Finite Group Databasebeta

← #35303834

Subgroups — #35303834

Order 21248.

396 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907214C211yesmaximal, characteristic, complemented
2#24906226C211yesmaximal, characteristic, complemented
3#24906078C211yesmaximal, characteristic, complemented
4#24906074C211yesmaximal, characteristic
5#24905975C211yesmaximal, characteristic, complemented
6#24905971C211yesmaximal, characteristic
7C83×#3167C211yesmaximal, characteristic
8#19127758C2211yescharacteristic
9#19127758C2211yescharacteristic
10C2×#14581703C2211yescharacteristic
11#19127547C2211yescharacteristic
12#19127455—21no—
13#19127455—21no—
14#19127454—21no—
15#19127452—21no—
16#19127443—22no—
17#19127441—22no—
18#19127438—22no—
19#19127435—22no—
20#533×C83C2211yescharacteristic
21#532×C83C2211yescharacteristic
22#498×C83C411yescharacteristic, complemented
23#492×C83C411yescharacteristic, complemented
24#487×C83C411yescharacteristic, complemented
25#486×C83C411yescharacteristic, complemented
26#465×C83—21no—
27#465×C83—21no—
28C4×SD16×C83—21no—
29C4×SD16×C83—21no—
30C2×C4×C8×C83C2211yescharacteristic
31C22×C83⋊C8—21no—
32C2×C4×Dic83—21no—
33C4⋊(C83⋊C8)—21no—
34C4⋊(C83⋊C8)—21no—
35C22⋊(C83⋊C8)—21no—
36C22⋊(C83⋊C8)—21no—
37#14581703—21no—
38#14581703—21no—
39#14581703—21no—
40#14581703D412yes—
41C83⋊#95—21no—
42C83⋊#95D412yes—
43C2×#41×C83—21no—
44C2×SD16×C83—21no—
45C2×SD16×C83—21no—
46C22×C8×C83D411yescharacteristic
47C22×C8×C83D411yescharacteristic
48C4⋊Q8×C83C2×C411yescharacteristic
49C4⋊Q8×C83C2×C411yescharacteristic
50C4⋊Q8×C83C2×C411yescharacteristic
51#127×C83C2×C411yescharacteristic
52#125×C83—21no—
53#124×C83—21no—
54C22⋊Q8×C83—21no—
55C22⋊Q8×C83—21no—
56#121×C83—21no—
57#121×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83—21no—
60C4×D4×C83—21no—
61C4×D4×C83—21no—
62#117×C83—21no—
63C2×C42×C83C2311yescharacteristic
64#106×C83—21no—
65#106×C83—21no—
66#103×C83—21no—
67#103×C83—21no—
68#102×C83—21no—
69#102×C83—21no—
70C4×C8×C83C2×C411yescharacteristic
71C4×C8×C83C2×C411yescharacteristic
72C4×C8×C83D411yescharacteristic
73C4×C8×C83D411yescharacteristic
74C22×Dic83—22no—
75C2×C83⋊C8—22no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8—41no—
78C2×C83⋊C8—41no—
79C2×C83⋊C8—41no—
80C4×Dic83—41no—
81C4×Dic83—41no—
82#41×C83—82no—
83C2×Q8×C83—21no—
84C2×Q8×C83—21no—
85C2×Q8×C83—21no—
86C2×D4×C83—21no—
87C2×D4×C83—21no—
88C2×D4×C83—21no—
89C2×D4×C83—41no—
90C22×C4×C83—41no—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83#4111yescharacteristic
94SD16×C83—42no—
95SD16×C83—42no—
96C2×C8×C83—41no—
97C2×C8×C83—41no—
98C2×C8×C83C22⋊C411yescharacteristic
99C2×C8×C83C22⋊C411yescharacteristic
100C2×C8×C83C22⋊C411yescharacteristic
101C2×C8×C83C22⋊C411yescharacteristic
102C4⋊C4×C83—21no—
103C4⋊C4×C83—21no—
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—21no—
108C4⋊C4×C83—41no—
109C4⋊C4×C83—41no—
110C22⋊C4×C83—41no—
111C22⋊C4×C83—41no—
112C22⋊C4×C83—41no—
113C42×C83—41no—
114C42×C83C22×C411yescharacteristic
115C42×C83C2×D411yescharacteristic
116C42×C83C2×D411yescharacteristic
117C42×C83#4111yescharacteristic
118C23×C83—41no—
119C23×C83#12111yescharacteristic
120Q8×C83—41no—
121Q8×C83—41no—
122Q8×C83—81no—
123Q8×C83—81no—
124D4×C83—41no—
125D4×C83—41no—
126D4×C83—81no—
127D4×C83—81no—
128D4×C83—82no—
129C2×C4×C83—22no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—82no—
137C2×C4×C83C2×C22⋊C411yescharacteristic
138C2×C4×C83C4×D411yescharacteristic
139C2×C4×C83C4×D411yescharacteristic
140C2×C4×C83#12111yescharacteristic
141C2×C4×C83#12411yescharacteristic
142C2×C4×C83#12711yescharacteristic
143C2×Dic83—42no—
144C2×Dic83—42no—
145C2×Dic83—42no—
146C8×C83—21no—
147C8×C83—21no—
148C8×C83—21no—
149C8×C83—21no—
150C83⋊C8—42no—
151C83⋊C8—81no—
152C83⋊C8—81no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—81no—
156C22×C83—81no—
157C22×C83—81no—
158C22×C83#39011yescharacteristic
159C22×C83#47112yes—
160C4×C83—21no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—81no—
166C4×C83—81no—
167C4×C83—81no—
168C4×C83#42111yescharacteristic
