Finite Group Databasebeta

← #35308948

Subgroups — #35308948

Order 21248.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24908390C211yesmaximal, characteristic
2#24908269C211yesmaximal, characteristic
3#24908221C211yesmaximal, characteristic, complemented
4C2×#19127454C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905347C211yesmaximal, characteristic, complemented
7C83×#1939C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9C2×#14581725—21no—
10C2×C4⋊(C83⋊C8)C2211yescharacteristic
11C2×C4⋊(C83⋊C8)C2211yescharacteristic
12C2×#14581714—21no—
13C2×#14581714—21no—
14C2×#14581712—21no—
15C2×#14581712—21no—
16#19127571C2211yescharacteristic
17#19127463—21no—
18#19127454—22no—
19#19127444—22no—
20C2×C4⋊C8×C83—21no—
21C2×C4⋊C8×C83C2211yescharacteristic
22C2×C4×C8×C83C2211yescharacteristic
23C4×C4⋊C4×C83C2211yescharacteristic
24#336×C83—21no—
25C22×D332—21no—
26C22×D332—21no—
27C22×D332—21no—
28C22×C83⋊C8—21no—
29C22×C83⋊C8—21no—
30#14581789—21no—
31#14581789—21no—
32#14581789—21no—
33#14581789Q812yes—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83—21no—
36C2×C4⋊Dic83—21no—
37#14581725—42no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)D412yes—
42#14581714—42no—
43#14581714—42no—
44#14581712—42no—
45#14581712—42no—
46C22×C8×C83—21no—
47C22×C8×C83D411yescharacteristic
48C22×C8×C83D411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83D411yescharacteristic
51C2×C42×C83—21no—
52C2×C42×C83C2311yescharacteristic
53C4⋊C8×C83—21no—
54C4⋊C8×C83—21no—
55C4⋊C8×C83—42no—
56C4×C8×C83—21no—
57C4×C8×C83—21no—
58#95×C83—21no—
59C23×D83—41no—
60C2×D332—22no—
61C2×D332—22no—
62C2×D332—22no—
63C2×D332—41no—
64C2×D332—41no—
65C2×D332—41no—
66C2×D332—41no—
67C2×D332—41no—
68C2×D332—41no—
69C2×D332—41no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—41no—
73C2×D332—41no—
74C2×D332—81no—
75C2×D332—81no—
76C22×Dic83—41no—
77C2×C83⋊C8—22no—
78C2×C83⋊C8—41no—
79C2×C83⋊C8—41no—
80C2×C83⋊C8—41no—
81C2×C83⋊C8—41no—
82C2×C83⋊C8—41no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C4⋊Dic83—41no—
86C4⋊Dic83—41no—
87C4⋊Dic83—81no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83#4111yescharacteristic
93C2×C8×C83—21no—
94C2×C8×C83—21no—
95C2×C8×C83—21no—
96C2×C8×C83—21no—
97C2×C8×C83—21no—
98C2×C8×C83—41no—
99C2×C8×C83—41no—
100C2×C8×C83—41no—
101C2×C8×C83D812yes—
102C4⋊C4×C83—21no—
103C4⋊C4×C83—21no—
104C4⋊C4×C83—21no—
105C4⋊C4×C83D812yes—
106C42×C83—41no—
107C42×C83—41no—
108C42×C83C2×D411yescharacteristic
109C42×C83C2×Q811yescharacteristic
110C42×C83#4111yescharacteristic
111C42×C83#4111yescharacteristic
112C23×C83#12111yescharacteristic
113C22×D83—42no—
114C22×D83—81no—
115C22×D83—81no—
116C22×D83—81no—
117C22×D83—81no—
118C22×D83—81no—
119C22×D83—81no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—22no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83#12111yescharacteristic
129C2×C4×C83#12111yescharacteristic
130C2×C4×C83C22⋊Q811yescharacteristic
131C2×C4×C83C22⋊Q811yescharacteristic
132C2×C4×C83C22⋊Q811yescharacteristic
133C2×C4×C83#12611yescharacteristic
134C2×C4×C83C2×D811yescharacteristic
