Finite Group Databasebeta

← #35302739

Subgroups — #35302739

Order 21248.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24908389C211yesmaximal, characteristic
2#24908269C211yesmaximal, characteristic
3#24908227C211yesmaximal, characteristic, complemented
4C8×C4⋊Dic83C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905347C211yesmaximal, characteristic, complemented
7C83×#1277C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9C2×#14581724—21no—
10C2×C4⋊(C83⋊C8)C2211yescharacteristic
11C2×#14581714—21no—
12C2×#14581714—21no—
13C2×#14581714—21no—
14C2×#14581714—21no—
15C2×C8×Dic83—21no—
16#19127571C2211yescharacteristic
17C4×C4⋊Dic83C2211yescharacteristic
18#19127464—21no—
19#19127455—21no—
20#19127444C414yescomplemented
21C2×C4⋊C8×C83C2211yescharacteristic
22C2×C4⋊C8×C83C2211yescharacteristic
23C2×C4×C8×C83C2211yescharacteristic
24C22×D332—21no—
25C22×D332—21no—
26C22×D332—21no—
27C22×C83⋊C8—21no—
28#14581789—21no—
29#14581789—21no—
30#14581789—21no—
31#14581789C2×C412yes—
32C2×C4⋊Dic83—21no—
33C2×C4⋊Dic83—21no—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83D411yescharacteristic
36C2×C4⋊Dic83D411yescharacteristic
37C2×C4×Dic83—21no—
38#14581724—42no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)C2×C412yes—
41#14581714—42no—
42#14581714—42no—
43#14581714—42no—
44#14581714—42no—
45C8×Dic83—42no—
46C83⋊#95—21no—
47C22×C8×C83—21no—
48C22×C8×C83—21no—
49C22×C8×C83D411yescharacteristic
50C22×C8×C83D411yescharacteristic
51C2×C42×C83C2311yescharacteristic
52C4⋊C8×C83—21no—
53C4⋊C8×C83—21no—
54C4⋊C8×C83—21no—
55C4⋊C8×C83C2×C412yes—
56C4×C8×C83—21no—
57C4×C8×C83—21no—
58C23×D83—41no—
59C2×D332—22no—
60C2×D332—22no—
61C2×D332—22no—
62C2×D332—41no—
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—81no—
74C2×D332—81no—
75C22×Dic83—21no—
76C22×Dic83—21no—
77C22×Dic83—41no—
78C2×C83⋊C8—22no—
79C2×C83⋊C8—41no—
80C2×C83⋊C8—41no—
81C4⋊Dic83—21no—
82C4⋊Dic83—21no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C4⋊Dic83—41no—
86C4⋊Dic83—41no—
87C4⋊Dic83—81no—
88C4⋊Dic83D812yes—
89C4⋊Dic83SD1612yes—
90C4×Dic83—41no—
91C4×Dic83—41no—
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83#4111yescharacteristic
95C2×C8×C83—21no—
96C2×C8×C83—21no—
97C2×C8×C83—22no—
98C2×C8×C83—21no—
99C2×C8×C83—21no—
100C2×C8×C83—21no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83SD1612yes—
107C42×C83C22×C411yescharacteristic
108C42×C83#4111yescharacteristic
109C42×C83#4111yescharacteristic
110C42×C83#4111yescharacteristic
111C23×C83#12111yescharacteristic
112C22×D83—42no—
113C22×D83—81no—
114C22×D83—81no—
115C22×D83—81no—
116C22×D83—81no—
117C22×D83—81no—
118C22×D83—81no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83#11711yescharacteristic
122C2×C4×C83C4×D411yescharacteristic
123C2×C4×C83C4×D411yescharacteristic
124C2×C4×C83#12311yescharacteristic
125C2×C4×C83#12411yescharacteristic
126C2×C4×C83#12611yescharacteristic
127C2×C4×C83C2×D811yescharacteristic
128C2×C4×C83C2×SD1611yescharacteristic
129C2×C4×C83C2×SD1611yescharacteristic
130C2×C4×C83#13511yescharacteristic
