Finite Group Databasebeta

← #35319589

Subgroups — #35319589

Order 21248.

464 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 113 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#24910181C211yesmaximal, characteristic
2C2×#19127499C212yesmaximal
3#101×Dic83C212yesmaximal
4#24905590C211yesmaximal, characteristic
5C83×#1394C211yesmaximal, characteristic
6C2×#14581790C2211yescharacteristic
7C2×Q8×Dic83C2211yescharacteristic
8C2×#14581757C411yescharacteristic
9C2×#14581711C2212yes—
10M16×Dic83—22no—
11#19127607C411yescharacteristic
12#19127588—22no—
13#19127583—22no—
14#19127566C411yescharacteristic
15#19127547C411yescharacteristic
16#19127499C418yescomplemented
17#19127469—22no—
18C2×#101×C83C2212yes—
19#391×C83C2211yescharacteristic
20C22×Dic166C2×C411yescharacteristic
21C2×#13050574—22no—
22#14581790—21no—
23#14581790—21no—
24#14581790C2×C414yes—
25Q8×Dic83—42no—
26#14581757—42no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×C4⋊Dic83D412yes—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—22no—
31C2×C4×Dic83—21no—
32C2×C4×Dic83C2×C411yescharacteristic
33C2×C4×Dic83C2×C411yescharacteristic
34C2×C4×Dic83D412yes—
35#14581711—22no—
36#14581711C2×C414yes—
37#14581702—42no—
38C8×Dic83—42no—
39C83⋊#95—22no—
40C83⋊#95—21no—
41C83⋊#95—22no—
42C83⋊#95—21no—
43C22×Q8×C83C2311yescharacteristic
44C2×M16×C83—22no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C42×C83C2×C411yescharacteristic
47#101×C83—22no—
48#101×C83C2×C414yes—
49#95×C83—22no—
50C2×Dic166—21no—
51C2×Dic166—41no—
52C2×Dic166—41no—
53C2×Dic166C4212yes—
54C22×Dic83—21no—
55C22×Dic83—21no—
56C22×Dic83C22⋊C412yes—
57C22×Dic83C22⋊C411yescharacteristic
58C22×Dic83C22⋊C411yescharacteristic
59#13050574—24no—
60#13050574—42no—
61C2×C83⋊C8—42no—
62C4⋊Dic83—41no—
63C4⋊Dic83—42no—
64Dic83⋊C4—24no—
65Dic83⋊C4—24no—
66Dic83⋊C4—42no—
67C4×Dic83—21no—
68C4×Dic83—22no—
69C4×Dic83—22no—
70C4×Dic83—22no—
71C4×Dic83—42no—
72C4×Dic83C4212yes—
73C2×Q8×C83—41no—
74C2×Q8×C83—41no—
75C2×Q8×C83C22×C412yes—
76C2×Q8×C83#4111yescharacteristic
77C2×Q8×C83#4111yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C2×D412yes—
82M16×C83—24no—
83M16×C83—42no—
84C2×C8×C83—42no—
85C4⋊C4×C83—41no—
86C42×C83—41no—
87C23×C83C2×C22⋊C411yescharacteristic
88Q8×C83—42no—
89Q8×C83—41no—
90Q8×C83—41no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—41no—
96C2×C4×C83—41no—
97C2×C4×C83—42no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83C2×C4211yescharacteristic
101C2×C4×C83#11711yescharacteristic
102C2×C4×C83C4×D412yes—
103C2×C4×C83C4×D412yes—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—21no—
107C2×Dic83—22no—
108C2×Dic83—41no—
109C2×Dic83—41no—
110C2×Dic83—41no—
111C2×Dic83—41no—
112C2×Dic83—41no—
113C2×Dic83—42no—
114C2×Dic83—41no—
115C2×Dic83#9912yes—
116Dic166—42no—
117Dic166—42no—
118C8×C83—44no—
119C83⋊C8—44no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83C4×C22⋊C411yescharacteristic
123C22×C83C2×#9911yescharacteristic
124C22×C83#41011yescharacteristic
125C4×C83—22no—
126C4×C83—22no—
127C4×C83—21no—
128C4×C83—21no—
129C4×C83—81no—
130C4×C83—81no—
131Dic83—22no—
132Dic83—42no—
133Dic83—41no—
134Dic83—42no—
