Finite Group Databasebeta

← #29689469

Subgroups — #29689469

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic
2#21448420C212yesmaximal
3#21446802C218yesmaximal
4#21445896C211yesmaximal, characteristic
5#21445894C212yesmaximal
6C61×#1820C211yesmaximal, characteristic
7#16840390—24no—
8#16840390—24no—
9C2×#14519777C412yescomplemented
10C2×#14519776C412yescomplemented
11C2×#14519769C412yescomplemented
12C2×#14519768C412yescomplemented
13C2×#14519754C2214yes—
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839647—24no—
17#16839645—24no—
18#16839625C2212yes—
19#16839620C2214yes—
20C22⋊C4×Dic61C2214yes—
21#16839529C2214yes—
22#16839528C2218yes—
23C2×#117×C61C2211yescharacteristic
24C61×#381C2212yes—
25C61×#342C2214yes—
26C22×Dic122C2×C414yes—
27C23×Dic61C2312yes—
28#14519777—24no—
29#14519777—24no—
30#14519776—28no—
31#14519769—24no—
32#14519769—24no—
33#14519768—28no—
34#14519754—28no—
35#14519754—28no—
36#14519736—22no—
37#14519736—22no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61C2×C414yes—
41C2×C4⋊Dic61C2×C412yes—
42C2×C4⋊Dic61C2×C412yes—
43C2×C4⋊Dic61D414yes—
44C2×C4⋊Dic61D414yes—
45C2×#11758196—24no—
46C2×#11758196C2×C414yes—
47C2×#11758196C2×C414yes—
48C2×C22⋊Dic61C2314yes—
49C2×C22⋊Dic61C2312yes—
50C2×C4×Dic61—24no—
51C2×C4×Dic61C2×C412yes—
52#14519684—22no—
53#14519684—22no—
54#14519684—24no—
55#14519684—24no—
56C23×C4×C61C2311yescharacteristic
57#117×C61—22no—
58#117×C61—22no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61C2314yes—
62C2×C22⋊C4×C61C2312yes—
63C2×C42×C61C2×C412yes—
64#95×C61—22no—
65#95×C61—22no—
66C24×C61C2411yescharacteristic
67C22×C4×C61—24no—
68C22×C4×C61—24no—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C22×C412yes—
72C22×C4×C61C22×C412yes—
73C22×C4×C61C2×D412yes—
74C22×C4×C61C2×D412yes—
75C22×Dic61—24no—
76C22×Dic61—82no—
77C22×Dic61C22×C412yes—
78C22×Dic61C22×C412yes—
79C22×Dic61C22×C414yes—
80C22×Dic61C2×D414yes—
81C22×Dic61C2×D414yes—
82C2×Dic122—28no—
83C2×Dic122—84no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—22no—
86C22⋊C4×C61—24no—
87C22⋊C4×C61—28no—
88C42×C61—22no—
89C42×C61—22no—
90C22⋊Dic61—24no—
91C22⋊Dic61—28no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61—21no—
96C4⋊Dic61—28no—
97C4⋊Dic61C22⋊C418yes—
98C4⋊Dic61C22⋊C414yes—
99C4⋊Dic61C22⋊C414yes—
100#11758196—28no—
101#11758196—28no—
102#11758196—84no—
103C4×Dic61—24no—
104C4×Dic61—84no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61C23×C411yescharacteristic
108C23×C61C22×D412yes—
109C23×C612−1+414yes—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—24no—
113C2×C4×C61—24no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—44no—
119C2×C4×C61—44no—
120C2×C4×C61—44no—
121C2×C4×C61C2×C22⋊C412yes—
122C2×C4×C61C2×C22⋊C411yescharacteristic
123C2×C4×C61C2×C22⋊C411yescharacteristic
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—44no—
127C2×Dic61—44no—
128C2×Dic61—44no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—44no—
132C2×Dic61—44no—
133C2×Dic61—44no—
134C2×Dic61C2×C22⋊C418yes—
135Dic122—88no—
136#21828—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—22no—
139C22×C61—21no—
140C22×C61—22no—
141C22×C61—44no—
142C22×C61C22×C22⋊C411yescharacteristic
143C22×C61#51912yes—
144C22×C61#53614yes—
145C4×C61—22no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—42no—
149C4×C61—42no—
150C4×C61—84no—
151Dic61—28no—
152Dic61—42no—
153Dic61—42no—
154Dic61—84no—
155C2×#544—611no—
156#2313—612no—
157#1996—618no—
