Finite Group Databasebeta

← #35298919

Subgroups — #35298919

Order 21248.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#24916564C211yesmaximal, characteristic
2#24908440C211yesmaximal, characteristic
3#24906252C211yesmaximal, characteristic
4#24906251C211yesmaximal, characteristic
5#24906127C211yesmaximal, characteristic
6#24905296C211yesmaximal, characteristic
7C83×#1276C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19127831—21no—
10#19127828—21no—
11C8×Dic166—21no—
12#19127825—21no—
13#19127759—22no—
14#19127754C2211yescharacteristic
15#19127754C2211yescharacteristic
16#19127725C2211yescharacteristic
17#19127461—22no—
18#19127454—21no—
19#19127453—21no—
20#19127452—21no—
21#19127451—21no—
22#19127448C411yescharacteristic, complemented
23#19127448C411yescharacteristic, complemented
24#19127446C411yescharacteristic, complemented
25#19127446C411yescharacteristic, complemented
26#431×C83C2211yescharacteristic
27#431×C83C2211yescharacteristic
28C43×C83C2211yescharacteristic
29C2×#13050574—21no—
30#14581792C2×C411yescharacteristic
31#14581792C2×C411yescharacteristic
32#14581784Q812yes—
33#14581771—21no—
34#14581771—21no—
35C4×Dic166—21no—
36C4×Dic166—21no—
37#14581751D412yes—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)C2×C411yescharacteristic
41C4⋊(C83⋊C8)C2×C411yescharacteristic
42C4⋊(C83⋊C8)D411yescharacteristic
43C4⋊(C83⋊C8)D411yescharacteristic
44Dic83⋊C8—22no—
45Dic83⋊C8—22no—
46C22⋊(C83⋊C8)—21no—
47C22⋊(C83⋊C8)—21no—
48#14581702—22no—
49C4×C83⋊C8—21no—
50C4×C83⋊C8—21no—
51C8×Dic83—22no—
52C2×C42×C83—21no—
53C2×C42×C83—21no—
54C2×C42×C83—21no—
55C2×C42×C83C2311yescharacteristic
56C4⋊C8×C83C2×C411yescharacteristic
57C4⋊C8×C83C2×C411yescharacteristic
58C4⋊C8×C83D411yescharacteristic
59C4⋊C8×C83D411yescharacteristic
60C22⋊C8×C83—21no—
61C22⋊C8×C83—21no—
62C4×C8×C83—21no—
63C4×C8×C83—21no—
64C2×Dic166—21no—
65C2×Dic166—21no—
66#13050574—42no—
67C2×C83⋊C8—21no—
68C2×C83⋊C8—21no—
69C2×C83⋊C8—21no—
70C2×C83⋊C8—21no—
71C2×C83⋊C8—41no—
72C4⋊Dic83—21no—
73C4⋊Dic83—21no—
74C4⋊Dic83C4⋊C412yes—
75C4⋊Dic83C4⋊C412yes—
76Dic83⋊C4—22no—
77Dic83⋊C4—22no—
78C22⋊Dic83—22no—
79C4×Dic83—22no—
80C22×C4×C83—41no—
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83#4111yescharacteristic
83C22×C4×C83#4111yescharacteristic
84C2×C8×C83—21no—
85C2×C8×C83—21no—
86C2×C8×C83—21no—
87C2×C8×C83—21no—
88C42×C83—22no—
89C42×C83—22no—
90C42×C83—22no—
91C42×C83—41no—
92C42×C83—41no—
93C42×C83—41no—
94C42×C83C22×C411yescharacteristic
95C42×C83C2×D411yescharacteristic
96C42×C83C2×D411yescharacteristic
97C42×C83C2×Q811yescharacteristic
98C23×C83#12311yescharacteristic
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83C2×C4⋊C411yescharacteristic
109C2×C4×C83C4×D411yescharacteristic
110C2×C4×C83C4×D411yescharacteristic
111C2×C4×C83#12011yescharacteristic
112C2×C4×C83C22⋊Q811yescharacteristic
113C2×C4×C83C22⋊Q811yescharacteristic
114C2×Dic83—22no—
115C2×Dic83—22no—
116Dic166—41no—
117Dic166—41no—
118Dic166—41no—
119Dic166—41no—
120C8×C83—41no—
121C8×C83—41no—
122C8×C83—41no—
123C8×C83—41no—
124C83⋊C8—41no—
125C83⋊C8—41no—
126C83⋊C8—41no—
127C83⋊C8—41no—
128C83⋊C8—81no—
129C22×C83—21no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83#38511yescharacteristic
133C4×C83—21no—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—44no—
137C4×C83#44311yescharacteristic
138C4×C83#44311yescharacteristic
139C4×C83#44411yescharacteristic
140C4×C83#44411yescharacteristic
141C4×C83#45411yescharacteristic
142C4×C83#45611yescharacteristic
