Finite Group Databasebeta

← #35309113

Subgroups — #35309113

Order 21248.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24908256C211yesmaximal, characteristic
2#24907622C211yesmaximal, characteristic, complemented
3#24907464C211yesmaximal, characteristic
4#24907460C211yesmaximal, characteristic
5#24905347C211yesmaximal, characteristic, complemented
6#24905297C211yesmaximal, characteristic, complemented
7C83×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D332C2211yescharacteristic
9C2×#14581725—21no—
10C2×#14581725—21no—
11C2×C4⋊(C83⋊C8)C2211yescharacteristic
12C2×#14581714—21no—
13C2×#14581712—21no—
14C2×(C2×D83)⋊C8—21no—
15C2×(C2×D83)⋊C8—21no—
16C2×C4×C83⋊C8C2211yescharacteristic
17#19127571C2211yescharacteristic
18#19127464—21no—
19#19127463—21no—
20C2×C4⋊C8×C83C2211yescharacteristic
21C2×C4⋊C8×C83C2211yescharacteristic
22#384×C83C2211yescharacteristic
23C22×D332D411yescharacteristic
24C22×C4×D83—21no—
25C22×C83⋊C8—21no—
26C22×C83⋊C8D411yescharacteristic
27C22×C83⋊C8D411yescharacteristic
28C4×D332—21no—
29C4×D332—21no—
30C4×D332—21no—
31C4×D332—21no—
32C2×C4⋊Dic83—21no—
33C2×C4⋊Dic83—21no—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83D411yescharacteristic
36C2×(C2×D83)⋊C4—21no—
37#14581725—42no—
38#14581725—42no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41#14581714—42no—
42#14581712—42no—
43(C2×D83)⋊C8—42no—
44(C2×D83)⋊C8—42no—
45C4×C83⋊C8—21no—
46C4×C83⋊C8—21no—
47C22×C8×C83—21no—
48C22×C8×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83—21no—
52C2×C42×C83C2311yescharacteristic
53C4⋊C8×C83—21no—
54C4⋊C8×C83—21no—
55C4⋊C8×C83—21no—
56C4⋊C8×C83—21no—
57C23×D83—21no—
58C2×D332—21no—
59C2×D332—21no—
60C2×D332—21no—
61C2×D332—21no—
62C2×D332—21no—
63C2×D332Q1612yes—
64C22×Dic83—21no—
65C22×Dic83—41no—
66C2×C4×D83—41no—
67C2×C4×D83—41no—
68C2×C4×D83—41no—
69C2×C4×D83—41no—
70C2×C4×D83—41no—
71C2×C4×D83—41no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—21no—
74C2×C83⋊C8—21no—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8—41no—
78C2×C83⋊C8—41no—
79C2×C83⋊C8D812yes—
80C2×C83⋊C8SD1612yes—
81C4⋊Dic83—21no—
82C4⋊Dic83—21no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C4⋊Dic83—41no—
86C4⋊Dic83—41no—
87C4⋊Dic83—81no—
88(C2×D83)⋊C4—41no—
89(C2×D83)⋊C4—41no—
90C22×C4×C83—41no—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83#4111yescharacteristic
94C2×C8×C83—41no—
95C2×C8×C83—41no—
96C2×C8×C83—41no—
97C2×C8×C83—41no—
98C2×C8×C83—41no—
99C2×C8×C83—41no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—81no—
105C42×C83#4111yescharacteristic
106C42×C83#4111yescharacteristic
107C42×C83#4111yescharacteristic
108C42×C83#4111yescharacteristic
109C23×C83#12111yescharacteristic
110C22×D83—22no—
111C22×D83—41no—
112C22×D83—41no—
113C22×D83—41no—
114C22×D83—41no—
115C22×D83—41no—
116C22×D83—41no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—81no—
120C2×C4×C83—81no—
121C2×C4×C83—81no—
122C2×C4×C83#12311yescharacteristic
123C2×C4×C83#12311yescharacteristic
124C2×C4×C83#12311yescharacteristic
125C2×C4×C83#12411yescharacteristic
