Finite Group Databasebeta

← #35299409

Subgroups — #35299409

Order 21248.

836 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 135 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#24910061C211yesmaximal, characteristic
2#24907398C212yesmaximal, complemented
3C2×#19127425C211yesmaximal, characteristic
4#24905342C212yesmaximal, complemented
5C2×#393×C83C211yesmaximal, characteristic
6C2×#14581716—24no—
7C2×C22⋊(C83⋊C8)C2212yes—
8#19127594—24no—
9#19127561C2212yes—
10#19127557—21no—
11C4×C22⋊Dic83—21no—
12#19127425—22no—
13C23×Q8×C83C2211yescharacteristic
14C22×C22⋊C4×C83—21no—
15C22×C22⋊C4×C83C2211yescharacteristic
16#393×C83—22no—
17#393×C83C414yescomplemented
18C2×#95×C83—21no—
19C2×#95×C83C2211yescharacteristic
20C22×C83⋊C8—22no—
21C2×C4⋊Dic83D414yes—
22C2×C22⋊Dic83—22no—
23C2×C22⋊Dic83—22no—
24C2×C22⋊Dic83—22no—
25C2×C4×Dic83—22no—
26#14581716—48no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—22no—
29C83⋊#95—22no—
30C83⋊#95—22no—
31C25×C83D411yescharacteristic
32C22×Q8×C83—21no—
33C22×Q8×C83—22no—
34C22×Q8×C83—22no—
35C22×Q8×C83—44no—
36C22×Q8×C83C2×C412yes—
37C23×C4×C83—21no—
38C23×C4×C83—21no—
39C23×C4×C83D411yescharacteristic
40C23×C4×C83C2311yescharacteristic
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—24no—
44C2×C22⋊C4×C83—44no—
45C2×C22⋊C4×C83C2×C412yes—
46C2×C22⋊C4×C83D412yes—
47#95×C83—24no—
48#95×C83—21no—
49#95×C83C2×C412yes—
50C22×Dic83—24no—
51C2×C83⋊C8—44no—
52C2×C83⋊C8—42no—
53C4⋊Dic83—44no—
54C22⋊Dic83—42no—
55C22⋊Dic83—42no—
56C22⋊Dic83—82no—
57C22⋊Dic83—82no—
58C4×Dic83—82no—
59C24×C83—21no—
60C24×C83—22no—
61C24×C83—21no—
62C24×C83—22no—
63C24×C83—42no—
64C24×C83—42no—
65C24×C83C22⋊C412yes—
66C24×C83C2×D411yescharacteristic
67C2×Q8×C83—28no—
68C2×Q8×C83—24no—
69C2×Q8×C83—44no—
70C2×Q8×C83—44no—
71C2×Q8×C83—44no—
72C2×Q8×C83—48no—
73C2×Q8×C83—42no—
74C22×C4×C83—24no—
75C22×C4×C83—22no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—22no—
79C22×C4×C83—42no—
80C22×C4×C83—42no—
81C22×C4×C83—41no—
82C22×C4×C83—41no—
83C22×C4×C83—42no—
84C22×C4×C83—42no—
85C22×C4×C83—42no—
86C22×C4×C83—42no—
87C22×C4×C83C22⋊C412yes—
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D412yes—
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83#4112yes—
92C22⋊C4×C83—24no—
93C22⋊C4×C83—24no—
94C22⋊C4×C83—48no—
95C23×C83—22no—
96C23×C83—22no—
97C23×C83—22no—
98C23×C83—22no—
99C23×C83—22no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—48no—
103C23×C83—42no—
104C23×C83—42no—
105C23×C83—42no—
106C23×C83—42no—
107C23×C83—81no—
108C23×C83—81no—
109C23×C83—84no—
110C23×C83C2×C22⋊C411yescharacteristic
111C23×C83#12011yescharacteristic
112C23×C83#12311yescharacteristic
113C23×C83#13512yes—
114C23×C83#13712yes—
115Q8×C83—28no—
116Q8×C83—48no—
117Q8×C83—44no—
118C2×C4×C83—24no—
119C2×C4×C83—24no—
120C2×C4×C83—22no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—44no—
126C2×C4×C83—42no—
127C2×C4×C83—42no—
128C2×C4×C83—44no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83—81no—
134C2×C4×C83—84no—
135C2×C4×C83C4×D412yes—
136C2×C4×C83#12112yes—
137C2×Dic83—44no—
138C2×Dic83—44no—
139C2×Dic83—44no—
140C83⋊C8—82no—
141C22×C83—24no—
142C22×C83—24no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—44no—
148C22×C83—42no—
149C22×C83—42no—
150C22×C83—42no—
151C22×C83—82no—
152C22×C83—84no—
153C22×C83—84no—
154C22×C83#38611yescharacteristic
155C22×C83#41611yescharacteristic
