Finite Group Databasebeta

← #29665201

Subgroups — #29665201

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456769C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840043C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445271C214yesmaximal, complemented
6#21445185C214yesmaximal, complemented
7#21444505C211yesmaximal, characteristic, complemented
8#21444477C211yesmaximal, characteristic, complemented
9C61×#1571C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11C2×#14519777C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840142C2212yescomplemented
14#16840113C2212yescomplemented
15#16840063—22no—
16#16840053C2212yes—
17#16840043—22no—
18#16840043C2212yes—
19#16840043C2214yes—
20#16839973—22no—
21#16839920—22no—
22C8×D244C2214yes—
23#16839827—22no—
24#16839815C2211yescharacteristic
25#16839804—21no—
26C2×#14519711—21no—
27#16839768—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839510C2212yescomplemented
31#16839507C2212yescomplemented
32#16839507C2214yescomplemented
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#335C2214yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40C2×Dic244—24no—
41C2×C8×D61—24no—
42#14519777—21no—
43#14519777—21no—
44#14519777—21no—
45#14519777C2312yes—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53C4×D244—21no—
54C4×D244—21no—
55C4×D244D412yes—
56C4×D244C2312yes—
57#14519736—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61D411yescharacteristic
61C2×#11758198—21no—
62#14519711—24no—
63#14519711—22no—
64#14519709—22no—
65C4⋊#92418C2312yes—
66#14519702—24no—
67#14519699—24no—
68#14519699—22no—
69#14519699—22no—
70C22⋊#92418—21no—
71#14519689—24no—
72C4×#92418—21no—
73C2×M16×C61—21no—
74C22×C8×C61—21no—
75C2×C42×C61C2311yescharacteristic
76C4⋊C8×C61—21no—
77C4⋊C8×C61—21no—
78C4⋊C8×C61D412yes—
79C4⋊C8×C61C2312yes—
80C4⋊C8×C61C2312yes—
81C22⋊C8×C61D412yes—
82C4×C8×C61C2312yes—
83C23×D61—21no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—41no—
89C2×D244—21no—
90C2×D244—22no—
91C2×D244Q1614yescomplemented
92C2×D244C2×D412yes—
93C2×C4×D61—22no—
94C2×C4×D61—22no—
95C2×C4×D61—82no—
96C2×Dic122—22no—
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—41no—
100C2×Dic122—82no—
101M16×C61—42no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61—42no—
105C2×C8×C61C2×D414yes—
106C42×C61C2×D411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61#4111yescharacteristic
109C42×C61C2411yescharacteristic
110C22⋊Dic61—41no—
111#11758198—22no—
112#11758198—22no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—22no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122C4⋊Dic61C2×D412yes—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—41no—
126C2×#92418—22no—
127Dic244—84no—
128C8×D61—84no—
129C23×C61#12111yescharacteristic
130C22×D61—22no—
131C22×D61—24no—
132C22×D61—81no—
133C22×D61—81no—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61C2×Q1612yes—
140C2×C4×C61#13512yes—
141C2×C4×C61C22×D411yescharacteristic
142C2×C4×C61C22×D411yescharacteristic
143C2×C4×C61C2×#4111yescharacteristic
144C2×Dic61—22no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148C2×Dic61—81no—
149C2×Dic61—82no—
150D244—22no—
151D244—24no—
152D244C2×Q1614yes—
153C4×D61—44no—
154C4×D61—82no—
155Dic122—42no—
156Dic122—44no—
157Dic122—82no—
158Dic122—81no—
159C8×C61—24no—
160C8×C61—42no—
161C8×C61—42no—
162#92418—42no—
163#19387—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#46712yes—
167C22×C61C2×#12111yescharacteristic
168C2×D61—24no—
169C2×D61—42no—
170C2×D61—42no—
171C2×D61—84no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61#46712yes—
176C4×C61C22×Q1611yescharacteristic
177C4×C61C2×#13511yescharacteristic
178C4×C61#57612yes—
179Dic61—42no—
180Dic61—82no—
181Dic61—81no—
182Dic61—81no—
183#3209—611no—
184#3077—611no—
185C2×#474—611no—
186#2972—611no—
187#1683—614no—
188#1669—614no—
189#1571D6111yescharacteristic, complemented
190#1549—611no—
191#1542—611no—
192C2×C61—22no—
193C2×C61#304111yescharacteristic
194C2×C61C2×#46711yescharacteristic
195C2×C61#306911yescharacteristic
196D61—44no—
197#533—611no—
198C2×C4⋊Q8—611no—
199C2×C4×D4—611no—
200#494—612no—
201#487—612no—
