Finite Group Databasebeta

← #29678150

Subgroups — #29678150

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839814C212yesmaximal
2C43×D61C211yesmaximal, characteristic
3#21445674C212yesmaximal, complemented
4#21445672C211yesmaximal, characteristic, complemented
5#21445582C212yesmaximal
6#21445580C212yesmaximal
7#21445563C212yesmaximal, complemented
8#97×Dic61C212yesmaximal, complemented
9C61×#1767C211yesmaximal, characteristic, complemented
10C2×C42×D61C412yes—
11C2×C42×D61C412yes—
12C2×C42×D61C2211yescharacteristic
13C2×C42×D61C2211yescharacteristic
14C2×C42×D61C2211yescharacteristic
15#16839814C4116yescomplemented
16C2×C4⋊#92418C2212yes—
17C2×#14519689C2214yes—
18C2×#14519688C2212yes—
19C2×#14519687C2214yes—
20C2×C8×Dic61C2214yes—
21C42×Dic61C412yes—
22C42×Dic61C412yes—
23C42×Dic61C2211yescharacteristic
24C42×Dic61C2211yescharacteristic
25C42×Dic61C2211yescharacteristic
26#16839517C2212yes—
27#16839516C2214yes—
28C2×C4⋊C8×C61C2212yes—
29C2×#97×C61C2212yes—
30C43×C61C2211yescharacteristic
31C61×#336C2212yes—
32C22×C4×D61C2×C412yes—
33C22×C4×D61C2×C412yes—
34C22×C4×D61C2×C411yescharacteristic
35C22×C4×D61C2×C411yescharacteristic
36C22×C4×D61C2311yescharacteristic
37C22×#92418C2314yes—
38C2×C4×Dic61C2×C412yes—
39C2×C4×Dic61C2×C412yes—
40C2×C4×Dic61C2×C412yes—
41C2×C4×Dic61C2×C412yes—
42C2×C4×Dic61C2×C412yes—
43C2×C4×Dic61C2×C412yes—
44C2×C4×Dic61C2×C412yes—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61C2×C411yescharacteristic
47C2×C4×Dic61C2×C412yes—
48C2×C4×Dic61C2311yescharacteristic
49C2×C4×Dic61C2311yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51C42×D61—28no—
52C42×D61—22no—
53C42×D61—22no—
54C42×D61—28no—
55C42×D61—24no—
56C42×D61C2×C418yes—
57C4⋊#92418C2×C418yescomplemented
58#14519689C2×C4116yes—
59#14519688—24no—
60#14519687—28no—
61C8×Dic61C2×C4116yescomplemented
62C22×C8×C61C2314yes—
63C2×C42×C61C2×C412yes—
64C2×C42×C61C2×C412yes—
65C2×C42×C61C2311yescharacteristic
66C2×C42×C61C2311yescharacteristic
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61C2×C418yescomplemented
69#97×C61—24no—
70C23×D61C22×C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C22×C412yes—
73C22×C4×C61C22×C412yes—
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2411yescharacteristic
76C22×Dic61C22×C411yescharacteristic
77C22×Dic61C22×C412yes—
78C22×Dic61C22×C412yes—
79C22×Dic61C22×C411yescharacteristic
80C22×Dic61C22×C411yescharacteristic
81C2×C4×D61—24no—
82C2×C4×D61—24no—
83C2×C4×D61—24no—
84C2×C4×D61—24no—
85C2×C4×D61—22no—
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—24no—
89C2×C4×D61—24no—
90C2×C4×D61—24no—
91C2×C4×D61C4214yes—
92C2×C4×D61C4214yes—
93C2×C4×D61M1618yes—
94C2×C4×D61C22×C414yes—
95C2×C8×C61—24no—
96C2×C8×C61—24no—
97C2×C8×C61C22×C418yes—
98C42×C61—24no—
99C42×C61—24no—
100C42×C61—22no—
101C42×C61C22×C414yes—
102C42×C61#4112yes—
103C42×C61#4112yes—
104C4×Dic61—24no—
105C4×Dic61—24no—
106C4×Dic61—21no—
107C4×Dic61—21no—
108C4×Dic61—24no—
109C4×Dic61—24no—
110C4×Dic61—24no—
111C4×Dic61—24no—
112C4×Dic61—24no—
113C4×Dic61C4214yes—
