Finite Group Databasebeta

← #35301079

Subgroups — #35301079

Order 21248.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#24918607C211yesmaximal, characteristic
2#24916400C211yesmaximal, characteristic
3#24908284C211yesmaximal, characteristic
4#24906028C211yesmaximal, characteristic
5#24905323C211yesmaximal, characteristic
6#24905293C211yesmaximal, characteristic
7C83×#1546C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19128097C412yescomplemented
10#19128053C412yescomplemented
11#19128031—22no—
12#19127983—22no—
13#19127830—22no—
14#19127830—22no—
15#19127754C2211yescharacteristic
16#19127729—21no—
17C2×#14581716—21no—
18C2×#14581716—21no—
19#19127659—21no—
20C2×#14581703C2211yescharacteristic
21#19127567C2211yescharacteristic
22#19127439—22no—
23C2×C4⋊Q8×C83C2211yescharacteristic
24#431×C83C2211yescharacteristic
25#423×C83C2211yescharacteristic
26#332×C83—22no—
27C22×C83⋊C8D411yescharacteristic
28C22×C83⋊C8D411yescharacteristic
29#14581792C2×C411yescharacteristic
30#14581792C2×C411yescharacteristic
31#14581784—21no—
32#14581771—21no—
33#14581771D412yes—
34C4×Dic166—21no—
35C4×Dic166D412yes—
36#14581751—21no—
37C2×C4⋊Dic83—21no—
38#14581723—24no—
39C4⋊(C83⋊C8)—21no—
40Dic83⋊C8—24no—
41#14581716—42no—
42#14581716—42no—
43#14581715—24no—
44C22⋊(C83⋊C8)—21no—
45#14581703—21no—
46#14581703—21no—
47#14581702—22no—
48C4×C83⋊C8C2×C412yes—
49C8×Dic83—22no—
50C83⋊#95—21no—
51C83⋊#95—21no—
52C22×Q8×C83—21no—
53C2×M16×C83—21no—
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83—21no—
57C4⋊Q8×C83C2×C412yes—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83—21no—
60C2×C42×C83C2311yescharacteristic
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83D412yes—
64C22⋊C8×C83—21no—
65C4×C8×C83—21no—
66C2×Dic166—21no—
67C2×Dic166C22⋊C412yes—
68C22×Dic83—41no—
69C2×C83⋊C8—21no—
70C2×C83⋊C8—21no—
71C2×C83⋊C8—22no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—21no—
74C2×C83⋊C8—21no—
75C2×C83⋊C8SD1612yes—
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83—21no—
79C4⋊Dic83—41no—
80C4⋊Dic83—41no—
81C4⋊Dic83C22⋊C412yes—
82Dic83⋊C4—22no—
83Dic83⋊C4—22no—
84C22⋊Dic83—22no—
85C4×Dic83—22no—
86C2×Q8×C83—22no—
87C2×Q8×C83—41no—
88C2×Q8×C83—41no—
89C2×Q8×C83—41no—
90C2×Q8×C83—41no—
91C2×Q8×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83#4111yescharacteristic
96M16×C83—41no—
97M16×C83—41no—
98C2×C8×C83—22no—
99C2×C8×C83—41no—
100C4⋊C4×C83—22no—
101C4⋊C4×C83—22no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C42×C83C22×C411yescharacteristic
105C42×C83C2×D411yescharacteristic
106C42×C83C2×D411yescharacteristic
107C42×C83#4111yescharacteristic
108C23×C83#12111yescharacteristic
109Q8×C83—42no—
110Q8×C83—42no—
111Q8×C83—81no—
112Q8×C83—81no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—42no—
116C2×C4×C83—81no—
117C2×C4×C83—81no—
118C2×C4×C83C2×C22⋊C411yescharacteristic
119C2×C4×C83C4×D411yescharacteristic
120C2×C4×C83C4×D411yescharacteristic
121C2×C4×C83#12011yescharacteristic
122C2×C4×C83#12111yescharacteristic
123C2×C4×C83#12311yescharacteristic
124C2×C4×C83C2×SD1611yescharacteristic
125C2×C4×C83#13511yescharacteristic
126C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C83#13711yescharacteristic
128C2×Dic83—22no—
129C2×Dic83—22no—
130C2×Dic83—81no—
131C2×Dic83—81no—
132C2×Dic83—81no—
133Dic166—42no—
134Dic166—81no—
135C8×C83—42no—
136C8×C83—81no—
137C83⋊C8—22no—
138C83⋊C8—42no—
139C83⋊C8—41no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83#38611yescharacteristic
143C22×C83#46011yescharacteristic
144C22×C83#46411yescharacteristic
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—82no—
150C4×C83C4×SD1611yescharacteristic
151C4×C83#44011yescharacteristic
152C4×C83#44811yescharacteristic
153C4×C83#44911yescharacteristic
154Dic83—42no—
