Finite Group Databasebeta

← #35315457

Subgroups — #35315457

Order 21248.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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#24912462C211yesmaximal, characteristic
2#24905787C211yesmaximal, characteristic, complemented
3#24905465C214yesmaximal, complemented
4C83×#1801C211yesmaximal, characteristic, complemented
5C2×#14581789C411yescharacteristic
6C2×#14581781—21no—
7#19127754—21no—
8#19127618C412yes—
9#19127618C2212yes—
10#19127603—21no—
11#19127603—21no—
12#19127602—21no—
13C4×C4⋊Dic83C411yescharacteristic
14#19127427—21no—
15#19127427C2212yes—
16C2×C4×Q8×C83C2211yescharacteristic
17#423×C83—21no—
18C2×#101×C83—21no—
19#343×C83C2212yes—
20C22×D332—21no—
21C22×D332—21no—
22C22×D332C2×C412yes—
23#14581789—41no—
24#14581789—41no—
25#14581781—42no—
26#14581781—42no—
27C2×C4⋊Dic83—21no—
28C2×(C2×D83)⋊C4—22no—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4—22no—
31C2×(C2×D83)⋊C4—22no—
32C2×C4×Dic83—21no—
33C4⋊(C83⋊C8)—41no—
34C22⋊(C83⋊C8)—22no—
35C4×C83⋊C8—41no—
36C83⋊#95C2×C412yes—
37C22×Q8×C83D411yescharacteristic
38C2×M16×C83—22no—
39C4×Q8×C83—41no—
40C4×Q8×C83—41no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C4⋊C4×C83C2311yescharacteristic
44C2×C42×C83C2×C411yescharacteristic
45C2×C42×C83D411yescharacteristic
46C2×C42×C83D411yescharacteristic
47C4⋊C8×C83—41no—
48C4⋊C8×C83—41no—
49#101×C83—42no—
50C23×D83—22no—
51C2×D332—42no—
52C2×D332—42no—
53C2×D332—42no—
54C2×D332—42no—
55C2×D332—42no—
56C2×D332—41no—
57C2×D332—41no—
58C2×D332—42no—
59C2×D332—82no—
60C22×Dic83—22no—
61C2×C83⋊C8—42no—
62C4⋊Dic83—42no—
63(C2×D83)⋊C4—44no—
64(C2×D83)⋊C4—42no—
65(C2×D83)⋊C4—42no—
66(C2×D83)⋊C4—82no—
67(C2×D83)⋊C4—82no—
68C4×Dic83—42no—
69C2×Q8×C83—41no—
70C2×Q8×C83—41no—
71C2×Q8×C83D812yes—
72C2×Q8×C83SD1612yes—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80M16×C83—42no—
81M16×C83—42no—
82C2×C8×C83—42no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—41no—
86C42×C83—21no—
87C42×C83—21no—
88C42×C83—21no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C23×C83C2×C22⋊C411yescharacteristic
93C22×D83—42no—
94C22×D83—42no—
95C22×D83—42no—
96C22×D83—42no—
97C22×D83—42no—
98C22×D83—42no—
99C22×D83—42no—
100Q8×C83—42no—
101Q8×C83—42no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—22no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83#9911yescharacteristic
116C2×C4×C83#9911yescharacteristic
117C2×C4×C83#10212yes—
118C2×C4×C83#10212yes—
119C2×C4×C83C2×D811yescharacteristic
120C2×C4×C83C2×SD1611yescharacteristic
121C2×Dic83—42no—
122C2×Dic83—42no—
123C2×Dic83—42no—
124D332—82no—
125D332—84no—
126D332—84no—
127D332—82no—
128C8×C83—82no—
129C83⋊C8—82no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83C2×#9911yescharacteristic
133C22×C83C2×#10211yescharacteristic
134C22×C83#42111yescharacteristic
135C2×D83—82no—
136C2×D83—82no—
137C2×D83—82no—
138C2×D83—82no—
139C2×D83—82no—
140C2×D83—82no—
141C2×D83—82no—
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—22no—
145C4×C83—22no—
146C4×C83—21no—
147C4×C83—41no—
148C4×C83—41no—
149C4×C83—41no—
150C4×C83#35512yes—
151Dic83—82no—
152#11882—831nomaximal, Sylow
153C2×C83—21no—
154C2×C83—21no—
155C2×C83#151911yescharacteristic
156C2×C83#212211yescharacteristic
157C2×C83C2×#35511yescharacteristic
158D83—162no—
159#2548—831no—
160#1801D8311yescharacteristic, complemented
161#1461—831no—
