊Dic83—82no—
74C22⋊Dic83—82no—
75C4×Dic83—21no—
76C4×Dic83—21no—
77C83⋊C16—41no—
78C24×C83—41no—
79C2×D4×C83—22no—
80C2×D4×C83—41no—
81C2×D4×C83—41no—
82C2×D4×C83—41no—
83C2×D4×C83—41no—
84C2×D4×C83—41no—
85C22×C4×C83—41no—
86C22×C4×C83C2×D411yescharacteristic
87D8×C83—22no—
88D8×C83—22no—
89D8×C83—41no—
90D8×C83—41no—
91C2×C8×C83C2×D411yescharacteristic
92C2×C8×C83C2×D411yescharacteristic
93C2×C8×C83C2×Q811yescharacteristic
94C2×C8×C83#4111yescharacteristic
95C2×C8×C83#4111yescharacteristic
96C2×C8×C83#4111yescharacteristic
97C4⋊C4×C83—22no—
98C4⋊C4×C83—41no—
99C16×C83—41no—
100C23×C83—42no—
101C23×C83—81no—
102C23×C83—81no—
103C23×C83—81no—
104C23×C83—81no—
105C23×C83—81no—
106C23×C83—81no—
107C23×C83C2×D811yescharacteristic
108D4×C83—42no—
109D4×C83—42no—
110D4×C83—81no—
111D4×C83—81no—
112C2×C4×C83—42no—
113C2×C4×C83—81no—
114C2×C4×C83—81no—
115C2×C4×C83#12011yescharacteristic
116C2×C4×C83C22⋊Q811yescharacteristic
117C2×C4×C83#12411yescharacteristic
118C2×C4×C83C2×SD1611yescharacteristic
119C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C83#13711yescharacteristic
121C2×Dic83—21no—
122C2×Dic83—21no—
123C2×Dic83—21no—
124C2×Dic83—21no—
125C2×Dic83—42no—
126C2×Dic83—81no—
127C2×Dic83—81no—
128C2×Dic83—81no—
129C2×Dic83—81no—
130C2×Dic83—81no—
131C2×Dic83D1612yes—
132C2×Dic83SD3212yes—
133Dic166—88no—
134C8×C83C22⋊Q811yescharacteristic
135C8×C83C22⋊Q811yescharacteristic
136C8×C83#12411yescharacteristic
137C8×C83#12411yescharacteristic
138C83⋊C8—22no—
139C22×C83—82no—
140C22×C83—82no—
141C22×C83—82no—
142C22×C83—161no—
143C22×C83—161no—
144C22×C83—161no—
145C22×C83—161no—
146C22×C83#44711yescharacteristic
147C22×C83#47411yescharacteristic
148C22×C83#48611yescharacteristic
149C22×C83C2×D1611yescharacteristic
150C22×C83C2×SD3211yescharacteristic
151C22×C83#50911yescharacteristic
152C22×C83#51011yescharacteristic
153C4×C83—82no—
154C4×C83#39311yescharacteristic
155C4×C83#44811yescharacteristic
156C4×C83#47611yescharacteristic
157C4×C83#48911yescharacteristic
158Dic83—22no—
159Dic83—22no—
160Dic83—82no—
161Dic83—161no—
162#13363—831nomaximal, Sylow
163C2×C83—162no—
164C2×C83#202311yescharacteristic
165C2×C83#218411yescharacteristic
166C2×C83#218511yescharacteristic
167C2×C83#222411yescharacteristic
168C2×C83#222611yescharacteristic
169C2×C83#224011yescharacteristic
170C2×C83#224311yescharacteristic
171C2×#500—831no—
172C2×#357—831no—
173C2×#357D8311yescharacteristic
174#1970—831no—
175#1380—831no—
176#1380—831no—
177#1374—831no—
178C83#1336311yescharacteristic, complemented, Sylow
179C22×D8C2×D8311yescharacteristic
180C2×C4⋊Q8—1661no—
181C22×C16—831no—
182C22×C16C2×D8311yescharacteristic
183#500—1662no—
184#500—1662no—
185C2×#107—831no—
186C2×#107—831no—
187C2×#107—1661no—
188C2×#107C2×D8311yescharacteristic
189C2×#102—1661no—
190C2×C4×C8—831no—
191#390—1661no—
192#357—22no—
193#357—1662no—
194C22×Q8—3321no—
195C22×D4—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×D8Dic16612yescomplemented
202C22×C8—831no—
203C22×C8C22×D8311yescharacteristic
204C4⋊Q8—3322no—
205C4⋊Q8—1664no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—3321no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4—1661no—
