Finite Group Databasebeta

← #35301066

Subgroups — #35301066

Order 21248.

422 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 113 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#24918408C211yesmaximal, characteristic
2#24916404C211yesmaximal, characteristic
3#24908435C211yesmaximal, characteristic
4#24906147C211yesmaximal, characteristic
5#24905321C211yesmaximal, characteristic
6#24905293C211yesmaximal, characteristic
7C83×#1518C211yesmaximal, characteristic
8C2×#14581792C2211yescharacteristic
9#19128080C412yescomplemented
10#19128053C412yescomplemented
11#19127911—22no—
12#19127856—22no—
13#19127832—22no—
14#19127832—22no—
15#19127754C2211yescharacteristic
16#19127728—21no—
17#19127725C2211yescharacteristic
18#19127657—21no—
19C2×#14581703C2211yescharacteristic
20#19127463—21no—
21#19127463—21no—
22#19127448—22no—
23#533×C83C2211yescharacteristic
24#431×C83C2211yescharacteristic
25#382×C83C2211yescharacteristic
26#328×C83—22no—
27C22×Dic166—21no—
28C2×#13050580—24no—
29C2×#13050574—21no—
30C22×C83⋊C8D411yescharacteristic
31C22×C83⋊C8D411yescharacteristic
32#14581792—21no—
33#14581792—21no—
34#14581792—21no—
35#14581792C2×C412yes—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83—21no—
38C4⋊(C83⋊C8)—24no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)D412yes—
42#14581715—24no—
43C22⋊(C83⋊C8)—21no—
44#14581703—22no—
45#14581703—21no—
46#14581703—21no—
47C4×C83⋊C8—22no—
48C4×C83⋊C8C2×C412yes—
49C4⋊Q8×C83C2×C411yescharacteristic
50C4⋊Q8×C83C2×C411yescharacteristic
51#125×C83—21no—
52C22⋊Q8×C83—21no—
53C22⋊Q8×C83D412yes—
54C4×Q8×C83—21no—
55C4×Q8×C83D412yes—
56#117×C83—21no—
57C2×C4⋊C4×C83—21no—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—21no—
60C22⋊C8×C83—21no—
61C4×C8×C83—21no—
62#95×C83—21no—
63#95×C83—21no—
64C2×Dic166—22no—
65C2×Dic166—41no—
66C2×Dic166—82no—
67C22×Dic83—41no—
68#13050580—84no—
69#13050574—41no—
70#13050574—41no—
71C2×C83⋊C8—21no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—22no—
74C2×C83⋊C8—21no—
75C2×C83⋊C8—22no—
76C2×C83⋊C8—21no—
77C2×C83⋊C8—21no—
78C2×C83⋊C8Q1612yescomplemented
79C4⋊Dic83—22no—
80C4⋊Dic83—22no—
81C4⋊Dic83—41no—
82C4⋊Dic83—41no—
83C2×Q8×C83—21no—
84C2×Q8×C83C22⋊C412yes—
85C22×C4×C83—41no—
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83#4111yescharacteristic
89C2×C8×C83—41no—
90C4⋊C4×C83—21no—
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83—22no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83C22⋊C412yes—
98C22⋊C4×C83—22no—
99C42×C83—22no—
100C42×C83C22×C411yescharacteristic
101C42×C83C2×D411yescharacteristic
102C42×C83C2×D411yescharacteristic
103C42×C83#4111yescharacteristic
104C23×C83#12111yescharacteristic
105Q8×C83—42no—
106Q8×C83—81no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
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×Q1611yescharacteristic
121C2×C4×C83#13511yescharacteristic
122C2×C4×C83#13711yescharacteristic
123C2×C4×C83#13711yescharacteristic
124C2×Dic83—42no—
125C2×Dic83—82no—
126Dic166—44no—
127Dic166—82no—
128C8×C83—81no—
129C83⋊C8—22no—
130C83⋊C8—42no—
131C83⋊C8—41no—
132C83⋊C8—42no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83#38611yescharacteristic
136C22×C83#46211yescharacteristic
137C22×C83#46411yescharacteristic
138C4×C83—21no—
139C4×C83—21no—
140C4×C83—21no—
141C4×C83—21no—
142C4×C83—42no—
143C4×C83—42no—
144C4×C83—161no—
145C4×C83C4×Q1611yescharacteristic
146C4×C83#44111yescharacteristic
147C4×C83#45111yescharacteristic
148C4×C83#45211yescharacteristic
149Dic83—82no—
150#9290—831nomaximal, Sylow
151C2×C83—21no—
152C2×C83—21no—
153C2×C83#187811yescharacteristic
