Finite Group Databasebeta

← #35315529

Subgroups — #35315529

Order 21248.

580 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 99 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#24910828C211yesmaximal, characteristic
2#24907451C211yesmaximal, characteristic
3#24905485C214yesmaximal, complemented
4C83×#1460C211yesmaximal, characteristic
5C23×Dic166C411yescharacteristic, complemented
6C2×#14581771—21no—
7C2×#14581707—21no—
8#19127621C412yescomplemented
9#19127613C2212yes—
10#19127607—21no—
11#19127581—21no—
12#19127573—21no—
13#19127561—21no—
14#19127561C411yescharacteristic, complemented
15#19127468C2212yes—
16C2×C4×D4×C83C2211yescharacteristic
17#431×C83—21no—
18#323×C83—21no—
19#323×C83C2212yes—
20C22×Dic166—21no—
21C22×Dic166—41no—
22C22×Dic166—41no—
23C22×Dic166—44no—
24C22×Dic166C2×C412yes—
25C23×Dic83—21no—
26#14581771—44no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—41no—
29C2×Dic83⋊C4—22no—
30C2×C22⋊Dic83—21no—
31C2×C22⋊Dic83—22no—
32C2×C22⋊Dic83—22no—
33#14581707—42no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—41no—
37C83⋊#95C2×C412yes—
38C22×D4×C83D411yescharacteristic
39C23×C4×C83C2×C411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C4×D4×C83—41no—
42C4×D4×C83—41no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C22⋊C4×C83D411yescharacteristic
45C2×C22⋊C4×C83C2311yescharacteristic
46C2×C42×C83D411yescharacteristic
47C4⋊C8×C83—41no—
48C22⋊C8×C83—22no—
49C4×C8×C83—41no—
50C2×Dic166—24no—
51C2×Dic166—42no—
52C2×Dic166—48no—
53C2×Dic166—44no—
54C2×Dic166—41no—
55C2×Dic166—41no—
56C2×Dic166—42no—
57C2×Dic166—44no—
58C2×Dic166—44no—
59C22×Dic83—22no—
60C22×Dic83—22no—
61C22×Dic83—44no—
62C22×Dic83—42no—
63C22×Dic83—42no—
64C4⋊Dic83—42no—
65C4⋊Dic83—82no—
66Dic83⋊C4—44no—
67C22⋊Dic83—44no—
68C22⋊Dic83—42no—
69C22⋊Dic83—42no—
70C22⋊Dic83—42no—
71C24×C83C22⋊C411yescharacteristic
72C24×C83C2×D411yescharacteristic
73C2×D4×C83—41no—
74C2×D4×C83—41no—
75C2×D4×C83SD1612yes—
76C2×D4×C83Q1612yes—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—42no—
82C22×C4×C83—42no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C2×C8×C83—42no—
89C4⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C42×C83—21no—
93C42×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83#10312yes—
103C23×C83C2×C22⋊C411yescharacteristic
104C23×C83C2×SD1611yescharacteristic
105C23×C83C2×Q1611yescharacteristic
106D4×C83—42no—
107D4×C83—42no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—22no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—81no—
121C2×C4×C83#9911yescharacteristic
122C2×C4×C83#9911yescharacteristic
123C2×C4×C83#10312yes—
124C2×Dic83—24no—
125C2×Dic83—42no—
126C2×Dic83—48no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—82no—
132C2×Dic83—82no—
133C2×Dic83—82no—
134Dic166—44no—
135Dic166—44no—
136Dic166—44no—
137Dic166—44no—
138C8×C83—82no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—22no—
143C22×C83—21no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—42no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—81no—
153C22×C83C2×#9911yescharacteristic
154C22×C83C2×#10311yescharacteristic
155C22×C83#42111yescharacteristic
156C4×C83—21no—
157C4×C83—22no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83AΣL(1,𝔽16)12yes—
162Dic83—44no—
163Dic83—82no—
164Dic83—162no—
165#11898—831nomaximal, Sylow
166C2×C83—21no—
167C2×C83—21no—
168C2×C83—41no—
169C2×C83—42no—
170C2×C83—41no—
171C2×C83#152011yescharacteristic
