Finite Group Databasebeta

← #35315022

Subgroups — #35315022

Order 21248.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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#24912636C211yesmaximal, characteristic
2C4×#14581707C211yesmaximal, characteristic
3#24905485C214yesmaximal, complemented
4C83×#1465C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6C2×#14581707—21no—
7#19127613C412yescomplemented
8#19127613C2212yes—
9#19127604—21no—
10#19127566—21no—
11#19127561—21no—
12#19127561—21no—
13C4×C4⋊Dic83C411yescharacteristic, complemented
14C4×C22⋊Dic83—21no—
15#19127468C2212yes—
16C2×C4×D4×C83C2211yescharacteristic
17#407×C83—21no—
18#324×C83—21no—
19#323×C83C2212yes—
20C22×Dic166C2×C412yes—
21#14581792—41no—
22#14581792—41no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C22⋊Dic83—22no—
27C2×C22⋊Dic83—22no—
28C2×C22⋊Dic83—22no—
29C2×C4×Dic83—21no—
30C2×C4×Dic83—41no—
31#14581707—24no—
32C83⋊#95—22no—
33C83⋊#95—22no—
34C83⋊#95—41no—
35C83⋊#95C2×C412yes—
36C22×D4×C83D411yescharacteristic
37C23×C4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C2×M16×C83—41no—
40C4×D4×C83—41no—
41C4×D4×C83—41no—
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C22⋊C4×C83C2×C411yescharacteristic
44C2×C22⋊C4×C83C2311yescharacteristic
45C2×C42×C83C2×C411yescharacteristic
46C22⋊C8×C83—22no—
47C22⋊C8×C83—41no—
48C2×Dic166—42no—
49C2×Dic166—82no—
50C22×Dic83—22no—
51C22×Dic83—22no—
52C22×Dic83—42no—
53C4⋊Dic83—42no—
54C4⋊Dic83—42no—
55C4⋊Dic83—41no—
56C4⋊Dic83—41no—
57C22⋊Dic83—24no—
58C22⋊Dic83—42no—
59C22⋊Dic83—82no—
60C22⋊Dic83—82no—
61C4×Dic83—42no—
62C4×Dic83—82no—
63C24×C83C22⋊C411yescharacteristic
64C24×C83C2×D411yescharacteristic
65C2×D4×C83—22no—
66C2×D4×C83—22no—
67C2×D4×C83—41no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—42no—
73C22×C4×C83—42no—
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79M16×C83—82no—
80C2×C8×C83—42no—
81C4⋊C4×C83—41no—
82C22⋊C4×C83—41no—
83C22⋊C4×C83—41no—
84C42×C83—21no—
85C42×C83—21no—
86C23×C83—21no—
87C23×C83—21no—
88C23×C83—21no—
89C23×C83—21no—
90C23×C83—21no—
91C23×C83—41no—
92C23×C83—41no—
93C23×C83—41no—
94C23×C83—41no—
95C23×C83C2×C22⋊C411yescharacteristic
96C23×C83#13712yes—
97D4×C83—44no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—22no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—81no—
110C2×C4×C83—81no—
111C2×C4×C83#9911yescharacteristic
112C2×C4×C83#9911yescharacteristic
113C2×C4×C83#10412yes—
114C2×Dic83—44no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117C2×Dic83—42no—
118C2×Dic83—42no—
119C2×Dic83—42no—
120C2×Dic83—42no—
121C2×Dic83—82no—
122C2×Dic83—82no—
123Dic166—84no—
124C8×C83—82no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—22no—
128C22×C83—42no—
129C22×C83—42no—
130C22×C83—41no—
131C22×C83—41no—
132C22×C83—41no—
133C22×C83—41no—
134C22×C83—41no—
135C22×C83—81no—
136C22×C83—81no—
137C22×C83C2×#9911yescharacteristic
138C22×C83#41911yescharacteristic
139C22×C83C2×#10411yescharacteristic
140C4×C83—21no—
141C4×C83—22no—
142C4×C83—41no—
143C4×C83—41no—
144C4×C83—41no—
145C4×C83#35412yes—
146Dic83—82no—
147Dic83—84no—
148Dic83—82no—
149#11787—831nomaximal, Sylow
150C2×C83—21no—