155Dic83—42no—
156Dic83—161no—
157#9292—831nomaximal, Sylow
158C2×C83—21no—
159C2×C83—21no—
160C2×C83#188011yescharacteristic
161C2×C83#190211yescharacteristic
162C2×C83#190911yescharacteristic
163#3229—831no—
164#2961—831no—
165#1960—831no—
166#1546D8311yescharacteristic
167#1518—831no—
168#1286—831no—
169#1275—831no—
170C83#929211yescharacteristic, complemented, Sylow
171#533—831no—
172C2×C4⋊Q8C2×D8311yescharacteristic
173#498—832no—
174#487—832no—
175#483—1662no—
176#475—1662no—
177#446—1662no—
178#446—1662no—
179#431—831no—
180#431C2×D8311yescharacteristic
181#424—1661no—
182#423C2×D8311yescharacteristic
183C2×#103—1661no—
184C2×#103—1661no—
185#405—1661no—
186C2×#97—831no—
187#382—831no—
188#332—22no—
189#328—1662no—
190C22×Q8—21no—
191C2×M16—21no—
192C22×C8—831no—
193C22×C8—831no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—831no—
198C4⋊Q8—831no—
199C4⋊Q8C4×D8312yes—
200#125—1661no—
201C22⋊Q8—832no—
202C22⋊Q8—1661no—
203C4×Q8—832no—
204C4×Q8—1661no—
205#117—1661no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—1661no—
209C2×C42C22×D8311yescharacteristic
210#106—1664no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—1664no—
214C4⋊C8—1661no—
215C4⋊C8C22⋊D8312yes—
216#103—3322no—
217#103—3322no—
218#103—1664no—
219C22⋊C8—21no—
220C22⋊C8—1661no—
221#97—1662no—
222#97—1661no—
223#97—1661no—
224C4×C8—21no—
225C4×C8—832no—
226C4×C8—1662no—
227#95—1661no—
228#95—1661no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—832no—
236C2×Q8—1661no—
237C22×C4—41no—
238C22×C4—3321no—
239C22×C4D4×D8311yescharacteristic
240C22×C4D4×D8311yescharacteristic
241C22×C4#1305059011yescharacteristic
242M16—41no—
243M16—41no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—832no—
247C2×C8—1661no—
248C2×C8—1661no—
249C2×C8—1661no—
250C2×C8—1662no—
251C2×C8—1661no—
252C2×C8—1661no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—3321no—
258C4⋊C4—3321no—
259C4⋊C4—832no—
260C4⋊C4—1662no—
261C4⋊C4—1662no—
262C4⋊C4—1661no—
263C4⋊C4—1661no—
264C4⋊C4—1661no—
265C22⋊C4—1662no—
266C42—1662no—
267C42C2×C4×D8311yescharacteristic
268C42D4×D8311yescharacteristic
269C42C2×C22⋊D8311yescharacteristic
270C42#1305058911yescharacteristic
271C23#1458176511yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8—3322no—
277Q8—6641no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—6641no—
284C2×C4—6641no—
285C2×C4—6641no—
286C2×C4—1662no—
287C2×C4—1662no—
288C2×C4C22⋊C4×D8311yescharacteristic
289C2×C4(C22⋊D83)⋊C411yescharacteristic
290C2×C4C4×C22⋊D8311yescharacteristic
291C2×C4#1458176011yescharacteristic
292C2×C4#1458176311yescharacteristic
293C2×C4#1458177411yescharacteristic
294C2×C4SD16×D8311yescharacteristic
295C2×C4#1458181111yescharacteristic
296C2×C4#1458181611yescharacteristic
297C2×C4#1458182111yescharacteristic
298C8—42no—
299C8—81no—
300C8—3322no—
301C8—3321no—
302C8—1662no—
303C22—21no—
304C22—21no—
305C22#1912757711yescharacteristic
306C22#1912790011yescharacteristic
307C22#1912792211yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—13281no—
314C4—3322no—
315C4—3322no—
316C4#1912779211yescharacteristic
317C4#1912779811yescharacteristic
318C4#1912783811yescharacteristic
319C4#1912784311yescharacteristic
320C2—21no—
321C2—21no—
322C2#2490781511yescharacteristic
323C2#2490793911yescharacteristic
324C2#2490798211yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1546C2
socleC22×C83#386
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1546D83
hypocenterC83#9292
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
9. #19128097C4semidirect
10. #19128053C4semidirect
170. C83#9292semidirect

Maximal subgroups

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

subgroupindex
#249186072
#249164002
#249082842
#249060282
#249053232
#249052932
C83×#15462
#929283