Finite Group Databasebeta

← #35301051

Subgroups — #35301051

Order 21248.

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

normal subgroups only

This group has 410 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#24918614C211yesmaximal, characteristic
2#24916400C211yesmaximal, characteristic
3#24908437C211yesmaximal, characteristic
4#24906028C211yesmaximal, characteristic
5#24905323C211yesmaximal, characteristic
6#24905293C211yesmaximal, characteristic
7C83×#1548C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19128104C412yescomplemented
10#19128053C412yescomplemented
11#19128036—22no—
12#19127993—22no—
13#19127830—22no—
14#19127830—22no—
15#19127754C2211yescharacteristic
16#19127729—21no—
17#19127659—21no—
18C2×#14581703C2211yescharacteristic
19#19127567C2211yescharacteristic
20#19127463—21no—
21#19127463—21no—
22#19127439—22no—
23C2×C4⋊Q8×C83C2211yescharacteristic
24#431×C83C2211yescharacteristic
25#423×C83C2211yescharacteristic
26#333×C83—22no—
27C22×C83⋊C8D411yescharacteristic
28C22×C83⋊C8D411yescharacteristic
29#14581792C2×C411yescharacteristic
30#14581792C2×C411yescharacteristic
31#14581784—21no—
32#14581771—21no—
33#14581771D412yes—
34C4×Dic166—21no—
35C4×Dic166D412yes—
36#14581751—21no—
37C2×C4⋊Dic83—21no—
38#14581725—24no—
39C4⋊(C83⋊C8)—21no—
40Dic83⋊C8—24no—
41#14581715—24no—
42C22⋊(C83⋊C8)—21no—
43#14581703—21no—
44#14581703—21no—
45#14581702—22no—
46C4×C83⋊C8C2×C412yes—
47C8×Dic83—22no—
48C83⋊#95—21no—
49C83⋊#95—21no—
50C22×Q8×C83—21no—
51C2×M16×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83—21no—
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83C2×C412yes—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83—21no—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83D412yes—
62C22⋊C8×C83—21no—
63C4×C8×C83—21no—
64C2×Dic166—21no—
65C2×Dic166C22⋊C412yes—
66C22×Dic83—41no—
67C2×C83⋊C8—21no—
68C2×C83⋊C8—21no—
69C2×C83⋊C8—22no—
70C2×C83⋊C8—21no—
71C2×C83⋊C8—21no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8Q1612yes—
74C4⋊Dic83—21no—
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83—41no—
78C4⋊Dic83—41no—
79C4⋊Dic83C22⋊C412yes—
80Dic83⋊C4—22no—
81Dic83⋊C4—22no—
82C22⋊Dic83—22no—
83C4×Dic83—22no—
84C2×Q8×C83—22no—
85C2×Q8×C83—41no—
86C2×Q8×C83—82no—
87C22×C4×C83—41no—
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83#4111yescharacteristic
91M16×C83—41no—
92M16×C83—41no—
93C2×C8×C83—22no—
94C2×C8×C83—41no—
95C4⋊C4×C83—22no—
96C4⋊C4×C83—22no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C42×C83C22×C411yescharacteristic
100C42×C83C2×D411yescharacteristic
101C42×C83C2×D411yescharacteristic
102C42×C83#4111yescharacteristic
103C23×C83#12111yescharacteristic
104Q8×C83—82no—
105Q8×C83—82no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—42no—
109C2×C4×C83—82no—
110C2×C4×C83C2×C22⋊C411yescharacteristic
111C2×C4×C83C4×D411yescharacteristic
112C2×C4×C83C4×D411yescharacteristic
113C2×C4×C83#12011yescharacteristic
114C2×C4×C83#12111yescharacteristic
115C2×C4×C83#12311yescharacteristic
116C2×C4×C83C2×Q1611yescharacteristic
117C2×C4×C83#13511yescharacteristic
118C2×C4×C83#13711yescharacteristic
119C2×C4×C83#13711yescharacteristic
120C2×Dic83—22no—
121C2×Dic83—22no—
122C2×Dic83—81no—
123C2×Dic83—81no—
124C2×Dic83—81no—
125Dic166—42no—
126Dic166—81no—
127C8×C83—42no—
128C8×C83—81no—
129C83⋊C8—22no—
130C83⋊C8—42no—
131C83⋊C8—41no—
132C22×C83—21no—
133C22×C83—21no—
134C22×C83#38611yescharacteristic
135C22×C83#46211yescharacteristic
136C22×C83#46411yescharacteristic
137C4×C83—21no—
138C4×C83—21no—
139C4×C83—21no—
140C4×C83—21no—
141C4×C83—82no—
142C4×C83C4×Q1611yescharacteristic
143C4×C83#44111yescharacteristic
144C4×C83#45111yescharacteristic
145C4×C83#45211yescharacteristic
146Dic83—42no—