156C22×C83#41911yescharacteristic
157C22×C83#45212yes—
158C22×C83#45512yes—
159C22×C83B(4,𝔽2)12yes—
160C22×C83#48512yes—
161C4×C83—22no—
162C4×C83—22no—
163C4×C83—42no—
164C4×C83—44no—
165C4×C83—82no—
166Dic83—84no—
167#9016—831nomaximal, Sylow
168C2×C83—21no—
169C2×C83—21no—
170C2×C83—22no—
171C2×C83—82no—
172C2×C83#160614yes—
173C2×C83#187311yescharacteristic
174C2×C83#188511yescharacteristic
175C2×C83#188911yescharacteristic
176C2×#393D8311yescharacteristic
177#2312—831no—
178#1787—832no—
179C2×#323—831no—
180#1294—832no—
181C83#901611yescharacteristic, complemented, Sylow
182C23×Q8C2×D8311yescharacteristic
183C22×C22⋊C4—21no—
184C22×C22⋊C4C2×D8311yescharacteristic
185C2×#103—1664no—
186C2×C22⋊C8—832no—
187#393—22no—
188#393Dic8314yescomplemented
189#388—1664no—
190#380—832no—
191#379—1661no—
192C4×C22⋊C4—1661no—
193C2×#95—21no—
194C2×#95C2×D8311yescharacteristic
195#323—1662no—
196C25C22⋊D8311yescharacteristic
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—44no—
201C22×Q8C2×Dic8312yes—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C22⋊D8311yescharacteristic
205C23×C4C22×D8311yescharacteristic
206C22×C8—1662no—
207C2×C4⋊C4—834no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—1662no—
213C2×C22⋊C4—1662no—
214C2×C22⋊C4—1662no—
215C2×C22⋊C4C2×Dic8312yes—
216C2×C22⋊C4C22⋊D8312yes—
217C2×C42—1662no—
218#103—3328no—
219C22⋊C8—1662no—
220C22⋊C8—1662no—
221#95—24no—
222#95—21no—
223#95—1662no—
224#95—1662no—
225#95C2×Dic8312yes—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24C22⋊Dic8312yes—
233C24C2×C22⋊D8311yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—1664no—
255C22×C4C22⋊Dic8312yes—
256C22×C4C22×Dic8311yescharacteristic
257C22×C4D4×D8312yes—
258C22×C4C2×C22⋊D8311yescharacteristic
259C22×C4#1305058912yes—
260C2×C8—3324no—
261C2×C8—3322no—
262C4⋊C4—3324no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—3322no—
267C22⋊C4—3322no—
268C22⋊C4—6642no—
269C22⋊C4—6642no—
270C42—6642no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×C22⋊Dic8311yescharacteristic
287C23#1458176011yescharacteristic
288C23#1458177311yescharacteristic
289C23#1458181212yes—
290C23#1458182312yes—
291Q8—28no—
292Q8—48no—
293Q8—44no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—84no—
311C2×C4—3324no—
312C2×C4—3324no—
313C2×C4—3324no—
314C2×C4D4×Dic8312yes—
315C2×C4#1458176412yes—
316C8—6642no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1912758111yescharacteristic
331C22#1912769711yescharacteristic
332C22#1912771011yescharacteristic
333C22#1912786112yes—
334C22#1912787812yes—
335C22#1912788712yes—
336C22#1912803912yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—44no—
341C4—82no—
342C4—6644no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#2490644114yes—
348C2#2490777611yescharacteristic
349C2#2490785211yescharacteristic
350C2#2490787211yescharacteristic
subgroupG/H
centerC23#14581760
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC2×#393×C83C2
socleC23×C83#120
prime radicalC25×C83D4
cosocleC23×C4×C83C23
hypercenterC2×#393D83
hypocenterC83#9016
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #24907398C2semidirect
4. #24905342C2semidirect
17. #393×C83C4semidirect
181. C83#9016semidirect
188. #393Dic83semidirect

Maximal subgroups

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

subgroupindex
#249100612
#249073982
C2×#191274252
#249053422
C2×#393×C832
#901683