Finite Group Databasebeta

← #29582355

Subgroups — #29582355

Order 15616.

712 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16840052C212yesmaximal, complemented
2#21446620C211yesmaximal, characteristic, complemented
3C2×D4⋊#92418C212yesmaximal, complemented
4#21443702C211yesmaximal, characteristic, complemented
5C61×#2537C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7#16840052—24no—
8#16840052—24no—
9C2×#14519708—22no—
10C2×C4⋊#92418C2212yes—
11C2×#14519700—22no—
12C2×#14519698—22no—
13C2×C4×#92418C2211yescharacteristic
14D4⋊#92418—24no—
15C2×#121×C61—22no—
16C2×C4×D4×C61C2212yes—
17C61×#394—22no—
18C61×#384C2211yescharacteristic
19C61×#380—22no—
20C22×Dic122—21no—
21C22×#92418—21no—
22C22×#92418—22no—
23#14519777—22no—
24#14519777—21no—
25#14519777D412yescomplemented
26C2×C4⋊Dic61—22no—
27#14519708—28no—
28C4⋊#92418—22no—
29C4⋊#92418D414yescomplemented
30#14519700—44no—
31#14519698—28no—
32C4×#92418—22no—
33C22×D4×C61—22no—
34C22×D4×C61D412yes—
35C23×C4×C61—22no—
36#121×C61—44no—
37#121×C61—44no—
38C4×D4×C61—22no—
39C4×D4×C61—22no—
40C4×D4×C61—22no—
41C4×D4×C61—22no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61D412yes—
44C2×C4⋊C4×C61D412yes—
45C2×C22⋊C4×C61—24no—
46C2×C22⋊C4×C61—22no—
47C2×C42×C61C2311yescharacteristic
48#95×C61—22no—
49C24×C61—22no—
50C2×D4×C61—22no—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61—48no—
56C2×D4×C61—82no—
57C2×D4×C61SD1614yes—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61—42no—
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×Dic61—41no—
69C2×Dic122—22no—
70C2×Dic122—42no—
71C2×Dic122—42no—
72C2×Dic122—41no—
73C4⋊C4×C61—22no—
74C4⋊C4×C61—22no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—42no—
78C22⋊C4×C61—48no—
79C22⋊C4×C61—42no—
80C22⋊C4×C61—42no—
81C42×C61C2×D411yescharacteristic
82C42×C61C2×D412yes—
83C42×C61#4111yescharacteristic
84C4⋊Dic61—24no—
85C4⋊Dic61—42no—
86C2×#92418—22no—
87C2×#92418—24no—
88C2×#92418—41no—
89C2×#92418—41no—
90C2×#92418—42no—
91C2×#92418—42no—
92C23×C61—24no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61#12011yescharacteristic
100D4×C61—24no—
101D4×C61—24no—
102D4×C61—42no—
103D4×C61—42no—
104D4×C61—88no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—44no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—82no—
113C2×C4×C61—82no—
114C2×C4×C61—82no—
115C2×C4×C61—82no—
116C2×C4×C61#12012yes—
117C2×C4×C61#12111yescharacteristic
118C2×C4×C61#12112yes—
119C2×C4×C61#12711yescharacteristic
120C2×C4×C61C2×SD1612yes—
121C2×C4×C61#13512yes—
122C2×C4×C61AGL(1,ℤ/8)12yes—
123C2×C4×C61#13712yes—
124C2×Dic61—42no—
125C2×Dic61—81no—
126C2×Dic61—81no—
127Dic122—44no—
128Dic122—82no—
129#92418—42no—
130#92418—44no—
131#10235—611nomaximal, Sylow
132C22×C61—44no—
133C22×C61—44no—
134C22×C61—44no—
135C22×C61—82no—
136C22×C61—82no—
137C22×C61—82no—
138C22×C61—82no—
139C22×C61#39211yescharacteristic
140C22×C61#45012yes—
141C22×C61#45212yes—
142C22×C61#45311yescharacteristic
143C22×C61#45411yescharacteristic
144C4×C61—21no—
145C4×C61—21no—
146C4×C61—82no—
147C4×C61—82no—
148C4×C61#44812yes—
149C4×C61#44912yes—
150C4×C61#46512yes—
151C4×C61#46812yes—
152Dic61—82no—
153C2×#476—612no—