147Dic83—42no—
148Dic83—161no—
149#9288—831nomaximal, Sylow
150C2×C83—21no—
151C2×C83—21no—
152C2×C83#187811yescharacteristic
153C2×C83#190311yescharacteristic
154C2×C83#190911yescharacteristic
155#3230—831no—
156#2961—831no—
157#1993—831no—
158#1548D8311yescharacteristic
159#1518—831no—
160#1286—831no—
161#1275—831no—
162C83#928811yescharacteristic, complemented, Sylow
163#533—831no—
164C2×C4⋊Q8C2×D8311yescharacteristic
165#500—832no—
166#487—832no—
167#484—1662no—
168#477—1662no—
169#446—1662no—
170#446—1662no—
171#431—831no—
172#431C2×D8311yescharacteristic
173#424—1661no—
174#423C2×D8311yescharacteristic
175#405—1661no—
176C2×#97—831no—
177#382—831no—
178#340—1661no—
179#340—1661no—
180#333—22no—
181#328—1662no—
182C22×Q8—21no—
183C2×M16—21no—
184C22×C8—831no—
185C22×C8—831no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—831no—
190C4⋊Q8—831no—
191C4⋊Q8C4×D8312yes—
192#125—1661no—
193C22⋊Q8—832no—
194C22⋊Q8—1661no—
195C4×Q8—832no—
196C4×Q8—1661no—
197#117—1661no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—1661no—
201C2×C42C22×D8311yescharacteristic
202#107—1664no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—1664no—
206C4⋊C8—1661no—
207C4⋊C8C22⋊D8312yes—
208#103—1664no—
209C22⋊C8—21no—
210C22⋊C8—1661no—
211#97—1662no—
212#97—1661no—
213#97—1661no—
214C4×C8—21no—
215C4×C8—832no—
216C4×C8—1662no—
217#95—1661no—
218#95—1661no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—82no—
222C2×Q8—832no—
223C2×Q8—1661no—
224C22×C4—41no—
225C22×C4—3321no—
226C22×C4D4×D8311yescharacteristic
227C22×C4D4×D8311yescharacteristic
228C22×C4#1305059011yescharacteristic
229M16—41no—
230M16—41no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—832no—
234C2×C8—1661no—
235C2×C8—1661no—
236C2×C8—1661no—
237C2×C8—1662no—
238C2×C8—1661no—
239C2×C8—1661no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—3321no—
245C4⋊C4—3321no—
246C4⋊C4—832no—
247C4⋊C4—1662no—
248C4⋊C4—1662no—
249C4⋊C4—1661no—
250C4⋊C4—1661no—
251C4⋊C4—1661no—
252C22⋊C4—1662no—
253C42—1662no—
254C42C2×C4×D8311yescharacteristic
255C42D4×D8311yescharacteristic
256C42C2×C22⋊D8311yescharacteristic
257C42#1305058911yescharacteristic
258C23#1458176511yescharacteristic
259Q8—82no—
260Q8—82no—
261Q8—3322no—
262Q8—6641no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—6641no—
268C2×C4—6641no—
269C2×C4—6641no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4C22⋊C4×D8311yescharacteristic
273C2×C4(C22⋊D83)⋊C411yescharacteristic
274C2×C4C4×C22⋊D8311yescharacteristic
275C2×C4#1458176011yescharacteristic
276C2×C4#1458176311yescharacteristic
277C2×C4#1458177411yescharacteristic
278C2×C4Q16×D8311yescharacteristic
279C2×C4#1458181311yescharacteristic
280C2×C4#1458182111yescharacteristic
281C2×C4#1458182411yescharacteristic
282C8—42no—
283C8—81no—
284C8—3322no—
285C8—3321no—
286C8—1662no—
287C22—21no—
288C22—21no—
289C22#1912757711yescharacteristic
290C22#1912791011yescharacteristic
291C22#1912792611yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—82no—
297C4—13281no—
298C4—3322no—
299C4—3322no—
300C4#1912779311yescharacteristic
301C4#1912780511yescharacteristic
302C4#1912785511yescharacteristic
303C4#1912786011yescharacteristic
304C2—21no—
305C2—21no—
306C2#2490780111yescharacteristic
307C2#2490794611yescharacteristic
308C2#2490798211yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1548C2
socleC22×C83#386
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1548D83
hypocenterC83#9288
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
9. #19128104C4semidirect
10. #19128053C4semidirect
162. C83#9288semidirect

Maximal subgroups

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

subgroupindex
#249186142
#249164002
#249084372
#249060282
#249053232
#249052932
C83×#15482
#928883