Finite Group Databasebeta

← #35313513

Subgroups — #35313513

Order 21248.

594 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#24910054C211yesmaximal, characteristic
2#324×D83C212yesmaximal, complemented
3#24905397C211yesmaximal, characteristic, complemented
4#24905394C212yesmaximal, complemented
5C83×#1319C211yesmaximal, characteristic, complemented
6C22×#13050591C411yescharacteristic
7C2×#14581749C411yescharacteristic
8C2×C4⋊C4×D83C2211yescharacteristic
9C22⋊C8×D83—22no—
10#19127608C411yescharacteristic
11#19127603C411yescharacteristic
12#19127602C2211yescharacteristic
13#19127579—22no—
14#19127560—22no—
15#19127427C2212yes—
16#19127426—22no—
17#391×C83C2211yescharacteristic
18#324×C83C2212yes—
19C2×#13050591—22no—
20C2×#13050591—42no—
21C2×#13050591—42no—
22C2×#13050591C814yescomplemented
23C22×D332—21no—
24C22×D332C2×C411yescharacteristic
25C22×C4×D83—21no—
26C22×C4×D83C2×C411yescharacteristic
27C22×C4×D83D412yes—
28C2×C8×D83—42no—
29#14581749—42no—
30C2×C4⋊Dic83C2×C411yescharacteristic
31C2×Dic83⋊C4—21no—
32C2×Dic83⋊C4D412yes—
33C4⋊C4×D83—42no—
34C2×(C2×D83)⋊C4—21no—
35C2×(C2×D83)⋊C4—22no—
36C2×(C2×D83)⋊C4—22no—
37C2×C4×Dic83C2×C411yescharacteristic
38C22⋊(C83⋊C8)—22no—
39(C2×D83)⋊C8—42no—
40C83⋊#95—22no—
41C22×Q8×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83C2311yescharacteristic
43C2×C42×C83C2×C411yescharacteristic
44C22⋊C8×C83—22no—
45#95×C83—22no—
46C23×D83—21no—
47C23×D83C22⋊C411yescharacteristic
48#13050591—48no—
49#13050591—48no—
50C2×D332—24no—
51C2×D332—21no—
52C2×D332—41no—
53C2×D332—41no—
54C2×D332—42no—
55C2×D332—42no—
56C2×D332C2×C812yescomplemented
57C22×Dic83—21no—
58C22×Dic83C22⋊C411yescharacteristic
59C22×Dic83C22⋊C412yes—
60C2×C4×D83—22no—
61C2×C4×D83—22no—
62C2×C4×D83—22no—
63C2×C4×D83—24no—
64C2×C4×D83—22no—
65C2×C4×D83—42no—
66C2×C4×D83—42no—
67C2×C4×D83—82no—
68C2×C4×D83C2×C812yes—
69C2×C4×D83M1612yes—
70C2×C83⋊C8—42no—
71C8×D83—84no—
72C4⋊Dic83—41no—
73Dic83⋊C4—42no—
74Dic83⋊C4—42no—
75(C2×D83)⋊C4—42no—
76(C2×D83)⋊C4—82no—
77(C2×D83)⋊C4—82no—
78C4×Dic83—41no—
79C2×Q8×C83—21no—
80C2×Q8×C83—41no—
81C2×Q8×C83—41no—
82C2×Q8×C83C2×C812yescomplemented
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83C22×C411yescharacteristic
86C22×C4×C83C2×D412yes—
87C2×C8×C83—42no—
88C4⋊C4×C83—41no—
89C42×C83—41no—
90C23×C83C2×C22⋊C411yescharacteristic
91C22×D83—22no—
92C22×D83—41no—
93C22×D83—41no—
94C22×D83—42no—
95C22×D83—42no—
96C22×D83—41no—
97C22×D83—41no—
98C22×D83C22⋊C812yes—
99C22×D83#9912yes—
100C22×D83#10012yes—
101Q8×C83—42no—
102Q8×C83—42no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—42no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83C22×C811yescharacteristic
114C2×C4×C83C2×M1611yescharacteristic
115C2×Dic83—21no—
116C2×Dic83—21no—
117C2×Dic83—22no—
118C2×Dic83—42no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83C22⋊C812yes—
123D332—42no—
124D332—42no—
125D332—48no—
126C4×D83—24no—
127C4×D83—24no—
128C4×D83—48no—
129C4×D83—44no—
130C4×D83—82no—
131C8×C83—82no—
132C83⋊C8—82no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83C2×C22⋊C811yescharacteristic
136C22×C83C2×#9911yescharacteristic
137C22×C83C2×#10011yescharacteristic
138C2×D83—22no—
139C2×D83—22no—
140C2×D83—42no—
141C2×D83—42no—
142C2×D83—44no—
143C2×D83—42no—
144C2×D83—82no—
145C2×D83—81no—
146C2×D83—81no—
147C2×D83#32314yes—
148C4×C83—21no—
149C4×C83—21no—
150C4×C83—42no—
151C4×C83—42no—
152C4×C83—81no—
153Dic83—22no—
154Dic83—42no—
155Dic83—81no—
156#11432—831nomaximal, Sylow
157C2×C83—21no—
158C2×C83—21no—
159C2×C83C2×#32311yescharacteristic
160C2×C83#212711yescharacteristic
