Finite Group Databasebeta

← #35313648

Subgroups — #35313648

Order 21248.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#24916156C211yesmaximal, characteristic
2#24909072C211yesmaximal, characteristic, complemented
3C22⋊C16×D83C211yesmaximal, characteristic, complemented
4#24905405C211yesmaximal, characteristic, complemented
5#24905388C211yesmaximal, characteristic, complemented
6#24905388C211yesmaximal, characteristic, complemented
7C83×#1323C211yesmaximal, characteristic, complemented
8C2×#14581800C411yescharacteristic
9C22×C8×D83C2211yescharacteristic
10#19128213C411yescharacteristic
11C2×#14581731—21no—
12C2×C16×D83—21no—
13#19127771—21no—
14#19127771—21no—
15#19127766—21no—
16#19127766—21no—
17#19127667C2211yescharacteristic
18#19127664C411yescharacteristic
19#19127657C411yescharacteristic
20C22⋊(C83⋊C16)C2211yescharacteristic
21C22⋊(C83⋊C16)C2211yescharacteristic
22(C2×D83)⋊C16—21no—
23(C2×D83)⋊C16—21no—
24#424×C83C2211yescharacteristic
25C22⋊C16×C83C2211yescharacteristic
26C22⋊C16×C83C2211yescharacteristic
27C2×#13050590C2×C411yescharacteristic
28C22×C4×D83C2×C411yescharacteristic
29#14581800—42no—
30C2×#13050574C2×C411yescharacteristic
31C2×#13050573—21no—
32C22×C83⋊C8C2×C411yescharacteristic
33C2×C8×D83—21no—
34C2×C8×D83—42no—
35C2×C8×D83D411yescharacteristic
36C2×C8×D83D411yescharacteristic
37C2×C8×D83D411yescharacteristic
38C2×C8×D83D411yescharacteristic
39#14581776—21no—
40#14581772C811yescharacteristic
41#14581772C811yescharacteristic
42#14581767C811yescharacteristic
43#14581767C811yescharacteristic
44#14581762—21no—
45C4×D332—21no—
46C4×D332—21no—
47#14581731—42no—
48C2×C83⋊C16—21no—
49C2×C83⋊C16—21no—
50C16×D83—42no—
51(C2×D83)⋊C8—21no—
52(C2×D83)⋊C8—21no—
53(C2×D83)⋊C8—21no—
54(C2×D83)⋊C8—21no—
55#14581703—21no—
56C4×C83⋊C8—21no—
57C2×M16×C83C2×C411yescharacteristic
58C22×C8×C83C2311yescharacteristic
59#117×C83C2×C411yescharacteristic
60C2×C16×C83—21no—
61C2×C16×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C23×D83M1611yescharacteristic
65#13050590—41no—
66#13050590—41no—
67C2×Dic166C2×C811yescharacteristic
68C2×D332—21no—
69C2×D332—21no—
70C2×D332C2×C811yescharacteristic
71C2×C22⋊D83—21no—
72C22×Dic83M1611yescharacteristic
73C2×C4×D83—21no—
74C2×C4×D83—41no—
75C2×C4×D83—41no—
76C2×C4×D83C22⋊C411yescharacteristic
77C2×C4×D83C22⋊C411yescharacteristic
78C2×C4×D83C2×C811yescharacteristic
79C2×C4×D83C2×C811yescharacteristic
80#13050574—41no—
81#13050573—42no—
82C2×C83⋊C8—22no—
83C2×C83⋊C8—21no—
84C2×C83⋊C8C22⋊C411yescharacteristic
85C2×C83⋊C8C22⋊C411yescharacteristic
86C8×D83—22no—
87C8×D83—22no—
88C8×D83—41no—
89C8×D83—41no—
90C8×D83—42no—
91C4⋊Dic83—21no—
92C4⋊Dic83—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97C83⋊C16—41no—
98C83⋊C16—41no—
99C22×C4×C83C22×C411yescharacteristic
100M16×C83—41no—
101C2×C8×C83—21no—
102C2×C8×C83—41no—
103C2×C8×C83C2×D411yescharacteristic
104C2×C8×C83C2×D411yescharacteristic
105C4⋊C4×C83C2×C811yescharacteristic
106C4⋊C4×C83C2×C811yescharacteristic
107C22⋊C4×C83—21no—
108C42×C83—21no—
109C16×C83—41no—
110C16×C83—41no—
111C23×C83C2×M1611yescharacteristic
112C22×D83—22no—
113C22×D83—21no—
114C22×D83—41no—
115C22×D83—41no—
116C22×D83C22⋊C811yescharacteristic
117C22×D83C22⋊C811yescharacteristic
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83C2×C22⋊C411yescharacteristic
123C2×C4×C83C22×C811yescharacteristic
124C22⋊D83—42no—
125C2×Dic83—21no—
126C2×Dic83—41no—
127C2×Dic83C22⋊C811yescharacteristic
128C2×Dic83C22⋊C811yescharacteristic
129D332—41no—
130D332—42no—
131D332—42no—
132C4×D83—21no—
133C4×D83—21no—
134C4×D83—21no—
135C4×D83—21no—
136C4×D83—41no—
137C4×D83—41no—
138C4×D83—41no—
139C4×D83—41no—
140Dic166—41no—
141C8×C83—21no—
142C8×C83—21no—
143C8×C83—41no—
144C83⋊C8—21no—
145C83⋊C8—41no—
146C83⋊C8#9912yes—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83C2×C22⋊C811yescharacteristic
151C2×D83—21no—
152C2×D83—21no—
153C2×D83—21no—
154C2×D83—21no—
155C2×D83—21no—
156C2×D83—44no—
157C2×D83—41no—
158C2×D83—41no—
159C2×D83—41no—
160C2×D83#35012yes—
161C4×C83—21no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83C2×#9911yescharacteristic
165C4×C83#41311yescharacteristic