135C2×C4×C83C2×D811yescharacteristic
136C2×C4×C83#13511yescharacteristic
137C2×C4×C83#13511yescharacteristic
138C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C83#13711yescharacteristic
141C2×C4×C83#13711yescharacteristic
142C2×Dic83—81no—
143C2×Dic83—81no—
144C2×Dic83—81no—
145D332—42no—
146D332—42no—
147D332—42no—
148D332—42no—
149D332—82no—
150D332—82no—
151D332—81no—
152D332—81no—
153D332—81no—
154D332—81no—
155C8×C83—22no—
156C8×C83—41no—
157C8×C83—81no—
158C83⋊C8—42no—
159C83⋊C8—81no—
160C22×C83#39611yescharacteristic
161C22×C83#45911yescharacteristic
162C22×C83#46411yescharacteristic
163C22×C83#46611yescharacteristic
164C22×C83#46811yescharacteristic
165C22×C83#47111yescharacteristic
166C22×C83#47211yescharacteristic
167C2×D83—82no—
168C2×D83—82no—
169C2×D83—82no—
170C2×D83—161no—
171C2×D83—161no—
172C2×D83—161no—
173C2×D83—161no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—42no—
177C4×C83—81no—
178C4×C83#45911yescharacteristic
179C4×C83#46411yescharacteristic
180C4×C83#46611yescharacteristic
181C4×C83#46811yescharacteristic
182C4×C83#47411yescharacteristic
183C4×C83#47411yescharacteristic
184C4×C83#47811yescharacteristic
185C4×C83#47811yescharacteristic
186Dic83—161no—
187#10591—831nomaximal, Sylow
188C2×C83#166711yescharacteristic
189C2×C83#168211yescharacteristic
190C2×C83#168511yescharacteristic
191C2×C83#169811yescharacteristic
192C2×C83#206311yescharacteristic
193C2×C83#206711yescharacteristic
194C2×C83#206711yescharacteristic
195D83—162no—
196#1982—831no—
197#1957—831no—
198#1947—831no—
199#1939D8311yescharacteristic, complemented
200C2×#335—831no—
201C2×#331—831no—
202#1296—831no—
203C83#1059111yescharacteristic, complemented, Sylow
204C2×#127—831no—
205C2×#107—1661no—
206C2×C4⋊C8—21no—
207C2×C4⋊C8—831no—
208C2×C4⋊C8—831no—
209C2×C4⋊C8C2×D8311yescharacteristic
210C2×#102—1661no—
211C2×#102—1661no—
212C2×#102—1661no—
213C2×#102—1661no—
214C2×C4×C8C2×D8311yescharacteristic
215#384—831no—
216C4×C4⋊C4C2×D8311yescharacteristic
217#340—1661no—
218#336—21no—
219#335—1662no—
220#331—1662no—
221C22×D4—1661no—
222C22×D4—1661no—
223C22×D4—1661no—
224C22×C8—21no—
225C22×C8—1661no—
226C22×C8—1661no—
227C22×C8C22⋊D8311yescharacteristic
228C22×C8C22⋊D8311yescharacteristic
229#127—832no—
230#127—1661no—
231#127—1661no—
232#127—1661no—
233C2×C4⋊C4—1661no—
234C2×C4⋊C4—1661no—
235C2×C4⋊C4—1661no—
236C2×C4⋊C4D33211yescharacteristic
237C2×C4⋊C4D33211yescharacteristic
238C2×C42—21no—
239C2×C42C22×D8311yescharacteristic
240#107—3322no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—832no—
245C4⋊C8—1661no—
246C4⋊C8—1661no—
247C4⋊C8—1661no—
248#102—3322no—
249#102—3322no—
250#102—3322no—
251#102—3322no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255C24—3321no—
256C2×D4—3321no—
257C2×D4—3321no—
258C2×D4—3321no—
259C2×D4—3321no—
260C2×D4—3321no—
261C2×D4—3321no—
262C2×D4—3321no—
263C2×D4—3321no—
264C2×D4—3321no—
265C2×D4—3321no—
266C2×D4—3321no—
267C2×D4—6641no—
268C2×D4—6641no—
269C2×D4—1662no—
270C2×D4—1662no—
271C2×D4—1662no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—3321no—
275C22×C4C2×C22⋊D8311yescharacteristic
