Finite Group Databasebeta

← #29580100

Subgroups — #29580100

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454171C211yesmaximal, characteristic
2#21444046C211yesmaximal, characteristic
3#21443699C214yesmaximal
4C61×#1749C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic, complemented
6C2×#14519756—21no—
7#16839668—21no—
8#16839660—21no—
9#16839660C412yescomplemented
10#16839632C411yescharacteristic, complemented
11#16839622—21no—
12#16839622C2212yes—
13C2×#14519684—21no—
14C2×#14519684—21no—
15C2×#14519684C2212yes—
16C22×C42×C61C2211yescharacteristic
17C2×C22⋊C8×C61C2212yes—
18C2×#97×C61—21no—
19C61×#336—21no—
20C22×Dic122—21no—
21C23×Dic61—22no—
22#14519777—42no—
23#14519756—44no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C412yes—
28C2×C4⋊Dic61C2×C412yes—
29C2×#11758196—22no—
30C2×C22⋊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—42no—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C2×C42×C61D411yescharacteristic
44C2×C42×C61D411yescharacteristic
45C22⋊C8×C61—22no—
46C22⋊C8×C61Q814yescomplemented
47#97×C61—42no—
48C24×C61C2×D411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×Q812yes—
64C22×C4×C61#4112yes—
65C22×Dic61—24no—
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—42no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C22×Dic61—42no—
73C22×Dic61—42no—
74C2×Dic122—42no—
75C2×Dic122—42no—
76C2×Dic122—42no—
77C2×C8×C61—24no—
78C2×C8×C61—42no—
79C2×C8×C61—42no—
80C42×C61—22no—
81C42×C61—22no—
82C42×C61—41no—
83C42×C61—41no—
84C22⋊Dic61—44no—
85C4⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4⋊Dic61—42no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90#11758196—44no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61C2×C22⋊C411yescharacteristic
96C23×C61#12411yescharacteristic
97C23×C61C4⋊Q811yescharacteristic
98C23×C61AGL(1,ℤ/8)12yes—
99C23×C61#13712yes—
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—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—44no—
117C2×C4×C61—44no—
118C2×C4×C61#9912yes—
119C2×C4×C61#10412yes—
120C2×C4×C61C4×Q812yes—
121C2×C4×C61C22⋊Q812yes—
122C2×Dic61—44no—
123C2×Dic61—44no—
124C2×Dic61—44no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—82no—
132C2×Dic61—82no—
133C2×Dic61—82no—
134C2×Dic61—82no—
135Dic122—84no—
136C8×C61—44no—
137#9889—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61#39111yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C22×C61#41011yescharacteristic
149C22×C61#41811yescharacteristic
150C22×C61#41911yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C22×C61#42111yescharacteristic
153C22×C61#48912yes—
154C22×C61#50112yes—
155C4×C61—22no—
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—41no—
160C4×C61—41no—
161Dic61—84no—
162Dic61—82no—
163Dic61—82no—
164#2857—611no—
165#1749D6111yescharacteristic
166#1434—611no—
167#1294—614no—
168C2×C61—22no—
169C2×C61—22no—
170C2×C61#150511yescharacteristic
171C2×C61#150811yescharacteristic
172C2×C61#151011yescharacteristic
173C2×C61#151511yescharacteristic
174C2×C61#198611yescharacteristic
175C2×C61#199311yescharacteristic
176C2×C61#199611yescharacteristic
177C2×C4⋊Q8—611no—
178C2×C22⋊Q8—1221no—
179C22×C42C2×D6111yescharacteristic
180C2×C22⋊C8C2×D6112yes—
181C2×#97—21no—
182#391—1221no—
183#389—612no—
184#389—1221no—
185#384—611no—
186#380—612no—
187#380—1221no—
188C2×#95—612no—
189C2×#95—1221no—
190C2×#95—1221no—
191#336—21no—
192C61#988911yescharacteristic, complemented, Sylow
193C22×Q8—1221no—
194C23×C4—1222no—
195C23×C4D24411yescharacteristic
196C23×C4D24411yescharacteristic
197C23×C4C22×D6111yescharacteristic
198C22×C8—22no—
199C4⋊Q8—2442no—
200C22⋊Q8—2444no—
201C2×C4⋊C4—612no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C22⋊C4—1222no—
208C2×C42—42no—
209C2×C42C4×D6111yescharacteristic
210C2×C42C4×D6111yescharacteristic
211C2×C42C22⋊D6111yescharacteristic
212C2×C42C22⋊D6111yescharacteristic
213C22⋊C8—22no—
214C22⋊C8Dic12214yescomplemented
215#97—42no—
216#95—1224no—
217#95—1222no—
218#95—1222no—
219#95—1224no—
220#95—2442no—
221C24C2×D24411yescharacteristic
222C2×Q8—2442no—
223C2×Q8—2442no—
224C2×Q8—2442no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—1224no—
234C22×C4—1222no—
235C22×C4—1222no—
236C22×C4—2442no—
237C22×C4—2442no—
238C22×C4—2442no—
239C22×C4—2442no—
240C22×C4—2442no—
241C22×C4—2442no—
242C22×C4#1175819811yescharacteristic
243C22×C4#1175819811yescharacteristic
244C22×C4#1175819811yescharacteristic
245C22×C4#1175819811yescharacteristic
246C22×C4C2×Dic12212yes—
247C22×C4C2×C4×D6111yescharacteristic
248C22×C4#1175822212yes—
249C22×C4C2×C22⋊D6111yescharacteristic
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—2442no—
254C4⋊C4—2442no—
255C4⋊C4—2444no—
256C4⋊C4—2442no—
257C4⋊C4—2442no—
258C4⋊C4—2442no—
259C22⋊C4—2444no—
260C42—22no—
261C42—22no—
262C42—41no—
263C42—41no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23C2×#1175819811yescharacteristic
269C23#1451976511yescharacteristic
270C23#1451977711yescharacteristic
271C23#1451980512yes—
272C23#1451981012yes—
273Q8—4884no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—4882no—
293C2×C4—4882no—
294C2×C4—4882no—
295C2×C4—4882no—
296C2×C4—2444no—
297C2×C4—2444no—
298C2×C4—2444no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4#1451969112yes—
306C2×C4#1451970312yes—
307C2×C4C4×Dic12212yes—
308C2×C4#1451975612yes—
309C8—44no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22#1683966811yescharacteristic
319C22C2×#1451969111yescharacteristic
320C22#1683973511yescharacteristic
321C22#1683976811yescharacteristic
322C22#1683977211yescharacteristic
323C22C2×#1451970311yescharacteristic
324C22#1683977811yescharacteristic
325C22#1684012212yes—
326C22#1684017112yes—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—4884no—
334C4—4882no—
335C4—4882no—
336C2—22no—
337C2—22no—
338C2#2144432811yescharacteristic
339C2#2144434011yescharacteristic
340C2#2144434911yescharacteristic
341C2#2144437111yescharacteristic
342C2#2144676311yescharacteristic
343C2#2144679211yescharacteristic
344C2#2144680411yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1749C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1749D61
hypocenterC61#9889
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
5. C2×#14519777C4semidirect
9. #16839660C4semidirect
10. #16839632C4semidirect
46. C22⋊C8×C61Q8semidirect
192. C61#9889semidirect
214. C22⋊C8Dic122semidirect

Maximal subgroups

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

subgroupindex
#214541712
#214440462
#214436992
C61×#17492
#988961