202#478—1222no—
203#476—612no—
204#474—612no—
205#474—614no—
206#474—1222no—
207#462—1222no—
208#451—1222no—
209C8×D4—614no—
210C8×D4—1222no—
211#431—611no—
212#431C2×D6111yescharacteristic
213#428—1221no—
214C2×#107—1221no—
215#423C2×D6111yescharacteristic
216C2×C4⋊C8C2×D6111yescharacteristic
217#418—1221no—
218C2×#102—1221no—
219C2×C22⋊C8—1222no—
220#335C2×D6114yescomplemented
221#333—612no—
222#332—612no—
223#332—614no—
224C61#1938711yescharacteristic, complemented, Sylow
225C22×Q8—1221no—
226C22×D4—611no—
227C23×C4—1221no—
228C2×Q16—1224no—
229C2×M16—21no—
230C22×C8—21no—
231C22×C8—1224no—
232C4⋊Q8—612no—
233C4⋊Q8—611no—
234C4⋊Q8—611no—
235C4⋊Q8—1221no—
236C4⋊Q8—1221no—
237C4⋊Q8—1221no—
238#125—1221no—
239C22⋊Q8—1221no—
240C22⋊Q8—1221no—
241C4×Q8—1221no—
242C4×Q8—1221no—
243C4×D4—612no—
244C4×D4—612no—
245C4×D4—1221no—
246C4×D4—1221no—
247#117—1221no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C42C22×D6111yescharacteristic
253#107—1224no—
254#107—1222no—
255#106—1222no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—612no—
259C4⋊C8D24412yes—
260C4⋊C8C22×D6112yes—
261C4⋊C8C22×D6112yes—
262#103—1224no—
263#102—1224no—
264#102—1222no—
265#102—1222no—
266C22⋊C8—1221no—
267C22⋊C8—1224no—
268C22⋊C8D24412yes—
269C4×C8—1221no—
270C4×C8C22×D6112yes—
271C24—1221no—
272C2×Q8—1222no—
273C2×Q8—1221no—
274C2×Q8—1221no—
275C2×Q8—4882no—
276C2×Q8—2441no—
277C2×D4—612no—
278C2×D4—614no—
279C2×D4—1221no—
280C2×D4—1222no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—4882no—
285C22×C4—2441no—
286C22×C4C2×D24411yescharacteristic
287C22×C4D4×D6111yescharacteristic
288C22×C4#1175822611yescharacteristic
289Q16—4884no—
290M16—42no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—1222no—
295C2×C8—4884no—
296C2×C8C2×D24414yes—
297C4⋊C4—612no—
298C4⋊C4—1221no—
299C4⋊C4—1221no—
300C4⋊C4—1222no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—1222no—
304C4⋊C4—1221no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C4⋊C4—2441no—
308C4⋊C4—2441no—
309C22⋊C4—1222no—
310C22⋊C4—1222no—
311C22⋊C4—2441no—
312C42—2441no—
313C42C2×D24411yescharacteristic
314C42D4×D6111yescharacteristic
315C42#1175822611yescharacteristic
316C42C23×D6111yescharacteristic
317C23—1224no—
318C23—1222no—
319C23—4881no—
320C23—4881no—
321C23#1451975211yescharacteristic
322Q8—4881no—
323Q8—4882no—
324Q8—2442no—
325Q8—2444no—
326D4—614no—
327D4—1224no—
328D4—1222no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—1222no—
332C2×C4—4882no—
333C2×C4—4882no—
334C2×C4—4881no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2442no—
338C2×C4—2444no—
339C2×C4#1451975211yescharacteristic
340C2×C4#1451975211yescharacteristic
341C2×C4#1451975211yescharacteristic
342C2×C4Q16×D6112yes—
343C2×C4#1451979612yes—
344C2×C4C2×D4×D6111yescharacteristic
345C2×C4C22×D24411yescharacteristic
346C2×C4C2×#1175822611yescharacteristic
347C8—24no—
348C8—42no—
349C8—42no—
350C8—2442no—
351C22—21no—
352C22—21no—
353C22—1224no—
354C22—4884no—
355C22—2442no—
356C22—2442no—
357C22#1684000412yes—
358C22C2×#1451975211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—4881no—
363C4—4882no—
364C4—4881no—
365C4—2442no—
366C4#1684000412yes—
367C4C2×Q16×D6111yescharacteristic
368C4C2×#1451979611yescharacteristic
369C4#1684056712yes—
370C2—22no—
371C2—2444no—
372C2#2145532311yescharacteristic
373C2C2×#1684000411yescharacteristic
374C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1571C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1571D61
hypocenterC61#19387
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
1. #21456769C2semidirect
5. #21445271C2semidirect
6. #21445185C2semidirect
7. #21444505C2semidirect
8. #21444477C2semidirect
9. C61×#1571C2semidirect
13. #16840142C22semidirect
14. #16840113C22semidirect
30. #16839510C22semidirect
31. #16839507C22semidirect
32. #16839507C22semidirect
36. C61×#335C22semidirect
91. C2×D244Q16semidirect
189. #1571D61semidirect
220. #335C2×D61semidirect
224. C61#19387semidirect

Maximal subgroups

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

subgroupindex
#214567692
#214556182
C2×#168400432
#214548422
#214452712
#214451852
#214445052
#214444772
C61×#15712
#1938761