Finite Group Databasebeta

← #35298037

Subgroups — #35298037

Order 21248.

466 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910436C211yesmaximal, characteristic
2C4×#14581707C211yesmaximal, characteristic
3#24905348C212yesmaximal, complemented
4#24905317C212yesmaximal, complemented
5C83×#1465C211yesmaximal, characteristic
6C2×#14581707—21no—
7#19127604—21no—
8#19127594—21no—
9#19127594—21no—
10#19127567C411yescharacteristic, complemented
11#19127564C411yescharacteristic, complemented
12#19127564C2211yescharacteristic
13C4⋊C4×Dic83—21no—
14C4×C4⋊Dic83C411yescharacteristic, complemented
15C4×C22⋊Dic83C411yescharacteristic, complemented
16C4×C22⋊Dic83C2212yes—
17C4×C22⋊Dic83C2211yescharacteristic
18#19127468—21no—
19C2×C4×D4×C83C2211yescharacteristic
20#407×C83C2212yes—
21#324×C83—21no—
22#323×C83—21no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C22⋊Dic83—21no—
26C2×C22⋊Dic83—21no—
27C2×C22⋊Dic83—22no—
28C2×C22⋊Dic83—21no—
29C2×C22⋊Dic83—21no—
30C2×C4×Dic83—21no—
31C2×C4×Dic83—21no—
32C2×C4×Dic83—21no—
33C2×C4×Dic83—22no—
34#14581707—24no—
35C83⋊#95—21no—
36C83⋊#95—21no—
37C83⋊#95—21no—
38C83⋊#95—22no—
39C83⋊#95C2×C411yescharacteristic
40C83⋊#95C2×C411yescharacteristic
41C83⋊#95C2×C411yescharacteristic
42C83⋊#95C2×C411yescharacteristic
43C22×D4×C83D411yescharacteristic
44C23×C4×C83C2×C411yescharacteristic
45C23×C4×C83C2311yescharacteristic
46C2×M16×C83—22no—
47C4×D4×C83—22no—
48C4×D4×C83—22no—
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C22⋊C4×C83D411yescharacteristic
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C42×C83C2×C411yescharacteristic
53C22⋊C8×C83—22no—
54C22⋊C8×C83—22no—
55C22×Dic83—21no—
56C22×Dic83—21no—
57C22×Dic83—21no—
58C22×Dic83—21no—
59C22×Dic83—42no—
60C4⋊Dic83—42no—
61C4⋊Dic83—42no—
62C22⋊Dic83—24no—
63C22⋊Dic83—42no—
64C22⋊Dic83—42no—
65C22⋊Dic83—42no—
66C22⋊Dic83—81no—
67C22⋊Dic83—81no—
68C4×Dic83—42no—
69C4×Dic83—81no—
70C4×Dic83—81no—
71C4×Dic83—82no—
72C24×C83C22⋊C411yescharacteristic
73C24×C83C2×D411yescharacteristic
74C2×D4×C83—22no—
75C2×D4×C83—22no—
76C2×D4×C83—41no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—81no—
80C22×C4×C83—81no—
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83#4112yes—
87C22×C4×C83#4112yes—
88M16×C83—82no—
89C2×C8×C83—42no—
90C4⋊C4×C83—22no—
91C22⋊C4×C83—22no—
92C22⋊C4×C83—22no—
93C42×C83—21no—
94C42×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83C2×C22⋊C411yescharacteristic
105C23×C83#12312yes—
106D4×C83—44no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—42no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—42no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—81no—
118C2×C4×C83—81no—
119C2×C4×C83#10412yes—
120C2×C4×C83#11711yescharacteristic
121C2×C4×C83#11711yescharacteristic
122C2×C4×C83#12312yes—
123C2×Dic83—22no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—44no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—82no—
137C2×Dic83—82no—
138C8×C83—82no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—22no—
142C22×C83—22no—
143C22×C83—22no—
144C22×C83—41no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—81no—
148C22×C83—81no—
149C22×C83—81no—
150C22×C83—81no—
151C22×C83#38111yescharacteristic
152C22×C83C2×#10411yescharacteristic
153C22×C83#42111yescharacteristic
154C4×C83—22no—
155C4×C83—22no—
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—41no—
159C4×C83—41no—
160Dic83—42no—
161Dic83—81no—
162Dic83—81no—
163Dic83—81no—
164Dic83—84no—
165#8778—831nomaximal, Sylow
166C2×C83—21no—
