Finite Group Databasebeta

← #35294548

Subgroups — #35294548

Order 21248.

874 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C2×#19127663C212yesmaximal
2#407×D83C211yesmaximal, characteristic
3#24905351C212yesmaximal, complemented
4#24905350C211yesmaximal, characteristic, complemented
5C83×#1298C211yesmaximal, characteristic, complemented
6C23×C4×D83C2211yescharacteristic
7C22×#13050573—22no—
8C2×M16×D83C412yes—
9#19127665C2211yescharacteristic
10#19127664C412yes—
11#19127663—22no—
12#19127663—22no—
13#19127663—22no—
14#19127663C414yes—
15#19127663C414yes—
16C2×C22⋊(C83⋊C8)C2212yes—
17C2×(C2×D83)⋊C8—22no—
18C22⋊C8×D83—22no—
19#407×C83C2211yescharacteristic
20C2×C22⋊C8×C83C2212yes—
21C24×D83Q811yescharacteristic
22C23×Dic83Q811yescharacteristic
23C22×C4×D83—44no—
24C22×C4×D83C2×C411yescharacteristic
25C22×C4×D83C2×C412yes—
26C22×C4×D83C2×C412yes—
27C22×C4×D83C2×C411yescharacteristic
28C22×C4×D83D412yes—
29C22×C4×D83D412yes—
30C22×C4×D83D411yescharacteristic
31C22×C4×D83D411yescharacteristic
32C2×#13050574C2×C412yes—
33C2×#13050573—22no—
34C2×#13050573—24no—
35C2×#13050573—24no—
36C2×#13050573—48no—
37M16×D83—44no—
38C22×C83⋊C8—22no—
39C2×C8×D83—22no—
40C22⋊(C83⋊C8)—22no—
41C22⋊(C83⋊C8)—22no—
42C22⋊(C83⋊C8)C2×C414yes—
43(C2×D83)⋊C8—24no—
44(C2×D83)⋊C8—24no—
45(C2×D83)⋊C8—24no—
46C23×C4×C83C2311yescharacteristic
47C2×M16×C83C2×C412yes—
48C22×C8×C83—22no—
49C22⋊C8×C83—22no—
50C22⋊C8×C83—22no—
51C22⋊C8×C83C2×C414yes—
52C23×D83—42no—
53C23×D83—44no—
54C23×D83C4⋊C412yes—
55C23×D83C4⋊C412yes—
56C23×D83C4⋊C411yescharacteristic
57C23×D83C4⋊C411yescharacteristic
58C22×Dic83—42no—
59C22×Dic83C4⋊C411yescharacteristic
60C22×Dic83C4⋊C412yes—
61C22×Dic83C4⋊C411yescharacteristic
62C22×Dic83C4⋊C412yes—
63C2×C4×D83—24no—
64C2×C4×D83—28no—
65C2×C4×D83—44no—
66C2×C4×D83—42no—
67C2×C4×D83—42no—
68C2×C4×D83—44no—
69C2×C4×D83—44no—
70C2×C4×D83—42no—
71C2×C4×D83C4211yescharacteristic
72C2×C4×D83C4211yescharacteristic
73C2×C4×D83C4212yes—
74C2×C4×D83C22⋊C412yes—
75C2×C4×D83C22⋊C412yes—
76C2×C4×D83C22⋊C411yescharacteristic
77C2×C4×D83C22⋊C412yes—
78C2×C4×D83C22⋊C412yes—
79C2×C4×D83C22⋊C412yes—
80C2×C4×D83C22⋊C411yescharacteristic
81#13050574—42no—
82#13050573—44no—
83#13050573—48no—
84#13050573—48no—
85C2×C83⋊C8—22no—
86C2×C83⋊C8—24no—
87C2×C83⋊C8—44no—
88C8×D83—44no—
89C24×C83C2×Q811yescharacteristic
90C22×C4×C83—42no—
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C22×C412yes—
93C22×C4×C83C2×D412yes—
94C22×C4×C83C2×D411yescharacteristic
95M16×C83—42no—
96C2×C8×C83—22no—
97C2×C8×C83—24no—
98C2×C8×C83—44no—
99C23×C83—24no—
100C23×C83—41no—