210C2×C22⋊C4—3321no—
211C2×C22⋊C4—3321no—
212C2×C42—831no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—1661no—
217C2×C16—1661no—
218C2×C16—1661no—
219#107—21no—
220#107—832no—
221#107—1664no—
222#107—1661no—
223#107—1661no—
224#107—1661no—
225#107C22⋊D8312yes—
226#102—3322no—
227C4×C8—1661no—
228C4×C8—1661no—
229C24—41no—
230C2×Q8—3324no—
231C2×Q8—6642no—
232C2×Q8—6642no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—3321no—
241C22×C4—3321no—
242C22×C4—831no—
243C22×C4—831no—
244C22×C4D4×D8311yescharacteristic
245D8—22no—
246D8—22no—
247D8—41no—
248D8—41no—
249C2×C8—832no—
250C2×C8—1661no—
251C2×C8—1661no—
252C2×C8D4×D8311yescharacteristic
253C2×C8C2×C22⋊D8311yescharacteristic
254C2×C8C2×Dic16611yescharacteristic
255C2×C8#1305058911yescharacteristic
256C2×C8#1305058911yescharacteristic
257C2×C8#1305059011yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—3324no—
261C4⋊C4—3321no—
262C4⋊C4—3321no—
263C4⋊C4—3321no—
264C4⋊C4—1662no—
265C22⋊C4—6642no—
266C22⋊C4—6642no—
267C42—1661no—
268C42—1661no—
269C16—41no—
270C16—3321no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23D8×D8311yescharacteristic
279Q8—6648no—
280D4—42no—
281D4—42no—
282D4—81no—
283D4—81no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—3322no—
288C2×C4—832no—
289C2×C4—832no—
290C2×C4—6641no—
291C2×C4—6641no—
292C2×C4—6641no—
293C2×C4—6641no—
294C2×C4—6641no—
295C2×C4—1661no—
296C2×C4—1661no—
297C2×C4—1661no—
298C2×C4—1661no—
299C2×C4#1458176011yescharacteristic
300C2×C4#1458176911yescharacteristic
301C2×C4#1458177811yescharacteristic
302C2×C4SD16×D8311yescharacteristic
303C2×C4#1458181611yescharacteristic
304C2×C4#1458182111yescharacteristic
305C8—1662no—
306C8#1458176911yescharacteristic
307C8#1458177111yescharacteristic
308C8#1458177811yescharacteristic
309C8#1458177911yescharacteristic
310C22—82no—
311C22—82no—
312C22—82no—
313C22—161no—
314C22—161no—
315C22—161no—
316C22—161no—
317C22#1912783311yescharacteristic
318C22#1912797911yescharacteristic
319C22#1912804611yescharacteristic
320C22D16×D8311yescharacteristic
321C22SD32×D8311yescharacteristic
322C22#1912813511yescharacteristic
323C22#1912814011yescharacteristic
324C4—82no—
325C4—13281no—
326C4—6642no—
327C4—1662no—
328C4—1662no—
329C4#1912761311yescharacteristic
330C4#1912783811yescharacteristic
331C4#1912798911yescharacteristic
332C4#1912805911yescharacteristic
333C2—162no—
334C2#2490859711yescharacteristic
335C2#2490937311yescharacteristic
336C2#2490937811yescharacteristic
337C2#2490959811yescharacteristic
338C2#2490960811yescharacteristic
339C2#2490968511yescharacteristic
340C2#2490969811yescharacteristic
subgroupG/H
centerC23D8×D83
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC2×#357×C83C2
socleC23×C83C2×D8
prime radicalC22×D8×C83C22
hypercenterC2×#357D83
hypocenterC83#13363
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. C2×#19128105C2semidirect
4. #24905555C2semidirect
5. #24905554C2semidirect
6. #24905540C2semidirect
50. C2×D8×C83Q8semidirect
178. C83#13363semidirect
201. C2×D8Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191281052
C2×#191275022
#249083302
#249055552
#249055542
#249055402
C2×#357×C832
#1336383