Finite Group Databasebeta

← #35307917

Subgroups — #35307917

Order 21248.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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#24907619C211yesmaximal, characteristic, complemented
2#24907463C212yesmaximal, complemented
3C2×D4⋊(C83⋊C8)C212yesmaximal, complemented
4#24905346C211yesmaximal, characteristic, complemented
5C83×#2537C211yesmaximal, characteristic
6C2×#14581723—22no—
7C2×C4⋊(C83⋊C8)C2212yes—
8C2×#14581713—22no—
9C2×C4×C83⋊C8C2211yescharacteristic
10#19127571C2211yescharacteristic
11#19127463—22no—
12D4⋊(C83⋊C8)—24no—
13C2×#121×C83—22no—
14C2×C4×D4×C83C2212yes—
15#394×C83—22no—
16#384×C83C2211yescharacteristic
17#380×C83—22no—
18C22×C83⋊C8—21no—
19C22×C83⋊C8—22no—
20C2×C4⋊Dic83—22no—
21C2×C4⋊Dic83—21no—
22#14581723—44no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)—22no—
25#14581713—44no—
26C4×C83⋊C8—21no—
27C4×C83⋊C8D412yescomplemented
28C22×D4×C83—22no—
29C22×D4×C83D412yes—
30C23×C4×C83—22no—
31#121×C83—44no—
32#121×C83—44no—
33C4×D4×C83—22no—
34C4×D4×C83—22no—
35C4×D4×C83—22no—
36C4×D4×C83—22no—
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83D412yes—
39C2×C4⋊C4×C83D412yes—
40C2×C22⋊C4×C83—24no—
41C2×C22⋊C4×C83—22no—
42C2×C42×C83C2311yescharacteristic
43#95×C83—22no—
44C22×Dic83—41no—
45C2×C83⋊C8—22no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8—22no—
48C2×C83⋊C8—42no—
49C2×C83⋊C8—42no—
50C2×C83⋊C8—42no—
51C4⋊Dic83—42no—
52C4⋊Dic83—42no—
53C4⋊Dic83—42no—
54C24×C83—22no—
55C2×D4×C83—22no—
56C2×D4×C83—22no—
57C2×D4×C83—22no—
58C2×D4×C83—22no—
59C2×D4×C83—22no—
60C2×D4×C83—48no—
61C2×D4×C83—82no—
62C2×D4×C83SD1614yes—
63C22×C4×C83—22no—
64C22×C4×C83—22no—
65C22×C4×C83—42no—
66C22×C4×C83—42no—
67C22×C4×C83—42no—
68C22×C4×C83—42no—
69C22×C4×C83—42no—
70C22×C4×C83—42no—
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83C2×D412yes—
73C4⋊C4×C83—22no—
74C4⋊C4×C83—22no—
75C4⋊C4×C83—42no—
76C4⋊C4×C83—42no—
77C4⋊C4×C83—42no—
78C22⋊C4×C83—48no—
79C22⋊C4×C83—42no—
80C22⋊C4×C83—42no—
81C42×C83C2×D411yescharacteristic
82C42×C83#4111yescharacteristic
83C42×C83#4112yes—
84C23×C83—24no—
85C23×C83—42no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83—42no—
90C23×C83—42no—
91C23×C83#12011yescharacteristic
92D4×C83—24no—
93D4×C83—24no—
94D4×C83—42no—
95D4×C83—42no—
96D4×C83—88no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—42no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—44no—
103C2×C4×C83—42no—
104C2×C4×C83—82no—
105C2×C4×C83—82no—
106C2×C4×C83—82no—
107C2×C4×C83—82no—
108C2×C4×C83#12111yescharacteristic
109C2×C4×C83#12112yes—
110C2×C4×C83#12312yes—
111C2×C4×C83#12411yescharacteristic
112C2×C4×C83C2×SD1612yes—
113C2×C4×C83#13512yes—
114C2×C4×C83AGL(1,ℤ/8)12yes—
115C2×C4×C83AGL(1,ℤ/8)12yes—
116C2×Dic83—82no—
117C2×Dic83—81no—
118C83⋊C8—24no—
119C83⋊C8—82no—
120C22×C83—44no—
121C22×C83—44no—
122C22×C83—44no—
123C22×C83—82no—
124C22×C83—82no—
125C22×C83—82no—
126C22×C83—82no—
127C22×C83#39411yescharacteristic
128C22×C83#44912yes—
129C22×C83#45012yes—
130C22×C83#45311yescharacteristic
131C22×C83#45611yescharacteristic
132C4×C83—21no—
133C4×C83—21no—
134C4×C83—82no—
135C4×C83—82no—
136C4×C83#46512yes—
