Finite Group Databasebeta

← #35324198

Subgroups — #35324198

Order 21248.

378 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
1C2×#19127503C211yesmaximal, characteristic
2C2×#19127501C211yesmaximal, characteristic
3#24908327C211yesmaximal, characteristic
4#24907621C211yesmaximal, characteristic, complemented
5#24905554C211yesmaximal, characteristic, complemented
6#24905540C211yesmaximal, characteristic, complemented
7C83×#1380C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C22×C83⋊C16C2211yescharacteristic
10C2×#14581726C2211yescharacteristic
11C2×#14581725—21no—
12C2×#14581712—21no—
13C2×C8×Dic83C2211yescharacteristic
14#19127602—21no—
15#19127570—21no—
16#19127503—22no—
17#19127501—22no—
18C22×C16×C83C2211yescharacteristic
19C2×#107×C83C2211yescharacteristic
20C2×#107×C83C2211yescharacteristic
21C22×D332—21no—
22C2×D664—21no—
23C2×D664—21no—
24C2×D664—21no—
25C2×D664—21no—
26C2×D664—21no—
27C2×D664D412yes—
28C22×C83⋊C8D411yescharacteristic
29C2×C4⋊Dic83—21no—
30C2×Dic83⋊C4—41no—
31C2×Dic83⋊C4—41no—
32C2×(C2×D83)⋊C4—41no—
33C2×(C2×D83)⋊C4—41no—
34C2×C4×Dic83D411yescharacteristic
35C2×C83⋊C16—21no—
36C2×C83⋊C16—21no—
37C2×C83⋊C16—21no—
38#14581726—21no—
39#14581726—21no—
40#14581725—42no—
41#14581712—42no—
42C8×Dic83—21no—
43C8×Dic83—21no—
44C22×C8×C83C2311yescharacteristic
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83—21no—
47C2×C16×C83—21no—
48C2×C16×C83—21no—
49C2×C16×C83—21no—
50#107×C83—21no—
51#107×C83—21no—
52#107×C83—21no—
53#107×C83—21no—
54C23×D83—41no—
55C2×D332—22no—
56C2×D332—41no—
57C2×D332—41no—
58C2×D332—41no—
59C2×D332—41no—
60C2×D332—41no—
61C22×Dic83—41no—
62C22×Dic83D811yescharacteristic
63C22×Dic83D811yescharacteristic
64D664—22no—
65D664—22no—
66D664—41no—
67D664—41no—
68C2×C83⋊C8—21no—
69C2×C83⋊C8—21no—
70C2×C83⋊C8SD1612yes—
71C4⋊Dic83—41no—
72C4⋊Dic83—41no—
73Dic83⋊C4—82no—
74Dic83⋊C4—82no—
75(C2×D83)⋊C4—82no—
76(C2×D83)⋊C4—82no—
77C4×Dic83—21no—
78C4×Dic83—21no—
79C83⋊C16—41no—
80C22×C4×C83—41no—
81C22×C4×C83—41no—
82C22×C4×C83C2×D411yescharacteristic
83C2×C8×C83C2×D411yescharacteristic
84C2×C8×C83#4111yescharacteristic
85C2×C8×C83#4111yescharacteristic
86C2×C8×C83#4111yescharacteristic
87C2×C8×C83#4111yescharacteristic
88C2×C8×C83#4111yescharacteristic
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C16×C83—41no—
94C23×C83C2×D811yescharacteristic
95C22×D83—42no—
96C22×D83—81no—
97C22×D83—81no—
98C22×D83—81no—
99C22×D83—81no—
100C22×D83—81no—
101C22×D83—81no—
102C2×C4×C83—81no—
103C2×C4×C83—81no—
104C2×C4×C83—81no—
105C2×C4×C83—81no—
106C2×C4×C83—81no—
107C2×C4×C83—81no—
108C2×C4×C83#12111yescharacteristic
109C2×C4×C83#12311yescharacteristic
110C2×C4×C83#12411yescharacteristic
111C2×C4×C83C2×SD1611yescharacteristic
112C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
113C2×C4×C83#13711yescharacteristic
114C2×Dic83—21no—
115C2×Dic83—21no—
116C2×Dic83—21no—
117C2×Dic83—21no—
118C2×Dic83—81no—
119C2×Dic83—81no—
120C2×Dic83—81no—
121C2×Dic83D1612yes—
122C2×Dic83SD3212yes—
123D332—42no—
124D332—42no—
125D332—81no—
126D332—81no—
127C8×C83#12311yescharacteristic
128C8×C83#12311yescharacteristic
129C8×C83#12611yescharacteristic
130C8×C83#12611yescharacteristic
131C83⋊C8—22no—
132C22×C83#45911yescharacteristic
133C22×C83#48011yescharacteristic
134C22×C83#48611yescharacteristic
135C22×C83C2×D1611yescharacteristic
136C22×C83C2×SD3211yescharacteristic
137C22×C83#50911yescharacteristic
138C22×C83#51011yescharacteristic
139C2×D83—82no—
140C2×D83—82no—
141C2×D83—82no—
142C2×D83—161no—
143C2×D83—161no—
144C2×D83—161no—
145C2×D83—161no—
146C4×C83—161no—
147C4×C83—161no—
148C4×C83#39711yescharacteristic
149C4×C83#46111yescharacteristic
150C4×C83#48111yescharacteristic
151C4×C83#48911yescharacteristic
152Dic83—22no—
153Dic83—22no—
154Dic83—161no—
155#13480—831nomaximal, Sylow
156C2×C83#207111yescharacteristic
157C2×C83#220611yescharacteristic
158C2×C83#220811yescharacteristic
159C2×C83#223211yescharacteristic
160C2×C83#223311yescharacteristic
161C2×C83#224011yescharacteristic