161C2×C83#213311yescharacteristic
162D83—24no—
163D83—82no—
164#2311—831no—
165C2×#324—832no—
166#1319—831no—
167#1319D8311yescharacteristic, complemented
168#1318—832no—
169C83#1143211yescharacteristic, complemented, Sylow
170C22×#41—831no—
171C2×#117—831no—
172C22×C4⋊C4—831no—
173C2×C22⋊C8—1662no—
174#391—831no—
175#391C2×D8311yescharacteristic
176#390—831no—
177#390—831no—
178#386—1662no—
179#380—1662no—
180#324—832no—
181#324—1662no—
182#324C2×D8312yes—
183C2×#41—3322no—
184C2×#41—3322no—
185C2×#41—834no—
186C2×#41—1662no—
187C22×Q8C4×D8311yescharacteristic
188C22×D4—831no—
189C22×D4—1661no—
190C23×C4—831no—
191C23×C4—832no—
192C23×C4—1661no—
193C22×C8—3322no—
194#117—3322no—
195C2×C4⋊C4—3322no—
196C2×C4⋊C4—831no—
197C2×C4⋊C4—832no—
198C2×C4⋊C4—1661no—
199C2×C4⋊C4C22×D8311yescharacteristic
200C2×C22⋊C4—1661no—
201C2×C22⋊C4—1662no—
202C2×C22⋊C4—1662no—
203C2×C42—831no—
204C2×C42C4×D8311yescharacteristic
205C22⋊C8—22no—
206C22⋊C8—3322no—
207C22⋊C8—1662no—
208#95—22no—
209#95—1662no—
210C24—831no—
211C24—1661no—
212#41—3328no—
213#41—3328no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8C8×D8312yescomplemented
218C2×D4—3321no—
219C2×D4—3321no—
220C2×D4—3322no—
221C2×D4—3322no—
222C2×D4—832no—
223C2×D4—1664no—
224C2×D4—1661no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—3322no—
228C22×C4—3322no—
229C22×C4—832no—
230C22×C4—831no—
231C22×C4—832no—
232C22×C4—832no—
233C22×C4—6642no—
234C22×C4—1664no—
235C22×C4—1662no—
236C22×C4—1662no—
237C22×C4—1662no—
238C22×C4—1662no—
239C22×C4—1661no—
240C22×C4C2×C4×D8311yescharacteristic
241C22×C4D4×D8312yes—
242C2×C8—42no—
243C2×C8—3322no—
244C2×C8—6644no—
245C4⋊C4—41no—
246C4⋊C4—3322no—
247C4⋊C4—3321no—
248C4⋊C4—3322no—
249C22⋊C4—3322no—
250C22⋊C4—6642no—
251C22⋊C4—6642no—
252C42—41no—
253C42—3321no—
254C23—3321no—
255C23—3321no—
256C23—3322no—
257C23—3322no—
258C23—3321no—
259C23—3321no—
260C23—832no—
261C23—832no—
262C23—832no—
263C23—1662no—
264C23C22⋊C4×D8311yescharacteristic
265Q8—42no—
266Q8—42no—
267D4—3322no—
268D4—3322no—
269D4—3328no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—3324no—
281C2×C4—3328no—
282C2×C4—3322no—
283C2×C4—3321no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—832no—
287C2×C4—6642no—
288C2×C4—1664no—
289C2×C4—1664no—
290C2×C4—1662no—
291C2×C4—1661no—
292C2×C4—1661no—
293C2×C4C2×C8×D8311yescharacteristic
294C2×C4M16×D8311yescharacteristic
295C8—82no—
296C8—6642no—
297C22—21no—
298C22—21no—
299C22—3324no—
300C22—3322no—
301C22—3322no—
302C22—3322no—
303C22—834no—
304C22—6642no—
305C22—6641no—
306C22—6641no—
307C22—1662no—
308C22—1662no—
309C22C22⋊C8×D8311yescharacteristic
310C22#99×D8311yescharacteristic
311C22#100×D8311yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—3322no—
318C4—6641no—
319C4—1662no—
320C2—21no—
321C2—21no—
322C2—6642no—
323C2—1664no—
324C2#323×D8311yescharacteristic
325C2#2490913111yescharacteristic
326C2#2490915111yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1319C2
socleC22×C83C2×#99
prime radicalC22×#13050591C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1319D83
hypocenterC83#11432
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
2. #324×D83C2semidirect
3. #24905397C2semidirect
4. #24905394C2semidirect
5. C83×#1319C2semidirect
22. C2×#13050591C8semidirect
56. C2×D332C2×C8semidirect
82. C2×Q8×C83C2×C8semidirect
167. #1319D83semidirect
169. C83#11432semidirect
217. C2×Q8C8×D83semidirect

Maximal subgroups

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

subgroupindex
#249100542
#324×D832
#249053972
#249053942
C83×#13192
#1143283