Finite Group Databasebeta

← #35303807

Subgroups — #35303807

Order 21248.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907149C211yesmaximal, 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
11C2×#14581702—21no—
12C2×C8×Dic83—21no—
13#19127547C2211yescharacteristic
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—
41#14581702—24no—
42C8×Dic83—24no—
43C83⋊#95—21no—
44C83⋊#95D412yes—
45C2×#41×C83—21no—
46C2×SD16×C83—21no—
47C2×SD16×C83—21no—
48C22×C8×C83D411yescharacteristic
49C22×C8×C83D411yescharacteristic
50C4⋊Q8×C83C2×C411yescharacteristic
51C4⋊Q8×C83C2×C411yescharacteristic
52C4⋊Q8×C83C2×C411yescharacteristic
53#127×C83C2×C411yescharacteristic
54#125×C83—21no—
55#124×C83—21no—
56C22⋊Q8×C83—21no—
57C22⋊Q8×C83—21no—
58#121×C83—21no—
59#121×C83—21no—
60C4×Q8×C83—21no—
61C4×Q8×C83—21no—
62C4×D4×C83—21no—
63C4×D4×C83—21no—
64#117×C83—21no—
65C2×C42×C83C2311yescharacteristic
66#106×C83—21no—
67#106×C83—21no—
68#103×C83—21no—
69#103×C83—21no—
70#102×C83—21no—
71#102×C83—21no—
72C4×C8×C83C2×C411yescharacteristic
73C4×C8×C83C2×C411yescharacteristic
74C4×C8×C83D411yescharacteristic
75C4×C8×C83D411yescharacteristic
76C22×Dic83—22no—
77C2×C83⋊C8—22no—
78C2×C83⋊C8—22no—
79C2×C83⋊C8—22no—
80C2×C83⋊C8—41no—
81C2×C83⋊C8—41no—
82C4×Dic83—22no—
83C4×Dic83—22no—
84#41×C83—82no—
85C2×Q8×C83—21no—
86C2×Q8×C83—21no—
87C2×Q8×C83—21no—
88C2×D4×C83—21no—
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83#4111yescharacteristic
96SD16×C83—42no—
97SD16×C83—42no—
98C2×C8×C83—22no—
99C2×C8×C83—22no—
100C2×C8×C83C22⋊C411yescharacteristic
101C2×C8×C83C22⋊C411yescharacteristic
102C2×C8×C83C22⋊C411yescharacteristic
103C2×C8×C83C22⋊C411yescharacteristic
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—21no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83—21no—
110C4⋊C4×C83—41no—
111C4⋊C4×C83—41no—
112C22⋊C4×C83—41no—
113C22⋊C4×C83—41no—
114C22⋊C4×C83—41no—
115C42×C83—41no—
116C42×C83C22×C411yescharacteristic
117C42×C83C2×D411yescharacteristic
118C42×C83C2×D411yescharacteristic
119C42×C83#4111yescharacteristic
120C23×C83—41no—
121C23×C83#12111yescharacteristic
122Q8×C83—41no—
123Q8×C83—41no—
124Q8×C83—81no—
125Q8×C83—81no—
126D4×C83—41no—
127D4×C83—41no—
128D4×C83—81no—
129D4×C83—81no—
130D4×C83—82no—
131C2×C4×C83—22no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—82no—
139C2×C4×C83C2×C22⋊C411yescharacteristic
140C2×C4×C83C4×D411yescharacteristic
141C2×C4×C83C4×D411yescharacteristic
142C2×C4×C83#12111yescharacteristic
143C2×C4×C83#12411yescharacteristic
144C2×C4×C83#12711yescharacteristic
145C2×Dic83—24no—
146C2×Dic83—42no—
147C2×Dic83—42no—
148C8×C83—21no—
149C8×C83—21no—
150C8×C83#10412yes—
151C8×C83#10412yes—
152C83⋊C8—24no—
153C83⋊C8—81no—
154C83⋊C8—81no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—81no—
158C22×C83—81no—
159C22×C83—81no—
160C22×C83#39011yescharacteristic
161C22×C83#47112yes—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—81no—
168C4×C83—81no—
169C4×C83—81no—
170C4×C83C2×#10411yescharacteristic
171C4×C83C2×#10411yescharacteristic
172Dic83—44no—
173#9761—831nomaximal, Sylow
174C2×C83—22no—
175C2×C83—161no—
176C2×C83#196211yescharacteristic
177C2×C83#197611yescharacteristic
178C2×C83#197711yescharacteristic
179#3167D8311yescharacteristic
180#1726—831no—
181#1568—831no—
182#1531—831no—
183#1530—831no—
184#1506—831no—
185#1505—831no—
186C83#976111yescharacteristic, complemented, Sylow
187#533C2×D8311yescharacteristic
188#532C2×D8311yescharacteristic
189#498Dic8311yescharacteristic, complemented
190#492Dic8311yescharacteristic, complemented