154C2×C83#190211yescharacteristic
155C2×C83#191011yescharacteristic
156#3209—831no—
157#2961—831no—
158#1993—831no—
159#1549—831no—
160#1518D8311yescharacteristic
161#1286—831no—
162#1275—831no—
163C83#929011yescharacteristic, complemented, Sylow
164#533C2×D8311yescharacteristic
165C2×C4⋊Q8—831no—
166#494—832no—
167#487—832no—
168#462—1662no—
169#451—1662no—
170#446—1662no—
171#446—1662no—
172#431—831no—
173#431C2×D8311yescharacteristic
174#424—1661no—
175#423—831no—
176#405—1661no—
177C2×#97—831no—
178#382C2×D8311yescharacteristic
179#340—1661no—
180#340—1661no—
181#332—1662no—
182#328—22no—
183C22×Q8—1661no—
184C2×Q16—1664no—
185C2×M16—1661no—
186C22×C8—831no—
187C22×C8—831no—
188C4⋊Q8—832no—
189C4⋊Q8—1661no—
190C4⋊Q8—1661no—
191C4⋊Q8—1661no—
192C4⋊Q8C4×D8311yescharacteristic
193C4⋊Q8C4×D8311yescharacteristic
194#125—21no—
195C22⋊Q8—21no—
196C22⋊Q8C22⋊D8312yes—
197C4×Q8—21no—
198C4×Q8D33212yes—
199#117—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1661no—
202C2×C4⋊C4—1661no—
203C2×C42C22×D8311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—832no—
206C4⋊C8—1664no—
207C4⋊C8—1661no—
208C4⋊C8—1661no—
209#103—1664no—
210C22⋊C8—21no—
211C22⋊C8—1661no—
212#97—1662no—
213#97—1661no—
214#97—1661no—
215C4×C8—21no—
216C4×C8—832no—
217C4×C8—1662no—
218#95—21no—
219#95—21no—
220C2×Q8—21no—
221C2×Q8—3321no—
222C2×Q8—6642no—
223C2×Q8—1662no—
224C2×Q8(C2×D83)⋊C412yes—
225C22×C4—41no—
226C22×C4—3321no—
227C22×C4D4×D8311yescharacteristic
228C22×C4C2×C22⋊D8311yescharacteristic
229C22×C4#1305058911yescharacteristic
230Q16—6644no—
231M16—3321no—
232M16—3321no—
233C2×C8—41no—
234C2×C8—832no—
235C2×C8—1662no—
236C2×C8—1661no—
237C2×C8—1661no—
238C2×C8—1661no—
239C2×C8—1662no—
240C2×C8—1661no—
241C2×C8—1661no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—3321no—
250C4⋊C4—3321no—
251C4⋊C4—1662no—
252C4⋊C4—1662no—
253C4⋊C4(C2×D83)⋊C412yes—
254C22⋊C4—22no—
255C42—22no—
256C42C2×C4×D8311yescharacteristic
257C42D4×D8311yescharacteristic
258C42C2×D33211yescharacteristic
259C42#1305058911yescharacteristic
260C23#1458176411yescharacteristic
261Q8—42no—
262Q8—81no—
263Q8—3324no—
264Q8—6642no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—3322no—
273C2×C4—6642no—
274C2×C4C2×(C2×D83)⋊C411yescharacteristic
275C2×C4(C22⋊D83)⋊C411yescharacteristic
276C2×C4(C22⋊D83)⋊C411yescharacteristic
277C2×C4#1458176211yescharacteristic
278C2×C4#1458176411yescharacteristic
279C2×C4#1458177611yescharacteristic
280C2×C4Q16×D8311yescharacteristic
281C2×C4#1458181311yescharacteristic
282C2×C4#1458182111yescharacteristic
283C2×C4#1458182411yescharacteristic
284C8—81no—
285C8—3322no—
286C8—3321no—
287C8—3322no—
288C8—1662no—
289C22—21no—
290C22—21no—
291C22#1912757811yescharacteristic
292C22#1912790911yescharacteristic
293C22#1912792111yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—42no—
299C4—42no—
300C4—161no—
301C4—6642no—
302C4#1912779311yescharacteristic
303C4#1912780511yescharacteristic
304C4#1912785811yescharacteristic
305C4#1912786611yescharacteristic
306C2—21no—
307C2—21no—
308C2#2490780211yescharacteristic
309C2#2490793411yescharacteristic
310C2#2490798811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1518C2
socleC22×C83#386
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1518D83
hypocenterC83#9290
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
9. #19128080C4semidirect
10. #19128053C4semidirect
78. C2×C83⋊C8Q16semidirect
163. C83#9290semidirect

Maximal subgroups

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

subgroupindex
#249184082
#249164042
#249084352
#249061472
#249053212
#249052932
C83×#15182
#929083