Finite Group Databasebeta

← #35301069

Subgroups — #35301069

Order 21248.

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

normal subgroups only

This group has 422 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#24918413C211yesmaximal, characteristic
2#24916400C211yesmaximal, characteristic
3#24908435C211yesmaximal, characteristic
4#24906149C211yesmaximal, characteristic
5#24906026C211yesmaximal, characteristic
6#24905293C211yesmaximal, characteristic
7C83×#1286C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9C2×#14581792C2211yescharacteristic
10#19128081C412yescomplemented
11#19128052C412yescomplemented
12#19127912—22no—
13#19127859—22no—
14#19127830—22no—
15#19127830—22no—
16#19127754C2211yescharacteristic
17#19127729—21no—
18#19127659—21no—
19C2×#14581703C2211yescharacteristic
20#19127463—21no—
21#19127463—21no—
22#19127446—22no—
23#19127437—22no—
24#431×C83C2211yescharacteristic
25#423×C83C2211yescharacteristic
26#382×C83C2211yescharacteristic
27C22×Dic166—21no—
28C2×Dic332—24no—
29C22×C83⋊C8D411yescharacteristic
30C22×C83⋊C8D411yescharacteristic
31#14581792—21no—
32#14581792—21no—
33#14581792—21no—
34#14581792C2×C412yes—
35#14581792C2×C411yescharacteristic
36#14581792C2×C411yescharacteristic
37#14581784—21no—
38#14581771—21no—
39#14581771D412yes—
40C4×Dic166—21no—
41C4×Dic166D412yes—
42#14581751—21no—
43C2×C4⋊Dic83—21no—
44C2×C4⋊Dic83—21no—
45C4⋊(C83⋊C8)—21no—
46Dic83⋊C8—24no—
47#14581717—24no—
48C22⋊(C83⋊C8)—21no—
49#14581703—21no—
50#14581703—21no—
51#14581702—22no—
52C4×C83⋊C8—21no—
53C8×Dic83—22no—
54C2×M16×C83—21no—
55C2×C4⋊C4×C83—21no—
56C2×C42×C83C2311yescharacteristic
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83D412yes—
60C22⋊C8×C83—21no—
61C4×C8×C83C2×C412yes—
62#95×C83—21no—
63#95×C83—21no—
64C2×Dic166—22no—
65C2×Dic166—21no—
66C2×Dic166—41no—
67C2×Dic166—82no—
68C2×Dic166C22⋊C412yes—
69C22×Dic83—41no—
70Dic332—84no—
71C2×C83⋊C8—21no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—21no—
74C2×C83⋊C8—21no—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8Q1612yescomplemented
78C4⋊Dic83—21no—
79C4⋊Dic83—21no—
80C4⋊Dic83—21no—
81C4⋊Dic83—22no—
82C4⋊Dic83—22no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C4⋊Dic83C22⋊C412yes—
86Dic83⋊C4—22no—
87Dic83⋊C4—22no—
88C22⋊Dic83—22no—
89C4×Dic83—22no—
90C22×C4×C83—41no—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83#4111yescharacteristic
94M16×C83—41no—
95M16×C83—41no—
96C2×C8×C83—22no—
97C2×C8×C83—22no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C42×C83C22×C411yescharacteristic
101C42×C83C2×D411yescharacteristic
102C42×C83C2×D411yescharacteristic
103C42×C83#4111yescharacteristic
104C23×C83#12111yescharacteristic
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83—81no—
108C2×C4×C83—81no—
109C2×C4×C83—81no—
110C2×C4×C83C2×C22⋊C411yescharacteristic
111C2×C4×C83C4×D411yescharacteristic
112C2×C4×C83C4×D411yescharacteristic
113C2×C4×C83#12011yescharacteristic
114C2×C4×C83#12111yescharacteristic
115C2×C4×C83#12311yescharacteristic
116C2×C4×C83C2×Q1611yescharacteristic
117C2×C4×C83#13511yescharacteristic
118C2×C4×C83#13711yescharacteristic
119C2×C4×C83#13711yescharacteristic
120C2×Dic83—22no—
121C2×Dic83—22no—
122C2×Dic83—42no—
123C2×Dic83—82no—
124Dic166—42no—
125Dic166—44no—
126Dic166—81no—
127Dic166—82no—
128C8×C83—42no—
129C8×C83—42no—
130C83⋊C8—22no—
131C83⋊C8—41no—
132C83⋊C8—81no—
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—161no—
