Finite Group Databasebeta

← #35302368

Subgroups — #35302368

Order 21248.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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#24911771C211yesmaximal, characteristic
2#24907541C212yesmaximal, complemented
3#24905801C211yesmaximal, characteristic
4#24905368C212yesmaximal, complemented
5C83×#2975C211yesmaximal, characteristic
6C2×#14581773C412yes—
7C22×C4⋊Dic83C2211yescharacteristic
8#19127665C2212yes—
9#19127631—21no—
10#19127626—21no—
11#19127599—21no—
12#19127562C412yes—
13#19127561—21no—
14#19127561C2211yescharacteristic
15#19127469—24no—
16#19127427—21no—
17#19127425—21no—
18C2×C4×D4×C83C2211yescharacteristic
19#436×C83—24no—
20#431×C83—21no—
21#407×C83—21no—
22#407×C83C2212yes—
23C23×Dic83C2×C412yes—
24C2×#13050574—22no—
25#14581773—44no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—22no—
30C2×C4⋊Dic83—22no—
31C2×Dic83⋊C4—22no—
32C2×C22⋊Dic83—22no—
33C2×C22⋊Dic83C2×C412yes—
34C22⋊(C83⋊C8)—22no—
35C22⋊(C83⋊C8)—22no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C83⋊#95—21no—
39C22×D4×C83C2×C411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C23×C4×C83C2311yescharacteristic
42C2×M16×C83—22no—
43C2×M16×C83D414yes—
44C4×D4×C83—22no—
45C4×D4×C83—22no—
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50C4⋊C8×C83—22no—
51C22⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53C22⋊C8×C83—22no—
54C4×C8×C83—22no—
55C22×Dic83—22no—
56C22×Dic83—22no—
57C22×Dic83—24no—
58C22×Dic83—42no—
59C22×Dic83—42no—
60#13050574—44no—
61C2×C83⋊C8—42no—
62C4⋊Dic83—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—42no—
65C4⋊Dic83—41no—
66Dic83⋊C4—44no—
67C22⋊Dic83—44no—
68C22⋊Dic83—42no—
69C24×C83C22⋊C411yescharacteristic
70C24×C83C22×C411yescharacteristic
71C2×D4×C83—22no—
72C2×D4×C83—22no—
73C2×D4×C83—41no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C2×D412yes—
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83#4112yes—
85M16×C83—44no—
86M16×C83—82no—
87C2×C8×C83—24no—
88C4⋊C4×C83—22no—
89C22⋊C4×C83—22no—
90C22⋊C4×C83—22no—
91C42×C83—21no—
92C42×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83C2×C22⋊C411yescharacteristic
103C23×C83C4×D412yes—
104D4×C83—44no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—24no—
108C2×C4×C83—22no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—41no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—81no—
117C2×C4×C83#10012yes—
118C2×C4×C83#12011yescharacteristic
119C2×C4×C83#12112yes—
120C2×C4×C83#12311yescharacteristic
121C2×Dic83—42no—
122C2×Dic83—42no—
123C2×Dic83—42no—
124C2×Dic83—42no—
125C2×Dic83—42no—
126C2×Dic83—42no—
127C2×Dic83—84no—
128C8×C83—44no—
129C83⋊C8—82no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83—22no—
133C22×C83—22no—
134C22×C83—22no—
135C22×C83—42no—
136C22×C83—42no—
137C22×C83—42no—
138C22×C83—41no—
139C22×C83—41no—
140C22×C83—41no—
141C22×C83—81no—
142C22×C83#38611yescharacteristic
143C22×C83C2×#10011yescharacteristic
144C22×C83#41311yescharacteristic
145C4×C83—22no—
146C4×C83—21no—
147C4×C83—22no—
148C4×C83—41no—
149C4×C83—41no—
150C4×C83#45812yes—
151Dic83—82no—
152Dic83—82no—
153#9508—831nomaximal, Sylow
154C2×C83—21no—
155C2×C83—21no—
156C2×C83—42no—
157C2×C83—81no—