276C22×C4C2×D33211yescharacteristic
277C22×C4#1305059011yescharacteristic
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—3321no—
287C2×C8—3321no—
288C2×C8—3321no—
289C2×C8—3321no—
290C2×C8—3321no—
291C2×C8—1662no—
292C2×C8#1305057512yes—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—3321no—
297C4⋊C4—3321no—
298C4⋊C4—3321no—
299C4⋊C4—3321no—
300C4⋊C4—6641no—
301C4⋊C4D66412yes—
302C42—41no—
303C42—41no—
304C42D4×D8311yescharacteristic
305C42Q8×D8311yescharacteristic
306C42#1305058911yescharacteristic
307C42#1305059111yescharacteristic
308C23—3322no—
309C23—6641no—
310C23—6641no—
311C23—6641no—
312C23—6641no—
313C23—6641no—
314C23—6641no—
315C23#1458176611yescharacteristic
316D4—3322no—
317D4—3322no—
318D4—3322no—
319D4—3322no—
320D4—6642no—
321D4—6642no—
322D4—6641no—
323D4—6641no—
324D4—6641no—
325D4—6641no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—6641no—
335C2×C4—6641no—
336C2×C4—6641no—
337C2×C4#1458176411yescharacteristic
338C2×C4#1458176711yescharacteristic
339C2×C4#1458176811yescharacteristic
340C2×C4#1458177011yescharacteristic
341C2×C4#1458177211yescharacteristic
342C2×C4#1458178711yescharacteristic
343C2×C4C2×#1305057511yescharacteristic
344C2×C4C2×D66411yescharacteristic
345C2×C4#1458181211yescharacteristic
346C2×C4#1458181411yescharacteristic
347C2×C4#1458181711yescharacteristic
348C2×C4#1458182011yescharacteristic
349C2×C4#1458182311yescharacteristic
350C2×C4#1458182511yescharacteristic
351C8—22no—
352C8—41no—
353C8—81no—
354C8—3322no—
355C8—6641no—
356C22—13281no—
357C22—13281no—
358C22—13281no—
359C22—13281no—
360C22—6642no—
361C22—6642no—
362C22—6642no—
363C22#1912762311yescharacteristic
364C22#1912789611yescharacteristic
365C22#1912792311yescharacteristic
366C22#1912793711yescharacteristic
367C22#1912794711yescharacteristic
368C22#1912796411yescharacteristic
369C22#1912797111yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—13281no—
375C4#1912789711yescharacteristic
376C4#1912792411yescharacteristic
377C4#1912793511yescharacteristic
378C4#1912794911yescharacteristic
379C4#1912797811yescharacteristic
380C4#1912798211yescharacteristic
381C4#1912799811yescharacteristic
382C4#1912800211yescharacteristic
383C2—13282no—
384C2#2490681811yescharacteristic
385C2#2490690711yescharacteristic
386C2#2490692811yescharacteristic
387C2#2490700311yescharacteristic
388C2#2490882211yescharacteristic
389C2#2490884811yescharacteristic
390C2#2490885011yescharacteristic
subgroupG/H
centerC23#14581766
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1939C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenter#1939D83
hypocenterC83#10591
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939

Complements

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

NH ≅ G/Ntype
3. #24908221C2semidirect
4. C2×#19127454C2semidirect
6. #24905347C2semidirect
7. C83×#1939C2semidirect
199. #1939D83semidirect
203. C83#10591semidirect

Maximal subgroups

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

subgroupindex
#249083902
#249082692
#249082212
C2×#191274542
C2×#191274442
#249053472
C83×#19392
#1059183