Finite Group Databasebeta

← #35300189

Subgroups — #35300189

Order 21248.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#24915794C211yesmaximal, characteristic
2#24907531C211yesmaximal, characteristic, complemented
3C2×#19127434C212yesmaximal
4#24905342C212yesmaximal, complemented
5C2×#383×C83C211yesmaximal, characteristic, complemented
6C2×#14581775—21no—
7C2×#14581766C2211yescharacteristic
8C2×C22⋊(C83⋊C8)C2212yes—
9#19127603C412yescomplemented
10#19127599C412yescomplemented
11#19127592—22no—
12#19127580—21no—
13#19127561C2211yescharacteristic
14#19127456—22no—
15#19127434—24no—
16C22×C42×C83C2211yescharacteristic
17#383×C83—24no—
18C2×#95×C83—21no—
19C2×#95×C83—21no—
20C2×#95×C83C2212yes—
21C22×D332C2×C411yescharacteristic
22C22×C22⋊D83—21no—
23C22×C83⋊C8—22no—
24#14581775—44no—
25#14581766—22no—
26#14581766—22no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×C4⋊Dic83C2×C412yes—
29C2×Dic83⋊C4—22no—
30C2×C22⋊Dic83—21no—
31C2×(C2×D83)⋊C4—22no—
32C2×(C2×D83)⋊C4—22no—
33C2×(C2×D83)⋊C4—21no—
34C22⋊(C83⋊C8)—22no—
35C22⋊(C83⋊C8)—22no—
36C83⋊#95—22no—
37C83⋊#95—21no—
38C23×C4×C83—22no—
39C23×C4×C83D411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C23×C4×C83C2311yescharacteristic
42C2×C42×C83—22no—
43C2×C42×C83—22no—
44C2×C42×C83C2×C412yes—
45C2×C42×C83D412yes—
46#95×C83—24no—
47#95×C83—24no—
48#95×C83—22no—
49#95×C83D414yes—
50C23×D83—21no—
51C2×D332—22no—
52C2×D332—42no—
53C2×C22⋊D83—41no—
54C2×C22⋊D83—41no—
55C2×C22⋊D83—42no—
56C2×C22⋊D83—42no—
57C22×Dic83—21no—
58C22×Dic83—22no—
59C2×C83⋊C8—44no—
60C2×C83⋊C8—42no—
61C4⋊Dic83—22no—
62C4⋊Dic83—42no—
63Dic83⋊C4—44no—
64C22⋊Dic83—42no—
65(C2×D83)⋊C4—44no—
66(C2×D83)⋊C4—44no—
67(C2×D83)⋊C4—42no—
68C24×C83C2×D411yescharacteristic
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—22no—
74C22×C4×C83—22no—
75C22×C4×C83—24no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83—44no—
79C22×C4×C83—44no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83C22⋊C412yes—
83C22×C4×C83C22⋊C412yes—
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D412yes—
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83#4112yes—
88C42×C83—24no—
89C42×C83—24no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83C2×C22⋊C411yescharacteristic
93C23×C83#12011yescharacteristic
94C23×C83#12311yescharacteristic
95C23×C83#13512yes—
96C23×C83AGL(1,ℤ/8)12yes—
97C22×D83—41no—
98C22×D83—42no—
99C22×D83—41no—
100C22×D83—41no—
101C22×D83—42no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—44no—
117C2×C4×C83—44no—
118C2×C4×C83—44no—
119C2×C4×C83—84no—
120C2×C4×C83—84no—
121C2×C4×C83C4×D412yes—
122C2×C4×C83#12112yes—
123C22⋊D83—82no—
124C22⋊D83—82no—
125C2×Dic83—41no—
126C2×Dic83—42no—
127C2×Dic83—41no—
128C2×Dic83—42no—
129C2×Dic83—41no—
130C2×Dic83—42no—
131D332—82no—
132C83⋊C8—82no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—24no—
138C22×C83—24no—
139C22×C83#38611yescharacteristic
140C22×C83#41012yes—
141C22×C83#41611yescharacteristic
142C22×C83#41811yescharacteristic
143C22×C83#42112yes—
144C22×C83#44912yes—
145C22×C83#48212yes—
146C2×D83—81no—
147C2×D83—81no—
148C2×D83—81no—
149C2×D83—82no—
150C2×D83—82no—
151C4×C83—22no—
152C4×C83—22no—
153C4×C83—42no—
154C4×C83—42no—
155C4×C83—84no—
156Dic83—81no—
157Dic83—82no—
158#9157—831nomaximal, Sylow
159C2×C83—22no—
160C2×C83—22no—
161C2×C83#151412yes—
162C2×C83#152112yes—
