Finite Group Databasebeta

← #35314035

Subgroups — #35314035

Order 21248.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#24912439C211yesmaximal, characteristic
2#24905787C211yesmaximal, characteristic, complemented
3#24905396C212yesmaximal, complemented
4#24905394C212yesmaximal
5C83×#1467C211yesmaximal, characteristic, complemented
6C2×#14581790C411yescharacteristic
7C2×#14581781C411yescharacteristic
8C2×#14581780C411yescharacteristic
9#19127754—21no—
10#19127623—21no—
11#19127623—21no—
12#19127606C2211yescharacteristic
13#19127602C2211yescharacteristic
14#19127592—21no—
15#19127592—21no—
16C4×C4⋊Dic83C411yescharacteristic
17#19127427—21no—
18#19127427C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#432×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C22×D332C2×C411yescharacteristic
25#14581790—42no—
26#14581781—42no—
27#14581780—41no—
28#14581780—41no—
29C2×C4⋊Dic83—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×(C2×D83)⋊C4—21no—
33C2×(C2×D83)⋊C4—21no—
34C2×(C2×D83)⋊C4—21no—
35C2×(C2×D83)⋊C4—21no—
36C2×(C2×D83)⋊C4—21no—
37C2×(C2×D83)⋊C4—21no—
38C2×C4×Dic83C2×C411yescharacteristic
39C2×C4×Dic83C2×C411yescharacteristic
40C4⋊(C83⋊C8)—41no—
41C22⋊(C83⋊C8)—22no—
42C4×C83⋊C8—41no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C22×Q8×C83C2×C411yescharacteristic
47C4×Q8×C83—41no—
48C4×Q8×C83—41no—
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83D411yescharacteristic
51C2×C4⋊C4×C83C2311yescharacteristic
52C2×C42×C83C2×C411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C4⋊C8×C83—41no—
56C22⋊C8×C83—22no—
57#97×C83—41no—
58C23×D83—21no—
59C2×Dic166—22no—
60C2×Dic166—41no—
61C2×Dic166—41no—
62C2×D332—22no—
63C2×D332—41no—
64C2×D332—41no—
65C22×Dic83—21no—
66C22×Dic83—21no—
67C22×Dic83—21no—
68C2×C83⋊C8—42no—
69C4⋊Dic83—42no—
70Dic83⋊C4—42no—
71Dic83⋊C4—42no—
72(C2×D83)⋊C4—42no—
73(C2×D83)⋊C4—42no—
74(C2×D83)⋊C4—42no—
75(C2×D83)⋊C4—42no—
76(C2×D83)⋊C4—41no—
77(C2×D83)⋊C4—41no—
78(C2×D83)⋊C4—42no—
79C4×Dic83—41no—
80C4×Dic83—41no—
81C2×Q8×C83—22no—
82C2×Q8×C83—22no—
83C2×Q8×C83—41no—
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C2×C8×C83—42no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—21no—
101C23×C83C2×C22⋊C411yescharacteristic
102C22×D83—41no—
103C22×D83—41no—
104C22×D83—41no—
105C22×D83—41no—
106C22×D83—41no—
107C22×D83—41no—
108C22×D83—41no—
109Q8×C83—44no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83#9911yescharacteristic
125C2×C4×C83#9911yescharacteristic
126C2×C4×C83#10012yes—
127C2×C4×C83#13112yes—
128C2×Dic83—22no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137D332—82no—
138Dic166—82no—
139C8×C83—82no—
140C83⋊C8—82no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83#40811yescharacteristic
144C22×C83C2×#9911yescharacteristic
145C22×C83C2×#10011yescharacteristic
146C2×D83—81no—
147C2×D83—81no—
148C2×D83—81no—
149C2×D83—81no—
150C2×D83—81no—
151C2×D83—81no—
152C2×D83—81no—
153C4×C83—21no—
154C4×C83—21no—
155C4×C83—22no—
156C4×C83—22no—
157C4×C83—41no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161Dic83—42no—
162Dic83—81no—
163Dic83—81no—
164#11565—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83#147111yescharacteristic
168C2×C83#212611yescharacteristic
169C2×C83#213311yescharacteristic
170D83—161no—
171#2545—831no—
172#1467D8311yescharacteristic, complemented