114C4×Dic61C4214yes—
115C4×Dic61M1618yes—
116C4×Dic61C22×C414yes—
117C4×Dic61#4114yes—
118C2×#92418—24no—
119C2×#92418—24no—
120C2×#92418C22×C418yes—
121C23×C61C23×C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—22no—
125C22×D61C2×C4212yes—
126C22×D61C2×M1614yes—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61C2×C4212yes—
133C2×C4×C61C2×C4212yes—
134C2×C4×C61#11714yes—
135C2×C4×C61C2×M1614yes—
136C2×C4×C61C2×M1614yes—
137C2×C4×C61C23×C412yes—
138C2×C4×C61C2×#4112yes—
139C2×C4×C61C2×#4111yescharacteristic
140C2×C4×C61C2×#4111yescharacteristic
141C2×Dic61—22no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—22no—
145C2×Dic61—22no—
146C2×Dic61—22no—
147C2×Dic61C2×C4212yes—
148C2×Dic61C2×C4212yes—
149C2×Dic61C2×C4212yes—
150C2×Dic61#11714yes—
151C2×Dic61C2×M1614yes—
152C2×Dic61C2×M1614yes—
153C2×Dic61C2×M1614yes—
154C4×D61—216no—
155C4×D61—216no—
156C4×D61—24no—
157C4×D61—24no—
158C4×D61—28no—
159C4×D61#97116yescomplemented
160C8×C61—28no—
161#92418—28no—
162#20463—611nomaximal, Sylow
163C22×C61C22×C4211yescharacteristic
164C22×C61C2×#11711yescharacteristic
165C22×C61C2×#11711yescharacteristic
166C22×C61C22×M1612yes—
167C22×C61C22×M1612yes—
168C2×D61—22no—
169C2×D61—24no—
170C2×D61—24no—
171C2×D61—22no—
172C2×D61C2×#9714yes—
173C4×C61—22no—
174C4×C61—24no—
175C4×C61—24no—
176C4×C61C2×#9714yes—
177C4×C61C4×#4112yes—
178C4×C61C4×#4112yes—
179Dic61—24no—
180Dic61—24no—
181Dic61—22no—
182Dic61C2×#9714yes—
183Dic61C4×M1614yes—
184C2×#431—612no—
185C2×C43—611no—
186#1767—612no—
187#1767—611no—
188#1767D6111yescharacteristic, complemented
189#1749—612no—
190#1749—612no—
191#1744—612no—
192C4×#97—612no—
193C2×C61C4×#11711yescharacteristic
194C2×C61C22×#9711yescharacteristic
195C2×C61C2×C4×M1611yescharacteristic
196C2×C61#292012yes—
197C2×C61#292012yes—
198D61—24no—
199C22×C42—611no—
200C22×C42—612no—
201C22×C42—612no—
202C22×C42—611no—
203C22×C42—611no—
204#431—6116no—
205C2×C4⋊C8—612no—
206C2×C4⋊C8C2×D6112yes—
207C2×C22⋊C8—614no—
208C2×#97—614no—
209C2×#97—612no—
210C2×#97C2×D6112yes—
211C2×C4×C8—614no—
212C43—612no—
213C43—612no—
214C43—611no—
215C43—611no—
216C43—611no—
217C43C2×D6111yescharacteristic
218#336—614no—
219#336—612no—
220#336C2×D6112yes—
221C61#2046311yescharacteristic, complemented, Sylow
222C23×C4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C23×C4—612no—
226C23×C4—612no—
227C22×C8—614no—
228C22×C8C22×D6114yes—
229C2×C42—611no—
230C2×C42—611no—
231C2×C42—612no—
232C2×C42—611no—
233C2×C42—611no—
234C2×C42—618no—
235C2×C42—612no—
236C2×C42—612no—
237C2×C42—612no—
238C2×C42—612no—
239C2×C42—612no—
240C2×C42—612no—
241C2×C42—612no—
242C2×C42—611no—
243C2×C42—1224no—
244C2×C42—1228no—
245C2×C42—1228no—
246C2×C42—1222no—
247C2×C42—1222no—
248C2×C42C4×D6112yes—
249C2×C42C4×D6112yes—
250C2×C42C22×D6111yescharacteristic
251C2×C42C22×D6111yescharacteristic
252C2×C42C22×D6111yescharacteristic
253C4⋊C8—618no—
254C4⋊C8C4×D6118yescomplemented
255C22⋊C8—6116no—
256#97—24no—
257#97—1228no—
258#97—1224no—
259C4×C8—6116no—
