Finite Group Databasebeta

← #29582354

Subgroups — #29582354

Order 15616.

712 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16840049C212yesmaximal, complemented
2#21450735C211yesmaximal, characteristic
3C2×#16839491C212yesmaximal, complemented
4#21443702C211yesmaximal, characteristic, complemented
5C61×#1956C211yesmaximal, characteristic, complemented
6C2×#14519752—22no—
7C2×C4×D244C2212yes—
8#16840049—24no—
9#16840049—24no—
10C2×#14519708—22no—
11C2×C4⋊#92418C2212yes—
12C2×#14519697—22no—
13#16839679—22no—
14#16839619—22no—
15#16839491—24no—
16C2×C4⋊Q8×C61C2211yescharacteristic
17C2×#103×C61—22no—
18C2×C4×C8×C61C2211yescharacteristic
19C61×#384C2211yescharacteristic
20C22×D244—22no—
21C22×D244D412yes—
22C22×C4×D61—22no—
23C22×#92418—22no—
24#14519752—44no—
25#14519752—44no—
26C4×D244—22no—
27C4×D244—22no—
28C4×D244—22no—
29C4×D244—22no—
30C2×C4⋊Dic61D412yes—
31C2×#11758198—24no—
32C2×#11758198—22no—
33#14519708—28no—
34C4⋊#92418—22no—
35C4⋊#92418D414yescomplemented
36#14519697—28no—
37#14519684—22no—
38C22×Q8×C61—21no—
39C22×C8×C61—21no—
40C4⋊Q8×C61—22no—
41C4⋊Q8×C61—21no—
42C4⋊Q8×C61D412yescomplemented
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61D412yes—
46C2×C42×C61C2311yescharacteristic
47#103×C61—44no—
48C4×C8×C61—22no—
49C23×D61—22no—
50C2×Q8×C61—22no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C2×Q8×C61—41no—
54C22×C4×C61—22no—
55C22×C4×C61—41no—
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D412yes—
58C22×Dic61—22no—
59C2×D244—22no—
60C2×D244—22no—
61C2×D244—22no—
62C2×D244—22no—
63C2×D244—22no—
64C2×D244—48no—
65C2×D244—82no—
66C2×D244SD1614yes—
67C2×C4×D61—42no—
68C2×C4×D61—42no—
69C2×C4×D61—42no—
70C2×C4×D61—42no—
71C2×C4×D61—42no—
72C2×C4×D61—42no—
73C2×C8×C61—22no—
74C2×C8×C61—41no—
75C2×C8×C61—41no—
76C4⋊C4×C61—24no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—42no—
81C42×C61C2×D411yescharacteristic
82C42×C61C2×D412yes—
83C42×C61#4111yescharacteristic
84#11758198—48no—
85#11758198—42no—
86#11758198—42no—
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C2×#92418—24no—
90C2×#92418—42no—
91C2×#92418—42no—
92C23×C61#12011yescharacteristic
93C22×D61—24no—
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98C22×D61—42no—
99C22×D61—42no—
100Q8×C61—44no—
101Q8×C61—82no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—42no—
105C2×C4×C61—44no—
106C2×C4×C61—42no—
107C2×C4×C61—81no—
108C2×C4×C61—81no—
109C2×C4×C61#12012yes—
110C2×C4×C61#12111yescharacteristic
111C2×C4×C61#12112yes—
112C2×C4×C61#12711yescharacteristic
113C2×C4×C61C2×SD1612yes—
114C2×C4×C61#13512yes—
115C2×C4×C61AGL(1,ℤ/8)12yes—
116C2×C4×C61#13712yes—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120D244—24no—
121D244—24no—
122D244—42no—
123D244—42no—
124D244—88no—
125C4×D61—82no—
126C4×D61—82no—
127C4×D61—82no—
128C4×D61—82no—
129C8×C61—42no—
130#92418—44no—
131#10235—611nomaximal, Sylow
132C22×C61#39211yescharacteristic
133C22×C61#45012yes—
134C22×C61#45212yes—
135C22×C61#45311yescharacteristic
136C22×C61#45411yescharacteristic
137C2×D61—44no—
138C2×D61—44no—
139C2×D61—44no—
140C2×D61—82no—
141C2×D61—82no—
142C2×D61—82no—
143C2×D61—82no—
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#2537—611no—
155#1956D6111yescharacteristic, complemented
156C2×D4⋊C8—612no—
157#1296—611no—
158C2×C61#162911yescharacteristic
159C2×C61#163411yescharacteristic
160C2×C61#163512yes—
