Finite Group Databasebeta

← #35315180

Subgroups — #35315180

Order 21248.

446 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 91 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#24907451C211yesmaximal, characteristic
2#24905801C211yesmaximal, characteristic
3#24905484C212yesmaximal, complemented
4#24905463C212yesmaximal, complemented
5C83×#2398C211yesmaximal, characteristic
6C2×#14581707—21no—
7#19127665—21no—
8#19127561—21no—
9#19127468C2212yes—
10#19127427—21no—
11#19127425C2212yes—
12C2×C4×Q8×C83C411yescharacteristic, complemented
13C2×C4×D4×C83C2211yescharacteristic
14#400×C83—21no—
15#397×C83C411yescharacteristic, complemented
16#397×C83C2211yescharacteristic
17#394×C83—21no—
18#393×C83C411yescharacteristic, complemented
19#393×C83C2211yescharacteristic
20C4×C22⋊C4×C83—21no—
21C4×C22⋊C4×C83—21no—
22#376×C83C411yescharacteristic, complemented
23C2×#13050574—41no—
24C2×C4⋊Dic83—41no—
25C2×C22⋊Dic83—22no—
26#14581707—42no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—41no—
29C83⋊#95—41no—
30C22×Q8×C83C2×C411yescharacteristic
31C22×D4×C83D411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C4×Q8×C83—41no—
35C4×Q8×C83—41no—
36C4×D4×C83—41no—
37C4×D4×C83—41no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83C2311yescharacteristic
47C2×C42×C83—21no—
48C2×C42×C83—21no—
49C2×C42×C83C2×C411yescharacteristic
50#95×C83—21no—
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57C22×Dic83—42no—
58#13050574—82no—
59C2×C83⋊C8—42no—
60C4⋊Dic83—82no—
61C22⋊Dic83—42no—
62C22⋊Dic83—42no—
63C24×C83C22⋊C411yescharacteristic
64C24×C83C2×D411yescharacteristic
65C2×Q8×C83—42no—
66C2×Q8×C83—41no—
67C2×D4×C83—21no—
68C2×D4×C83—21no—
69C2×D4×C83—41no—
70C2×D4×C83—41no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—42no—
80C22×C4×C83—42no—
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—42no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—42no—
99C42×C83—41no—
100C42×C83—41no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83C2×C22⋊C411yescharacteristic
111C23×C83AGL(1,ℤ/8)11yescharacteristic
112C23×C83#13711yescharacteristic
113Q8×C83—42no—
114Q8×C83—42no—
115D4×C83—42no—
116D4×C83—42no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—81no—
142C2×C4×C83—81no—
143C2×C4×C83#9911yescharacteristic
144C2×C4×C83#9911yescharacteristic
145C2×Dic83—82no—
146C2×Dic83—82no—
147C2×Dic83—82no—
148C83⋊C8—82no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—42no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—81no—
162C22×C83—81no—
163C22×C83C2×#9911yescharacteristic
164C22×C83#41711yescharacteristic
165C22×C83#42111yescharacteristic
166C4×C83—21no—
167C4×C83—42no—
168C4×C83—41no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—81no—
173C4×C83—81no—
174C4×C83—81no—
175C4×C83#35412yes—
176Dic83—162no—
177#11823—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—41no—
181C2×C83—41no—
182C2×C83—81no—
183C2×C83#151311yescharacteristic
184C2×C83#212011yescharacteristic
185C2×C83C2×#35411yescharacteristic
186#2398D8311yescharacteristic
187#1799—831no—
188#1465—831no—
189#1351—832no—
190#1344—832no—
191C83#1182311yescharacteristic, complemented, Sylow
192C2×C4×Q8Dic8311yescharacteristic, complemented
193C2×C4×D4C2×D8311yescharacteristic
194C2×#99—1661no—
195#407—1661no—
196#400—21no—
197#397Dic8311yescharacteristic, complemented
198#397C2×D8311yescharacteristic
199#394—21no—
200#393Dic8311yescharacteristic, complemented
201#393C2×D8311yescharacteristic
202#380—1661no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205#376Dic8311yescharacteristic, complemented
206#342—832no—
207#324—1661no—
208#323—832no—
209C22×Q8C2×Dic8311yescharacteristic
210C22×D4C22⋊D8311yescharacteristic
211C23×C4C22⋊D8311yescharacteristic
212C23×C4C22⋊D8311yescharacteristic
213C2×M16—3321no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—3321no—
220C2×C4⋊C4C2×Dic8311yescharacteristic
221C2×C4⋊C4C22⋊D8311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1662no—
227C2×C22⋊C4C2×Dic8311yescharacteristic
228C2×C22⋊C4C22×D8311yescharacteristic
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C2×Dic8311yescharacteristic
232#99—3322no—
233C22⋊C8—3321no—
234C22⋊C8—1662no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—3321no—
241#95C2×Dic8311yescharacteristic
242#95C2×Dic8311yescharacteristic
243C24C22⋊Dic8311yescharacteristic
244C24C2×C22⋊D8311yescharacteristic
245C2×Q8—42no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—3322no—
262C22×C4C22⋊Dic8311yescharacteristic
263C22×C4C22⋊Dic8311yescharacteristic
264C22×C4C22⋊Dic8311yescharacteristic
265C22×C4C22×Dic8311yescharacteristic
266C22×C4C2×C22⋊D8311yescharacteristic
267M16—6642no—
268C2×C8—3322no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—6642no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—3322no—
281C22⋊C4—3322no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23C2×C22⋊Dic8311yescharacteristic
297C23#1458181711yescharacteristic
298C23#1458182211yescharacteristic
299Q8—42no—
300Q8—42no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—6642no—
330C2×C4—6642no—
331C2×C4—6642no—
332C2×C4#1458170711yescharacteristic
333C2×C4#1458170711yescharacteristic
334C8—6642no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#1458170711yescharacteristic
350C22#1912770211yescharacteristic
351C22#1912771811yescharacteristic
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—13282no—
362C4#1912749612yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#2490600611yescharacteristic
369C2#2490910411yescharacteristic
370C2C2×#1912749611yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2398C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#2398D83
hypocenterC83#11823
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #24905484C2semidirect
4. #24905463C2semidirect
12. C2×C4×Q8×C83C4semidirect
15. #397×C83C4semidirect
18. #393×C83C4semidirect
22. #376×C83C4semidirect
191. C83#11823semidirect
192. C2×C4×Q8Dic83semidirect
197. #397Dic83semidirect
200. #393Dic83semidirect
205. #376Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074512
#249058012
#249054842
#249054632
C83×#23982
#1182383