Finite Group Databasebeta

← #35301083

Subgroups — #35301083

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#24918609C211yesmaximal, characteristic
2#24916400C211yesmaximal, characteristic
3#24908282C211yesmaximal, characteristic
4#24906136C211yesmaximal, characteristic
5#24906026C211yesmaximal, characteristic
6#24905293C211yesmaximal, characteristic
7C83×#1286C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9C2×#14581792C2211yescharacteristic
10#19128099C412yescomplemented
11#19128052C412yescomplemented
12#19128032—22no—
13#19127987—22no—
14#19127830—22no—
15#19127830—22no—
16#19127754C2211yescharacteristic
17#19127729—21no—
18C2×#14581715—21no—
19C2×#14581715—21no—
20#19127659—21no—
21C2×#14581703C2211yescharacteristic
22#19127448—22no—
23#19127437—22no—
24#431×C83C2211yescharacteristic
25#423×C83C2211yescharacteristic
26#382×C83C2211yescharacteristic
27C22×Dic166—21no—
28C22×C83⋊C8D411yescharacteristic
29C22×C83⋊C8D411yescharacteristic
30#14581792—21no—
31#14581792—21no—
32#14581792—21no—
33#14581792C2×C412yes—
34#14581792C2×C411yescharacteristic
35#14581792C2×C411yescharacteristic
36#14581784—21no—
37#14581771—21no—
38#14581771D412yes—
39C4×Dic166—21no—
40C4×Dic166D412yes—
41#14581751—21no—
42C2×C4⋊Dic83—21no—
43C2×C4⋊Dic83—21no—
44#14581724—24no—
45C4⋊(C83⋊C8)—21no—
46Dic83⋊C8—24no—
47#14581717—24no—
48#14581715—42no—
49#14581715—42no—
50C22⋊(C83⋊C8)—21no—
51#14581703—21no—
52#14581703—21no—
53#14581702—22no—
54C4×C83⋊C8—21no—
55C8×Dic83—22no—
56C2×M16×C83—21no—
57C2×C4⋊C4×C83—21no—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83D412yes—
62C22⋊C8×C83—21no—
63C4×C8×C83C2×C412yes—
64#95×C83—21no—
65#95×C83—21no—
66C2×Dic166—22no—
67C2×Dic166—21no—
68C2×Dic166—41no—
69C2×Dic166—41no—
70C2×Dic166—41no—
71C2×Dic166—41no—
72C2×Dic166—41no—
73C2×Dic166C22⋊C412yes—
74C22×Dic83—41no—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—21no—
77C2×C83⋊C8—21no—
78C2×C83⋊C8—21no—
79C2×C83⋊C8—21no—
80C2×C83⋊C8—41no—
81C2×C83⋊C8SD1612yes—
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83—22no—
86C4⋊Dic83—22no—
87C4⋊Dic83—41no—
88C4⋊Dic83—41no—
89C4⋊Dic83C22⋊C412yes—
90Dic83⋊C4—22no—
91Dic83⋊C4—22no—
92C22⋊Dic83—22no—
93C4×Dic83—22no—
94C22×C4×C83—41no—
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83#4111yescharacteristic
98M16×C83—41no—
99M16×C83—41no—
100C2×C8×C83—22no—
101C2×C8×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
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—81no—
112C2×C4×C83—81no—
113C2×C4×C83—81no—
114C2×C4×C83C2×C22⋊C411yescharacteristic
115C2×C4×C83C4×D411yescharacteristic
116C2×C4×C83C4×D411yescharacteristic
117C2×C4×C83#12011yescharacteristic
118C2×C4×C83#12111yescharacteristic
119C2×C4×C83#12311yescharacteristic
120C2×C4×C83C2×SD1611yescharacteristic
121C2×C4×C83#13511yescharacteristic
122C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C83#13711yescharacteristic
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—42no—
127C2×Dic83—81no—
128C2×Dic83—81no—
129Dic166—42no—
130Dic166—42no—
131Dic166—42no—
132Dic166—81no—
133Dic166—81no—
134Dic166—81no—
135C8×C83—42no—
136C8×C83—42no—
137C83⋊C8—22no—
138C83⋊C8—41no—
139C83⋊C8—81no—
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—161no—
