Finite Group Databasebeta

← #35320409

Subgroups — #35320409

Order 21248.

480 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 101 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#24910176C211yesmaximal, characteristic
2C2×#19127498C211yesmaximal, characteristic
3#24907457C212yesmaximal, complemented
4#24905589C212yesmaximal, complemented
5C2×#355×C83C211yesmaximal, characteristic
6C2×Q8×Dic83C2211yescharacteristic
7C2×D4×Dic83C411yescharacteristic
8#19127724—22no—
9C2×#14581711C2212yes—
10#19127608C2211yescharacteristic
11#19127604C411yescharacteristic
12#19127594—22no—
13#19127576—22no—
14#19127571C411yescharacteristic
15#19127547C411yescharacteristic
16#19127498—22no—
17#19127469—22no—
18C2×#124×C83C2211yescharacteristic
19C2×#101×C83C2212yes—
20#355×C83—22no—
21C23×Dic83—21no—
22C2×#13050574—22no—
23Q8×Dic83—42no—
24D4×Dic83—42no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×C4⋊Dic83D412yes—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—22no—
31C2×C4×Dic83C2×C411yescharacteristic
32C2×C4×Dic83C2×C411yescharacteristic
33C2×C4×Dic83D412yes—
34Dic83⋊C8—44no—
35#14581711—22no—
36#14581711—22no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—22no—
40C22×Q8×C83C2311yescharacteristic
41C22×D4×C83C2×C411yescharacteristic
42C2×M16×C83—22no—
43#124×C83—22no—
44#124×C83D412yes—
45#124×C83Q812yes—
46C2×C22⋊C4×C83—22no—
47C2×C42×C83C2×C411yescharacteristic
48#101×C83—22no—
49#101×C83—22no—
50C22×Dic83—21no—
51C22×Dic83—21no—
52C22×Dic83—42no—
53C22×Dic83—82no—
54C22×Dic83C22⋊C411yescharacteristic
55C22×Dic83C22⋊C412yes—
56C22×Dic83C22⋊C411yescharacteristic
57#13050574—42no—
58#13050574—42no—
59C2×C83⋊C8—42no—
60C4⋊Dic83—41no—
61C4⋊Dic83—42no—
62C4⋊Dic83—41no—
63C4⋊Dic83—82no—
64C22⋊Dic83—42no—
65C22⋊Dic83—82no—
66C4×Dic83—24no—
67C4×Dic83—22no—
68C4×Dic83—41no—
69C24×C83—21no—
70C2×Q8×C83—41no—
71C2×Q8×C83—41no—
72C2×Q8×C83C2×D411yescharacteristic
73C2×Q8×C83C2×Q811yescharacteristic
74C2×Q8×C83#4112yes—
75C2×D4×C83—21no—
76C2×D4×C83—41no—
77C2×D4×C83—41no—
78C2×D4×C83C4⋊C412yes—
79C22×C4×C83—21no—
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C2×D412yes—
82M16×C83—42no—
83M16×C83—42no—
84C2×C8×C83—42no—
85C22⋊C4×C83—24no—
86C22⋊C4×C83—24no—
87C42×C83—21no—
88C42×C83C4⋊C412yes—
89C23×C83—22no—
90C23×C83—42no—
91C23×C83—42no—
92C23×C83—41no—
93C23×C83—41no—
94C23×C83C2×C22⋊C411yescharacteristic
95Q8×C83—41no—
96Q8×C83—41no—
97Q8×C83—42no—
98D4×C83—42no—
99D4×C83—42no—
100C2×C4×C83—22no—
101C2×C4×C83—21no—
102C2×C4×C83—22no—
103C2×C4×C83—21no—
104C2×C4×C83—22no—
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83C2×C4⋊C411yescharacteristic
108C2×C4×C83#11711yescharacteristic
109C2×C4×C83#12112yes—
110C2×C4×C83C22⋊Q812yes—
111C2×Dic83—21no—
112C2×Dic83—22no—
113C2×Dic83—41no—
114C2×Dic83—42no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117C2×Dic83—42no—
118C2×Dic83—41no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—84no—
122C2×Dic83#9912yes—
123C8×C83—82no—
124C83⋊C8—82no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—44no—
128C22×C83—42no—
129C22×C83—82no—
130C22×C83—81no—
131C22×C83—81no—
132C22×C83#38011yescharacteristic
133C22×C83C2×#9911yescharacteristic
134C22×C83#41011yescharacteristic
135C4×C83—22no—
136C4×C83—21no—
137C4×C83—21no—
138C4×C83—22no—
139C4×C83—42no—
140Dic83—22no—
141Dic83—42no—
