Finite Group Databasebeta

← #29592114

Subgroups — #29592114

Order 15616.

476 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456594C211yesmaximal, characteristic, complemented
2#21447748C212yesmaximal, complemented
3#21443830C212yesmaximal, complemented
4#21443743C211yesmaximal, characteristic, complemented
5C61×#1999C211yesmaximal, characteristic
6C2×#14519756—21no—
7C2×#14519721—22no—
8#16840098—22no—
9#16840003—21no—
10#16840003C2212yes—
11#16839920—22no—
12#16839563—22no—
13C22⋊#11758185C2212yes—
14C22×D8×C61C2211yescharacteristic
15C2×#121×C61—21no—
16C2×#102×C61—21no—
17C2×C22⋊C8×C61C2211yescharacteristic
18C61×#340C2211yescharacteristic
19C22×Dic122—21no—
20C2×Dic244—21no—
21C2×Dic244D412yescomplemented
22#14519756—22no—
23#14519756—22no—
24#14519756—41no—
25#14519756—41no—
26C2×C4⋊Dic61—21no—
27C2×#11758196—41no—
28C2×C22⋊Dic61—41no—
29#14519721—44no—
30C2×#11758185—22no—
31#14519711—21no—
32#14519711D412yescomplemented
33#14519702—22no—
34#14519702—22no—
35C22×D4×C61—21no—
36C22×D4×C61—41no—
37C23×C4×C61D411yescharacteristic
38C2×D8×C61—22no—
39C2×D8×C61—42no—
40C22×C8×C61D411yescharacteristic
41C22×C8×C61C2311yescharacteristic
42#121×C61—42no—
43#121×C61—42no—
44C2×C4⋊C4×C61—21no—
45C2×C22⋊C4×C61—41no—
46#102×C61—42no—
47C22⋊C8×C61—22no—
48C24×C61—41no—
49C24×C61D811yescharacteristic
50C2×D4×C61—22no—
51C2×D4×C61—41no—
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61—44no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61—82no—
58C2×D4×C61—82no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—41no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61D811yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×Dic61—41no—
67C22×Dic61—41no—
68C2×Dic122—22no—
69C2×Dic122—41no—
70C2×Dic122—81no—
71C2×Dic122—81no—
72D8×C61—24no—
73D8×C61—42no—
74C2×C8×C61—21no—
75C2×C8×C61—41no—
76C2×C8×C61—41no—
77C2×C8×C61C2×D412yes—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—44no—
81C22⋊Dic61—42no—
82C22⋊Dic61—42no—
83C4⋊Dic61—22no—
84C4⋊Dic61—41no—
85#11758196—42no—
86#11758196—42no—
87Dic244—42no—
88Dic244—81no—
89#11758185—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—42no—
95C23×C61—81no—
96C23×C61—81no—
97C23×C61—81no—
98C23×C61—81no—
99C23×C61—81no—
100C23×C61—81no—
101C23×C61SD3212yescomplemented
102C23×C61C2×D811yescharacteristic
103D4×C61—42no—
104D4×C61—42no—
105D4×C61—81no—
106D4×C61—88no—
107D4×C61—81no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—42no—
115C2×C4×C61—81no—
116C2×C4×C61—81no—
117C2×C4×C61—81no—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—82no—
123C2×Dic61—82no—
124Dic122—42no—
125Dic122—42no—
126Dic122—82no—
127C8×C61—22no—
128C8×C61—41no—
129#12103—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—42no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61—82no—
138C22×C61—82no—
139C22×C61—82no—
140C22×C61—161no—
141C22×C61—161no—
142C22×C61—161no—
143C22×C61—161no—
144C22×C61#44711yescharacteristic
145C22×C61C2×SD3211yescharacteristic
146C22×C61#51011yescharacteristic
147C4×C61—21no—
148C4×C61—41no—
149C4×C61—82no—
150C4×C61#45311yescharacteristic
151C4×C61B(4,𝔽2)11yescharacteristic
152Dic61—82no—
153Dic61—82no—
154#3189—611no—
155#2187—612no—
156#1999D6111yescharacteristic
157#1348—612no—
158#1314—611no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—42no—
162C2×C61—162no—
163C2×C61#159511yescharacteristic
164C2×C61#218511yescharacteristic
165C2×C61#219011yescharacteristic
166C22×D8C2×D6111yescharacteristic
167C2×C22⋊Q8—1221no—
168C2×#121—21no—
169C2×SD32—1222no—
170#484—1222no—
171#467—612no—
172#467—1221no—
173#451—1222no—
174C2×#102—21no—
175C2×C22⋊C8C2×D6111yescharacteristic
176#357—1222no—
177C22⋊C16—612no—
178#340C2×D6111yescharacteristic
179C61#1210311yescharacteristic, complemented, Sylow
180C22×Q8—1221no—
181C22×D4—21no—
182C22×D4—41no—
183C23×C4C22⋊D6111yescharacteristic
184C2×Q16—612no—
185C2×Q16—1221no—
186C2×D8—22no—
187C2×D8—42no—
188C22×C8C22⋊D6111yescharacteristic
189C22×C8C22×D6111yescharacteristic
190C22⋊Q8—1222no—
191C22⋊Q8—1222no—
192C22⋊Q8—2441no—
193C22⋊Q8—2441no—
194#121—42no—
195#121—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—2441no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—2441no—
201SD32—2444no—
202C2×C16—1222no—
203#107—612no—
204#107—1221no—
205#103—1222no—
206#103—1222no—
207#102—42no—
208C22⋊C8—22no—
209C24—41no—
210C24#1175819911yescharacteristic
211C2×Q8—1222no—
212C2×Q8—4881no—
213C2×Q8—4881no—
214C2×Q8—2441no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—2441no—
230C22×C4—2441no—
231C22×C4#1175819911yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233Q16—4881no—
234Q16—2442no—
235D8—24no—
236D8—42no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8D4×D6112yes—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—1222no—
244C4⋊C4—2441no—
245C4⋊C4—2442no—
246C4⋊C4—2442no—
247C22⋊C4—44no—
248C22⋊C4—2442no—
249C22⋊C4—2442no—
250C16—2442no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23#1451972112yescomplemented
263C23C2×#1175819911yescharacteristic
264Q8—4882no—
265Q8—2442no—
266Q8—2442no—
267D4—42no—
268D4—42no—
269D4—81no—
270D4—88no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—4882no—
283C2×C4—4882no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4#1451974511yescharacteristic
287C2×C4#1451980211yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22#1683989511yescharacteristic
305C22C2×#1451972111yescharacteristic
306C22#1684020211yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—4882no—
311C4—4882no—
312C4#1683992911yescharacteristic
313C4#1683994811yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—162no—
318C2#2144473611yescharacteristic
319C2#2144773511yescharacteristic
320C2#2144776411yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
Fitting, prime radicalC61×#1999C2
socleC22×C61#447
hypercenter#1999D61
hypocenterC61#12103
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. #21456594C2semidirect
2. #21447748C2semidirect
3. #21443830C2semidirect
4. #21443743C2semidirect
21. C2×Dic244D4semidirect
32. #14519711D4semidirect
101. C23×C61SD32semidirect
179. C61#12103semidirect
262. C23#14519721semidirect

Maximal subgroups

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

subgroupindex
#214565942
#214477482
#214438302
#214437432
C61×#19992
#1210361