151C2×C83—21no—
152C2×C83—42no—
153C2×C83—81no—
154C2×C83#150811yescharacteristic
155C2×C83#212211yescharacteristic
156C2×C83C2×#35411yescharacteristic
157#2563—831no—
158C4×#99—831no—
159#1465D8311yescharacteristic
160#1351—834no—
161C83#1178711yescharacteristic, complemented, Sylow
162C2×C4⋊Q8—831no—
163C2×C4×D4C2×D8311yescharacteristic
164C2×#99—1661no—
165#407—21no—
166#393—832no—
167#393—832no—
168#390—1661no—
169#382—1661no—
170#380—1661no—
171#380—1661no—
172C4×C4⋊C4—831no—
173C4×C22⋊C4—1661no—
174#342—832no—
175#324—21no—
176#323C2×D8312yes—
177C22×Q8—832no—
178C22×D4D33211yescharacteristic
179C23×C4D33211yescharacteristic
180C23×C4C22⋊D8311yescharacteristic
181C2×M16—41no—
182C4⋊Q8—3321no—
183C4⋊Q8—3321no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—1661no—
187C2×C4⋊C4—1661no—
188C2×C4⋊C4—1661no—
189C2×C4⋊C4C22⋊D8311yescharacteristic
190C2×C22⋊C4—1662no—
191C2×C22⋊C4—1662no—
192C2×C22⋊C4—1662no—
193C2×C22⋊C4C4×D8311yescharacteristic
194C2×C22⋊C4C22×D8311yescharacteristic
195C2×C42—3321no—
196C2×C42—1661no—
197C2×C42C4×D8311yescharacteristic
198#99—1664no—
199C22⋊C8—22no—
200C22⋊C8—41no—
201#95—3321no—
202#95—832no—
203#95—1662no—
204#95—1662no—
205C24(C2×D83)⋊C411yescharacteristic
206C24C2×D33211yescharacteristic
207C2×Q8—3322no—
208C2×Q8—6642no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—3322no—
219C22×C4—1662no—
220C22×C4—1662no—
221C22×C4(C2×D83)⋊C411yescharacteristic
222C22×C4(C2×D83)⋊C411yescharacteristic
223C22×C4(C2×D83)⋊C411yescharacteristic
224C22×C4C2×C4×D8311yescharacteristic
225C22×C4C2×C22⋊D8311yescharacteristic
226M16—82no—
227C2×C8—42no—
228C4⋊C4—41no—
229C4⋊C4—3322no—
230C4⋊C4—3322no—
231C4⋊C4—3321no—
232C4⋊C4—3321no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—3322no—
236C22⋊C4—6642no—
237C22⋊C4—6642no—
238C22⋊C4—1664no—
239C42—21no—
240C42—21no—
241C42—3322no—
242C42—6642no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C2×(C2×D83)⋊C411yescharacteristic
253C23#1458182512yes—
254Q8—6644no—
255D4—44no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—3324no—
270C2×C4—3322no—
271C2×C4—3322no—
272C2×C4—3322no—
273C2×C4—3322no—
274C2×C4—3322no—
275C2×C4—3322no—
276C2×C4—6642no—
277C2×C4—6642no—
278C2×C4#1458170611yescharacteristic
279C2×C4#1458170611yescharacteristic
280C2×C4#1458171812yes—
281C8—82no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#1458170611yescharacteristic
295C22#1912771111yescharacteristic
296C22C2×#1458171811yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—6642no—
303C4—6644no—
304C4—6642no—
305C4#1912749512yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2490598411yescharacteristic
311C2#2490911411yescharacteristic
312C2C2×#1912749511yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1465C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1465D83
hypocenterC83#11787
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #24905485C2semidirect
5. C2×#14581792C4semidirect
7. #19127613C4semidirect
13. C4×C4⋊Dic83C4semidirect
161. C83#11787semidirect

Maximal subgroups

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

subgroupindex
#249126362
C4×#145817072
#249054852
C83×#14652
#1178783