Finite Group Databasebeta

← #35302259

Subgroups — #35302259

Order 21248.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#24916544C211yesmaximal, characteristic, complemented
2#24907459C211yesmaximal, characteristic, complemented
3#24905888C212yesmaximal
4#24905321C212yesmaximal, complemented
5C83×#2536C211yesmaximal, characteristic
6Q8×C83⋊C8—24no—
7#19127754—22no—
8#19127754C2212yes—
9#19127725C2212yes—
10C2×#14581711—21no—
11#19127470—21no—
12D4⋊(C83⋊C8)—24no—
13C2×C22⋊Q8×C83—21no—
14C2×C4×Q8×C83C2211yescharacteristic
15C2×C4×D4×C83C2211yescharacteristic
16#398×C83C412yes—
17#394×C83C412yes—
18#388×C83—22no—
19#382×C83C2211yescharacteristic
20#380×C83—21no—
21C2×#13050574—22no—
22C4⋊(C83⋊C8)—24no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)—22no—
25C4⋊(C83⋊C8)—22no—
26#14581711—42no—
27C22⋊(C83⋊C8)—22no—
28C4×C83⋊C8—24no—
29C4×C83⋊C8D414yes—
30C22×Q8×C83D411yescharacteristic
31C22×D4×C83C2×C411yescharacteristic
32C23×C4×C83—21no—
33C22⋊Q8×C83—42no—
34C22⋊Q8×C83—42no—
35C4×Q8×C83—22no—
36C4×Q8×C83—22no—
37C4×D4×C83—22no—
38C4×D4×C83—22no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83C2×C412yes—
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—21no—
47C2×C42×C83D412yes—
48C2×C42×C83C2311yescharacteristic
49#95×C83—22no—
50#95×C83—21no—
51#95×C83C2×C412yes—
52#13050574—42no—
53#13050574—42no—
54C2×C83⋊C8—24no—
55C2×C83⋊C8—42no—
56C24×C83—21no—
57C2×Q8×C83—22no—
58C2×Q8×C83—22no—
59C2×Q8×C83D812yes—
60C2×Q8×C83SD1612yes—
61C2×D4×C83—22no—
62C2×D4×C83—42no—
63C22×C4×C83—22no—
64C22×C4×C83—21no—
65C22×C4×C83—42no—
66C22×C4×C83—41no—
67C22×C4×C83—41no—
68C22×C4×C83—42no—
69C22×C4×C83C22⋊C412yes—
70C22×C4×C83C22⋊C412yes—
71C22×C4×C83C22×C411yescharacteristic
72C22×C4×C83C2×D411yescharacteristic
73C22×C4×C83C2×D411yescharacteristic
74C4⋊C4×C83—24no—
75C4⋊C4×C83—22no—
76C4⋊C4×C83—22no—
77C4⋊C4×C83—44no—
78C4⋊C4×C83—41no—
79C4⋊C4×C83—41no—
80C22⋊C4×C83—44no—
81C22⋊C4×C83—44no—
82C22⋊C4×C83—41no—
83C22⋊C4×C83—41no—
84C42×C83—22no—
85C42×C83—22no—
86C42×C83C2×D412yes—
87C42×C83#4112yes—
88C23×C83—22no—
89C23×C83—41no—
90C23×C83—41no—
91C23×C83—41no—
92C23×C83—41no—
93C23×C83—41no—
94C23×C83—41no—
95C23×C83C2×C22⋊C411yescharacteristic
96Q8×C83—24no—
97Q8×C83—24no—
98D4×C83—24no—
99D4×C83—42no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—41no—
109C2×C4×C83—42no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—82no—
113C2×C4×C83—82no—
114C2×C4×C83#10214yes—
115C2×C4×C83C4×D412yes—
116C2×C4×C83#12011yescharacteristic
117C2×C4×C83#12112yes—
118C2×C4×C83#12311yescharacteristic
119C2×C4×C83C2×D811yescharacteristic
120C2×C4×C83C2×SD1611yescharacteristic
121C2×C4×C83#13512yes—
122C83⋊C8—44no—
123C83⋊C8—82no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—81no—
130C22×C83—81no—
131C22×C83—81no—
132C22×C83—81no—
133C22×C83#38611yescharacteristic
134C22×C83C2×#10211yescharacteristic
135C22×C83#41611yescharacteristic
136C4×C83—22no—
137C4×C83—24no—
138C4×C83—21no—
139C4×C83—21no—
140C4×C83—42no—
141C4×C83—82no—
142C4×C83—81no—
143C4×C83#46512yes—
144C4×C83#46612yes—
145#9486—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83—82no—
149C2×C83#189111yescharacteristic
150C2×C83#189311yescharacteristic
151C2×C83#191211yescharacteristic
152#2982—831no—
153#2536D8311yescharacteristic
154#1801—831no—
155#1486—832no—
156#1286—832no—
157C83#948611yescharacteristic, complemented, Sylow
158C2×C22⋊Q8—21no—
159C2×C4×Q8C2×D8311yescharacteristic
160C2×C4×D4C2×D8311yescharacteristic
161C8×Q8—1664no—
162#431—832no—
163#431—1662no—
164#423—832no—
165C2×#101—1661no—
166#398Dic8312yes—
167#394Dic8312yes—
168#388—22no—
169#382C2×D8311yescharacteristic
170#380—21no—
171#343—1661no—
172D4⋊C8—1664no—
173C22×Q8C22⋊D8311yescharacteristic
174C22×D4C2×Dic8311yescharacteristic
175C23×C4—21no—
176C2×M16—1662no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic8311yescharacteristic
186C2×C4⋊C4C2×Dic8312yes—
187C2×C4⋊C4C22⋊D8311yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C42C22⋊D8312yes—
192C2×C42C22×D8311yescharacteristic
193C4⋊C8—1664no—
194C4⋊C8—1662no—
195C4⋊C8—1662no—
196C4⋊C8—1662no—
197#101—3322no—
198C22⋊C8—1662no—
199C4×C8—834no—
200C4×C8—1664no—
201#95—22no—
202#95—21no—
203#95C2×Dic8312yes—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1305057512yes—
208C2×Q8#1305057812yes—
209C2×D4—22no—
210C2×D4—42no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊Dic8312yes—
218C22×C4C22⋊Dic8312yes—
219C22×C4C22×Dic8311yescharacteristic
220C22×C4C2×C22⋊D8311yescharacteristic
221C22×C4C2×C22⋊D8311yescharacteristic
222M16—3322no—
223M16—3322no—
224C2×C8—3322no—
225C2×C8—1664no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—44no—
236C42—22no—
237C42—22no—
238C42D4×D8312yes—
239C42#1305058912yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic8311yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—24no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#1458171314yes—
267C2×C4D4×Dic8312yes—
268C2×C4#1458176011yescharacteristic
269C2×C4#1458176412yes—
270C2×C4#1458177311yescharacteristic
271C2×C4C2×#1305057511yescharacteristic
272C2×C4C2×#1305057811yescharacteristic
273C2×C4#1458181212yes—
274C8—3324no—
275C8—6642no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#1912758111yescharacteristic
286C22C2×#1458171311yescharacteristic
287C22#1912769711yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#1912792812yes—
296C4#1912793512yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2490788211yescharacteristic
301C2#2490789211yescharacteristic
302C2#2490800111yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2536C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenter#2536D83
hypocenterC83#9486
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #24916544C2semidirect
2. #24907459C2semidirect
4. #24905321C2semidirect
157. C83#9486semidirect

Maximal subgroups

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

subgroupindex
#249165442
#249074592
#249058882
#249053212
C83×#25362
#948683