Finite Group Databasebeta

← #35300170

Subgroups — #35300170

Order 21248.

586 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#19127569C211yesmaximal, characteristic
2#24907531C211yesmaximal, characteristic
3C2×#19127438C212yesmaximal
4#24905341C212yesmaximal
5C83×#2855C211yesmaximal, characteristic
6C2×C22⋊(C83⋊C8)C2212yes—
7#19127569—24no—
8C2×C83⋊#95—21no—
9C2×C83⋊#95—21no—
10C2×C83⋊#95C2212yes—
11#19127456—22no—
12#19127438—24no—
13C2×C22⋊Q8×C83C2211yescharacteristic
14C22×C42×C83C2211yescharacteristic
15#391×C83C412yescomplemented
16#389×C83C412yescomplemented
17#387×C83—22no—
18#385×C83—21no—
19#381×C83—21no—
20#380×C83C2211yescharacteristic
21C23×Dic83—22no—
22C22×C83⋊C8—22no—
23C22⋊(C83⋊C8)—22no—
24C22⋊(C83⋊C8)—22no—
25C83⋊#95—24no—
26C83⋊#95—24no—
27C83⋊#95—22no—
28C83⋊#95D414yes—
29C22×Q8×C83C2×C411yescharacteristic
30C23×C4×C83D411yescharacteristic
31C23×C4×C83D411yescharacteristic
32C23×C4×C83C2311yescharacteristic
33C22⋊Q8×C83—22no—
34C22⋊Q8×C83—22no—
35C2×C4⋊C4×C83—22no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83C2×C411yescharacteristic
38C2×C4⋊C4×C83C2×C412yes—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C42×C83—22no—
42C2×C42×C83—22no—
43C2×C42×C83C2×C412yes—
44C2×C42×C83D412yes—
45#95×C83—22no—
46#95×C83—22no—
47#95×C83—22no—
48#95×C83—21no—
49C22×Dic83—24no—
50C22×Dic83—44no—
51C22×Dic83—44no—
52C22×Dic83—42no—
53C22×Dic83—42no—
54C2×C83⋊C8—44no—
55C2×C83⋊C8—42no—
56C24×C83C2×D411yescharacteristic
57C2×Q8×C83—22no—
58C2×Q8×C83—42no—
59C22×C4×C83—21no—
60C22×C4×C83—21no—
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83—22no—
64C22×C4×C83—22no—
65C22×C4×C83—22no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—42no—
69C22×C4×C83—42no—
70C22×C4×C83C22⋊C412yes—
71C22×C4×C83C22⋊C412yes—
72C22×C4×C83C22×C411yescharacteristic
73C22×C4×C83C2×D412yes—
74C22×C4×C83C2×D411yescharacteristic
75C22×C4×C83#4112yes—
76C4⋊C4×C83—22no—
77C4⋊C4×C83—42no—
78C4⋊C4×C83—44no—
79C4⋊C4×C83—42no—
80C22⋊C4×C83—41no—
81C22⋊C4×C83—41no—
82C22⋊C4×C83—42no—
83C42×C83—24no—
84C42×C83—24no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83C2×C22⋊C411yescharacteristic
88C23×C83#12011yescharacteristic
89C23×C83#12311yescharacteristic
90C23×C83#13512yes—
91C23×C83#13712yes—
92Q8×C83—82no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—42no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83C4×D412yes—
115C2×C4×C83#12112yes—
116C2×Dic83—44no—
117C2×Dic83—44no—
118C2×Dic83—44no—
119C2×Dic83—84no—
120C2×Dic83—84no—
121C83⋊C8—82no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—24no—
127C22×C83—24no—
128C22×C83#38611yescharacteristic
129C22×C83#41012yes—
130C22×C83#41611yescharacteristic
131C22×C83#41911yescharacteristic
132C22×C83#42112yes—
133C22×C83#45212yes—
134C22×C83#48512yes—
135C4×C83—22no—
136C4×C83—22no—
137C4×C83—42no—
138C4×C83—42no—
139C4×C83—81no—
140C4×C83—81no—
141C4×C83—82no—
142Dic83—84no—
143#9153—831nomaximal, Sylow
144C2×C83—22no—
145C2×C83—22no—
146C2×C83#151512yes—
147C2×C83#152112yes—
148C2×C83#187311yescharacteristic
149C2×C83#189711yescharacteristic
150C2×C83#189911yescharacteristic
151#2855D8311yescharacteristic
152C2×#383—831no—
153#1817—831no—
154C2×#328—832no—
155#1294—832no—
156C83#915311yescharacteristic, complemented, Sylow
157C2×C22⋊Q8C2×D8311yescharacteristic
158C22×C42C2×D8311yescharacteristic
159C2×C22⋊C8—832no—
160#391Dic8312yescomplemented
161#389Dic8312yescomplemented
162#387—22no—
163#385—21no—
164#383—1664no—
165#381—21no—
166#380C2×D8311yescharacteristic
167C2×#95—832no—
168C2×#95—1661no—
169C2×#95—1661no—
170#336—1662no—
171#328—1664no—
172C22×Q8C2×Dic8311yescharacteristic
173C23×C4—1662no—
174C23×C4C22⋊D8311yescharacteristic
175C23×C4C22⋊D8311yescharacteristic
176C23×C4C22×D8311yescharacteristic
177C22×C8—1662no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×Dic8311yescharacteristic
183C2×C4⋊C4C2×Dic8312yes—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C2×Dic8312yes—
189C2×C42C22⋊D8312yes—
190C22⋊C8—1662no—
191C22⋊C8—1662no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95—834no—
197#95—1664no—
198#95—1664no—
199#95—1662no—
200C24C2×C22⋊D8311yescharacteristic
201C2×Q8—22no—
202C2×Q8—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—3324no—
215C22×C4—3324no—
216C22×C4—3322no—
217C22×C4—3322no—
218C22×C4—1664no—
219C22×C4C22⋊Dic8312yes—
220C22×C4C22⋊Dic8312yes—
221C22×C4C22×Dic8311yescharacteristic
222C22×C4D4×D8312yes—
223C22×C4C2×C22⋊D8311yescharacteristic
224C22×C4#1305058912yes—
225C2×C8—3324no—
226C2×C8—3322no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×C22⋊Dic8311yescharacteristic
239C23#1458176011yescharacteristic
240C23#1458177311yescharacteristic
241C23#1458181212yes—
242C23#1458182312yes—
243Q8—82no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—3324no—
266C2×C4—3324no—
267C2×C4—3324no—
268C2×C4—6644no—
269C2×C4—6644no—
270C2×C4D4×Dic8312yes—
271C2×C4#1458176412yes—
272C8—6642no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1912758111yescharacteristic
280C22#1912767512yes—
281C22#1912769711yescharacteristic
282C22#1912771011yescharacteristic
283C22#1912771812yes—
284C22#1912786112yes—
285C22#1912803912yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—82no—
291C4—81no—
292C4—81no—
293C4—6644no—
294C2—22no—
295C2—22no—
296C2#2490601412yes—
297C2#2490604012yes—
298C2#2490777611yescharacteristic
299C2#2490791211yescharacteristic
300C2#2490792211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2855C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#2855D83
hypocenterC83#9153
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. #391×C83C4semidirect
16. #389×C83C4semidirect
156. C83#9153semidirect
160. #391Dic83semidirect
161. #389Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275692
#249075312
C2×#191274382
#249053412
C83×#28552
#915383