Finite Group Databasebeta

← #35315145

Subgroups — #35315145

Order 21248.

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

normal subgroups only

This group has 388 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#24910731C211yesmaximal, characteristic
2#24905797C211yesmaximal, characteristic
3#24905492C212yesmaximal, complemented
4#24905489C212yesmaximal, complemented
5C83×#1822C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic
7#19127758—21no—
8#19127640C2211yescharacteristic
9#19127637C411yescharacteristic
10#19127634—21no—
11#19127624—21no—
12#19127621C2211yescharacteristic
13#19127613C411yescharacteristic
14C4×Dic83⋊C4—21no—
15C4×C4⋊Dic83C411yescharacteristic
16C4×C22⋊Dic83—21no—
17#19127427C2212yes—
18#19127425—21no—
19C2×#117×C83C2211yescharacteristic
20#407×C83—21no—
21#343×C83—21no—
22#343×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C4×Dic166—41no—
25C4×Dic166—41no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35C4⋊(C83⋊C8)—41no—
36C22⋊(C83⋊C8)—22no—
37#14581703—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C2×M16×C83—22no—
46#117×C83—41no—
47#117×C83—41no—
48C2×C4⋊C4×C83D411yescharacteristic
49C2×C4⋊C4×C83C2311yescharacteristic
50C2×C22⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C42×C83C2×C411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C22⋊C8×C83—41no—
55C22⋊C8×C83—41no—
56C2×Dic166—41no—
57C2×Dic166—42no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—42no—
64C4⋊Dic83—41no—
65Dic83⋊C4—42no—
66Dic83⋊C4—42no—
67Dic83⋊C4—42no—
68C22⋊Dic83—42no—
69C22⋊Dic83—42no—
70C4×Dic83—42no—
71C4×Dic83—41no—
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
84M16×C83—42no—
85M16×C83—42no—
86C2×C8×C83—42no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C42×C83—21no—
92C42×C83—21no—
93C42×C83—21no—
94C42×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83C2×C22⋊C411yescharacteristic
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—21no—
108C2×C4×C83—21no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83#9911yescharacteristic
116C2×C4×C83#9911yescharacteristic
117C2×C4×C83#13712yes—
118C2×Dic83—22no—
119C2×Dic83—22no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130Dic166—42no—
131Dic166—42no—
132C8×C83—82no—
133C83⋊C8—82no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—42no—
137C22×C83—41no—
138C22×C83—41no—
139C22×C83—41no—
140C22×C83—81no—
141C22×C83C2×#9911yescharacteristic
142C22×C83#41911yescharacteristic
143C22×C83#42111yescharacteristic
144C4×C83—22no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—41no—
148C4×C83—41no—
149C4×C83—41no—
150C4×C83—41no—
151Dic83—42no—
152Dic83—42no—
153Dic83—81no—
154Dic83—81no—
155#11816—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83—81no—
159C2×C83#151111yescharacteristic
160C2×C83#212011yescharacteristic
161C2×C83#213211yescharacteristic
162#2396—831no—
163#1822D8311yescharacteristic
164#1464—831no—
165#1354—832no—
166#1353—832no—
167C83#1181611yescharacteristic, complemented, Sylow
168C2×C4×Q8—831no—
169C2×#117C2×D8311yescharacteristic
170#432—1661no—
171#407—21no—
172#400—831no—
173#399—831no—
174#398—1661no—
175#396—1661no—
176#395—831no—
177#393—831no—
178C4×C4⋊C4—831no—
179C4×C4⋊C4—1661no—
180C4×C22⋊C4—1661no—
181#343—21no—
182#343C2×D8312yes—
183#324—832no—
184#323—1661no—
185C22×Q8—831no—
186C23×C4C22⋊D8311yescharacteristic
187C2×M16—22no—
188C4×Q8—3321no—
189C4×Q8—3321no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—831no—
193C2×C4⋊C4—1661no—
194C2×C4⋊C4—1661no—
195C2×C4⋊C4—1661no—
196C2×C4⋊C4—1661no—
197C2×C4⋊C4C22⋊D8311yescharacteristic
198C2×C4⋊C4C22×D8311yescharacteristic
199C2×C22⋊C4—1661no—
200C2×C22⋊C4—1661no—
201C2×C22⋊C4C4×D8311yescharacteristic
202C2×C22⋊C4D33211yescharacteristic
203C2×C42—1661no—
204C2×C42—1661no—
205C2×C42C4×D8311yescharacteristic
206C2×C42D33211yescharacteristic
207C4⋊C8—3321no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—1662no—
211#97—3321no—
212#95—831no—
213#95—831no—
214#95—1661no—
215#95—1661no—
216#95—1661no—
217#95—1661no—
218C24(C2×D83)⋊C411yescharacteristic
219C2×Q8—3321no—
220C2×Q8—3322no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—1661no—
226C22×C4—1661no—
227C22×C4—1661no—
228C22×C4—1661no—
229C22×C4(C2×D83)⋊C411yescharacteristic
230C22×C4(C2×D83)⋊C411yescharacteristic
231C22×C4(C2×D83)⋊C411yescharacteristic
232C22×C4C2×C4×D8311yescharacteristic
233C22×C4C2×C22⋊D8311yescharacteristic
234C22×C4C2×D33211yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—42no—
238C2×C8—3322no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—3321no—
242C4⋊C4—3322no—
243C4⋊C4—3322no—
244C4⋊C4—3322no—
245C4⋊C4—3322no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—3322no—
249C22⋊C4—3322no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—3322no—
255C42—3321no—
256C42—3321no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×(C2×D83)⋊C411yescharacteristic
263Q8—3322no—
264Q8—3322no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—3321no—
280C2×C4—3321no—
281C2×C4—3321no—
282C2×C4—3321no—
283C2×C4—3321no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—3321no—
287C2×C4—3321no—
288C2×C4—3321no—
289C2×C4—1662no—
290C2×C4—1662no—
291C2×C4#1458170611yescharacteristic
292C2×C4#1458170611yescharacteristic
293C2×C4#1458182212yes—
294C8—82no—
295C8—6642no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1458170611yescharacteristic
304C22#1912770911yescharacteristic
305C22#1912771911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3322no—
314C4—3322no—
315C4—6641no—
316C4—6641no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2490599511yescharacteristic
321C2#2490910511yescharacteristic
322C2#2490914611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1822C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1822D83
hypocenterC83#11816
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #24905492C2semidirect
4. #24905489C2semidirect
167. C83#11816semidirect

Maximal subgroups

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

subgroupindex
#249107312
#249057972
#249054922
#249054892
C83×#18222
#1181683