260C24—611no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—614no—
264C22×C4—618no—
265C22×C4—611no—
266C22×C4—612no—
267C22×C4—612no—
268C22×C4—614no—
269C22×C4—614no—
270C22×C4—1224no—
271C22×C4—1224no—
272C22×C4—1224no—
273C22×C4—1224no—
274C22×C4—1224no—
275C22×C4—1222no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1224no—
279C22×C4—1224no—
280C22×C4C2×C4×D6111yescharacteristic
281C22×C4C2×C4×D6112yes—
282C22×C4C2×C4×D6112yes—
283C22×C4C2×C4×D6111yescharacteristic
284C22×C4C23×D6111yescharacteristic
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—618no—
288C2×C8—1224no—
289C2×C8—1224no—
290C2×C8C2×C4×D6118yes—
291C42—22no—
292C42—24no—
293C42—24no—
294C42—614no—
295C42—614no—
296C42—614no—
297C42—614no—
298C42—618no—
299C42—1224no—
300C42—1224no—
301C42—1224no—
302C42—1224no—
303C42—1224no—
304C42—1221no—
305C42—1221no—
306C42—1224no—
307C42—1224no—
308C42C2×C4×D6114yes—
309C42#1175822412yes—
310C42#1175822612yes—
311C23—612no—
312C23—614no—
313C23—1221no—
314C23—1221no—
315C23—1222no—
316C23C22×C4×D6111yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—612no—
323C2×C4—612no—
324C2×C4—614no—
325C2×C4—614no—
326C2×C4—6116no—
327C2×C4—612no—
328C2×C4—614no—
329C2×C4—614no—
330C2×C4—1228no—
331C2×C4—12216no—
332C2×C4—12216no—
333C2×C4—1221no—
334C2×C4—1222no—
335C2×C4—1221no—
336C2×C4—1222no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—1224no—
340C2×C4—1224no—
341C2×C4C42×D6112yes—
342C2×C4C42×D6112yes—
343C2×C4#1451973514yes—
344C2×C4M16×D6114yes—
345C2×C4M16×D6114yes—
346C2×C4C22×C4×D6112yes—
347C2×C4#41×D6112yes—
348C2×C4C2×#1175822411yescharacteristic
349C2×C4C2×#1175822611yescharacteristic
350C8—28no—
351C8—1228no—
352C22—614no—
353C22—1224no—
354C22—1224no—
355C22—1222no—
356C22—1222no—
357C22C2×C42×D6111yescharacteristic
358C22#117×D6111yescharacteristic
359C22C2×#1451973511yescharacteristic
360C22C2×M16×D6112yes—
361C22C2×M16×D6112yes—
362C4—24no—
363C4—24no—
364C4—22no—
365C4—614no—
366C4—614no—
367C4—1224no—
368C4—1224no—
369C4—1222no—
370C4#97×D6114yes—
371C4C4×#1175822412yes—
372C4C4×#1175822612yes—
373C2—1224no—
374C2C4×#1451973511yescharacteristic
375C2C2×#97×D6111yescharacteristic
376C2C4×M16×D6111yescharacteristic
377C2#2145444912yes—
378C2#2145445112yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutatorC22×C61C22×C42
FittingC61×#1767C2
socleC23×C61C23×C4
prime radicalC23×D61C22×C4
cosocleC22×C4×C61C24
hypercenter#1767D61
hypocenterC61#20463
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1767

Complements

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

NH ≅ G/Ntype
3. #21445674C2semidirect
4. #21445672C2semidirect
7. #21445563C2semidirect
8. #97×Dic61C2semidirect
9. C61×#1767C2semidirect
15. #16839814C4semidirect
57. C4⋊#92418C2×C4semidirect
61. C8×Dic61C2×C4semidirect
68. C4⋊C8×C61C2×C4semidirect
159. C4×D61#97semidirect
188. #1767D61semidirect
221. C61#20463semidirect
254. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398142
C43×D612
#214456742
#214456722
#214455822
#214455802
#214455632
#97×Dic612
C61×#17672
#2046361