158C2×C83#189011yescharacteristic
159C2×C83#190011yescharacteristic
160C2×C83#191511yescharacteristic
161#2975D8311yescharacteristic
162#2486—831no—
163#1821—832no—
164#1465—831no—
165#1305—832no—
166C83#950811yescharacteristic, complemented, Sylow
167C2×#123—832no—
168C2×C4×D4C2×D8311yescharacteristic
169C22×C4⋊C4—831no—
170#436—24no—
171#431—21no—
172#407—21no—
173#407—832no—
174#407C2×D8312yes—
175#398—1661no—
176#397—1661no—
177#389—1661no—
178#381—832no—
179#380—831no—
180#380—1661no—
181#343—1664no—
182#324—1661no—
183#323—1661no—
184C22×D4C4×D8311yescharacteristic
185C23×C4—832no—
186C23×C4C22⋊D8311yescharacteristic
187C23×C4C22×D8311yescharacteristic
188C2×M16—22no—
189C2×M16—1662no—
190C2×M16C22⋊D8314yes—
191#123—3324no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1662no—
195C2×C4⋊C4—1662no—
196C2×C4⋊C4—1661no—
197C2×C4⋊C4—1661no—
198C2×C4⋊C4—1661no—
199C2×C4⋊C4—1662no—
200C2×C4⋊C4D33211yescharacteristic
201C2×C22⋊C4—832no—
202C2×C22⋊C4—1662no—
203C2×C22⋊C4C4×D8311yescharacteristic
204C2×C22⋊C4D33211yescharacteristic
205C2×C42C22⋊D8311yescharacteristic
206C4⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—1662no—
211C22⋊C8—1662no—
212C4×C8—22no—
213#95—1662no—
214#95—1662no—
215#95—1661no—
216C24(C2×D83)⋊C411yescharacteristic
217C24C2×C4×D8311yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—3322no—
226C22×C4—3322no—
227C22×C4—1662no—
228C22×C4—1662no—
229C22×C4—1664no—
230C22×C4(C2×D83)⋊C411yescharacteristic
231C22×C4(C2×D83)⋊C411yescharacteristic
232C22×C4(C2×D83)⋊C411yescharacteristic
233C22×C4C2×C22⋊D8312yes—
234C22×C4C2×C22⋊D8311yescharacteristic
235C22×C4C2×D33211yescharacteristic
236C22×C4#1305059012yes—
237M16—44no—
238M16—82no—
239M16—3324no—
240C2×C8—24no—
241C2×C8—3322no—
242C4⋊C4—22no—
243C4⋊C4—3322no—
244C4⋊C4—3321no—
245C4⋊C4—3322no—
246C4⋊C4—3321no—
247C4⋊C4—3324no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—3324no—
251C22⋊C4—3322no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×(C2×D83)⋊C411yescharacteristic
264C23C4×C22⋊D8312yes—
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—3322no—
279C2×C4—3322no—
280C2×C4—3322no—
281C2×C4—3322no—
282C2×C4—3322no—
283C2×C4—3322no—
284C2×C4—6644no—
285C2×C4#1458170812yes—
286C2×C4#1458176111yescharacteristic
287C2×C4#1458176612yes—
288C2×C4#1458177511yescharacteristic
289C8—44no—
290C8—6642no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1912758011yescharacteristic
304C22C2×#1458170811yescharacteristic
305C22#1912768311yescharacteristic
306C4—22no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—6642no—
312C4—6642no—
313C4#1912789112yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2#2490787611yescharacteristic
319C2#2490792611yescharacteristic
320C2#2490801511yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC24×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2975C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC23×C4×C83C23
hypercenter#2975D83
hypocenterC83#9508
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #24907541C2semidirect
4. #24905368C2semidirect
166. C83#9508semidirect

Maximal subgroups

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

subgroupindex
#249117712
#249075412
#249058012
#249053682
C83×#29752
#950883