101C23×C83C2×C4⋊C411yescharacteristic
102C23×C83C2×C4⋊C412yes—
103C22×D83—24no—
104C22×D83—24no—
105C22×D83—24no—
106C22×D83—22no—
107C22×D83—42no—
108C22×D83—41no—
109C22×D83—42no—
110C22×D83—42no—
111C22×D83—42no—
112C22×D83—42no—
113C22×D83—416no—
114C22×D83#9511yescharacteristic
115C22×D83#9512yes—
116C22×D83#9511yescharacteristic
117C2×C4×C83—24no—
118C2×C4×C83—22no—
119C2×C4×C83—42no—
120C2×C4×C83—41no—
121C2×C4×C83C2×C4211yescharacteristic
122C2×C4×C83C2×C22⋊C412yes—
123C2×C4×C83C2×C22⋊C411yescharacteristic
124C2×Dic83—24no—
125C2×Dic83—22no—
126C2×Dic83—42no—
127C2×Dic83—41no—
128C2×Dic83#9511yescharacteristic
129C2×Dic83#9512yes—
130C2×Dic83#9511yescharacteristic
131C4×D83—24no—
132C4×D83—22no—
133C4×D83—22no—
134C4×D83—24no—
135C4×D83—24no—
136C4×D83—22no—
137C4×D83—42no—
138C4×D83—42no—
139C4×D83—42no—
140C4×D83#9914yes—
141C8×C83—44no—
142C8×C83—42no—
143C83⋊C8—44no—
144C83⋊C8—42no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—22no—
148C22×C83—21no—
149C22×C83—44no—
150C22×C83C2×#9511yescharacteristic
151C22×C83#43014yes—
152C2×D83—22no—
153C2×D83—22no—
154C2×D83—22no—
155C2×D83—22no—
156C2×D83—21no—
157C2×D83—22no—
158C2×D83—48no—
159C2×D83—44no—
160C2×D83—48no—
161C2×D83—44no—
162C2×D83#34212yes—
163C4×C83—22no—
164C4×C83—21no—
165C4×C83C2×#9912yes—
166Dic83—22no—
167Dic83—21no—
168Dic83#34212yes—
169#8055—831nomaximal, Sylow
170C2×C83—24no—
171C2×C83—41no—
172C2×C83#173812yes—
173C2×C83C2×#34211yescharacteristic
174D83—22no—
175D83—42no—
176D83—41no—
177C2×#407—831no—
178C2×#407—832no—
179#1298—831no—
180#1298—832no—
181#1298D8311yescharacteristic, complemented
182C83#805511yescharacteristic, complemented, Sylow
183C24×C4—831no—
184C22×M16—832no—
185C22×M16—1662no—
186#407—834no—
187#407—834no—
188#407—831no—
189#407—832no—
190#407—1662no—
191#407—1662no—
192#407—1662no—
193#407C2×D8311yescharacteristic
194C2×C22⋊C8—832no—
195C2×C22⋊C8—1662no—
196C2×C22⋊C8—1662no—
197C2×C22⋊C8C2×D8312yes—
198C25—831no—
199C23×C4—3324no—
200C23×C4—831no—
201C23×C4—831no—
202C23×C4—832no—
203C23×C4—832no—
204C23×C4—832no—
205C23×C4—831no—
206C23×C4—831no—
207C23×C4—831no—
208C23×C4—832no—
209C23×C4C22×D8311yescharacteristic
210C2×M16—3328no—
211C2×M16—3324no—
212C2×M16—832no—
213C2×M16—1662no—
214C2×M16—1664no—
215C2×M16—1664no—
216C2×M16C4×D8312yes—
217C22×C8—22no—
218C22×C8—1662no—
219C22×C8—1662no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—834no—
223C22⋊C8—1664no—
224C22⋊C8—1664no—
225C22⋊C8—1664no—
226C22⋊C8—1662no—
227C22⋊C8—1662no—
228C22⋊C8C4×D8314yes—
229C24—3324no—
230C24—3322no—
231C24—831no—
232C24—832no—