166Dic83—21no—
167Dic83—21no—
168Dic83—41no—
169#11470—831nomaximal, Sylow
170C2×C83—41no—
171C2×C83#145911yescharacteristic
172C2×C83C2×#35011yescharacteristic
173C2×C83#211711yescharacteristic
174D83—22no—
175D83—41no—
176D83—81no—
177#2929—831no—
178#2112—831no—
179C2×C22⋊C16—831no—
180#1323—831no—
181#1323D8311yescharacteristic, complemented
182#1315—831no—
183#1315—831no—
184C83#1147011yescharacteristic, complemented, Sylow
185C2×#131—831no—
186C23×C8—831no—
187#525—831no—
188C2×M32—1661no—
189C22×C16—1661no—
190#435—1661no—
191#435—1661no—
192C8×D4—1661no—
193C8×D4—1661no—
194#424C2×D8311yescharacteristic
195#408—831no—
196#407—831no—
197#405—831no—
198C22⋊C16—831no—
199C22⋊C16—831no—
200C22⋊C16—1661no—
201C22⋊C16—1661no—
202C22⋊C16C2×D8311yescharacteristic
203C22⋊C16C2×D8311yescharacteristic
204C2×#41—831no—
205C23×C4—831no—
206#131—3322no—
207C2×M16—831no—
208C2×M16—1661no—
209C2×M16C4×D8311yescharacteristic
210C22×C8—3322no—
211C22×C8—831no—
212C22×C8—831no—
213C22×C8—831no—
214C22×C8—831no—
215C22×C8—831no—
216C22×C8—1661no—
217C22×C8C22×D8311yescharacteristic
218#123—1661no—
219C22⋊Q8—831no—
220C22⋊Q8—831no—
221#121—831no—
222#121—831no—
223#120—1661no—
224C4×D4—1661no—
225C4×D4—1661no—
226#117C4×D8311yescharacteristic
227M32—3322no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—3322no—
231C2×C16—1661no—
232C2×C16—1661no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C22⋊C8—1661no—
236C22⋊C8—1661no—
237C22⋊C8—1661no—
238C22⋊C8—1661no—
239#97—1661no—
240C4×C8—1661no—
241C24—831no—
242#41—3321no—
243#41—3321no—
244C2×Q8—831no—
245C2×D4—831no—
246C2×D4—1661no—
247C2×D4—1661no—
248C2×D4—1661no—
249C22×C4—3321no—
250C22×C4—3321no—
251C22×C4—831no—
252C22×C4—831no—
253C22×C4—831no—
254C22×C4—831no—
255C22×C4—831no—
256C22×C4—1661no—
257C22×C4C2×C4×D8311yescharacteristic
258M16—41no—
259M16—3322no—
260M16—3321no—
261C2×C8—21no—
262C2×C8—41no—
263C2×C8—3321no—
264C2×C8—3321no—
265C2×C8—3322no—
266C2×C8—831no—
267C2×C8—831no—
268C2×C8—1662no—
269C2×C8—1662no—
270C2×C8—1661no—
271C2×C8—1662no—
272C2×C8D4×D8311yescharacteristic
273C2×C8D4×D8311yescharacteristic
274C4⋊C4—1661no—
275C4⋊C4—1661no—
276C4⋊C4C8×D8311yescharacteristic
277C4⋊C4C8×D8311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—1661no—
280C22⋊C4—1661no—
281C22⋊C4—1661no—
282C22⋊C4—1661no—
283C42—21no—
284C16—41no—
285C16—41no—
286C16—3321no—
287C16—3321no—
288C23—3321no—
289C23—3321no—
290C23—831no—
291C23—831no—
292C23—1662no—
293C23—1661no—
294C23M16×D8311yescharacteristic
295Q8—3321no—
296D4—3321no—
297D4—3322no—
298D4—3322no—
299D4—3322no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—3321no—
305C2×C4—3321no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—831no—
310C2×C4—831no—
311C2×C4—1661no—
312C2×C4—1661no—
313C2×C4—1661no—
314C2×C4—1661no—
315C2×C4—1661no—
316C2×C4C22⋊C4×D8311yescharacteristic
317C2×C4C2×C8×D8311yescharacteristic
318C8—21no—
319C8—21no—
320C8—41no—
321C8—3321no—
322C8—832no—
323C8—1661no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—3324no—
328C22—3321no—
329C22—3321no—
330C22—3321no—
331C22—832no—
332C22—1661no—
333C22—1661no—
334C22—1661no—
335C22—1661no—
336C22—1661no—
337C22C22⋊C8×D8311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—3321no—
342C4—1661no—
343C4—1661no—
344C4#99×D8311yescharacteristic
345C4#1912768411yescharacteristic
346C2—41no—
347C2—3321no—
348C2—6641no—
349C2—1662no—
350C2#2490578011yescharacteristic
351C2#350×D8311yescharacteristic
352C2#2490908911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC2×C4×C83C22×C8
FrattiniC22×C8C22×D83
FittingC83×#1323C2
socleC22×C83C2×C22⋊C8
prime radical#19128213C4
cosocleC22×C8×C83C23
hypercenter#1323D83
hypocenterC83#11470
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #24909072C2semidirect
3. C22⋊C16×D83C2semidirect
4. #24905405C2semidirect
5. #24905388C2semidirect
6. #24905388C2semidirect
7. C83×#1323C2semidirect
181. #1323D83semidirect
184. C83#11470semidirect

Maximal subgroups

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

subgroupindex
#249161562
#249090722
C22⋊C16×D832
#249054052
#249053882
C83×#13232
#1147083