131C2×C4×C83#13511yescharacteristic
132C2×C4×C83#13511yescharacteristic
133C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
135C2×Dic83—22no—
136C2×Dic83—22no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—81no—
142C2×Dic83—81no—
143C2×Dic83—81no—
144D332—42no—
145D332—42no—
146D332—42no—
147D332—42no—
148D332—82no—
149D332—82no—
150D332—81no—
151D332—81no—
152D332—81no—
153D332—81no—
154C8×C83—22no—
155C8×C83—42no—
156C8×C83—41no—
157C8×C83—81no—
158C83⋊C8—42no—
159C22×C83#38811yescharacteristic
160C22×C83#46011yescharacteristic
161C22×C83#46311yescharacteristic
162C22×C83#46511yescharacteristic
163C22×C83#46611yescharacteristic
164C22×C83#47111yescharacteristic
165C22×C83#47211yescharacteristic
166C2×D83—82no—
167C2×D83—82no—
168C2×D83—82no—
169C2×D83—161no—
170C2×D83—161no—
171C2×D83—161no—
172C2×D83—161no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83C4×D811yescharacteristic
176C4×C83C4×SD1611yescharacteristic
177C4×C83C4×SD1611yescharacteristic
178C4×C83#44211yescharacteristic
179C4×C83#48111yescharacteristic
180C4×C83#48211yescharacteristic
181C4×C83#48611yescharacteristic
182C4×C83#48811yescharacteristic
183Dic83—42no—
184Dic83—42no—
185Dic83—161no—
186#9564—831nomaximal, Sylow
187C2×C83#166611yescharacteristic
188C2×C83#166811yescharacteristic
189C2×C83#167711yescharacteristic
190C2×C83#168111yescharacteristic
191C2×C83#192611yescharacteristic
192C2×C83#193111yescharacteristic
193C2×C83#193411yescharacteristic
194D83—162no—
195#1982—831no—
196#1957—831no—
197#1948—831no—
198C8×C4⋊C4—831no—
199C2×#331—831no—
200#1296—831no—
201#1277D8311yescharacteristic, complemented
202C83#956411yescharacteristic, complemented, Sylow
203C2×#127—831no—
204C2×#106—1661no—
205C2×C4⋊C8—831no—
206C2×C4⋊C8C2×D8311yescharacteristic
207C2×C4⋊C8C2×D8311yescharacteristic
208C2×#102—1661no—
209C2×#102—1661no—
210C2×#102—1661no—
211C2×#102—1661no—
212C2×C4×C8—1661no—
213C2×C4×C8C2×D8311yescharacteristic
214#384—831no—
215C4×C4⋊C4—831no—
216#340—1661no—
217#336—1661no—
218#331—834no—
219C22×D4—1661no—
220C22×D4—1661no—
221C22×D4—1661no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—1661no—
225C22×C8D33211yescharacteristic
226C22×C8D33211yescharacteristic
227#127—832no—
228#127—1661no—
229#127—1661no—
230#127—1661no—
231C2×C4⋊C4—831no—
232C2×C4⋊C4—831no—
233C2×C4⋊C4—1661no—
234C2×C4⋊C4—1661no—
235C2×C4⋊C4—1661no—
236C2×C42—1661no—
237C2×C42C22×D8311yescharacteristic
238#106—3322no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—832no—
243C4⋊C8—1661no—
244C4⋊C8C4×D8312yes—
245#102—3322no—
246#102—3322no—
247#102—3322no—
248#102—3322no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—3322no—
252#95—1661no—
253C24—3321no—
254C2×D4—3321no—
255C2×D4—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—6641no—
266C2×D4—6641no—
267C2×D4—1662no—
268C2×D4—1662no—
269C2×D4—1662no—
270C22×C4—3321no—
271C22×C4—1661no—
272C22×C4—1661no—
273C22×C4D4×D8311yescharacteristic
274C22×C4C2×D33211yescharacteristic
275C22×C4#1305059111yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—3321no—