126C2×C4×C83#12611yescharacteristic
127C2×C4×C83#12611yescharacteristic
128C2×C4×C83C2×D811yescharacteristic
129C2×C4×C83C2×SD1611yescharacteristic
130C2×C4×C83C2×Q1611yescharacteristic
131C2×C4×C83#13511yescharacteristic
132C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C83#13711yescharacteristic
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—81no—
140C2×Dic83—81no—
141C2×Dic83—81no—
142D332—22no—
143D332—22no—
144D332—41no—
145D332—41no—
146C4×D83—81no—
147C4×D83—81no—
148C4×D83—81no—
149C4×D83—81no—
150C8×C83—81no—
151C8×C83—81no—
152C83⋊C8—22no—
153C83⋊C8—22no—
154C83⋊C8—81no—
155C22×C83#39711yescharacteristic
156C22×C83#45911yescharacteristic
157C22×C83#46111yescharacteristic
158C22×C83#46711yescharacteristic
159C22×C83#46911yescharacteristic
160C22×C83#47211yescharacteristic
161C22×C83#47311yescharacteristic
162C2×D83—42no—
163C2×D83—42no—
164C2×D83—42no—
165C2×D83—81no—
166C2×D83—81no—
167C2×D83—81no—
168C2×D83—81no—
169C4×C83—21no—
170C4×C83—21no—
171C4×C83—161no—
172C4×C83#48011yescharacteristic
173C4×C83#48111yescharacteristic
174C4×C83#48211yescharacteristic
175C4×C83#48211yescharacteristic
176C4×C83#48411yescharacteristic
177C4×C83#48411yescharacteristic
178C4×C83#48611yescharacteristic
179C4×C83#48911yescharacteristic
180Dic83—81no—
181Dic83—161no—
182#10616—831nomaximal, Sylow
183C2×C83#166911yescharacteristic
184C2×C83#167511yescharacteristic
185C2×C83#168711yescharacteristic
186C2×C83#169311yescharacteristic
187C2×C83#207111yescharacteristic
188C2×C83#207611yescharacteristic
189C2×C83#207811yescharacteristic
190D83—82no—
191#1954—831no—
192#1845—831no—
193#1802—831no—
194#1802—831no—
195#1296—831no—
196#1296D8311yescharacteristic, complemented
197#1277—831no—
198C83#1061611yescharacteristic, complemented, Sylow
199C2×C4×D4—831no—
200C2×#107—1661no—
201C2×#107—1661no—
202C2×C4⋊C8—831no—
203C2×C4⋊C8C2×D8311yescharacteristic
204C2×C4⋊C8C2×D8311yescharacteristic
205C2×#102—1661no—
206C2×#102—1661no—
207C2×C22⋊C8—1661no—
208C2×C22⋊C8—1661no—
209C2×C4×C8—831no—
210#384—831no—
211#384C2×D8311yescharacteristic
212#340—1661no—
213#340—1661no—
214C22×D4—831no—
215C23×C4—1661no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—831no—
219C22×C8—831no—
220C22×C8—1661no—
221C4×D4—1661no—
222C4×D4—1661no—
223C4×D4—1661no—
224C4×D4—1661no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—831no—
229C2×C4⋊C4—1661no—
230C2×C4⋊C4—1661no—
231C2×C4⋊C4—1661no—
232C2×C22⋊C4—1661no—
233C2×C42C22×D8311yescharacteristic
234#107—3322no—
235#107—3322no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—1661no—
241C4⋊C8—1661no—
242#102—3322no—
243#102—3322no—
244C22⋊C8—3322no—
245C22⋊C8—3322no—
246C4×C8—1661no—
247C4×C8—1661no—
248C24—1661no—
249C2×D4—832no—
250C2×D4—1661no—
251C2×D4—1661no—
252C2×D4—1661no—
253C2×D4—1661no—
254C2×D4—1661no—
255C22×C4—41no—
256C22×C4—3321no—
257C22×C4—3321no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—3321no—
263C22×C4—1661no—
264C22×C4D4×D8311yescharacteristic
265C22×C4D4×D8311yescharacteristic
266C22×C4#1305059011yescharacteristic