173#1461—831no—
174#1319—832no—
175#1318—832no—
176C83#1156511yescharacteristic, complemented, Sylow
177C2×C4⋊Q8—831no—
178C2×#124—831no—
179C2×#124—831no—
180C2×C4×Q8C2×D8311yescharacteristic
181#432—21no—
182#431—1661no—
183#396—1661no—
184#396—1661no—
185#391—831no—
186#390—831no—
187#388—1661no—
188#388—1661no—
189C4×C4⋊C4—831no—
190#324—21no—
191#324—832no—
192#324—1661no—
193#324C2×D8312yes—
194C22×Q8—831no—
195C22×Q8C4×D8311yescharacteristic
196C22×D4—831no—
197C4⋊Q8—3322no—
198#124—3321no—
199#124—3321no—
200#124—3322no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4D33211yescharacteristic
207C2×C4⋊C4C22⋊D8311yescharacteristic
208C2×C4⋊C4C22×D8311yescharacteristic
209C2×C22⋊C4—1661no—
210C2×C22⋊C4—1661no—
211C2×C22⋊C4—1661no—
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1661no—
214C2×C22⋊C4—1661no—
215C2×C42—831no—
216C2×C42—831no—
217C2×C42C4×D8311yescharacteristic
218C2×C42D33211yescharacteristic
219C2×C42C22⋊D8311yescharacteristic
220C4⋊C8—41no—
221C4⋊C8—3321no—
222C22⋊C8—22no—
223C22⋊C8—1662no—
224#97—41no—
225C4×C8—3321no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229C24—1661no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—3321no—
234C2×Q8—3321no—
235C2×Q8—1662no—
236C2×D4—3321no—
237C2×D4—3321no—
238C2×D4—1662no—
239C22×C4—1661no—
240C22×C4—1661no—
241C22×C4—1661no—
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4(C2×D83)⋊C411yescharacteristic
246C22×C4C2×C4×D8311yescharacteristic
247C22×C4C2×C22⋊D8311yescharacteristic
248C22×C4C2×D33211yescharacteristic
249C2×C8—42no—
250C2×C8—3322no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—3322no—
255C4⋊C4—3322no—
256C4⋊C4—3322no—
257C22⋊C4—3322no—
258C22⋊C4—3322no—
259C22⋊C4—3322no—
260C22⋊C4—3322no—
261C22⋊C4—3321no—
262C22⋊C4—3321no—
263C22⋊C4—3322no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—3321no—
271C42—3321no—
272C23—3321no—
273C23—3321no—
274C23—3321no—
275C23—3321no—
276C23—3321no—
277C23—3321no—
278C23—3321no—
279C23C2×(C2×D83)⋊C411yescharacteristic
280Q8—44no—
281Q8—6642no—
282D4—6642no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—3321no—
298C2×C4—3321no—
299C2×C4—3321no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—3321no—
305C2×C4—1662no—
306C2×C4#1458170611yescharacteristic
307C2×C4#1458170611yescharacteristic
308C2×C4#1458170812yes—
309C2×C4#1458180012yes—
310C8—82no—
311C8—6642no—
312C22—21no—
313C22—21no—
314C22—6641no—
315C22—6641no—
316C22—6641no—
317C22—6641no—
318C22—6641no—
319C22—6641no—
320C22—6641no—
321C22#1912766911yescharacteristic
322C22C2×#1458170611yescharacteristic
323C22C2×#1458170811yescharacteristic
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—3322no—
333C4—6641no—
334C4—6641no—
335C2—21no—
336C2—21no—
337C2—13281no—
338C2#2490582411yescharacteristic
339C2#2490912611yescharacteristic
340C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1467C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1467D83
hypocenterC83#11565
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #24905787C2semidirect
3. #24905396C2semidirect
5. C83×#1467C2semidirect
172. #1467D83semidirect
176. C83#11565semidirect

Maximal subgroups

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

subgroupindex
#249124392
#249057872
#249053962
#249053942
C83×#14672
#1156583