288C2×C8—3321no—
289C2×C8—1662no—
290C2×C8#1305057912yes—
291C4⋊C4—3321no—
292C4⋊C4—3321no—
293C4⋊C4—3321no—
294C4⋊C4—3321no—
295C4⋊C4—832no—
296C4⋊C4—832no—
297C4⋊C4—6641no—
298C4⋊C4—1661no—
299C4⋊C4—1661no—
300C42—3321no—
301C42—3321no—
302C42C2×C4×D8311yescharacteristic
303C42#1305058911yescharacteristic
304C42#1305058911yescharacteristic
305C42#1305059011yescharacteristic
306C23—3322no—
307C23—6641no—
308C23—6641no—
309C23—6641no—
310C23—6641no—
311C23—6641no—
312C23—6641no—
313C23#1458176711yescharacteristic
314D4—3322no—
315D4—3322no—
316D4—3322no—
317D4—3322no—
318D4—6642no—
319D4—6642no—
320D4—6641no—
321D4—6641no—
322D4—6641no—
323D4—6641no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—3321no—
327C2×C4—3321no—
328C2×C4—3321no—
329C2×C4—3321no—
330C2×C4—6641no—
331C2×C4—6641no—
332C2×C4—6641no—
333C2×C4—1662no—
334C2×C4—1662no—
335C2×C4#1458175011yescharacteristic
336C2×C4(C22⋊D83)⋊C411yescharacteristic
337C2×C4C4×D33211yescharacteristic
338C2×C4#1458177611yescharacteristic
339C2×C4#1458177811yescharacteristic
340C2×C4#1458178711yescharacteristic
341C2×C4D8×D8311yescharacteristic
342C2×C4SD16×D8311yescharacteristic
343C2×C4C2×#1305057911yescharacteristic
344C2×C4#1458181311yescharacteristic
345C2×C4#1458181411yescharacteristic
346C2×C4#1458181511yescharacteristic
347C2×C4#1458182011yescharacteristic
348C2×C4#1458182011yescharacteristic
349C8—22no—
350C8—41no—
351C8—42no—
352C8—81no—
353C8—3322no—
354C22—13281no—
355C22—13281no—
356C22—13281no—
357C22—13281no—
358C22—6642no—
359C22—6642no—
360C22—6642no—
361C22#1912759311yescharacteristic
362C22#1912790211yescharacteristic
363C22#1912791711yescharacteristic
364C22#1912793111yescharacteristic
365C22#1912793811yescharacteristic
366C22#1912796511yescharacteristic
367C22#1912797211yescharacteristic
368C4—21no—
369C4—21no—
370C4—13281no—
371C4—3322no—
372C4—3322no—
373C4#1912778111yescharacteristic
374C4#1912778611yescharacteristic
375C4C4×#1305057911yescharacteristic
376C4#1912781411yescharacteristic
377C4#1912802011yescharacteristic
378C4#1912802611yescharacteristic
379C4#1912804611yescharacteristic
380C4#1912805711yescharacteristic
381C2—13282no—
382C2#2490681211yescharacteristic
383C2#2490682411yescharacteristic
384C2#2490687311yescharacteristic
385C2#2490690111yescharacteristic
386C2#2490809111yescharacteristic
387C2#2490812611yescharacteristic
388C2#2490814711yescharacteristic
subgroupG/H
centerC23#14581767
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1277C2
socleC23×C83#121
prime radicalC2×#14581789C22
cosocleC2×C42×C83C23
hypercenter#1277D83
hypocenterC83#9564
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
3. #24908227C2semidirect
4. C8×C4⋊Dic83C2semidirect
6. #24905347C2semidirect
7. C83×#1277C2semidirect
20. #19127444C4semidirect
201. #1277D83semidirect
202. C83#9564semidirect

Maximal subgroups

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

subgroupindex
#249083892
#249082692
#249082272
C8×C4⋊Dic832
C2×#191274442
#249053472
C83×#12772
#956483