150C4×C83C4×SD1611yescharacteristic
151C4×C83#44011yescharacteristic
152C4×C83#44811yescharacteristic
153C4×C83#44911yescharacteristic
154Dic83—42no—
155Dic83—42no—
156Dic83—82no—
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#1546—831no—
167#1518—831no—
168#1286D8311yescharacteristic
169#1275—831no—
170C83#929211yescharacteristic, complemented, Sylow
171#533—831no—
172C2×C4⋊Q8—831no—
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#382C2×D8311yescharacteristic
188#332—1662no—
189#328—1662no—
190C22×Q8—1661no—
191C2×M16—21no—
192C22×C8—831no—
193C22×C8—831no—
194C4⋊Q8—831no—
195C4⋊Q8—831no—
196C4⋊Q8—832no—
197C4⋊Q8—1661no—
198C4⋊Q8—1661no—
199C4⋊Q8—1661no—
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—1661no—
208C2×C4⋊C4—1661no—
209C2×C42C22×D8311yescharacteristic
210#106—1664no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—1664no—
214C4⋊C8—1661no—
215C4⋊C8D33212yes—
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—1662no—
225C4×C8—1661no—
226C4×C8C4×D8312yes—
227#95—21no—
228#95—21no—
229C2×Q8—3321no—
230C2×Q8—3321no—
231C2×Q8—3321no—
232C2×Q8—3321no—
233C2×Q8—3321no—
234C2×Q8—832no—
235C2×Q8—1662no—
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—22no—
246C2×C8—3321no—
247C2×C8—832no—
248C2×C8—1661no—
249C2×C8—1661no—
250C2×C8—1661no—
251C2×C8—1661no—
252C2×C8—1661no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—3321no—
256C4⋊C4—3321no—
257C4⋊C4—832no—
258C4⋊C4—1662no—
259C4⋊C4—1662no—
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×D33211yescharacteristic
270C42#1305059111yescharacteristic
271C23#1458176511yescharacteristic
272Q8—3322no—
273Q8—3322no—
274Q8—3322no—
275Q8—6641no—
276Q8—6641no—
277Q8—6641no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—3322no—
284C2×C4—6641no—
285C2×C4—6641no—
286C2×C4—1662no—
287C2×C4—1662no—
288C2×C4C22⋊C4×D8311yescharacteristic
289C2×C4#1458175411yescharacteristic
290C2×C4C4×D33211yescharacteristic
291C2×C4#1458176211yescharacteristic
292C2×C4#1458176711yescharacteristic
293C2×C4#1458177711yescharacteristic
294C2×C4SD16×D8311yescharacteristic
295C2×C4#1458181311yescharacteristic
296C2×C4#1458181611yescharacteristic
297C2×C4#1458182411yescharacteristic
298C8—42no—
299C8—42no—
300C8—3321no—
301C8—6641no—
302C8—1662no—
303C22—21no—
304C22—21no—
305C22#1912757911yescharacteristic
306C22#1912790011yescharacteristic
307C22#1912792611yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—161no—
313C4—3322no—
314C4—3322no—
315C4—6642no—
316C4#1912778811yescharacteristic
317C4#1912780011yescharacteristic
318C4#1912784111yescharacteristic
319C4#1912784711yescharacteristic
320C2—21no—
321C2—21no—
322C2#2490781911yescharacteristic
323C2#2490793511yescharacteristic
324C2#2490798411yescharacteristic
subgroupG/H
centerC22#19127579
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1286C2
socleC22×C83#386
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1286D83
hypocenterC83#9292
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
10. #19128099C4semidirect
11. #19128052C4semidirect
170. C83#9292semidirect

Maximal subgroups

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

subgroupindex
#249186092
#249164002
#249082822
#249061362
#249060262
#249052932
C83×#12862
#929283