163C2×C83#187511yescharacteristic
164C2×C83#189711yescharacteristic
165C2×C83#189911yescharacteristic
166D83—161no—
167#2854—831no—
168C2×#383D8311yescharacteristic, complemented
169#1817—831no—
170C2×#327—832no—
171#1294—832no—
172C83#915711yescharacteristic, complemented, Sylow
173C2×#123—1661no—
174C2×#121—831no—
175C22×C42C2×D8311yescharacteristic
176C2×C22⋊C8—832no—
177#390—832no—
178#389—832no—
179#388—1662no—
180#386—1661no—
181#383—24no—
182#380—831no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95C2×D8312yes—
186#336—1662no—
187#327—1664no—
188C22×D4—831no—
189C22×D4—1661no—
190C23×C4—22no—
191C23×C4C22⋊D8311yescharacteristic
192C23×C4C22⋊D8311yescharacteristic
193C23×C4C22×D8311yescharacteristic
194C22×C8—1662no—
195#123—3324no—
196#121—1662no—
197#121—1662no—
198C2×C4⋊C4—832no—
199C2×C4⋊C4—831no—
200C2×C4⋊C4—1662no—
201C2×C22⋊C4—1661no—
202C2×C22⋊C4—1662no—
203C2×C22⋊C4—1662no—
204C2×C22⋊C4—1661no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D8312yes—
208C2×C42D33212yes—
209C22⋊C8—1662no—
210C22⋊C8—1662no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—1662no—
215#95—1661no—
216#95C22⋊D8314yes—
217C24—1661no—
218C24C2×C22⋊D8311yescharacteristic
219C2×D4—3322no—
220C2×D4—3321no—
221C2×D4—3321no—
222C2×D4—3322no—
223C2×D4—3322no—
224C2×D4—1662no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—1662no—
239C22×C4—1661no—
240C22×C4(C2×D83)⋊C412yes—
241C22×C4(C2×D83)⋊C412yes—
242C22×C4C2×C4×D8311yescharacteristic
243C22×C4C2×C22⋊D8312yes—
244C22×C4C2×D33211yescharacteristic
245C22×C4#1305059012yes—
246C2×C8—3324no—
247C2×C8—3322no—
248C4⋊C4—3322no—
249C4⋊C4—3324no—
250C4⋊C4—1662no—
251C22⋊C4—3324no—
252C22⋊C4—3324no—
253C22⋊C4—3322no—
254C22⋊C4—3322no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—3321no—
260C23—3322no—
261C23—3321no—
262C23—3321no—
263C23—3322no—
264C23C2×(C2×D83)⋊C411yescharacteristic
265C23#1458176111yescharacteristic
266C23#1458177511yescharacteristic
267C23#1458181212yes—
268C23#1458181712yes—
269D4—6642no—
270D4—6642no—
271D4—6642no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—3321no—
292C2×C4—3322no—
293C2×C4—3321no—
294C2×C4—3322no—
295C2×C4—3321no—
296C2×C4—3322no—
297C2×C4C4×C22⋊D8312yes—
298C2×C4#1458176612yes—
299C8—6642no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—6641no—
307C22—6642no—
308C22—6642no—
309C22—6641no—
310C22—6641no—
311C22#1912758011yescharacteristic
312C22#1912767412yes—
313C22#1912769511yescharacteristic
314C22#1912770511yescharacteristic
315C22#1912771912yes—
316C22#1912784612yes—
317C22#1912802512yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—84no—
323C4—6641no—
324C4—6642no—
325C2—22no—
326C2—22no—
327C2—13281no—
328C2#2490600912yes—
329C2#2490603812yes—
330C2#2490778511yescharacteristic
331C2#2490791111yescharacteristic
332C2#2490792111yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC2×#383×C83C2
socleC23×C83C2×C22⋊C4
prime radicalC2×#14581766C22
cosocleC23×C4×C83C23
hypercenterC2×#383D83
hypocenterC83#9157
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
2. #24907531C2semidirect
4. #24905342C2semidirect
5. C2×#383×C83C2semidirect
9. #19127603C4semidirect
10. #19127599C4semidirect
168. C2×#383D83semidirect
172. C83#9157semidirect

Maximal subgroups

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

subgroupindex
#249157942
#249075312
C2×#191274342
#249053422
C2×#383×C832
#915783