Finite Group Databasebeta

← #35313996

Subgroups — #35313996

Order 21248.

366 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 79 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#24911912C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905406C212yesmaximal, complemented
4#24905396C212yesmaximal, complemented
5C83×#1464C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic
7#19127754—21no—
8#19127629—21no—
9#19127629—21no—
10#19127606C411yescharacteristic
11#19127606C2211yescharacteristic
12#19127585—21no—
13#19127585—21no—
14#19127570C411yescharacteristic
15#19127566C2211yescharacteristic
16C4×C22⋊Dic83C411yescharacteristic
17#19127427C2212yes—
18#19127425—21no—
19C2×#117×C83C2211yescharacteristic
20#432×C83—21no—
21#324×C83C2212yes—
22#323×C83—21no—
23C22×Dic166C2×C411yescharacteristic
24#14581771—41no—
25#14581771—41no—
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×C22⋊Dic83—21no—
31C2×C22⋊Dic83—21no—
32C2×C4×Dic83C2×C411yescharacteristic
33C2×C4×Dic83C2×C411yescharacteristic
34C4⋊(C83⋊C8)—41no—
35C22⋊(C83⋊C8)—22no—
36C4×C83⋊C8—41no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C23×C4×C83C2×C411yescharacteristic
45#117×C83—41no—
46#117×C83—41no—
47C2×C4⋊C4×C83C2×C411yescharacteristic
48C2×C4⋊C4×C83C2311yescharacteristic
49C2×C22⋊C4×C83D411yescharacteristic
50C2×C22⋊C4×C83D411yescharacteristic
51C2×C42×C83D411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C4⋊C8×C83—41no—
54C22⋊C8×C83—22no—
55#97×C83—41no—
56C2×Dic166—22no—
57C2×Dic166—41no—
58C2×Dic166—41no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C2×C83⋊C8—42no—
64C4⋊Dic83—41no—
65Dic83⋊C4—42no—
66Dic83⋊C4—42no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69C22⋊Dic83—42no—
70C22⋊Dic83—42no—
71C4×Dic83—22no—
72C4×Dic83—41no—
73C24×C83C22⋊C411yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—22no—
77C22×C4×C83—41no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C2×C8×C83—42no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C22⋊C4×C83—41no—
88C22⋊C4×C83—41no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C42×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83C2×C22⋊C411yescharacteristic
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—21no—
106C2×C4×C83—21no—
107C2×C4×C83—42no—
108C2×C4×C83—42no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83#9911yescharacteristic
114C2×C4×C83#9911yescharacteristic
115C2×C4×C83#13112yes—
116C2×Dic83—22no—
117C2×Dic83—41no—
118C2×Dic83—41no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128Dic166—82no—
129C8×C83—82no—
130C83⋊C8—82no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83—41no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—81no—
137C22×C83—81no—
138C22×C83#40811yescharacteristic
139C22×C83C2×#9911yescharacteristic
140C22×C83#41311yescharacteristic
141C4×C83—22no—
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—41no—
145C4×C83—41no—
146C4×C83—41no—
147C4×C83—41no—
148Dic83—42no—
149Dic83—81no—
150Dic83—81no—
151Dic83—81no—
152#11556—831nomaximal, Sylow
153C2×C83—21no—
154C2×C83—21no—
155C2×C83—81no—
156C2×C83#146811yescharacteristic
157C2×C83#212911yescharacteristic
158C2×C83#213211yescharacteristic
159#2499—831no—
160#1464D8311yescharacteristic
161#1458—831no—
162#1324—832no—
163#1319—832no—
164C83#1155611yescharacteristic, complemented, Sylow
165C2×C22⋊Q8—831no—
166C2×#117C2×D8311yescharacteristic
167#432—21no—
168#431—1661no—
169#397—1661no—
170#397—1661no—
171#391—831no—
172#391—831no—
173#387—1661no—
174#387—1661no—
175#384—831no—
176#382—831no—
177C4×C22⋊C4—831no—
178#324—832no—
179#324C2×D8312yes—
180#323—21no—
181#323—1661no—
182C22×Q8—831no—
183C23×C4C4×D8311yescharacteristic
184C22⋊Q8—3321no—
185C22⋊Q8—3321no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—831no—
189C2×C4⋊C4—1661no—
190C2×C4⋊C4—1661no—
191C2×C4⋊C4—1661no—
192C2×C4⋊C4C4×D8311yescharacteristic
193C2×C4⋊C4C22×D8311yescharacteristic
194C2×C22⋊C4—1661no—
195C2×C22⋊C4—1661no—
196C2×C22⋊C4D33211yescharacteristic
197C2×C22⋊C4C22⋊D8311yescharacteristic
198C2×C42—831no—
199C2×C42—831no—
200C2×C42D33211yescharacteristic
201C2×C42C22⋊D8311yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—3321no—
204C22⋊C8—22no—
205C22⋊C8—1662no—
206#97—41no—
207C4×C8—3321no—
208#95—1661no—
209#95—1661no—
210#95—1661no—
211#95—1661no—
212#95—1661no—
213#95—1661no—
214#95—1661no—
215C24(C2×D83)⋊C411yescharacteristic
216C2×Q8—3321no—
217C2×Q8—3321no—
218C2×Q8—1662no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—1661no—
224C22×C4—1661no—
225C22×C4—1661no—
226C22×C4—1661no—
227C22×C4(C2×D83)⋊C411yescharacteristic
228C22×C4(C2×D83)⋊C411yescharacteristic
229C22×C4(C2×D83)⋊C411yescharacteristic
230C22×C4C2×C4×D8311yescharacteristic
231C22×C4C2×C22⋊D8311yescharacteristic
232C22×C4C2×D33211yescharacteristic
233C2×C8—42no—
234C2×C8—3322no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—3321no—
238C4⋊C4—3322no—
239C4⋊C4—3322no—
240C4⋊C4—3321no—
241C4⋊C4—3321no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—3322no—
245C22⋊C4—3322no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—3321no—
251C42—1662no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×(C2×D83)⋊C411yescharacteristic
258Q8—6642no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—3321no—
274C2×C4—3321no—
275C2×C4—3321no—
276C2×C4—3321no—
277C2×C4—3321no—
278C2×C4—3321no—
279C2×C4—3321no—
280C2×C4—3321no—
281C2×C4—3321no—
282C2×C4—3321no—
283C2×C4—3321no—
284C2×C4—1662no—
285C2×C4#1458170611yescharacteristic
286C2×C4#1458170611yescharacteristic
287C2×C4#1458180012yes—
288C8—82no—
289C8—6642no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1912766911yescharacteristic
298C22C2×#1458170611yescharacteristic
299C22#1912768311yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3322no—
308C4—6641no—
309C4—6641no—
310C4—6641no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2490581211yescharacteristic
315C2#2490913711yescharacteristic
316C2#2490914611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1464C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1464D83
hypocenterC83#11556
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #24905406C2semidirect
4. #24905396C2semidirect
164. C83#11556semidirect

Maximal subgroups

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

subgroupindex
#249119122
#249057772
#249054062
#249053962
C83×#14642
#1155683