154#2537D6111yescharacteristic
155#1956—611no—
156C2×D4⋊C8—612no—
157#1296—611no—
158C2×C61—84no—
159C2×C61#162911yescharacteristic
160C2×C61#163411yescharacteristic
161C2×C61#163512yes—
162C2×C61#200012yes—
163C2×C61#200611yescharacteristic
164C2×C4⋊Q8—611no—
165C2×#121—22no—
166C2×C4×D4C2×D6112yes—
167#476—1224no—
168#476—1224no—
169C2×#106—1222no—
170C2×C4⋊C8—612no—
171C2×#103—1222no—
172C2×#102—1222no—
173C2×C4×C8—611no—
174#394—22no—
175#384C2×D6111yescharacteristic
176#380—22no—
177D4⋊C8—1224no—
178C61#1023511yescharacteristic, complemented, Sylow
179C22×Q8—1221no—
180C22×D4—22no—
181C22×D4C22⋊D6112yes—
182C23×C4—22no—
183C22×C8—1221no—
184C22×C8—1222no—
185C4⋊Q8—612no—
186C4⋊Q8—1222no—
187C4⋊Q8—1221no—
188#121—44no—
189#121—44no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4C22⋊D6112yes—
197C2×C4⋊C4C22⋊D6112yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C42C22×D6111yescharacteristic
201#106—1228no—
202C4⋊C8—614no—
203C4⋊C8—1222no—
204#103—2444no—
205#102—1228no—
206C4×C8—1222no—
207#95—22no—
208C24—22no—
209C2×Q8—1222no—
210C2×Q8—2442no—
211C2×Q8—2442no—
212C2×Q8—2441no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—48no—
219C2×D4—82no—
220C2×D4#1175820014yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—2441no—
230C22×C4C2×C22⋊D6111yescharacteristic
231C22×C4C2×C22⋊D6112yes—
232C2×C8—1222no—
233C2×C8—1224no—
234C2×C8—2441no—
235C2×C8—2441no—
236C2×C8—2442no—
237C2×C8—2442no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—1224no—
244C4⋊C4—2442no—
245C22⋊C4—48no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42D4×D6111yescharacteristic
249C42D4×D6112yes—
250C42#1175822411yescharacteristic
251C23—24no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23#1451974611yescharacteristic
259Q8—4882no—
260Q8—2444no—
261D4—24no—
262D4—24no—
263D4—42no—
264D4—42no—
265D4—88no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—4881no—
278C2×C4—4881no—
279C2×C4—2442no—
280C2×C4#1451974512yes—
281C2×C4#1451974811yescharacteristic
282C2×C4#1451974912yes—
283C2×C4#1451977311yescharacteristic
284C2×C4C2×#1175820012yes—
285C2×C4#1451979712yes—
286C2×C4#1451980312yes—
287C2×C4#1451980712yes—
288C8—2442no—
289C8—2444no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22#1683967011yescharacteristic
298C22#1683991312yes—
299C22#1683992412yes—
300C22#1683993011yescharacteristic
301C22#1683993511yescharacteristic
302C4—21no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—4882no—
307C4#1683990012yes—
308C4#1683990512yes—
309C4#1683998912yes—
310C4#1684000612yes—
311C2—84no—
312C2#2144493511yescharacteristic
313C2#2144496411yescharacteristic
314C2#2144497012yes—
315C2#2144682212yes—
316C2#2144686111yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#2537C2
socleC23×C61#120
hypercenter#2537D61
hypocenterC61#10235
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16840052C2semidirect
2. #21446620C2semidirect
3. C2×D4⋊#92418C2semidirect
4. #21443702C2semidirect
25. #14519777D4semidirect
29. C4⋊#92418D4semidirect
178. C61#10235semidirect

Maximal subgroups

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

subgroupindex
C2×#168400522
#214466202
C2×D4⋊#924182
#214437022
C61×#25372
#1023561