Finite Group Databasebeta

← #35300122

Subgroups — #35300122

Order 21248.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915802C211yesmaximal, characteristic
2C2×C4×C4⋊Dic83C211yesmaximal, characteristic
3C2×#19127439C212yesmaximal
4#24905343C212yesmaximal
5C83×#1817C211yesmaximal, characteristic
6C2×#14581771C2211yescharacteristic
7C22×C4⋊Dic83—21no—
8C22×C4×Dic83—21no—
9#19127607C412yescomplemented
10#19127599C412yescomplemented
11#19127585—22no—
12#19127574—21no—
13#19127564—21no—
14#19127561C2211yescharacteristic
15C4×C4⋊Dic83—24no—
16C2×C83⋊#95C2212yes—
17#19127439—24no—
18C22×C42×C83C2211yescharacteristic
19C2×C22⋊C8×C83C2212yes—
20#336×C83—22no—
21C22×Dic166C2×C411yescharacteristic
22C23×Dic83—22no—
23#14581771—22no—
24#14581771—22no—
25C2×C4⋊Dic83—24no—
26C2×C4⋊Dic83—28no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×C4⋊Dic83C2×C412yes—
29C2×Dic83⋊C4—22no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—24no—
34C2×C4×Dic83—42no—
35C2×C4×Dic83—42no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C83⋊#95—22no—
39C83⋊#95—22no—
40C83⋊#95—21no—
41C83⋊#95D414yes—
42C23×C4×C83D411yescharacteristic
43C23×C4×C83D411yescharacteristic
44C23×C4×C83C2311yescharacteristic
45C22×C8×C83—22no—
46C2×C42×C83—22no—
47C2×C42×C83—22no—
48C2×C42×C83C2×C412yes—
49C2×C42×C83D412yes—
50C22⋊C8×C83—22no—
51C22⋊C8×C83—22no—
52C2×Dic166—22no—
53C2×Dic166—42no—
54C22×Dic83—28no—
55C22×Dic83—24no—
56C22×Dic83—21no—
57C22×Dic83—22no—
58C22×Dic83—21no—
59C22×Dic83—44no—
60C22×Dic83—42no—
61C22×Dic83—42no—
62C4⋊Dic83—216no—
63C4⋊Dic83—22no—
64C4⋊Dic83—42no—
65Dic83⋊C4—44no—
66Dic83⋊C4—42no—
67C22⋊Dic83—41no—
68C22⋊Dic83—41no—
69C22⋊Dic83—41no—
70C22⋊Dic83—41no—
71C4×Dic83—48no—
72C24×C83C2×D411yescharacteristic
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—24no—
80C22×C4×C83—42no—
81C22×C4×C83C22⋊C412yes—
82C22×C4×C83C22⋊C412yes—
83C22×C4×C83SD1612yes—
84C22×C4×C83Q1612yes—
85C22×C4×C83C22×C411yescharacteristic
86C22×C4×C83C2×D412yes—
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83#4112yes—
89C2×C8×C83—44no—
90C2×C8×C83—42no—
91C42×C83—24no—
92C42×C83—24no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83C2×C22⋊C411yescharacteristic
96C23×C83#12011yescharacteristic
97C23×C83#12311yescharacteristic
98C23×C83C2×SD1611yescharacteristic
99C23×C83C2×Q1611yescharacteristic
100C23×C83AGL(1,ℤ/8)11yescharacteristic
101C23×C83#13711yescharacteristic
102C2×C4×C83—24no—
103C2×C4×C83—24no—
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—22no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83#10314yes—
117C2×C4×C83#10414yes—
118C2×C4×C83C4×D412yes—
119C2×C4×C83#12112yes—
120C2×Dic83—28no—
121C2×Dic83—44no—
122C2×Dic83—48no—
123C2×Dic83—48no—
124C2×Dic83—44no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132Dic166—82no—
133C8×C83—82no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—22no—
139C22×C83—22no—
140C22×C83—22no—
141C22×C83—22no—
142C22×C83#38611yescharacteristic
143C22×C83#41012yes—
144C22×C83C2×#10311yescharacteristic
145C22×C83#41711yescharacteristic
146C22×C83C2×#10412yes—
147C22×C83#45011yescharacteristic
148C22×C83#45111yescharacteristic
149C22×C83#48311yescharacteristic
150C22×C83#48411yescharacteristic
151C4×C83—24no—
152C4×C83—24no—
153C4×C83—22no—
154C4×C83—22no—
155Dic83—48no—
156Dic83—81no—
157Dic83—81no—
158Dic83—82no—
159#9144—831nomaximal, Sylow
160C2×C83—21no—
161C2×C83—21no—
162C2×C83—21no—
163C2×C83—21no—
164C2×C83#150312yes—
165C2×C83#150912yes—
166C2×C83#187211yescharacteristic
167C2×C83#189611yescharacteristic
