Finite Group Databasebeta

← #35314984

Subgroups — #35314984

Order 21248.

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

normal subgroups only

This group has 446 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#24912368C211yesmaximal, characteristic
2C4×#14581711C211yesmaximal, characteristic, complemented
3#24905470C212yesmaximal, complemented
4#24905467C212yesmaximal
5C83×#1467C211yesmaximal, characteristic, complemented
6C2×#14581768—21no—
7C2×C4×D332C411yescharacteristic
8C2×#14581711—21no—
9C4×#13050574—21no—
10#19127631C411yescharacteristic
11#19127631C2211yescharacteristic
12#19127618C411yescharacteristic
13#19127618C2211yescharacteristic
14#19127592—21no—
15#19127592—21no—
16#19127567C411yescharacteristic
17#19127558—21no—
18#19127470C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#432×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×D332C2×C411yescharacteristic
24C22×C4×D83—21no—
25C2×#13050574—22no—
26#14581768—44no—
27C4×D332—41no—
28C4×D332—41no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83C2×C411yescharacteristic
31C2×Dic83⋊C4—21no—
32C2×Dic83⋊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—
38#14581711—24no—
39#14581703—41no—
40C4×C83⋊C8—41no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95C2×C411yescharacteristic
45C83⋊#95C2×C411yescharacteristic
46C22×Q8×C83D411yescharacteristic
47C4×Q8×C83—41no—
48C4×Q8×C83—41no—
49C2×C4⋊C4×C83C2×C411yescharacteristic
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×D332—41no—
60C2×D332—42no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C2×C4×D83—42no—
65C2×C4×D83—42no—
66C2×C4×D83—42no—
67#13050574—24no—
68#13050574—42no—
69C2×C83⋊C8—42no—
70C4⋊Dic83—42no—
71C4⋊Dic83—41no—
72Dic83⋊C4—42no—
73Dic83⋊C4—42no—
74(C2×D83)⋊C4—42no—
75(C2×D83)⋊C4—42no—
76(C2×D83)⋊C4—42no—
77(C2×D83)⋊C4—41no—
78(C2×D83)⋊C4—41no—
79(C2×D83)⋊C4—42no—
80C2×Q8×C83—22no—
81C2×Q8×C83—22no—
82C2×Q8×C83—41no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C2×C8×C83—42no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—21no—
100C23×C83C2×C22⋊C411yescharacteristic
101C22×D83—22no—
102C22×D83—41no—
103C22×D83—41no—
104C22×D83—41no—
105C22×D83—41no—
106C22×D83—41no—
107C22×D83—41no—
108Q8×C83—44no—
109C2×C4×C83—21no—
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—41no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83#9911yescharacteristic
124C2×C4×C83#9911yescharacteristic
125C2×C4×C83#10412yes—
126C2×C4×C83AGL(1,ℤ/8)12yes—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136D332—42no—
137D332—42no—
138C4×D83—84no—
139C8×C83—82no—
140C83⋊C8—44no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83C2×#9911yescharacteristic
144C22×C83#41811yescharacteristic
145C22×C83C2×#10411yescharacteristic
146C2×D83—42no—
147C2×D83—42no—
148C2×D83—42no—
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—81no—
162Dic83—81no—
163Dic83—81no—
164#11778—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83#150711yescharacteristic
168C2×C83#212011yescharacteristic
169C2×C83#213311yescharacteristic
170D83—82no—
171#2536—831no—
172C4×#101—831no—
173#1467D8311yescharacteristic, complemented
174#1346—832no—
175#1345—832no—
176C83#1177811yescharacteristic, complemented, Sylow
177C2×C22⋊Q8—1661no—
178C2×C4×Q8C2×D8311yescharacteristic
179C2×C4×D4—831no—
180#432—21no—
181C2×#101—1661no—
182C4×M16—1661no—
183#398—831no—
184#398—831no—
185#394—831no—
186#394—831no—
187#388—1661no—
188#388—1661no—
189#382—831no—
190#380—1661no—
191#343—832no—
192#324—21no—
193#324C2×D8312yes—
194C22×Q8D33211yescharacteristic
195C22×D4—831no—
196C23×C4—1661no—
197C2×M16—1662no—
198C22⋊Q8—3324no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×D4—3321no—
202C4×D4—3321no—
203C2×C4⋊C4—831no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4C4×D8311yescharacteristic
208C2×C4⋊C4C22⋊D8311yescharacteristic
209C2×C4⋊C4C22×D8311yescharacteristic
210C2×C22⋊C4—1661no—
211C2×C22⋊C4—1661no—
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1661no—
214C2×C22⋊C4—1661no—
215C2×C42C4×D8311yescharacteristic
216C2×C42D33211yescharacteristic
217C2×C42C22⋊D8311yescharacteristic
218C4⋊C8—41no—
219#101—1664no—
220C22⋊C8—22no—
221#97—41no—
222#97—3321no—
223C4×C8—3321no—
224#95—831no—
225#95—831no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229C24—1661no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×D4—3321no—
234C2×D4—3322no—
235C22×C4—3322no—
236C22×C4—3322no—
237C22×C4—3322no—
238C22×C4—1661no—
239C22×C4—1661no—
240C22×C4—1661no—
241C22×C4(C2×D83)⋊C411yescharacteristic
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4C2×C4×D8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4C2×D33211yescharacteristic
248M16—3322no—
249M16—1664no—
250C2×C8—42no—
251C2×C8—3322no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—3321no—
256C4⋊C4—3322no—
257C4⋊C4—3322no—
258C4⋊C4—3322no—
259C22⋊C4—3322no—
260C22⋊C4—3322no—
261C22⋊C4—3322no—
262C22⋊C4—3322no—
263C22⋊C4—3321no—
264C22⋊C4—3321no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—3321no—
272C23—3321no—
273C23—3321no—
274C23—3321no—
275C23—3321no—
276C23—3321no—
277C23—1662no—
278C23C2×(C2×D83)⋊C411yescharacteristic
279Q8—44no—
280D4—3322no—
281D4—3322no—
282C2×C4—21no—
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—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—3321no—
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—6644no—
306C2×C4#1458170611yescharacteristic
307C2×C4#1458170611yescharacteristic
308C2×C4#1458171812yes—
309C2×C4#1458182012yes—
310C8—82no—
311C8—3324no—
312C22—21no—
313C22—21no—
314C22—3322no—
315C22—3322no—
316C22—3322no—
317C22—6641no—
318C22—6641no—
319C22—6641no—
320C22—6641no—
321C22C2×#1458170611yescharacteristic
322C22#1912770711yescharacteristic
323C22C2×#1458171811yescharacteristic
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—6641no—
333C4—6641no—
334C4—6641no—
335C2—21no—
336C2—21no—
337C2—6642no—
338C2#2490598011yescharacteristic
339C2#2490910211yescharacteristic
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#11778
LCSG ▷ C22×C4×C83 ▷ C23×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. C4×#14581711C2semidirect
3. #24905470C2semidirect
5. C83×#1467C2semidirect
173. #1467D83semidirect
176. C83#11778semidirect

Maximal subgroups

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

subgroupindex
#249123682
C4×#145817112
#249054702
#249054672
C83×#14672
#1177883