161C2×C61#200012yes—
162C2×C61#200611yescharacteristic
163D61—84no—
164C2×C4⋊Q8C2×D6111yescharacteristic
165C2×#121—1222no—
166C2×C4×D4—612no—
167#476—1224no—
168#476—1224no—
169C2×#106—1222no—
170C2×C4⋊C8—612no—
171C2×#103—22no—
172C2×#102—1222no—
173C2×C4×C8C2×D6111yescharacteristic
174#394—1222no—
175#384C2×D6111yescharacteristic
176#380—1222no—
177D4⋊C8—1224no—
178C61#1023511yescharacteristic, complemented, Sylow
179C22×Q8—21no—
180C22×D4—612no—
181C22×D4—1222no—
182C23×C4—1222no—
183C22×C8—21no—
184C22×C8—1222no—
185C4⋊Q8—21no—
186C4⋊Q8—22no—
187C4⋊Q8D24412yescomplemented
188#121—2444no—
189#121—2444no—
190C4×D4—1222no—
191C4×D4—1222no—
192C4×D4—1222no—
193C4×D4—1222no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—612no—
197C2×C4⋊C4C22⋊D6112yes—
198C2×C22⋊C4—1224no—
199C2×C22⋊C4—1222no—
200C2×C42C22×D6111yescharacteristic
201#106—1228no—
202C4⋊C8—614no—
203C4⋊C8—1222no—
204#103—44no—
205#102—1228no—
206C4×C8—22no—
207#95—1222no—
208C24—1222no—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×D4—614no—
214C2×D4—1222no—
215C2×D4—1222no—
216C2×D4—1222no—
217C2×D4—1222no—
218C2×D4—1222no—
219C2×D4—4882no—
220C2×D4—2448no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—1222no—
224C22×C4—2442no—
225C22×C4—2442no—
226C22×C4—2442no—
227C22×C4—2442no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4D4×D6112yes—
231C22×C4C2×C22⋊D6111yescharacteristic
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—1224no—
236C2×C8—2442no—
237C2×C8—2442no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—1222no—
244C4⋊C4—1222no—
245C22⋊C4—2448no—
246C22⋊C4—2442no—
247C22⋊C4—2442no—
248C42C2×D24411yescharacteristic
249C42#1175822211yescharacteristic
250C42D4×D6112yes—
251C23—1224no—
252C23—2442no—
253C23—2442no—
254C23—2442no—
255C23—2442no—
256C23—2442no—
257C23—2442no—
258C23#1451974511yescharacteristic
259Q8—44no—
260Q8—82no—
261D4—1224no—
262D4—1224no—
263D4—4888no—
264D4—2442no—
265D4—2442no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—4882no—
274C2×C4—4882no—
275C2×C4—4882no—
276C2×C4—4882no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4#1451974712yes—
281C2×C4#1451975012yes—
282C2×C4#1451975111yescharacteristic
283C2×C4#1451977311yescharacteristic
284C2×C4SD16×D6112yes—
285C2×C4#1451980012yes—
286C2×C4#1451980412yes—
287C2×C4#1451980912yes—
288C8—42no—
289C8—2444no—
290C22—4882no—
291C22—4882no—
292C22—4882no—
293C22—4882no—
294C22—2444no—
295C22—2444no—
296C22—2444no—
297C22#1683967111yescharacteristic
298C22#1683991112yes—
299C22#1683992312yes—
300C22#1683992911yescharacteristic
301C22#1683993411yescharacteristic
302C4—21no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—4882no—
307C4#1683990212yes—
308C4#1683991012yes—
309C4#1683999412yes—
310C4#1684001112yes—
311C2—4884no—
312C2#2144493311yescharacteristic
313C2#2144496211yescharacteristic
314C2#2144496712yes—
315C2#2144682112yes—
316C2#2144686011yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1956C2
socleC23×C61#120
prime radical#21450735C2
hypercenter#1956D61
hypocenterC61#10235
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
3. C2×#16839491C2semidirect
4. #21443702C2semidirect
5. C61×#1956C2semidirect
35. C4⋊#92418D4semidirect
42. C4⋊Q8×C61D4semidirect
155. #1956D61semidirect
178. C61#10235semidirect
187. C4⋊Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
#214507352
C2×#168394912
#214437022
C61×#19562
#1023561