233C24—831no—
234C24—832no—
235C24Q8×D8311yescharacteristic
236C22×C4—42no—
237C22×C4—3324no—
238C22×C4—3322no—
239C22×C4—3324no—
240C22×C4—3322no—
241C22×C4—3324no—
242C22×C4—3322no—
243C22×C4—3322no—
244C22×C4—831no—
245C22×C4—832no—
246C22×C4—832no—
247C22×C4—832no—
248C22×C4—832no—
249C22×C4—831no—
250C22×C4—831no—
251C22×C4—832no—
252C22×C4—831no—
253C22×C4—831no—
254C22×C4—832no—
255C22×C4—832no—
256C22×C4—832no—
257C22×C4—831no—
258C22×C4—1668no—
259C22×C4—1664no—
260C22×C4C2×C4×D8311yescharacteristic
261C22×C4C2×C4×D8312yes—
262C22×C4D4×D8312yes—
263C22×C4D4×D8311yescharacteristic
264M16—42no—
265M16—3328no—
266M16—3328no—
267M16—3324no—
268M16—3322no—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—44no—
272C2×C8—3324no—
273C2×C8—3324no—
274C2×C8—1664no—
275C2×C8—1662no—
276C23—24no—
277C23—41no—
278C23—3322no—
279C23—3321no—
280C23—3322no—
281C23—3322no—
282C23—3322no—
283C23—3322no—
284C23—33216no—
285C23—831no—
286C23—832no—
287C23—831no—
288C23—1664no—
289C23—1662no—
290C23—1664no—
291C23—1664no—
292C23C4⋊C4×D8311yescharacteristic
293C23C4⋊C4×D8312yes—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—3322no—
299C2×C4—3322no—
300C2×C4—3322no—
301C2×C4—3322no—
302C2×C4—3321no—
303C2×C4—831no—
304C2×C4—834no—
305C2×C4—832no—
306C2×C4—831no—
307C2×C4—1664no—
308C2×C4—1662no—
309C2×C4—1664no—
310C2×C4—1662no—
311C2×C4—1664no—
312C2×C4—1662no—
313C2×C4—1662no—
314C2×C4—1664no—
315C2×C4C42×D8311yescharacteristic
316C2×C4C22⋊C4×D8312yes—
317C2×C4C22⋊C4×D8311yescharacteristic
318C8—44no—
319C8—42no—
320C8—3324no—
321C8—3322no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—44no—
327C22—3328no—
328C22—3328no—
329C22—3324no—
330C22—3324no—
331C22—832no—
332C22—1662no—
333C22—1662no—
334C22—1662no—
335C22—1662no—
336C22—1661no—
337C22—1662no—
338C22#95×D8311yescharacteristic
339C22#1912774814yes—
340C4—22no—
341C4—21no—
342C4—832no—
343C4—1662no—
344C4—1661no—
345C4#99×D8312yes—
346C2—24no—
347C2—41no—
348C2—3321no—
349C2—3322no—
350C2—1662no—
351C2#2490718812yes—
352C2#342×D8311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC2×C4×C83C2×C42
FrattiniC23×C4C22×D83
FittingC83×#1298C2
socleC22×C83C2×#95
prime radicalC24×D83Q8
cosocleC23×C4×C83C23
hypercenter#1298D83
hypocenterC83#8055
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
3. #24905351C2semidirect
4. #24905350C2semidirect
5. C83×#1298C2semidirect
181. #1298D83semidirect
182. C83#8055semidirect

Maximal subgroups

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

subgroupindex
C2×#191276632
#407×D832
#249053512
#249053502
C83×#12982
#805583