Finite Group Databasebeta

← #35320410

Subgroups — #35320410

Order 21248.

480 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 93 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#24910177C211yesmaximal, characteristic
2C2×#19127497C212yesmaximal
3#24907457C212yesmaximal, complemented
4#24905590C211yesmaximal, characteristic, complemented
5C83×#1394C211yesmaximal, characteristic, complemented
6C2×#14581781C2211yescharacteristic
7C2×Q8×Dic83C2211yescharacteristic
8C2×#14581754C411yescharacteristic
9#19127724—22no—
10C2×#14581711C2212yes—
11#19127603C411yescharacteristic
12#19127590—22no—
13#19127575—22no—
14#19127570C411yescharacteristic
15#19127547C411yescharacteristic
16#19127497—24no—
17#19127469—22no—
18C2×#101×C83C2212yes—
19#391×C83C2211yescharacteristic
20C22×D332C2×C411yescharacteristic
21C22×C4×D83—21no—
22C2×#13050574—22no—
23#14581781—22no—
24#14581781D412yes—
25#14581781Q812yes—
26Q8×Dic83—42no—
27#14581754—42no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×C4⋊Dic83D412yes—
30C2×Dic83⋊C4—22no—
31C2×(C2×D83)⋊C4—21no—
32C2×(C2×D83)⋊C4—22no—
33C2×(C2×D83)⋊C4—22no—
34C2×C4×Dic83C2×C411yescharacteristic
35C2×C4×Dic83C2×C411yescharacteristic
36C2×C4×Dic83D412yes—
37Dic83⋊C8—44no—
38#14581711—22no—
39#14581711—22no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C22×Q8×C83C2311yescharacteristic
43C2×M16×C83—22no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C42×C83C2×C411yescharacteristic
46#101×C83—22no—
47#101×C83—22no—
48#95×C83—22no—
49C23×D83—21no—
50C2×D332—21no—
51C2×D332—41no—
52C2×D332—41no—
53C2×D332C4⋊C412yes—
54C22×Dic83—21no—
55C22×Dic83C22⋊C411yescharacteristic
56C22×Dic83C22⋊C412yes—
57C22×Dic83C22⋊C411yescharacteristic
58C2×C4×D83—42no—
59C2×C4×D83—82no—
60#13050574—42no—
61#13050574—42no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—42no—
65Dic83⋊C4—82no—
66(C2×D83)⋊C4—24no—
67(C2×D83)⋊C4—24no—
68(C2×D83)⋊C4—42no—
69(C2×D83)⋊C4—82no—
70C4×Dic83—21no—
71C4×Dic83—24no—
72C4×Dic83—22no—
73C4×Dic83C4⋊C412yes—
74C2×Q8×C83—41no—
75C2×Q8×C83—41no—
76C2×Q8×C83C2×D411yescharacteristic
77C2×Q8×C83C2×Q811yescharacteristic
78C2×Q8×C83#4112yes—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D412yes—
83M16×C83—42no—
84M16×C83—42no—
85C2×C8×C83—42no—
86C4⋊C4×C83—41no—
87C42×C83—41no—
88C23×C83C2×C22⋊C411yescharacteristic
89C22×D83—22no—
90C22×D83—42no—
91C22×D83—42no—
92C22×D83—41no—
93C22×D83—41no—
94Q8×C83—41no—
95Q8×C83—41no—
96Q8×C83—42no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—21no—
100C2×C4×C83—22no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—42no—
104C2×C4×C83—41no—
105C2×C4×C83—41no—
106C2×C4×C83C2×C4⋊C411yescharacteristic
107C2×C4×C83#11711yescharacteristic
108C2×C4×C83#12112yes—
109C2×C4×C83C22⋊Q812yes—
110C2×Dic83—22no—
111C2×Dic83—21no—
112C2×Dic83—22no—
113C2×Dic83—41no—
114C2×Dic83—41no—
115C2×Dic83—41no—
116C2×Dic83—42no—
117C2×Dic83—41no—
118C2×Dic83#9912yes—
119D332—42no—
120D332—42no—
121C4×D83—84no—
122C8×C83—82no—
123C83⋊C8—82no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83#38011yescharacteristic
127C22×C83C2×#9911yescharacteristic
128C22×C83#41011yescharacteristic
129C2×D83—44no—
130C2×D83—42no—
131C2×D83—82no—
132C2×D83—81no—
133C2×D83—81no—
134C4×C83—22no—
135C4×C83—21no—
136C4×C83—21no—
137C4×C83—22no—
138C4×C83—81no—
139C4×C83—81no—
140Dic83—22no—
141Dic83—42no—
142Dic83—41no—
143Dic83—42no—