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—3321no—
274C2×C8—3321no—
275C2×C8—3321no—
276C2×C8—832no—
277C2×C8—832no—
278C2×C8—1661no—
279C2×C8—1661no—
280C2×C8—1661no—
281C2×C8—1661no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—3321no—
288C4⋊C4—3321no—
289C4⋊C4—3321no—
290C4⋊C4—3321no—
291C4⋊C4—6641no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C22⋊C4—3321no—
295C22⋊C4—3321no—
296C42#1305058911yescharacteristic
297C42#1305059011yescharacteristic
298C42#1305059011yescharacteristic
299C42#1305059111yescharacteristic
300C23—3321no—
301C23—3321no—
302C23—3321no—
303C23—3321no—
304C23—3321no—
305C23—3321no—
306C23—1662no—
307C23#1458176511yescharacteristic
308D4—3321no—
309D4—3321no—
310D4—1662no—
311D4—1662no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—3321no—
318C2×C4—3321no—
319C2×C4—3321no—
320C2×C4—6641no—
321C2×C4—6641no—
322C2×C4—6641no—
323C2×C4—6641no—
324C2×C4—6641no—
325C2×C4—6641no—
326C2×C4—6641no—
327C2×C4#1458177411yescharacteristic
328C2×C4#1458177711yescharacteristic
329C2×C4#1458177711yescharacteristic
330C2×C4#1458177811yescharacteristic
331C2×C4#1458178611yescharacteristic
332C2×C4#1458178711yescharacteristic
333C2×C4D8×D8311yescharacteristic
334C2×C4SD16×D8311yescharacteristic
335C2×C4Q16×D8311yescharacteristic
336C2×C4#1458181311yescharacteristic
337C2×C4#1458181611yescharacteristic
338C2×C4#1458181911yescharacteristic
339C2×C4#1458181911yescharacteristic
340C2×C4#1458182411yescharacteristic
341C8—81no—
342C8—81no—
343C8—6641no—
344C8—1662no—
345C8—1662no—
346C22—3322no—
347C22—3322no—
348C22—3322no—
349C22—6641no—
350C22—6641no—
351C22—6641no—
352C22—6641no—
353C22#1912763011yescharacteristic
354C22#1912789511yescharacteristic
355C22#1912790511yescharacteristic
356C22#1912794111yescharacteristic
357C22#1912795311yescharacteristic
358C22#1912797011yescharacteristic
359C22#1912797511yescharacteristic
360C4—21no—
361C4—21no—
362C4—161no—
363C4—13281no—
364C4—6641no—
365C4#1912801611yescharacteristic
366C4#1912802111yescharacteristic
367C4#1912802411yescharacteristic
368C4#1912802711yescharacteristic
369C4#1912803511yescharacteristic
370C4#1912803811yescharacteristic
371C4#1912804611yescharacteristic
372C4#1912806211yescharacteristic
373C2—6642no—
374C2#2490682711yescharacteristic
375C2#2490686111yescharacteristic
376C2#2490694111yescharacteristic
377C2#2490697511yescharacteristic
378C2#2490887011yescharacteristic
379C2#2490889511yescharacteristic
380C2#2490890711yescharacteristic
subgroupG/H
centerC23#14581765
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1296C2
socleC23×C83#121
prime radicalC22×D332D4
hypercenter#1296D83
hypocenterC83#10616
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #24907622C2semidirect
5. #24905347C2semidirect
6. #24905297C2semidirect
7. C83×#1296C2semidirect
196. #1296D83semidirect
198. C83#10616semidirect

Maximal subgroups

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

subgroupindex
#249082562
#249076222
#249074642
#249074602
#249053472
#249052972
C83×#12962
#1061683