162#1345—834no—
163C83#1188211yescharacteristic, complemented, Sylow
164C2×#127—831no—
165C2×#124—1661no—
166C2×C4×Q8C2×D8311yescharacteristic
167#431—1661no—
168#423—21no—
169C2×#101—21no—
170#394—832no—
171#394—832no—
172#390—1661no—
173#390—1661no—
174#390—1661no—
175C4×C4⋊C4—831no—
176#343C2×D8312yes—
177#324—832no—
178#324—1661no—
179C22×Q8C22⋊D8311yescharacteristic
180C22×D4—832no—
181C22×D4—1661no—
182C22×D4—1661no—
183C2×M16—22no—
184#127—3321no—
185#127—3321no—
186#124—3322no—
187#124—3322no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—1661no—
191C2×C4⋊C4C4×D8311yescharacteristic
192C2×C4⋊C4D33211yescharacteristic
193C2×C4⋊C4C22×D8311yescharacteristic
194C2×C22⋊C4—1662no—
195C2×C22⋊C4—1662no—
196C2×C22⋊C4—1662no—
197C2×C22⋊C4—1662no—
198C2×C42—1661no—
199C2×C42C4×D8311yescharacteristic
200C2×C42D33211yescharacteristic
201C2×C42C22⋊D8311yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—3321no—
205#101—42no—
206C22⋊C8—1662no—
207C4×C8—3321no—
208#95—832no—
209C24—1662no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1305057512yes—
213C2×Q8#1305057712yes—
214C2×D4—3322no—
215C2×D4—3322no—
216C2×D4—3322no—
217C2×D4—3322no—
218C2×D4—3322no—
219C2×D4—3321no—
220C2×D4—3321no—
221C2×D4—3322no—
222C2×D4—6642no—
223C22×C4—1662no—
224C22×C4(C2×D83)⋊C411yescharacteristic
225C22×C4(C2×D83)⋊C411yescharacteristic
226C22×C4(C2×D83)⋊C411yescharacteristic
227C22×C4(C2×D83)⋊C411yescharacteristic
228C22×C4C2×C4×D8311yescharacteristic
229C22×C4C2×C22⋊D8311yescharacteristic
230C22×C4C2×D33211yescharacteristic
231M16—42no—
232M16—42no—
233C2×C8—42no—
234C2×C8—3322no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—3322no—
239C22⋊C4—3322no—
240C22⋊C4—3322no—
241C22⋊C4—3324no—
242C22⋊C4—6642no—
243C22⋊C4—6642no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—3322no—
251C23—3322no—
252C23—3322no—
253C23—3322no—
254C23—3322no—
255C23—3322no—
256C23—3322no—
257C23—3322no—
258C23C2×(C2×D83)⋊C411yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—6644no—
262D4—6642no—
263D4—6644no—
264D4—6642no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—3322no—
279C2×C4—3322no—
280C2×C4—3322no—
281C2×C4#1458170611yescharacteristic
282C2×C4#1458170611yescharacteristic
283C2×C4#1458171212yes—
284C2×C4#1458171212yes—
285C2×C4C2×#1305057511yescharacteristic
286C2×C4C2×#1305057711yescharacteristic
287C8—82no—
288C8—6642no—
289C22—21no—
290C22—21no—
291C22—6642no—
292C22—6642no—
293C22—6642no—
294C22—6642no—
295C22—6642no—
296C22—6642no—
297C22—6642no—
298C22C2×#1458170611yescharacteristic
299C22C2×#1458171211yescharacteristic
300C22#1912771911yescharacteristic
301C4—21no—
302C4—21no—
303C4—22no—
304C4—22no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—6642no—
310C4#1912749712yes—
311C2—21no—
312C2—21no—
313C2—13282no—
314C2#2490603011yescharacteristic
315C2#2490911411yescharacteristic
316C2C2×#1912749711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1801C2
socleC22×C83C2×#99
prime radicalC2×#14581789C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1801D83
hypocenterC83#11882
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #24905787C2semidirect
3. #24905465C2semidirect
4. C83×#1801C2semidirect
160. #1801D83semidirect
163. C83#11882semidirect

Maximal subgroups

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

subgroupindex
#249124622
#249057872
#249054652
C83×#18012
#1188283