137C4×C83#46912yes—
138C4×C83#48112yes—
139C4×C83#48212yes—
140Dic83—161no—
141#10432—831nomaximal, Sylow
142C2×C83—84no—
143C2×C83#162411yescharacteristic
144C2×C83#162911yescharacteristic
145C2×C83#165412yes—
146C2×C83#203412yes—
147C2×C83#203911yescharacteristic
148#2537D8311yescharacteristic
149#1844—831no—
150#1802—832no—
151C2×D4⋊C8—832no—
152#1296—831no—
153C83#1043211yescharacteristic, complemented, Sylow
154C2×#121—22no—
155C2×C4×D4C2×D8312yes—
156C2×#106—1662no—
157C2×C4⋊C8—832no—
158C2×#102—1662no—
159C2×C4×C8—831no—
160#394—22no—
161#384—831no—
162#384C2×D8311yescharacteristic
163#380—22no—
164#340—1662no—
165D4⋊C8—1664no—
166C22×D4—22no—
167C22×D4C22⋊D8312yes—
168C23×C4—22no—
169C22×C8—1662no—
170C22×C8—1661no—
171#121—44no—
172#121—44no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—1662no—
179C2×C4⋊C4—1661no—
180C2×C4⋊C4C22⋊D8312yes—
181C2×C4⋊C4C22⋊D8312yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C42C22×D8311yescharacteristic
185#106—3324no—
186C4⋊C8—1662no—
187C4⋊C8—1662no—
188#102—3324no—
189C4×C8—832no—
190C4×C8—1661no—
191#95—22no—
192C24—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—48no—
199C2×D4—82no—
200C2×D4#1305057814yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—3321no—
210C22×C4C2×C22⋊D8311yescharacteristic
211C22×C4C2×C22⋊D8312yes—
212C2×C8—3322no—
213C2×C8—3322no—
214C2×C8—3322no—
215C2×C8—1662no—
216C2×C8—1662no—
217C2×C8—1662no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—3322no—
224C4⋊C4—3322no—
225C4⋊C4—3322no—
226C22⋊C4—48no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42D4×D8311yescharacteristic
230C42#1305058911yescharacteristic
231C42#1305058912yes—
232C23—24no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23#1458176111yescharacteristic
240D4—24no—
241D4—24no—
242D4—42no—
243D4—42no—
244D4—88no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—6642no—
257C2×C4—6641no—
258C2×C4#1458176311yescharacteristic
259C2×C4#1458176412yes—
260C2×C4#1458177312yes—
261C2×C4#1458177911yescharacteristic
262C2×C4C2×#1305057812yes—
263C2×C4#1458181212yes—
264C2×C4#1458181712yes—
265C2×C4#1458181812yes—
266C8—6642no—
267C8—1664no—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22#1912761511yescharacteristic
276C22#1912784612yes—
277C22#1912785212yes—
278C22#1912786911yescharacteristic
279C22#1912788411yescharacteristic
280C4—21no—
281C4—21no—
282C4—82no—
283C4—82no—
284C4—13281no—
285C4#1912792812yes—
286C4#1912795212yes—
287C4#1912801712yes—
288C4#1912802212yes—
289C2—84no—
290C2#2490654611yescharacteristic
291C2#2490657911yescharacteristic
292C2#2490674712yes—
293C2#2490864912yes—
294C2#2490868411yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
Fitting, prime radicalC83×#2537C2
socleC23×C83#120
hypercenter#2537D83
hypocenterC83#10432
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #24907619C2semidirect
2. #24907463C2semidirect
3. C2×D4⋊(C83⋊C8)C2semidirect
4. #24905346C2semidirect
27. C4×C83⋊C8D4semidirect
153. C83#10432semidirect

Maximal subgroups

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

subgroupindex
#249076192
#249074632
C2×D4⋊(C83⋊C8)2
#249053462
C83×#25372
#1043283