135Dic83—42no—
136#12631—831nomaximal, Sylow
137C2×C83—21no—
138C2×C83—21no—
139C2×C83C4×#9911yescharacteristic
140C2×C83#213311yescharacteristic
141C2×C83#213511yescharacteristic
142#2327—831no—
143C2×#356—832no—
144C4×#101—832no—
145#1394—831no—
146#1394D8311yescharacteristic
147C83#1263111yescharacteristic, complemented, Sylow
148C2×C4⋊Q8—831no—
149C2×C4×Q8—831no—
150C2×C4×Q8—831no—
151C2×#101—832no—
152C2×#101C2×D8312yes—
153C4×M16—1662no—
154#391—831no—
155#391C2×D8311yescharacteristic
156#387—1662no—
157#387—1662no—
158#382—831no—
159#376—831no—
160#356—838no—
161#343—1662no—
162C22×Q8—831no—
163C22×Q8C22×D8311yescharacteristic
164C2×M16—22no—
165C2×M16—1662no—
166C4⋊Q8—834no—
167C4⋊Q8—1661no—
168C4⋊Q8—1661no—
169C4×Q8—3322no—
170C4×Q8—3322no—
171C2×C4⋊C4—831no—
172C2×C4⋊C4—832no—
173C2×C4⋊C4—1661no—
174C2×C4⋊C4—1662no—
175C2×C4⋊C4C4×D8311yescharacteristic
176C2×C42—831no—
177C2×C42—831no—
178C2×C42—832no—
179C2×C42—1661no—
180C2×C42C4×D8311yescharacteristic
181#101—22no—
182#101—834no—
183#101—1662no—
184#101C4×D8314yes—
185#97—3322no—
186C4×C8—3322no—
187#95—22no—
188#95—1661no—
189#95—1661no—
190#95—1662no—
191#95—1662no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—3321no—
195C2×Q8—3321no—
196C2×Q8—832no—
197C2×Q8—1661no—
198C2×Q8C2×C4×D8312yes—
199C2×Q8#1305058911yescharacteristic
200C2×Q8#1305059111yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—831no—
204C22×C4—832no—
205C22×C4—831no—
206C22×C4—1661no—
207C22×C4—1661no—
208C22×C4C2×C4×D8311yescharacteristic
209C22×C4D4×D8312yes—
210M16—24no—
211M16—42no—
212M16—3322no—
213M16—1664no—
214C2×C8—42no—
215C2×C8—3322no—
216C4⋊C4—41no—
217C4⋊C4—3321no—
218C4⋊C4—3322no—
219C4⋊C4—3322no—
220C4⋊C4—1664no—
221C4⋊C4—1664no—
222C42—41no—
223C42—3322no—
224C42—832no—
225C42—1662no—
226C42—1662no—
227C42—1662no—
228C42—1661no—
229C23C22⋊C4×D8311yescharacteristic
230Q8—42no—
231Q8—41no—
232Q8—41no—
233Q8—3322no—
234Q8—3322no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—3321no—
245C2×C4—3321no—
246C2×C4—3321no—
247C2×C4—3321no—
248C2×C4—3322no—
249C2×C4—3321no—
250C2×C4—3321no—
251C2×C4—832no—
252C2×C4—1661no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4C42×D8311yescharacteristic
257C2×C4#1458175011yescharacteristic
258C2×C4(C22⋊D83)⋊C412yes—
259C2×C4#1458175412yes—
260C8—44no—
261C8—3324no—
262C22—21no—
263C22—21no—
264C22#1912754811yescharacteristic
265C22#99×D8311yescharacteristic
266C22#1912767311yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—81no—
272C4—81no—
273C4—3322no—
274C4—3321no—
275C4—3322no—
276C4—3322no—
277C4—1662no—
278C2—21no—
279C2—21no—
280C2#2490725011yescharacteristic
281C2#2490914911yescharacteristic
282C2#2490915611yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C2×C42
FrattiniC22×Q8C22×D83
FittingC83×#1394C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC22×Q8×C83C23
hypercenter#1394D83
hypocenterC83#12631
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
16. #19127499C4semidirect
147. C83#12631semidirect

Maximal subgroups

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

subgroupindex
#249101812
C2×#191274992
#101×Dic832
#249055902
C83×#13942
#1263183