169C4×C83#42111yescharacteristic
170Dic83—82no—
171#9766—831nomaximal, Sylow
172C2×C83—22no—
173C2×C83—161no—
174C2×C83#196311yescharacteristic
175C2×C83#197611yescharacteristic
176C2×C83#197711yescharacteristic
177#3167D8311yescharacteristic
178#1746—831no—
179#1568—831no—
180#1531—831no—
181#1530—831no—
182#1506—831no—
183#1505—831no—
184C83#976611yescharacteristic, complemented, Sylow
185#533C2×D8311yescharacteristic
186#532C2×D8311yescharacteristic
187#498Dic8311yescharacteristic, complemented
188#492Dic8311yescharacteristic, complemented
189#487Dic8311yescharacteristic, complemented
190#486Dic8311yescharacteristic, complemented
191#465—21no—
192#465—21no—
193C4×SD16—21no—
194C4×SD16—21no—
195#432—831no—
196#432—831no—
197C2×#97—831no—
198C2×C4×C8C2×D8311yescharacteristic
199#376—831no—
200#336—1661no—
201#336—1661no—
202#335—1661no—
203#334—1661no—
204#330—1662no—
205#329—1662no—
206#328—1662no—
207#327—1662no—
208C2×#41—21no—
209C2×SD16—21no—
210C2×SD16—21no—
211C22×C8—1661no—
212C22×C8C22⋊D8311yescharacteristic
213C22×C8C22⋊D8311yescharacteristic
214C4⋊Q8C2×Dic8311yescharacteristic
215C4⋊Q8C2×Dic8311yescharacteristic
216C4⋊Q8C2×Dic8311yescharacteristic
217#127C2×Dic8311yescharacteristic
218#125—21no—
219#124—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×D4—21no—
227C4×D4—21no—
228#117—21no—
229C2×C42—1661no—
230C2×C42C22×D8311yescharacteristic
231#106—21no—
232#106—21no—
233C4⋊C8—1661no—
234C4⋊C8—1661no—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C22⋊C8—1661no—
240C22⋊C8—1661no—
241#97—832no—
242#97—1661no—
243#97—1661no—
244#97—1661no—
245C4×C8C2×Dic8311yescharacteristic
246C4×C8C2×Dic8311yescharacteristic
247C4×C8C22⋊D8311yescharacteristic
248C4×C8C22⋊D8311yescharacteristic
249#95—832no—
250#95—1661no—
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—1662no—
261C22×C4D4×D8311yescharacteristic
262C22×C4C2×C22⋊D8311yescharacteristic
263C22×C4#1305058911yescharacteristic
264SD16—42no—
265SD16—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—3321no—
269C2×C8—3321no—
270C2×C8—3321no—
271C2×C8—3321no—
272C2×C8—1662no—
273C2×C8C22⋊Dic8311yescharacteristic
274C2×C8C22⋊Dic8311yescharacteristic
275C2×C8C22⋊Dic8311yescharacteristic
276C2×C8C22⋊Dic8311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—41no—
289C42—3321no—
290C42—3321no—
291C42C22×Dic8311yescharacteristic
292C42D4×D8311yescharacteristic
293C42C2×C22⋊D8311yescharacteristic
294C42#1305058911yescharacteristic
295C23—41no—
296C23#1458176311yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—81no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—82no—
314C2×C4—3322no—
315C2×C4—3322no—
316C2×C4—3322no—
317C2×C4C2×C22⋊Dic8311yescharacteristic
318C2×C4D4×Dic8311yescharacteristic
319C2×C4D4×Dic8311yescharacteristic
320C2×C4#1458176311yescharacteristic
321C2×C4#1458177911yescharacteristic
322C2×C4#1458178811yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C8—3322no—
328C8—6641no—
329C8—6641no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#1912760411yescharacteristic
336C22#1912796112yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—6642no—
346C4#1912771811yescharacteristic
347C4#1912771811yescharacteristic
348C2—22no—
349C2—161no—
350C2#2490829611yescharacteristic
351C2#2490836211yescharacteristic
352C2#2490836811yescharacteristic
subgroupG/H
centerC22#19127604
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3167C2
socleC22×C83#390
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#3167D83
hypocenterC83#9766
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #24907214C2semidirect
2. #24906226C2semidirect
3. #24906078C2semidirect
5. #24905975C2semidirect
22. #498×C83C4semidirect
23. #492×C83C4semidirect
24. #487×C83C4semidirect
25. #486×C83C4semidirect
184. C83#9766semidirect
187. #498Dic83semidirect
188. #492Dic83semidirect
189. #487Dic83semidirect
190. #486Dic83semidirect

Maximal subgroups

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

subgroupindex
#249072142
#249062262
#249060782
#249060742
#249059752
#249059712
C83×#31672
#976683