158#1820—612no—
159#1820—611no—
160#1820D6111yescharacteristic
161C2×C61—24no—
162C2×C61—22no—
163C2×C61#231011yescharacteristic
164C2×C61#288311yescharacteristic
165C2×C61#288411yescharacteristic
166#544—1224no—
167#544—1224no—
168C2×C4⋊Q8—612no—
169C2×C4⋊Q8—612no—
170C2×#125—612no—
171C2×#125—612no—
172C2×C22⋊Q8—614no—
173C2×#117—611no—
174C2×#117C2×D6111yescharacteristic
175C22×C4⋊C4—611no—
176#387—1224no—
177#387—1224no—
178#381—612no—
179#381C2×D6112yes—
180#380—614no—
181C4×C22⋊C4—614no—
182#342—618no—
183#342—614no—
184#342C2×D6114yes—
185C61#2182811yescharacteristic, complemented, Sylow
186C22×Q8—614no—
187C23×C4—612no—
188C23×C4C22×D6111yescharacteristic
189C4⋊Q8—1224no—
190C4⋊Q8—1224no—
191C4⋊Q8—1228no—
192#125—1224no—
193#125—1224no—
194#125—1228no—
195C22⋊Q8—1228no—
196C22⋊Q8—1228no—
197#117—22no—
198#117—22no—
199#117—1222no—
200#117—1222no—
201C2×C4⋊C4—612no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—614no—
204C2×C4⋊C4—614no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—614no—
208C2×C4⋊C4—614no—
209C2×C4⋊C4—614no—
210C2×C4⋊C4—1224no—
211C2×C4⋊C4C4×D6111yescharacteristic
212C2×C4⋊C4C4×D6111yescharacteristic
213C2×C22⋊C4—612no—
214C2×C22⋊C4—614no—
215C2×C22⋊C4C22×D6114yes—
216C2×C22⋊C4C22×D6112yes—
217C2×C42—612no—
218C2×C42—1224no—
219C2×C42C4×D6112yes—
220#95—22no—
221#95—22no—
222#95—1224no—
223#95—1224no—
224#95—1222no—
225#95—1222no—
226C24C23×D6111yescharacteristic
227C2×Q8—1228no—
228C2×Q8—4884no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—614no—
232C22×C4—614no—
233C22×C4—614no—
234C22×C4—612no—
235C22×C4—612no—
236C22×C4—1224no—
237C22×C4—4882no—
238C22×C4C2×C4×D6111yescharacteristic
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4C2×C4×D6112yes—
241C22×C4C2×C4×D6112yes—
242C22×C4D4×D6112yes—
243C22×C4D4×D6112yes—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—614no—
247C4⋊C4—614no—
248C4⋊C4—618no—
249C4⋊C4—1228no—
250C4⋊C4—1228no—
251C4⋊C4—1228no—
252C4⋊C4—1221no—
253C4⋊C4—1221no—
254C4⋊C4—1221no—
255C4⋊C4—1221no—
256C4⋊C4—4884no—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—1224no—
260C22⋊C4—1228no—
261C42—22no—
262C42—22no—
263C42—1224no—
264C42—4884no—
265C23—41no—
266C23—41no—
267C23C22×C4×D6111yescharacteristic
268C23C2×D4×D6112yes—
269C23#1451982414yes—
270Q8—4888no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—618no—
283C2×C4—1222no—
284C2×C4—1222no—
285C2×C4—2444no—
286C2×C4—2444no—
287C2×C4—2444no—
288C2×C4—2444no—
289C2×C4—2444no—
290C2×C4—2444no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4C22⋊C4×D6111yescharacteristic
294C2×C4C22⋊C4×D6111yescharacteristic
295C2×C4C22⋊C4×D6112yes—
296C22—21no—
297C22—22no—
298C22—21no—
299C22—22no—
300C22—44no—
301C22C2×C22⋊C4×D6111yescharacteristic
302C22#1684023912yes—
303C22#1684033414yes—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—84no—
310C4—1228no—
311C4—4884no—
312C4—2442no—
313C4—2442no—
314C2—24no—
315C2—22no—
316C2#2144840411yescharacteristic
317C2#2145427811yescharacteristic
318C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1820C2
socleC22×C61C22×C22⋊C4
prime radical, cosocleC24×C61C24
hypercenter#1820D61
hypocenterC61#21828
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
9. C2×#14519777C4semidirect
10. C2×#14519776C4semidirect
11. C2×#14519769C4semidirect
12. C2×#14519768C4semidirect
185. C61#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
#214484202
#214468022
#214458962
#214458942
C61×#18202
#2182861