Finite Group Databasebeta

← #35323998

Subgroups — #35323998

Order 21248.

482 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
1C22×#14581733C211yesmaximal, characteristic
2C2×C4⋊(C83⋊C16)C211yesmaximal, characteristic, complemented
3C2×#19127502C211yesmaximal, characteristic, complemented
4C2×#19127501C211yesmaximal, characteristic, complemented
5#24908329C211yesmaximal, characteristic, complemented
6#24905554C211yesmaximal, characteristic, complemented
7C83×#1970C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C2×#14581733—22no—
10C2×#14581733—22no—
11C2×#14581733—22no—
12C22×C83⋊C16C2211yescharacteristic
13C22×C83⋊C16C2211yescharacteristic
14C2×#14581726C2211yescharacteristic
15C2×#14581714—21no—
16#19127603—21no—
17C4⋊(C83⋊C16)—22no—
18#19127502—22no—
19#19127501—22no—
20C22×D8×C83C2211yescharacteristic
21C2×#107×C83C2211yescharacteristic
22C2×#102×C83—21no—
23C2×C4×C8×C83C2211yescharacteristic
24#390×C83—21no—
25C22×D332—21no—
26C2×D664—21no—
27C2×D664—21no—
28C2×D664—21no—
29C2×D664—21no—
30C2×D664—21no—
31C2×D664D412yes—
32C2×C4⋊Dic83—21no—
33C2×(C2×D83)⋊C4—41no—
34C2×(C2×D83)⋊C4—41no—
35#14581733—28no—
36C2×C83⋊C16—21no—
37C2×C83⋊C16—21no—
38C2×C83⋊C16—21no—
39C2×C83⋊C16—21no—
40C2×C83⋊C16—21no—
41C2×C83⋊C16D412yes—
42#14581726—21no—
43#14581726—21no—
44#14581714—42no—
45C22×D4×C83—21no—
46C2×D8×C83—21no—
47C2×D8×C83—21no—
48C2×D8×C83—21no—
49C2×D8×C83—21no—
50C2×D8×C83—21no—
51C2×D8×C83D412yes—
52C22×C8×C83D411yescharacteristic
53C22×C8×C83C2311yescharacteristic
54C2×C4⋊C4×C83—21no—
55C2×C22⋊C4×C83—41no—
56C2×C22⋊C4×C83—41no—
57C2×C42×C83D411yescharacteristic
58#107×C83—21no—
59#107×C83—21no—
60#102×C83—42no—
61C4×C8×C83—21no—
62C4×C8×C83—21no—
63C23×D83—41no—
64C2×D332—22no—
65C2×D332—41no—
66C2×D332—41no—
67C2×D332—41no—
68C2×D332—41no—
69C2×D332—41no—
70C22×Dic83—41no—
71D664—22no—
72D664—22no—
73D664—41no—
74D664—41no—
75C4⋊Dic83—41no—
76C4⋊Dic83—41no—
77(C2×D83)⋊C4—82no—
78(C2×D83)⋊C4—82no—
79C83⋊C16—22no—
80C83⋊C16—41no—
81C24×C83—41no—
82C2×D4×C83—22no—
83C2×D4×C83—41no—
84C2×D4×C83—41no—
85C2×D4×C83—41no—
86C2×D4×C83—41no—
87C2×D4×C83—41no—
88C22×C4×C83—41no—
89C22×C4×C83D811yescharacteristic
90C22×C4×C83D811yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92D8×C83—22no—
93D8×C83—22no—
94D8×C83—41no—
95D8×C83—41no—
96C2×C8×C83—21no—
97C2×C8×C83—21no—
98C2×C8×C83—21no—
99C2×C8×C83C2×D411yescharacteristic
100C2×C8×C83C2×D411yescharacteristic
101C2×C8×C83C2×D411yescharacteristic
102C2×C8×C83#4111yescharacteristic
103C2×C8×C83#4111yescharacteristic
104C2×C8×C83#4111yescharacteristic
105C4⋊C4×C83—41no—
106C4⋊C4×C83—41no—
107C22⋊C4×C83—82no—
108C22⋊C4×C83—82no—
109C42×C83—21no—
110C42×C83—21no—
111C23×C83—42no—
112C23×C83—81no—
113C23×C83—81no—
114C23×C83—81no—
115C23×C83—81no—
116C23×C83—81no—
117C23×C83—81no—
118C23×C83C2×D811yescharacteristic
119C22×D83—42no—
120C22×D83—81no—
121C22×D83—81no—
122C22×D83—81no—
123C22×D83—81no—
124C22×D83—81no—
125C22×D83—81no—
126D4×C83—42no—
127D4×C83—42no—
128D4×C83—81no—