167C2×C83—21no—
168C2×C83—42no—
169C2×C83—81no—
170C2×C83#182611yescharacteristic
171C2×C83#182711yescharacteristic
172C2×C83#182811yescharacteristic
173#2357—831no—
174C4×#99—831no—
175#1465D8311yescharacteristic
176#1297—832no—
177#1284—832no—
178C83#877811yescharacteristic, complemented, Sylow
179C2×C4×D4C2×D8311yescharacteristic
180C2×#99—1661no—
181#407C2×D8312yes—
182#390—1661no—
183#388—1661no—
184#388—1661no—
185#382—831no—
186#381—831no—
187#381—831no—
188C4×C4⋊C4—831no—
189C4×C4⋊C4—1661no—
190C4×C22⋊C4—831no—
191C4×C22⋊C4—831no—
192C4×C22⋊C4—832no—
193#342—1661no—
194#324—21no—
195#323—21no—
196C22×D4D33211yescharacteristic
197C23×C4C4×D8311yescharacteristic
198C23×C4C22×D8311yescharacteristic
199C2×M16—22no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—1661no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4C22⋊D8311yescharacteristic
205C2×C22⋊C4—1661no—
206C2×C22⋊C4—1661no—
207C2×C22⋊C4—1662no—
208C2×C22⋊C4—1661no—
209C2×C22⋊C4—1661no—
210C2×C22⋊C4D33211yescharacteristic
211C2×C22⋊C4C22⋊D8311yescharacteristic
212C2×C42—1661no—
213C2×C42—1662no—
214C2×C42—1661no—
215C2×C42—1661no—
216C2×C42C4×D8311yescharacteristic
217#99—1664no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220#95—831no—
221#95—831no—
222#95—831no—
223#95—831no—
224#95—1662no—
225#95—1661no—
226#95—1661no—
227#95—1661no—
228C24(C2×D83)⋊C411yescharacteristic
229C24C2×D33211yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—3322no—
238C22×C4—1661no—
239C22×C4—1661no—
240C22×C4—1661no—
241C22×C4—1661no—
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4C2×C4×D8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4#1305058912yes—
248C22×C4#1305058912yes—
249M16—82no—
250C2×C8—42no—
251C4⋊C4—22no—
252C4⋊C4—3322no—
253C4⋊C4—3322no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—3322no—
257C22⋊C4—3322no—
258C22⋊C4—3322no—
259C22⋊C4—6641no—
260C22⋊C4—6641no—
261C22⋊C4—1664no—
262C42—21no—
263C42—21no—
264C42—3322no—
265C42—6641no—
266C42—6641no—
267C42—6642no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×(C2×D83)⋊C411yescharacteristic
278C23#1458177612yes—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—3321no—
293C2×C4—3321no—
294C2×C4—3321no—
295C2×C4—3321no—
296C2×C4—3321no—
297C2×C4—3321no—
298C2×C4—3324no—
299C2×C4—3321no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—6642no—
305C2×C4—6642no—
306C2×C4—1662no—
307C2×C4#1458171812yes—
308C2×C4#1458174811yescharacteristic
309C2×C4#1458174811yescharacteristic
310C2×C4#1458177312yes—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1912756211yescharacteristic
325C22C2×#1458171811yescharacteristic
326C22#1912771711yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—3322no—
334C4—6641no—
335C4—6641no—
336C4—6641no—
337C4—6644no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2490756011yescharacteristic
343C2#2490756811yescharacteristic
344C2#2490756911yescharacteristic
subgroupG/H
centerC22#19127562
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1465C2
socleC22×C83#381
prime radicalC22×D4×C83D4
cosocleC23×C4×C83C23
hypercenter#1465D83
hypocenterC83#8778
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #24905348C2semidirect
4. #24905317C2semidirect
10. #19127567C4semidirect
11. #19127564C4semidirect
14. C4×C4⋊Dic83C4semidirect
15. C4×C22⋊Dic83C4semidirect
178. C83#8778semidirect

Maximal subgroups

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

subgroupindex
#249104362
C4×#145817072
#249053482
#249053172
C83×#14652
#877883