191#487Dic8311yescharacteristic, complemented
192#486Dic8311yescharacteristic, complemented
193#465—21no—
194#465—21no—
195C4×SD16—21no—
196C4×SD16—21no—
197#432—831no—
198#432—831no—
199C2×#97—831no—
200C2×#97—1661no—
201C2×C4×C8—1661no—
202C2×C4×C8C2×D8311yescharacteristic
203#376—831no—
204#335—1661no—
205#334—1661no—
206#330—1662no—
207#329—1662no—
208#328—1662no—
209#327—1662no—
210C2×#41—21no—
211C2×SD16—21no—
212C2×SD16—21no—
213C22×C8—1661no—
214C22×C8C22⋊D8311yescharacteristic
215C22×C8C22⋊D8311yescharacteristic
216C4⋊Q8C2×Dic8311yescharacteristic
217C4⋊Q8C2×Dic8311yescharacteristic
218C4⋊Q8C2×Dic8311yescharacteristic
219#127C2×Dic8311yescharacteristic
220#125—21no—
221#124—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224#121—21no—
225#121—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×D4—21no—
229C4×D4—21no—
230#117—21no—
231C2×C42—1661no—
232C2×C42C22×D8311yescharacteristic
233#106—21no—
234#106—21no—
235C4⋊C8—1661no—
236C4⋊C8—1661no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C22⋊C8—1661no—
242C22⋊C8—1661no—
243#97—832no—
244#97—1664no—
245#97—1661no—
246#97—1661no—
247#97—1661no—
248C4×C8—1664no—
249C4×C8C2×Dic8311yescharacteristic
250C4×C8C2×Dic8311yescharacteristic
251C4×C8C22⋊D8311yescharacteristic
252C4×C8C22⋊D8311yescharacteristic
253#95—832no—
254#95—1661no—
255#41—82no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—41no—
264C22×C4—1662no—
265C22×C4D4×D8311yescharacteristic
266C22×C4C2×C22⋊D8311yescharacteristic
267C22×C4#1305058911yescharacteristic
268SD16—42no—
269SD16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—3321no—
273C2×C8—3321no—
274C2×C8—1662no—
275C2×C8—1662no—
276C2×C8—1662no—
277C2×C8C22⋊Dic8311yescharacteristic
278C2×C8C22⋊Dic8311yescharacteristic
279C2×C8C22⋊Dic8311yescharacteristic
280C2×C8C22⋊Dic8311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C42—41no—
293C42—1662no—
294C42—1662no—
295C42C22×Dic8311yescharacteristic
296C42D4×D8311yescharacteristic
297C42C2×C22⋊D8311yescharacteristic
298C42#1305058911yescharacteristic
299C23—41no—
300C23#1458176311yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—82no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—82no—
318C2×C4—3322no—
319C2×C4—3322no—
320C2×C4—1664no—
321C2×C4C2×C22⋊Dic8311yescharacteristic
322C2×C4D4×Dic8311yescharacteristic
323C2×C4D4×Dic8311yescharacteristic
324C2×C4#1458176311yescharacteristic
325C2×C4#1458177911yescharacteristic
326C2×C4#1458178811yescharacteristic
327C8—21no—
328C8—21no—
329C8—6641no—
330C8—6641no—
331C8—1664no—
332C8#1458171912yes—
333C8#1458171912yes—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1912760411yescharacteristic
340C22#1912796112yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—3324no—
350C4C2×#1458171911yescharacteristic
351C4C2×#1458171911yescharacteristic
352C2—22no—
353C2—161no—
354C2#2490829211yescharacteristic
355C2#2490836211yescharacteristic
356C2#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#9761
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. #24907149C2semidirect
2. #24906226C2semidirect
3. #24906078C2semidirect
5. #24905975C2semidirect
22. #498×C83C4semidirect
23. #492×C83C4semidirect
24. #487×C83C4semidirect
25. #486×C83C4semidirect
186. C83#9761semidirect
189. #498Dic83semidirect
190. #492Dic83semidirect
191. #487Dic83semidirect
192. #486Dic83semidirect

Maximal subgroups

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

subgroupindex
#249071492
#249062262
#249060782
#249060742
#249059752
#249059712
C83×#31672
#976183