172C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
173C2×C83#212911yescharacteristic
174#2409—831no—
175#1799—831no—
176#1460D8311yescharacteristic
177#1351—834no—
178C83#1189811yescharacteristic, complemented, Sylow
179C23×Q8—831no—
180C2×C22⋊Q8—1661no—
181C2×C4×D4C2×D8311yescharacteristic
182#431—21no—
183C2×#99—1661no—
184#395—832no—
185#393—832no—
186#391—1661no—
187#386—1661no—
188#385—1661no—
189#380—831no—
190#380—1661no—
191#342—832no—
192#323—21no—
193#323C2×D8312yes—
194C22×Q8—3321no—
195C22×Q8—3321no—
196C22×Q8—3324no—
197C22×Q8—832no—
198C22×Q8—1661no—
199C22×D4D33211yescharacteristic
200C23×C4—1661no—
201C23×C4C4×D8311yescharacteristic
202C23×C4D33211yescharacteristic
203C22⋊Q8—3324no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—3321no—
207C2×C4⋊C4—1662no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4C4×D8311yescharacteristic
210C2×C22⋊C4—1662no—
211C2×C22⋊C4—1661no—
212C2×C22⋊C4—1662no—
213C2×C22⋊C4C22⋊D8311yescharacteristic
214C2×C22⋊C4C22×D8311yescharacteristic
215C2×C42C22⋊D8311yescharacteristic
216C4⋊C8—41no—
217#99—3322no—
218C22⋊C8—22no—
219C4×C8—41no—
220#95—3321no—
221#95—832no—
222#95—1662no—
223#95—1662no—
224C24(C2×D83)⋊C411yescharacteristic
225C24C2×D33211yescharacteristic
226C2×Q8—3322no—
227C2×Q8—3322no—
228C2×Q8—3324no—
229C2×Q8—3328no—
230C2×Q8—3324no—
231C2×Q8—3324no—
232C2×Q8—3321no—
233C2×Q8—3321no—
234C2×Q8—1664no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4#1305057912yes—
238C2×D4Dic33212yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—3322no—
246C22×C4—3324no—
247C22×C4—3322no—
248C22×C4—1662no—
249C22×C4—1662no—
250C22×C4(C2×D83)⋊C411yescharacteristic
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4(C2×D83)⋊C411yescharacteristic
253C22×C4C2×C4×D8311yescharacteristic
254C22×C4C2×C22⋊D8311yescharacteristic
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—3322no—
258C4⋊C4—3324no—
259C4⋊C4—6642no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—3322no—
263C22⋊C4—3324no—
264C22⋊C4—3322no—
265C22⋊C4—3322no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23#1458171712yes—
277C23C2×(C2×D83)⋊C411yescharacteristic
278C23C2×#1305057911yescharacteristic
279C23C2×Dic33211yescharacteristic
280Q8—3324no—
281Q8—3324no—
282Q8—3324no—
283Q8—3324no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—3322no—
300C2×C4—3328no—
301C2×C4—3322no—
302C2×C4—3322no—
303C2×C4—3322no—
304C2×C4—3322no—
305C2×C4—6642no—
306C2×C4—6642no—
307C2×C4—6642no—
308C2×C4—1664no—
309C2×C4#1458170611yescharacteristic
310C2×C4#1458170611yescharacteristic
311C2×C4#1458171712yes—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#1458170611yescharacteristic
328C22C2×#1458171711yescharacteristic
329C22#1912771711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—13282no—
336C4—3324no—
337C4—6642no—
338C4#1912748912yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2#2490603611yescharacteristic
345C2C2×#1912748911yescharacteristic
346C2#2490913711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1460C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1460D83
hypocenterC83#11898
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #24905485C2semidirect
5. C23×Dic166C4semidirect
8. #19127621C4semidirect
14. #19127561C4semidirect
178. C83#11898semidirect

Maximal subgroups

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

subgroupindex
#249108282
#249074512
#249054852
C83×#14602
#1189883