Finite Group Databasebeta

← #35304284

Subgroups — #35304284

Order 21248.

478 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 125 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#24918045C211yesmaximal, characteristic
2#24908440C211yesmaximal, characteristic
3C42×C83⋊C8C211yesmaximal, characteristic
4#24906221C211yesmaximal, characteristic
5#24905918C211yesmaximal, characteristic
6#24905917C211yesmaximal, characteristic
7C83×#1994C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19128104C411yescharacteristic, complemented
10#19128080C411yescharacteristic, complemented
11#19128053C411yescharacteristic, complemented
12#19128051C411yescharacteristic, complemented
13#19127940—22no—
14C4×#13050580—22no—
15#19127725C2211yescharacteristic
16C2×C4×C83⋊C8—22no—
17C2×C4×C83⋊C8C2211yescharacteristic
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127461—22no—
20#19127453—22no—
21Q8⋊(C83⋊C8)—22no—
22#19127431—22no—
23#533×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25C43×C83C2211yescharacteristic
26#339×C83—22no—
27C2×#13050580—22no—
28C2×#13050574—21no—
29C22×C83⋊C8—21no—
30C22×C83⋊C8D412yes—
31#14581792C2×C411yescharacteristic
32#14581792C2×C411yescharacteristic
33#14581784—21no—
34#14581771—21no—
35#14581771—21no—
36C4×Dic166—21no—
37C4×Dic166—21no—
38#14581751—21no—
39#14581725—22no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42#14581716—22no—
43#14581715—22no—
44C4×C83⋊C8—28no—
45C4×C83⋊C8—21no—
46C4×C83⋊C8C2×C411yescharacteristic
47C4×C83⋊C8C2×C411yescharacteristic
48C4×C83⋊C8D412yes—
49C4×C83⋊C8D412yes—
50C2×M16×C83—21no—
51C4⋊Q8×C83C2×C411yescharacteristic
52C4⋊Q8×C83C2×C411yescharacteristic
53#125×C83—21no—
54C22⋊Q8×C83—21no—
55C22⋊Q8×C83—21no—
56C4×Q8×C83—21no—
57C4×Q8×C83—21no—
58#117×C83—21no—
59C2×C42×C83—21no—
60C2×C42×C83—21no—
61C2×C42×C83D412yes—
62C2×C42×C83C2311yescharacteristic
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C2×Dic166—21no—
66C2×Dic166—21no—
67#13050580—44no—
68#13050574—42no—
69C2×C83⋊C8—24no—
70C2×C83⋊C8—22no—
71C2×C83⋊C8—24no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8C22⋊C412yes—
74C2×C83⋊C8C22⋊C412yes—
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83—21no—
79Dic83⋊C4—41no—
80Dic83⋊C4—41no—
81C22⋊Dic83—41no—
82C4×Dic83—41no—
83C2×Q8×C83—21no—
84C2×Q8×C83—21no—
85C22×C4×C83—22no—
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83#4111yescharacteristic
89M16×C83—42no—
90C2×C8×C83—41no—
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83—21no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C22⋊C4×C83—41no—
98C42×C83—24no—
99C42×C83—24no—
100C42×C83—22no—
101C42×C83—41no—
102C42×C83Q1614yescomplemented
103C42×C83C22×C411yescharacteristic
104C42×C83C2×D411yescharacteristic
105C42×C83C2×D411yescharacteristic
106C42×C83#4111yescharacteristic
107C23×C83#12111yescharacteristic
108Q8×C83—42no—
109Q8×C83—42no—
110C2×C4×C83—24no—
111C2×C4×C83—24no—
112C2×C4×C83—24no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83C2×C22⋊C411yescharacteristic
118C2×C4×C83C4×D411yescharacteristic
119C2×C4×C83C4×D411yescharacteristic
120C2×C4×C83#12111yescharacteristic
121C2×C4×C83#12411yescharacteristic
122C2×C4×C83#12711yescharacteristic
123C2×C4×C83C2×Q1612yes—
124C2×C4×C83#13512yes—
125C2×Dic83—41no—
126C2×Dic83—41no—
127Dic166—42no—
128Dic166—42no—
129C8×C83—81no—
130C83⋊C8—24no—
131C83⋊C8—22no—
132C83⋊C8—81no—
133C83⋊C8#10414yes—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83#39011yescharacteristic
137C22×C83#46712yes—
138C4×C83—28no—
139C4×C83—21no—
140C4×C83—21no—
141C4×C83—81no—
142C4×C83—81no—
143C4×C83C2×#10412yes—
144C4×C83C4×Q1612yes—
145C4×C83#44312yes—
146C4×C83#49411yescharacteristic
147C4×C83#49511yescharacteristic
148Dic83—81no—
149Dic83—81no—