162C2×C83#224311yescharacteristic
163D83—162no—
164C2×#357—831no—
165C2×#357—831no—
166#1970—831no—
167#1845—831no—
168#1380—831no—
169#1380D8311yescharacteristic, complemented
170#1374—831no—
171C83#1348011yescharacteristic, complemented, Sylow
172C22×D8—831no—
173C22×C16—831no—
174C22×C16C2×D8311yescharacteristic
175C2×#107—831no—
176C2×#107—1661no—
177C2×#107C2×D8311yescharacteristic
178C2×#107C2×D8311yescharacteristic
179C2×#102—1661no—
180C2×C4×C8—831no—
181#390—1661no—
182#384—1661no—
183#357—1662no—
184#357—1662no—
185C22×D4—1661no—
186C2×D8—832no—
187C2×D8—1661no—
188C2×D8—1661no—
189C2×D8—1661no—
190C2×D8—1661no—
191C2×D8—1661no—
192C22×C8—831no—
193C22×C8C22×D8311yescharacteristic
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—3321no—
197C2×C4⋊C4—3321no—
198C2×C4⋊C4—1661no—
199C2×C22⋊C4—3321no—
200C2×C22⋊C4—3321no—
201C2×C42—831no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—1661no—
206C2×C16—1661no—
207C2×C16—1661no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—3322no—
213#107—1661no—
214#107—1661no—
215#102—3322no—
216C4×C8—1661no—
217C4×C8—1661no—
218C24—3321no—
219C2×D4—3321no—
220C2×D4—3321no—
221C2×D4—3321no—
222C2×D4—3321no—
223C2×D4—3321no—
224C2×D4—1662no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—3321no—
228C22×C4—831no—
229C22×C4—831no—
230C22×C4D4×D8311yescharacteristic
231D8—3321no—
232D8—3321no—
233D8—1662no—
234D8—1662no—
235C2×C8—832no—
236C2×C8—1661no—
237C2×C8—1661no—
238C2×C8D4×D8311yescharacteristic
239C2×C8#1305058911yescharacteristic
240C2×C8#1305059011yescharacteristic
241C2×C8#1305059011yescharacteristic
242C2×C8#1305059011yescharacteristic
243C2×C8#1305059111yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—3321no—
249C4⋊C4—3321no—
250C4⋊C4—6642no—
251C4⋊C4—6642no—
252C22⋊C4—6642no—
253C22⋊C4—6642no—
254C42—1661no—
255C42—1661no—
256C16—41no—
257C16—3321no—
258C23—3322no—
259C23—6641no—
260C23—6641no—
261C23—6641no—
262C23—6641no—
263C23—6641no—
264C23—6641no—
265C23D8×D8311yescharacteristic
266D4—3322no—
267D4—3322no—
268D4—6641no—
269D4—6641no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—832no—
277C2×C4—832no—
278C2×C4—6641no—
279C2×C4—6641no—
280C2×C4—6641no—
281C2×C4—1661no—
282C2×C4—1661no—
283C2×C4—1661no—
284C2×C4—1661no—
285C2×C4#1458176511yescharacteristic
286C2×C4#1458177711yescharacteristic
287C2×C4#1458177811yescharacteristic
288C2×C4SD16×D8311yescharacteristic
289C2×C4#1458181911yescharacteristic
290C2×C4#1458182411yescharacteristic
291C8—1662no—
292C8#1458177411yescharacteristic
293C8#1458177711yescharacteristic
294C8#1458178611yescharacteristic
295C8#1458178711yescharacteristic
296C22—13281no—
297C22—13281no—
298C22—13281no—
299C22—13281no—
300C22—6642no—
301C22—6642no—
302C22—6642no—
303C22#1912789511yescharacteristic
304C22#1912801611yescharacteristic
305C22#1912804611yescharacteristic
306C22D16×D8311yescharacteristic
307C22SD32×D8311yescharacteristic
308C22#1912813811yescharacteristic
309C22#1912814311yescharacteristic
310C4—161no—
311C4—161no—
312C4—13281no—
313C4—1662no—
314C4—1662no—
315C4#1912763011yescharacteristic
316C4#1912790511yescharacteristic
317C4#1912802111yescharacteristic
318C4#1912806211yescharacteristic
319C2—13282no—
320C2#2490887011yescharacteristic
321C2#2490949711yescharacteristic
322C2#2490950711yescharacteristic
323C2#2490964511yescharacteristic
324C2#2490965011yescharacteristic
325C2#2490968511yescharacteristic
326C2#2490970111yescharacteristic
subgroupG/H
centerC23D8×D83
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1380C2
socleC23×C83C2×D8
prime radicalC22×D664C22
hypercenter#1380D83
hypocenterC83#13480
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
4. #24907621C2semidirect
5. #24905554C2semidirect
6. #24905540C2semidirect
7. C83×#1380C2semidirect
169. #1380D83semidirect
171. C83#13480semidirect

Maximal subgroups

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

subgroupindex
C2×#191275032
C2×#191275012
#249083272
#249076212
#249055542
#249055402
C83×#13802
#1348083