129D4×C83—81no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—81no—
136C2×C4×C83—81no—
137C2×C4×C83—81no—
138C2×C4×C83D1612yescomplemented
139C2×C4×C83#12111yescharacteristic
140C2×C4×C83#12111yescharacteristic
141C2×C4×C83#12111yescharacteristic
142C2×C4×C83#13511yescharacteristic
143C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
145C2×Dic83—81no—
146C2×Dic83—81no—
147C2×Dic83—81no—
148D332—42no—
149D332—42no—
150D332—81no—
151D332—81no—
152C8×C83—41no—
153C8×C83#12011yescharacteristic
154C8×C83#12311yescharacteristic
155C8×C83#12311yescharacteristic
156C8×C83#12411yescharacteristic
157C22×C83—82no—
158C22×C83—82no—
159C22×C83—82no—
160C22×C83—161no—
161C22×C83—161no—
162C22×C83—161no—
163C22×C83—161no—
164C22×C83#45911yescharacteristic
165C22×C83#45911yescharacteristic
166C22×C83#46611yescharacteristic
167C22×C83C2×D1611yescharacteristic
168C22×C83#50811yescharacteristic
169C22×C83#50911yescharacteristic
170C22×C83#50911yescharacteristic
171C2×D83—82no—
172C2×D83—82no—
173C2×D83—82no—
174C2×D83—161no—
175C2×D83—161no—
176C2×D83—161no—
177C2×D83—161no—
178C4×C83—22no—
179C4×C83—41no—
180C4×C83—161no—
181C4×C83#39411yescharacteristic
182C4×C83#46311yescharacteristic
183C4×C83#46311yescharacteristic
184C4×C83#46911yescharacteristic
185Dic83—161no—
186#13448—831nomaximal, Sylow
187C2×C83—162no—
188C2×C83#205411yescharacteristic
189C2×C83#220611yescharacteristic
190C2×C83#220611yescharacteristic
191C2×C83#221011yescharacteristic
192C2×C83#221011yescharacteristic
193C2×C83#221511yescharacteristic
194C2×C83#221811yescharacteristic
195D83—162no—
196C22×D16—831no—
197C2×C4⋊C16—831no—
198C2×#357—831no—
199C2×#357—831no—
200#1970—831no—
201#1970D8311yescharacteristic, complemented
202#1380—831no—
203C83#1344811yescharacteristic, complemented, Sylow
204C22×D8—831no—
205C22×D8C2×D8311yescharacteristic
206C2×D16—1662no—
207C2×D16—1662no—
208C2×D16—1662no—
209C22×C16—831no—
210C22×C16—831no—
211C2×#107—831no—
212C2×#107C2×D8311yescharacteristic
213C2×#102—21no—
214C2×#102—1661no—
215C2×C4×C8C2×D8311yescharacteristic
216#390—21no—
217#390—1661no—
218C4⋊C16—1662no—
219#357—1662no—
220#357—1662no—
221C22×D4—21no—
222C22×D4—1661no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—832no—
229C2×D8—1661no—
230C2×D8—1661no—
231C2×D8—1661no—
232C2×D8—1661no—
233C2×D8—1661no—
234C2×D8D33212yes—
235C22×C8C22⋊D8311yescharacteristic
236C22×C8C22×D8311yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1661no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—3321no—
242C2×C22⋊C4—3321no—
243C2×C42C22⋊D8311yescharacteristic
244D16—1668no—
245C2×C16—832no—
246C2×C16—1661no—
247C2×C16—1661no—
248C2×C16—1661no—
249C2×C16—1661no—
250C2×C16—1661no—
251#107—21no—
252#107—21no—
253#107—1661no—
254#107—1661no—
255#102—42no—
256#102—3322no—
257C4×C8—21no—
258C4×C8—21no—
259C24—41no—
260C24—3321no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—3321no—
268C2×D4—3321no—
269C2×D4—3321no—
270C2×D4—3321no—
271C2×D4—3321no—
272C2×D4—1662no—
273C22×C4—41no—
274C22×C4—3321no—
275C22×C4#1305057511yescharacteristic