143Dic83—42no—
144Dic83—42no—
145#8936—831nomaximal, Sylow
146C2×C83—41no—
147C2×C83#185711yescharacteristic
148C2×C83#186611yescharacteristic
149C2×C83#186611yescharacteristic
150#2986—831no—
151#1994—831no—
152#1574—831no—
153#1574—831no—
154#1544—831no—
155#1276—831no—
156#1276D8311yescharacteristic
157C83#893611yescharacteristic, complemented, Sylow
158#533—831no—
159#446—1661no—
160#446—1661no—
161C8×Q8—1661no—
162C8×Q8—1661no—
163#433—1662no—
164#431—831no—
165#431—831no—
166#431C2×D8311yescharacteristic
167#431C2×D8311yescharacteristic
168#423—831no—
169C43C2×D8311yescharacteristic
170#339—1662no—
171#335—1661no—
172#335—1661no—
173#334—1661no—
174#334—1661no—
175#332—831no—
176#332—831no—
177#332—831no—
178#332—831no—
179C2×M16—1661no—
180C4⋊Q8—831no—
181C4⋊Q8—831no—
182#125—832no—
183C22⋊Q8—1661no—
184C22⋊Q8—1661no—
185C4×Q8—1661no—
186C4×Q8—1661no—
187#117—832no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C22×D8311yescharacteristic
192C4⋊C8—831no—
193C4⋊C8—831no—
194C4⋊C8—831no—
195C4⋊C8—831no—
196C4⋊C8—1662no—
197C4⋊C8—1662no—
198C4⋊C8—1661no—
199C4⋊C8—1661no—
200C4⋊C8C4×D8311yescharacteristic
201C4⋊C8C4×D8311yescharacteristic
202C4⋊C8D33211yescharacteristic
203C4⋊C8D33211yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—1661no—
207C22⋊C8—1661no—
208#97—1662no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—1662no—
212C4×C8—1661no—
213C4×C8—1661no—
214C2×Q8—1661no—
215C2×Q8—1661no—
216C22×C4—41no—
217C22×C4D4×D8311yescharacteristic
218C22×C4#1305058911yescharacteristic
219C22×C4#1305059011yescharacteristic
220M16—3322no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—3321no—
226C2×C8—1661no—
227C2×C8—1661no—
228C2×C8—1661no—
229C2×C8—1661no—
230C4⋊C4—832no—
231C4⋊C4—832no—
232C4⋊C4—1662no—
233C4⋊C4—1662no—
234C4⋊C4—1661no—
235C4⋊C4—1661no—
236C22⋊C4—1662no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—1662no—
244C42C2×C4×D8311yescharacteristic
245C42D4×D8311yescharacteristic
246C42C2×D33211yescharacteristic
247C42Q8×D8311yescharacteristic
248C23#1458177411yescharacteristic
249Q8—3321no—
250Q8—3321no—
251Q8—3321no—
252Q8—3321no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4C4⋊C4×D8311yescharacteristic
265C2×C4(C22⋊D83)⋊C411yescharacteristic
266C2×C4C4×D33211yescharacteristic
267C2×C4#1458176211yescharacteristic
268C2×C4#1458177011yescharacteristic
269C2×C4#1458177211yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—3321no—
275C8—3321no—
276C8—3321no—
277C8—3321no—
278C8—6641no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1912757511yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—44no—
287C4—3322no—
288C4—3322no—
289C4#1912781511yescharacteristic
290C4#1912781911yescharacteristic
291C4#1912782011yescharacteristic
292C4#1912782411yescharacteristic
293C4#1912787511yescharacteristic
294C4#1912788511yescharacteristic
295C2—41no—
296C2#2490767911yescharacteristic
297C2#2490773911yescharacteristic
298C2#2490774111yescharacteristic
subgroupG/H
centerC2×C4#14581762
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1276C2
socleC22×C83#385
prime radicalC23×C83#123
cosocleC2×C42×C83C23
hypercenter#1276D83
hypocenterC83#8936
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
22. #19127448C4semidirect
23. #19127448C4semidirect
24. #19127446C4semidirect
25. #19127446C4semidirect
157. C83#8936semidirect

Maximal subgroups

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

subgroupindex
#249165642
#249084402
#249062522
#249062512
#249061272
#249052962
C83×#12762
#893683