Finite Group Databasebeta

← #29573405

Subgroups — #29573405

Order 15616.

588 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448226C211yesmaximal, characteristic
2#21444046C211yesmaximal, characteristic
3#21443699C214yesmaximal
4C61×#1817C211yesmaximal, characteristic
5C2×#14519736—21no—
6#16839632C411yescharacteristic, complemented
7#16839632C411yescharacteristic, complemented
8#16839628—21no—
9#16839622—21no—
10#16839622C2212yes—
11C4×C4⋊Dic61—21no—
12C4×C4⋊Dic61C412yescomplemented
13C2×#14519684—21no—
14C2×#14519684—21no—
15C2×#14519684C2212yes—
16C22×C42×C61C2211yescharacteristic
17C2×C22⋊C8×C61C2212yes—
18C61×#336—21no—
19C61×#336—21no—
20C23×Dic61—22no—
21#14519736—42no—
22#14519736—42no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C412yes—
28C2×C4⋊Dic61C2×C412yes—
29C2×C22⋊Dic61—22no—
30C2×C4×Dic61—22no—
31#14519684—24no—
32#14519684—22no—
33#14519684—22no—
34#14519684—24no—
35#14519684—42no—
36C23×C4×C61D411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61C2311yescharacteristic
39C22×C8×C61—22no—
40C2×C42×C61—24no—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C2×C42×C61D411yescharacteristic
44C2×C42×C61D411yescharacteristic
45C22⋊C8×C61—24no—
46C24×C61C2×D411yescharacteristic
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61D812yes—
59C22×C4×C61SD1612yes—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61#4114yes—
63C22×Dic61—24no—
64C22×Dic61—22no—
65C22×Dic61—22no—
66C22×Dic61—42no—
67C22×Dic61—42no—
68C22×Dic61—42no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C2×C8×C61—42no—
73C2×C8×C61—42no—
74C2×C8×C61—42no—
75C42×C61—22no—
76C42×C61—22no—
77C42×C61—22no—
78C42×C61—22no—
79C22⋊Dic61—42no—
80C22⋊Dic61—42no—
81C4⋊Dic61—42no—
82C4⋊Dic61—42no—
83C4⋊Dic61—42no—
84C4⋊Dic61—42no—
85C4⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4×Dic61—44no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61C2×C22⋊C411yescharacteristic
93C23×C61#12312yes—
94C23×C61C2×D811yescharacteristic
95C23×C61C2×SD1611yescharacteristic
96C23×C61#13712yes—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—24no—
108C2×C4×C61—24no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—44no—
113C2×C4×C61—44no—
114C2×C4×C61#10212yes—
115C2×C4×C61#10212yes—
116C2×C4×C61#10412yes—
117C2×C4×C61#11712yes—
118C2×C4×C61#12312yes—
119C2×Dic61—44no—
120C2×Dic61—44no—
121C2×Dic61—44no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—82no—
129C2×Dic61—82no—
130C2×Dic61—82no—
131C2×Dic61—82no—
132C8×C61—82no—
133#8716—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#38111yescharacteristic
143C22×C61#41011yescharacteristic
144C22×C61#41011yescharacteristic
145C22×C61C2×#10211yescharacteristic
146C22×C61#41911yescharacteristic
147C22×C61C2×#10411yescharacteristic
148C22×C61#42111yescharacteristic
149C22×C61#48012yes—
150C22×C61#48312yes—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—41no—
157Dic61—84no—
158Dic61—82no—
159Dic61—82no—
160#2285—611no—
161#1817D6111yescharacteristic
162#1434—611no—
163#1294—614no—
164C2×C61—22no—
165C2×C61—22no—
166C2×C61#150211yescharacteristic
167C2×C61#150611yescharacteristic
168C2×C61#151511yescharacteristic
169C2×C61#151711yescharacteristic
170C2×C61#182011yescharacteristic
171C2×C61#182311yescharacteristic
172C2×C61#182611yescharacteristic
173C2×#117—1221no—
174C22×C42C2×D6111yescharacteristic
175C2×C22⋊C8C2×D6112yes—
176#384—611no—
177#384—611no—
178#382—1221no—
179#380—612no—
180#380—1221no—
181C4×C4⋊C4—612no—
182C4×C4⋊C4—1221no—
183C2×#95—612no—
184C2×#95—1221no—
185C2×#95—1221no—
186#336—21no—
187#336—21no—
188C61#871611yescharacteristic, complemented, Sylow
189C23×C4—1222no—
190C23×C4D24411yescharacteristic
191C23×C4D24411yescharacteristic
192C23×C4C22×D6111yescharacteristic
193C22×C8—22no—
194#117—2442no—
195#117—2442no—
196C2×C4⋊C4—612no—
197C2×C4⋊C4—612no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—1221no—
202C2×C22⋊C4—1222no—
203C2×C42—24no—
204C2×C42—1222no—
205C2×C42C4×D6111yescharacteristic
206C2×C42C4×D6111yescharacteristic
207C2×C42C22⋊D6111yescharacteristic
208C2×C42C22⋊D6111yescharacteristic
209C22⋊C8—24no—
210#95—1224no—
211#95—1222no—
212#95—1222no—
213#95—1224no—
214#95—2442no—
215C24C2×D24411yescharacteristic
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—1224no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—2442no—
227C22×C4—2442no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4—2442no—
231C22×C4—2442no—
232C22×C4#1175819012yes—
233C22×C4D48812yes—
234C22×C4#1175819811yescharacteristic
235C22×C4#1175819811yescharacteristic
236C22×C4#1175819811yescharacteristic
237C22×C4#1175819811yescharacteristic
238C22×C4C2×C4×D6111yescharacteristic
239C22×C4#1175822414yes—
240C22×C4C2×C22⋊D6111yescharacteristic
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—2442no—
245C4⋊C4—2442no—
246C4⋊C4—2442no—
247C4⋊C4—2442no—
248C4⋊C4—2442no—
249C4⋊C4—2442no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—2444no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23C2×#1175819811yescharacteristic
262C23#1451976112yes—
263C23C2×D48811yescharacteristic
264C23C2×#1175819011yescharacteristic
265C23#1451981012yes—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—4882no—
284C2×C4—4882no—
285C2×C4—4882no—
286C2×C4—4882no—
287C2×C4—2444no—
288C2×C4—2444no—
289C2×C4—2444no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4#1451969912yes—
297C2×C4#1451969912yes—
298C2×C4#1451970312yes—
299C2×C4#1451973312yes—
300C2×C4#1451975812yes—
301C8—82no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22#1683962311yescharacteristic
311C22#1683973511yescharacteristic
312C22#1683973511yescharacteristic
313C22C2×#1451969911yescharacteristic
314C22#1683977211yescharacteristic
315C22C2×#1451970311yescharacteristic
316C22#1683977811yescharacteristic
317C22#1684007612yes—
318C22#1684009312yes—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—4884no—
326C4—4882no—
327C4—4882no—
328C2—22no—
329C2—22no—
330C2#2144431511yescharacteristic
331C2#2144433211yescharacteristic
332C2#2144437111yescharacteristic
333C2#2144438011yescharacteristic
334C2#2144589411yescharacteristic
335C2#2144590611yescharacteristic
336C2#2144591611yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1817C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1817D61
hypocenterC61#8716
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
6. #16839632C4semidirect
7. #16839632C4semidirect
12. C4×C4⋊Dic61C4semidirect
188. C61#8716semidirect

Maximal subgroups

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

subgroupindex
#214482262
#214440462
#214436992
C61×#18172
#871661