276C22×C4#1305057511yescharacteristic
277C22×C4C2×C22⋊D8311yescharacteristic
278D8—22no—
279D8—22no—
280D8—41no—
281D8—41no—
282D8—3321no—
283D8—3321no—
284D8—1662no—
285D8—1662no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8D4×D8311yescharacteristic
290C2×C8D4×D8311yescharacteristic
291C2×C8C2×D33211yescharacteristic
292C2×C8#1305058911yescharacteristic
293C2×C8#1305058911yescharacteristic
294C2×C8#1305059011yescharacteristic
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—3321no—
298C4⋊C4—3321no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—6642no—
302C22⋊C4—6642no—
303C42—21no—
304C42—21no—
305C16—3321no—
306C16—1662no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—3322no—
315C23—6641no—
316C23—6641no—
317C23—6641no—
318C23—6641no—
319C23—6641no—
320C23—6641no—
321C23C2×#1305057511yescharacteristic
322D4—42no—
323D4—42no—
324D4—81no—
325D4—81no—
326D4—3322no—
327D4—3322no—
328D4—6641no—
329D4—6641no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—6641no—
339C2×C4—6641no—
340C2×C4—6641no—
341C2×C4#1458173312yescomplemented
342C2×C4#1458176311yescharacteristic
343C2×C4#1458176411yescharacteristic
344C2×C4#1458176611yescharacteristic
345C2×C4#1458181211yescharacteristic
346C2×C4#1458181711yescharacteristic
347C2×C4#1458181811yescharacteristic
348C8—41no—
349C8#1458176211yescharacteristic
350C8#1458177411yescharacteristic
351C8#1458177611yescharacteristic
352C8#1458177811yescharacteristic
353C22—82no—
354C22—82no—
355C22—82no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—13281no—
361C22—13281no—
362C22—13281no—
363C22—13281no—
364C22—6642no—
365C22—6642no—
366C22—6642no—
367C22#1912789311yescharacteristic
368C22#1912789611yescharacteristic
369C22#1912793511yescharacteristic
370C22C2×#1458173311yescharacteristic
371C22#1912813111yescharacteristic
372C22#1912813611yescharacteristic
373C22#1912813711yescharacteristic
374C4—22no—
375C4—41no—
376C4—161no—
377C4—13281no—
378C4#1912761711yescharacteristic
379C4#1912791311yescharacteristic
380C4#1912791611yescharacteristic
381C4#1912795211yescharacteristic
382C2—162no—
383C2—13282no—
384C2#2490877311yescharacteristic
385C2#2490949511yescharacteristic
386C2#2490949811yescharacteristic
387C2#2490951511yescharacteristic
388C2#2490951811yescharacteristic
389C2#2490954711yescharacteristic
390C2#2490956411yescharacteristic
subgroupG/H
centerC23C2×#13050575
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1970C2
socleC23×C83C2×D8
prime radicalC22×#14581733C2
hypercenter#1970D83
hypocenterC83#13448
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C83⋊C16)C2semidirect
3. C2×#19127502C2semidirect
4. C2×#19127501C2semidirect
5. #24908329C2semidirect
6. #24905554C2semidirect
7. C83×#1970C2semidirect
138. C2×C4×C83D16semidirect
201. #1970D83semidirect
203. C83#13448semidirect
341. C2×C4#14581733semidirect

Maximal subgroups

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

subgroupindex
C22×#145817332
C2×C4⋊(C83⋊C16)2
C2×#191275022
C2×#191275012
#249083292
#249055542
C83×#19702
#1344883