150#9845—831nomaximal, Sylow
151C2×C83—22no—
152C2×C83#196211yescharacteristic
153C2×C83#196911yescharacteristic
154C2×C83#197411yescharacteristic
155#3170—831no—
156#1994—831no—
157#1994D8311yescharacteristic
158C42×C8—831no—
159#1567—831no—
160#1493—831no—
161#1493—831no—
162C83#984511yescharacteristic, complemented, Sylow
163#533—831no—
164#533C2×D8311yescharacteristic
165#500—831no—
166#494—831no—
167#487—831no—
168#487—831no—
169#467—1662no—
170C4×Q16—1662no—
171#423—831no—
172#423C2×D8311yescharacteristic
173C2×C4×C8—831no—
174C2×C4×C8—831no—
175C2×C4×C8—1662no—
176C43C2×D8311yescharacteristic
177#339—22no—
178#339—1662no—
179#335—1662no—
180Q8⋊C8—1662no—
181Q8⋊C8—1662no—
182C2×Q16—1662no—
183C2×M16—21no—
184C2×M16—1661no—
185C22×C8—832no—
186C22×C8—1661no—
187C4⋊Q8—831no—
188C4⋊Q8—831no—
189C4⋊Q8C4×D8311yescharacteristic
190C4⋊Q8C4×D8311yescharacteristic
191#125—21no—
192#125—1661no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—1661no—
196C22⋊Q8—1661no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—1661no—
200C4×Q8—1661no—
201#117—21no—
202#117—1661no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C22⋊D8312yes—
206C2×C42C22×D8311yescharacteristic
207#107—1662no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—1661no—
211C4⋊C8—1661no—
212#103—1662no—
213#103—1662no—
214C4×C8—831no—
215C4×C8—831no—
216C4×C8—832no—
217C4×C8—832no—
218C4×C8—1668no—
219C4×C8—1661no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—1661no—
223C2×Q8—1661no—
224C22×C4—22no—
225C22×C4D4×D8311yescharacteristic
226C22×C4C2×C22⋊D8311yescharacteristic
227C22×C4#1305058911yescharacteristic
228Q16—3324no—
229M16—42no—
230M16—3322no—
231C2×C8—41no—
232C2×C8—3321no—
233C2×C8—832no—
234C2×C8—832no—
235C2×C8—1664no—
236C2×C8—1664no—
237C2×C8—1662no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—3321no—
245C4⋊C4—3321no—
246C4⋊C4—1661no—
247C4⋊C4—1661no—
248C4⋊C4—1661no—
249C4⋊C4—1661no—
250C22⋊C4—41no—
251C22⋊C4—3321no—
252C42—24no—
253C42—24no—
254C42—22no—
255C42—41no—
256C42—3321no—
257C42#1305058014yescomplemented
258C42C2×C4×D8311yescharacteristic
259C42D4×D8311yescharacteristic
260C42D4×D8311yescharacteristic
261C42#1305059011yescharacteristic
262C23#1458176411yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—3322no—
266Q8—3322no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—3321no—
275C2×C4—3321no—
276C2×C4C22⋊C4×D8311yescharacteristic
277C2×C4(C22⋊D83)⋊C411yescharacteristic
278C2×C4C4×C22⋊D8311yescharacteristic
279C2×C4#1458176511yescharacteristic
280C2×C4#1458177811yescharacteristic
281C2×C4#1458178811yescharacteristic
282C2×C4C2×#1305058012yes—
283C2×C4#1458181212yes—
284C8—81no—
285C8—834no—
286C8—6641no—
287C8—1664no—
288C8—1662no—
289C22—21no—
290C22—21no—
291C22#1912760111yescharacteristic
292C22#1912794012yes—
293C4—28no—
294C4—21no—
295C4—21no—
296C4—81no—
297C4—81no—
298C4—6641no—
299C4—6641no—
300C4#104×D8312yes—
301C4C4×#1305058012yes—
302C4#1912781512yes—
303C4#1912808011yescharacteristic
304C4#1912808311yescharacteristic
305C2—22no—
306C2#2490828911yescharacteristic
307C2#2490832311yescharacteristic
308C2#2490834811yescharacteristic
subgroupG/H
centerC2×C4#14581788
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1994C2
socleC22×C83#390
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1994D83
hypocenterC83#9845
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
9. #19128104C4semidirect
10. #19128080C4semidirect
11. #19128053C4semidirect
12. #19128051C4semidirect
102. C42×C83Q16semidirect
162. C83#9845semidirect
257. C42#13050580semidirect

Maximal subgroups

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

subgroupindex
#249180452
#249084402
C42×C83⋊C82
#249062212
#249059182
#249059172
C83×#19942
#984583