Finite Group Databasebeta

← #35314743

Subgroups — #35314743

Order 21248.

470 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14581707C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905484C212yesmaximal, complemented
4#24905463C212yesmaximal, complemented
5C83×#2398C211yesmaximal, characteristic
6C2×#14581707—21no—
7C2×C22⋊(C83⋊C8)—21no—
8C4×C22⋊Dic83—21no—
9#19127468C2212yes—
10#19127427—21no—
11#19127425C2212yes—
12C2×C4×Q8×C83C411yescharacteristic, complemented
13C2×C4×D4×C83C2211yescharacteristic
14#400×C83—21no—
15#397×C83C411yescharacteristic, complemented
16#397×C83C2211yescharacteristic
17#394×C83—21no—
18#393×C83C411yescharacteristic, complemented
19#393×C83C2211yescharacteristic
20C4×C22⋊C4×C83—21no—
21C4×C22⋊C4×C83—21no—
22#376×C83C411yescharacteristic, complemented
23C22×C83⋊C8—41no—
24C2×C22⋊Dic83—22no—
25C2×C4×Dic83—41no—
26#14581707—24no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—41no—
29C83⋊#95—41no—
30C22×Q8×C83C2×C411yescharacteristic
31C22×D4×C83D411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C4×Q8×C83—41no—
35C4×Q8×C83—41no—
36C4×D4×C83—41no—
37C4×D4×C83—41no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83C2311yescharacteristic
47C2×C42×C83—21no—
48C2×C42×C83—21no—
49C2×C42×C83C2×C411yescharacteristic
50#95×C83—21no—
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57C22×Dic83—42no—
58C2×C83⋊C8—42no—
59C2×C83⋊C8—82no—
60C22⋊Dic83—24no—
61C22⋊Dic83—42no—
62C4×Dic83—82no—
63C24×C83C22⋊C411yescharacteristic
64C24×C83C2×D411yescharacteristic
65C2×Q8×C83—42no—
66C2×Q8×C83—41no—
67C2×D4×C83—22no—
68C2×D4×C83—22no—
69C2×D4×C83—41no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—42no—
79C22×C4×C83—42no—
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—42no—
88C22⋊C4×C83—41no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—42no—
97C42×C83—41no—
98C42×C83—41no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83#10412yes—
108C23×C83C2×C22⋊C411yescharacteristic
109C23×C83#13512yes—
110Q8×C83—42no—
111Q8×C83—42no—
112D4×C83—44no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—81no—
138C2×C4×C83—81no—
139C2×C4×C83#9911yescharacteristic
140C2×C4×C83#9911yescharacteristic
141C2×Dic83—44no—
142C2×Dic83—82no—
143C2×Dic83—82no—
144C83⋊C8—82no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—22no—
148C22×C83—22no—
149C22×C83—42no—
150C22×C83—42no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—81no—
156C22×C83—81no—
157C22×C83C2×#9911yescharacteristic
158C22×C83#41611yescharacteristic
159C22×C83C2×#10411yescharacteristic
160C4×C83—21no—
161C4×C83—42no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—81no—
167C4×C83—81no—
168C4×C83—81no—
169C4×C83#35412yes—
170Dic83—84no—
171#11728—831nomaximal, Sylow
172C2×C83—21no—
173C2×C83—21no—
174C2×C83—42no—
175C2×C83—42no—
176C2×C83#150111yescharacteristic
177C2×C83#212011yescharacteristic
178C2×C83C2×#35411yescharacteristic
179#2398D8311yescharacteristic
180C4×#99—831no—
181#1459—831no—
182#1351—832no—
183#1344—832no—
184C83#1172811yescharacteristic, complemented, Sylow
185C2×C4×Q8Dic8311yescharacteristic, complemented
186C2×C4×D4C2×D8311yescharacteristic
187C2×#99—1661no—
188C2×C22⋊C8—1661no—
189#400—21no—
190#397Dic8311yescharacteristic, complemented
191#397C2×D8311yescharacteristic
192#394—21no—
193#393Dic8311yescharacteristic, complemented
194#393C2×D8311yescharacteristic
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—1661no—
198#376Dic8311yescharacteristic, complemented
199#342—832no—
200#324—1661no—
201#323—832no—
202C22×Q8C2×Dic8311yescharacteristic
203C22×D4C22⋊D8311yescharacteristic
204C23×C4C22⋊D8311yescharacteristic
205C23×C4C22⋊D8311yescharacteristic
206C22×C8—3321no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×Dic8311yescharacteristic
213C2×C4⋊C4C22⋊D8311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—1662no—
219C2×C22⋊C4C2×Dic8311yescharacteristic
220C2×C22⋊C4C22×D8311yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—3321no—
224C2×C42C2×Dic8311yescharacteristic
225#99—1664no—
226C22⋊C8—3321no—
227C22⋊C8—1662no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—3321no—
234#95C2×Dic8311yescharacteristic
235#95C2×Dic8311yescharacteristic
236C24C22⋊Dic8311yescharacteristic
237C24C2×C22⋊D8311yescharacteristic
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—3322no—
254C22×C4C22⋊Dic8311yescharacteristic
255C22×C4C22⋊Dic8311yescharacteristic
256C22×C4C22⋊Dic8311yescharacteristic
257C22×C4C22×Dic8311yescharacteristic
258C22×C4C2×C22⋊D8311yescharacteristic
259C2×C8—3322no—
260C2×C8—6642no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—3322no—
271C22⋊C4—1664no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—6642no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1458171912yes—
287C23C2×C22⋊Dic8311yescharacteristic
288C23#1458181212yes—
289Q8—42no—
290Q8—42no—
291D4—44no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3324no—
319C2×C4—6642no—
320C2×C4—6642no—
321C2×C4#1458170711yescharacteristic
322C2×C4#1458170711yescharacteristic
323C8—6642no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1458170711yescharacteristic
337C22#1912769711yescharacteristic
338C22C2×#1458171911yescharacteristic
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—6644no—
349C4#1912749612yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2490595411yescharacteristic
355C2#2490910411yescharacteristic
356C2C2×#1912749611yescharacteristic
subgroupG/H
centerC2×C4#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2398C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#2398D83
hypocenterC83#11728
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #24905484C2semidirect
4. #24905463C2semidirect
12. C2×C4×Q8×C83C4semidirect
15. #397×C83C4semidirect
18. #393×C83C4semidirect
22. #376×C83C4semidirect
184. C83#11728semidirect
185. C2×C4×Q8Dic83semidirect
190. #397Dic83semidirect
193. #393Dic83semidirect
198. #376Dic83semidirect

Maximal subgroups

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

subgroupindex
C4×#145817072
#249057812
#249054842
#249054632
C83×#23982
#1172883