142Dic83—41no—
143Dic83—81no—
144Dic83—81no—
145#12803—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83—82no—
149C2×C83#179911yescharacteristic
150C2×C83#213311yescharacteristic
151C2×C83#213411yescharacteristic
152#2326—831no—
153C2×#355—831no—
154C2×#355D8311yescharacteristic
155#1801—832no—
156#1394—832no—
157C83#1280311yescharacteristic, complemented, Sylow
158C2×#124C2×D8311yescharacteristic
159C2×C4×Q8—831no—
160C2×C4×D4—831no—
161#423—1662no—
162C2×#101—832no—
163C2×#101C2×D8312yes—
164#391—831no—
165#390—831no—
166#388—1662no—
167#385—1662no—
168#384—831no—
169#376—831no—
170#355—22no—
171#355—1662no—
172#343—1662no—
173C22×Q8C22×D8311yescharacteristic
174C22×D4C4×D8311yescharacteristic
175C23×C4—1661no—
176C2×M16—22no—
177C2×M16—1662no—
178#124—22no—
179#124Dic16612yes—
180#124C22⋊D8312yes—
181C4×Q8—3322no—
182C4×D4—3322no—
183C2×C4⋊C4—831no—
184C2×C4⋊C4—832no—
185C2×C4⋊C4—831no—
186C2×C4⋊C4—1662no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1661no—
189C2×C22⋊C4—1662no—
190C2×C42—831no—
191C2×C42—831no—
192C2×C42—832no—
193C2×C42C4×D8311yescharacteristic
194C4⋊C8—3324no—
195#101—22no—
196#101—22no—
197#101—1662no—
198#101—1662no—
199#95—1661no—
200#95—1661no—
201#95—1662no—
202C24—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×C22⋊D8311yescharacteristic
206C2×Q8C2×Dic16611yescharacteristic
207C2×Q8#1305058912yes—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4Dic83⋊C412yes—
212C22×C4—21no—
213C22×C4—3322no—
214C22×C4—831no—
215C22×C4—831no—
216C22×C4—832no—
217C22×C4—6642no—
218C22×C4—1661no—
219C22×C4—1661no—
220C22×C4C2×C4×D8311yescharacteristic
221C22×C4D4×D8312yes—
222M16—42no—
223M16—42no—
224M16—3322no—
225M16—3322no—
226C2×C8—42no—
227C2×C8—3322no—
228C4⋊C4—3321no—
229C4⋊C4—3321no—
230C4⋊C4—3322no—
231C4⋊C4—6642no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—3322no—
235C22⋊C4—6642no—
236C42—21no—
237C42—3321no—
238C42—1664no—
239C42—1662no—
240C42Dic83⋊C412yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C22⋊C4×D8311yescharacteristic
247Q8—41no—
248Q8—41no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3322no—
260C2×C4—3321no—
261C2×C4—3321no—
262C2×C4—3322no—
263C2×C4—3321no—
264C2×C4—3321no—
265C2×C4—3321no—
266C2×C4—3321no—
267C2×C4—832no—
268C2×C4—6644no—
269C2×C4—1661no—
270C2×C4—1662no—
271C2×C4C2×Dic83⋊C411yescharacteristic
272C2×C4#1458174811yescharacteristic
273C2×C4#1458176412yes—
274C2×C4#1458176912yes—
275C8—82no—
276C8—6642no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1912755911yescharacteristic
285C22#99×D8311yescharacteristic
286C22#1912767311yescharacteristic
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—3321no—
293C4—3322no—
294C4—6641no—
295C4—6641no—
296C4—1662no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2490744911yescharacteristic
301C2#2490914911yescharacteristic
302C2#2490915311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC2×#355×C83C2
socleC22×C83C2×#99
prime radicalC22×D4×C83C2×C4
cosocleC22×Q8×C83C23
hypercenterC2×#355D83
hypocenterC83#12803
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #24907457C2semidirect
4. #24905589C2semidirect
157. C83#12803semidirect

Maximal subgroups

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

subgroupindex
#249101762
C2×#191274982
#249074572
#249055892
C2×#355×C832
#1280383