143C4×C83C4×Q1611yescharacteristic
144C4×C83#44111yescharacteristic
145C4×C83#45111yescharacteristic
146C4×C83#45211yescharacteristic
147Dic83—42no—
148Dic83—42no—
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#1518—831no—
161#1286D8311yescharacteristic
162#1275—831no—
163C83#929011yescharacteristic, complemented, Sylow
164#533—831no—
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#423C2×D8311yescharacteristic
176#405—1661no—
177C2×#97—831no—
178#382C2×D8311yescharacteristic
179#340—1661no—
180#340—1661no—
181#332—1662no—
182#328—1662no—
183C22×Q8—1661no—
184C2×Q16—1664no—
185C2×M16—21no—
186C22×C8—831no—
187C22×C8—831no—
188C4⋊Q8—831no—
189C4⋊Q8—831no—
190C4⋊Q8—832no—
191C4⋊Q8—1661no—
192C4⋊Q8—1661no—
193C4⋊Q8—1661no—
194#125—1661no—
195C22⋊Q8—832no—
196C22⋊Q8—1661no—
197C4×Q8—832no—
198C4×Q8—1661no—
199#117—1661no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1661no—
202C2×C4⋊C4—1661no—
203C2×C42C22×D8311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1664no—
207C4⋊C8—1661no—
208C4⋊C8D33212yes—
209#103—1664no—
210C22⋊C8—21no—
211C22⋊C8—1661no—
212#97—1662no—
213#97—1661no—
214#97—1661no—
215C4×C8—1662no—
216C4×C8—1661no—
217C4×C8C4×D8312yes—
218#95—21no—
219#95—21no—
220C2×Q8—3321no—
221C2×Q8—832no—
222C2×Q8—6642no—
223C2×Q8—1662no—
224C2×Q8—1661no—
225C22×C4—41no—
226C22×C4—3321no—
227C22×C4D4×D8311yescharacteristic
228C22×C4D4×D8311yescharacteristic
229C22×C4#1305059011yescharacteristic
230Q16—6644no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—3321no—
236C2×C8—832no—
237C2×C8—1661no—
238C2×C8—1661no—
239C2×C8—1661no—
240C2×C8—1661no—
241C2×C8—1661no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—3321no—
245C4⋊C4—3321no—
246C4⋊C4—832no—
247C4⋊C4—1662no—
248C4⋊C4—1662no—
249C4⋊C4—1662no—
250C4⋊C4—1662no—
251C4⋊C4—1661no—
252C4⋊C4—1661no—
253C4⋊C4—1661no—
254C22⋊C4—1662no—
255C42—1662no—
256C42C2×C4×D8311yescharacteristic
257C42D4×D8311yescharacteristic
258C42C2×D33211yescharacteristic
259C42#1305059111yescharacteristic
260C23#1458176511yescharacteristic
261Q8—3322no—
262Q8—3324no—
263Q8—6641no—
264Q8—6642no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—3322no—
271C2×C4—6642no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4C22⋊C4×D8311yescharacteristic
275C2×C4#1458175411yescharacteristic
276C2×C4C4×D33211yescharacteristic
277C2×C4#1458176211yescharacteristic
278C2×C4#1458176711yescharacteristic
279C2×C4#1458177711yescharacteristic
280C2×C4Q16×D8311yescharacteristic
281C2×C4#1458181111yescharacteristic
282C2×C4#1458182111yescharacteristic
283C2×C4#1458182111yescharacteristic
284C8—42no—
285C8—42no—
286C8—3321no—
287C8—6641no—
288C8—1662no—
289C22—21no—
290C22—21no—
291C22#1912757911yescharacteristic
292C22#1912791011yescharacteristic
293C22#1912792211yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—161no—
299C4—3322no—
300C4—3322no—
301C4—6642no—
302C4#1912779511yescharacteristic
303C4#1912780711yescharacteristic
304C4#1912785811yescharacteristic
305C4#1912786411yescharacteristic
306C2—21no—
307C2—21no—
308C2#2490780511yescharacteristic
309C2#2490793511yescharacteristic
310C2#2490798911yescharacteristic
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#9290
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. #19128081C4semidirect
11. #19128052C4semidirect
77. 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
#249184132
#249164002
#249084352
#249061492
#249060262
#249052932
C83×#12862
#929083