144#12803—831nomaximal, Sylow
145C2×C83—21no—
146C2×C83—21no—
147C2×C83#179911yescharacteristic
148C2×C83#213311yescharacteristic
149C2×C83#213411yescharacteristic
150D83—82no—
151#2326—831no—
152C2×#355—832no—
153#1801—832no—
154#1394—831no—
155#1394D8311yescharacteristic, complemented
156C83#1280311yescharacteristic, complemented, Sylow
157C2×#124—831no—
158C2×C4×Q8—831no—
159C2×C4×D4—831no—
160#423—1662no—
161C2×#101—832no—
162C2×#101C2×D8312yes—
163#391C2×D8311yescharacteristic
164#390—831no—
165#388—1662no—
166#385—1662no—
167#384—831no—
168#376—831no—
169#355—1664no—
170#343—1662no—
171C22×Q8C22×D8311yescharacteristic
172C22×D4—831no—
173C23×C4—1661no—
174C2×M16—22no—
175C2×M16—1662no—
176#124—832no—
177#124—832no—
178#124—1662no—
179C4×Q8—3322no—
180C4×D4—3322no—
181C2×C4⋊C4—831no—
182C2×C4⋊C4—832no—
183C2×C4⋊C4—1662no—
184C2×C4⋊C4C4×D8311yescharacteristic
185C2×C22⋊C4—1661no—
186C2×C22⋊C4—1662no—
187C2×C22⋊C4—1662no—
188C2×C42—831no—
189C2×C42—831no—
190C2×C42—832no—
191C2×C42C4×D8311yescharacteristic
192C4⋊C8—3324no—
193#101—22no—
194#101—22no—
195#101—1662no—
196#101—1662no—
197#95—22no—
198#95—1661no—
199#95—1661no—
200C24—1661no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8D4×D8311yescharacteristic
204C2×Q8Q8×D8311yescharacteristic
205C2×Q8#1305059012yes—
206C2×D4—3321no—
207C2×D4—3321no—
208C2×D4—832no—
209C2×D4—1661no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—3322no—
213C22×C4—831no—
214C22×C4—831no—
215C22×C4—832no—
216C22×C4—6642no—
217C22×C4—1661no—
218C22×C4C2×C4×D8311yescharacteristic
219C22×C4D4×D8312yes—
220M16—42no—
221M16—42no—
222M16—3322no—
223M16—3322no—
224C2×C8—42no—
225C2×C8—3322no—
226C4⋊C4—41no—
227C4⋊C4—3321no—
228C4⋊C4—3322no—
229C4⋊C4—6642no—
230C22⋊C4—3322no—
231C22⋊C4—6642no—
232C22⋊C4—1664no—
233C22⋊C4—1664no—
234C42—41no—
235C42—832no—
236C42—1664no—
237C42—1662no—
238C42—1661no—
239C23—3322no—
240C23—3322no—
241C23—3321no—
242C23—3321no—
243C23—1662no—
244C23C22⋊C4×D8311yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—3322no—
249D4—3322no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3321no—
260C2×C4—3321no—
261C2×C4—3322no—
262C2×C4—3321no—
263C2×C4—3321no—
264C2×C4—832no—
265C2×C4—6644no—
266C2×C4—1662no—
267C2×C4—1661no—
268C2×C4—1662no—
269C2×C4C4⋊C4×D8311yescharacteristic
270C2×C4#1458174911yescharacteristic
271C2×C4#1458176512yes—
272C2×C4#1458177012yes—
273C8—82no—
274C8—6642no—
275C22—21no—
276C22—21no—
277C22—3324no—
278C22—3322no—
279C22—6642no—
280C22—6641no—
281C22—6641no—
282C22#1912756011yescharacteristic
283C22#99×D8311yescharacteristic
284C22#1912767311yescharacteristic
285C4—22no—
286C4—22no—
287C4—21no—
288C4—21no—
289C4—81no—
290C4—81no—
291C4—3322no—
292C4—3321no—
293C4—3322no—
294C4—1662no—
295C2—21no—
296C2—21no—
297C2—6642no—
298C2#2490745011yescharacteristic
299C2#2490915111yescharacteristic
300C2#2490915311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC83×#1394C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC22×Q8×C83C23
hypercenter#1394D83
hypocenterC83#12803
LCSG ▷ C22×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
3. #24907457C2semidirect
4. #24905590C2semidirect
5. C83×#1394C2semidirect
155. #1394D83semidirect
156. C83#12803semidirect

Maximal subgroups

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

subgroupindex
#249101772
C2×#191274972
#249074572
#249055902
C83×#13942
#1280383