Finite Group Databasebeta

← #35302084

Subgroups — #35302084

Order 21248.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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#24916492C211yesmaximal, characteristic
2#24906189C212yesmaximal, complemented
3#24905804C211yesmaximal, characteristic
4#24905319C212yesmaximal, complemented
5C83×#2547C211yesmaximal, characteristic
6#19127773—24no—
7#19127758C2212yes—
8#19127758C2212yes—
9#19127665—21no—
10C2×C22⋊(C83⋊C8)—21no—
11#19127450—24no—
12#19127425—21no—
13#19127425—21no—
14C2×C4⋊Q8×C83C2211yescharacteristic
15C2×#124×C83C412yes—
16C2×C4×D4×C83C2211yescharacteristic
17#397×C83—21no—
18#397×C83—21no—
19#391×C83C412yes—
20#380×C83—21no—
21#380×C83—21no—
22C4×C4⋊C4×C83C2211yescharacteristic
23C2×#13050574—22no—
24C22×C83⋊C8—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)D414yes—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—22no—
29C22⋊(C83⋊C8)—22no—
30C22⋊(C83⋊C8)—22no—
31#14581703—22no—
32#14581703—22no—
33C22×Q8×C83C2×C412yes—
34C22×D4×C83C2×C411yescharacteristic
35C23×C4×C83D411yescharacteristic
36C23×C4×C83D411yescharacteristic
37C4⋊Q8×C83—22no—
38C4⋊Q8×C83—22no—
39#124×C83—44no—
40C4×D4×C83—22no—
41C4×D4×C83—22no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—22no—
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83D411yescharacteristic
50C2×C42×C83C2×C412yes—
51C2×C42×C83C2311yescharacteristic
52#95×C83—22no—
53#95×C83—22no—
54#95×C83—21no—
55#13050574—82no—
56C2×C83⋊C8—24no—
57C2×C83⋊C8—42no—
58C2×C83⋊C8—82no—
59C24×C83C22⋊C411yescharacteristic
60C24×C83C22⋊C411yescharacteristic
61C2×Q8×C83—24no—
62C2×Q8×C83—42no—
63C2×Q8×C83—42no—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83—41no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—22no—
72C22×C4×C83—22no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C4⋊C4×C83—22no—
81C4⋊C4×C83—42no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—42no—
84C4⋊C4×C83—41no—
85C22⋊C4×C83—22no—
86C22⋊C4×C83—22no—
87C22⋊C4×C83—44no—
88C22⋊C4×C83—44no—
89C42×C83—42no—
90C42×C83C2×D412yes—
91C42×C83#4112yes—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83—41no—
101C23×C83#10012yes—
102C23×C83C2×C22⋊C411yescharacteristic
103Q8×C83—84no—
104D4×C83—44no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—41no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83—42no—
120C2×C4×C83—42no—
121C2×C4×C83—81no—
122C2×C4×C83—81no—
123C2×C4×C83C4×D412yes—
124C2×C4×C83#12011yescharacteristic
125C2×C4×C83#12112yes—
126C2×C4×C83#12311yescharacteristic
127C83⋊C8—44no—
128C83⋊C8—82no—
129C22×C83—21no—
130C22×C83—21no—
131C22×C83—22no—
132C22×C83—42no—
133C22×C83—42no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—41no—
137C22×C83—41no—
138C22×C83—41no—
139C22×C83—81no—
140C22×C83—81no—
141C22×C83#38611yescharacteristic
142C22×C83C2×#10011yescharacteristic
143C22×C83#41311yescharacteristic
144C4×C83—21no—
145C4×C83—22no—
146C4×C83—22no—
147C4×C83—41no—
148C4×C83—41no—
149C4×C83—82no—
150C4×C83—82no—
151C4×C83#45512yes—
152#9451—831nomaximal, Sylow
153C2×C83—21no—
154C2×C83—21no—
155C2×C83—42no—
156C2×C83—81no—
157C2×C83#188511yescharacteristic
158C2×C83#190011yescharacteristic
159C2×C83#190911yescharacteristic
160#2973—831no—
161#2547D8311yescharacteristic
162#1558—832no—
163#1466—831no—
164#1285—832no—
165C83#945111yescharacteristic, complemented, Sylow
166C2×C4⋊Q8C2×D8311yescharacteristic
167C2×#124Dic8312yes—
168C2×C4×D4C2×D8311yescharacteristic
169#435—1664no—
170#432—832no—
171#432—832no—
172#407—1661no—
173C2×C22⋊C8—1661no—
174#397—21no—
175#397—21no—
176#391Dic8312yes—
177#380—21no—
178#380—21no—
179C4×C4⋊C4C2×D8311yescharacteristic
180#333—1664no—
181#323—1661no—
182#323—1661no—
183C22×Q8C2×Dic8312yes—
184C22×D4C2×Dic8311yescharacteristic
185C23×C4C22⋊D8311yescharacteristic
186C23×C4C22⋊D8311yescharacteristic
187C2×M16—1662no—
188C22×C8—1662no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191#124—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic8311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊D8311yescharacteristic
201C2×C22⋊C4C22⋊D8311yescharacteristic
202C2×C42C2×Dic8312yes—
203C2×C42C22×D8311yescharacteristic
204C4⋊C8—834no—
205C4⋊C8—1662no—
206C22⋊C8—1662no—
207C22⋊C8—1662no—
208C22⋊C8—1662no—
209C22⋊C8—1662no—
210#97—1662no—
211#97—1662no—
212#95—22no—
213#95—22no—
214#95—21no—
215C24C22⋊Dic8311yescharacteristic
216C24C22⋊Dic8311yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊Dic8311yescharacteristic
232C22×C4C22⋊Dic8311yescharacteristic
233C22×C4C22×Dic8311yescharacteristic
234C22×C4C2×C22⋊D8311yescharacteristic
235C22×C4C2×C22⋊D8311yescharacteristic
236M16—6642no—
237C2×C8—3322no—
238C2×C8—6642no—
239C2×C8—1664no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42D4×D8312yes—
251C42#1305058912yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1458170912yes—
262C23C2×C22⋊Dic8311yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4D4×Dic8312yes—
284C2×C4#1458176011yescharacteristic
285C2×C4#1458176412yes—
286C2×C4#1458177311yescharacteristic
287C8—3324no—
288C8—6642no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1912758111yescharacteristic
302C22C2×#1458170911yescharacteristic
303C22#1912768511yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4#1912787812yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2490785211yescharacteristic
317C2#2490792711yescharacteristic
318C2#2490798611yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2547C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenter#2547D83
hypocenterC83#9451
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. #24906189C2semidirect
4. #24905319C2semidirect
165. C83#9451semidirect

Maximal subgroups

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

subgroupindex
#249164922
#249061892
#249058042
#249053192
C83×#25472
#945183