168C2×C83#189911yescharacteristic
169#2855—831no—
170C2×C4×C4⋊C4—831no—
171#1817D8311yescharacteristic
172C2×#328—832no—
173#1294—832no—
174C83#914411yescharacteristic, complemented, Sylow
175C2×C22⋊Q8—831no—
176C22×C4⋊C4—1661no—
177C22×C42—1661no—
178C22×C42C2×D8311yescharacteristic
179C2×C22⋊C8C2×D8312yes—
180#391—832no—
181#389—832no—
182#387—1662no—
183#385—1661no—
184#381—1661no—
185#380—831no—
186C4×C4⋊C4—1664no—
187C2×#95—832no—
188#336—22no—
189#328—1664no—
190C22×Q8—831no—
191C23×C4—1662no—
192C23×C4D33211yescharacteristic
193C23×C4D33211yescharacteristic
194C23×C4C22×D8311yescharacteristic
195C22×C8—22no—
196C22⋊Q8—1662no—
197C22⋊Q8—1662no—
198C2×C4⋊C4—832no—
199C2×C4⋊C4—831no—
200C2×C4⋊C4—1662no—
201C2×C4⋊C4—1661no—
202C2×C4⋊C4—1664no—
203C2×C4⋊C4—1668no—
204C2×C22⋊C4—1661no—
205C2×C22⋊C4—1661no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—3322no—
209C2×C42—3322no—
210C2×C42—1664no—
211C2×C42C4×D8312yes—
212C2×C42C22⋊D8312yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—834no—
216#95—1662no—
217#95—1662no—
218#95—1662no—
219#95—1662no—
220#95—1661no—
221C24C2×D33211yescharacteristic
222C2×Q8—3322no—
223C2×Q8—1662no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—3324no—
233C22×C4—3322no—
234C22×C4—3322no—
235C22×C4—1662no—
236C22×C4—1661no—
237C22×C4—1661no—
238C22×C4—1668no—
239C22×C4—1664no—
240C22×C4(C2×D83)⋊C412yes—
241C22×C4(C2×D83)⋊C412yes—
242C22×C4#1305057912yes—
243C22×C4Dic33212yes—
244C22×C4C2×C4×D8311yescharacteristic
245C22×C4D4×D8312yes—
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4#1305058912yes—
248C2×C8—44no—
249C2×C8—42no—
250C4⋊C4—3322no—
251C4⋊C4—3322no—
252C4⋊C4—3324no—
253C4⋊C4—16616no—
254C4⋊C4—1662no—
255C22⋊C4—3321no—
256C22⋊C4—3321no—
257C22⋊C4—3321no—
258C22⋊C4—3321no—
259C42—24no—
260C42—24no—
261C42—3328no—
262C23—22no—
263C23—22no—
264C23C2×(C2×D83)⋊C411yescharacteristic
265C23#1458176211yescharacteristic
266C23#1458177611yescharacteristic
267C23C2×#1305057911yescharacteristic
268C23C2×Dic33211yescharacteristic
269C23#1458182011yescharacteristic
270C23#1458182511yescharacteristic
271Q8—6642no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—3328no—
287C2×C4—3324no—
288C2×C4—3328no—
289C2×C4—3324no—
290C2×C4—3321no—
291C2×C4—3321no—
292C2×C4—3322no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4—3322no—
296C2×C4—3322no—
297C2×C4—1668no—
298C2×C4#1458171714yes—
299C2×C4#1458171814yes—
300C2×C4(C22⋊D83)⋊C412yes—
301C2×C4#1458176412yes—
302C8—82no—
303C22—22no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22#1912757811yescharacteristic
312C22#1912767412yes—
313C22C2×#1458171711yescharacteristic
314C22#1912770311yescharacteristic
315C22C2×#1458171812yes—
316C22#1912785411yescharacteristic
317C22#1912785911yescharacteristic
318C22#1912803211yescharacteristic
319C22#1912803711yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—3328no—
325C4—6642no—
326C4—6641no—
327C4—6641no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2490596312yes—
333C2#2490598912yes—
334C2#2490776811yescharacteristic
335C2#2490790411yescharacteristic
336C2#2490791911yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1817C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#1817D83
hypocenterC83#9144
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
9. #19127607C4semidirect
10. #19127599C4semidirect
174. C83#9144semidirect

Maximal subgroups

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

subgroupindex
#249158022
C2×C4×C4⋊Dic832
C2×#191274392
#249053432
C83×#18172
#914483