Finite Group Databasebeta

← #29665243

Subgroups — #29665243

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456944C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3#21455279C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445227C214yesmaximal, complemented
6#21445193C214yesmaximal, complemented
7#21444589C211yesmaximal, characteristic, complemented
8#21444499C211yesmaximal, characteristic, complemented
9C61×#1540C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×#14519777C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840158C2212yescomplemented
14#16840108C2212yescomplemented
15#16840078—22no—
16#16840053—22no—
17#16840053C2214yes—
18#16840052—22no—
19#16839974—22no—
20#16839964C2212yes—
21#16839954C2212yes—
22C8×D244—22no—
23#16839827C2214yes—
24#16839815C2211yescharacteristic
25C2×#14519709—21no—
26#16839786C2211yescharacteristic
27#16839763—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839512C2214yescomplemented
31#16839510C2214yescomplemented
32C61×#532C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#332C2212yescomplemented
36C61×#331C2212yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40#14519802—44no—
41C2×#11758201—22no—
42C2×#11758199—22no—
43C2×#11758194—21no—
44C2×C8×D61—24no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777C2312yes—
49C4×D244—21no—
50C4×D244—21no—
51C4×D244D412yes—
52C4×D244C2312yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758198—21no—
57#14519709—24no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519699—24no—
64#14519698—22no—
65#14519698—24no—
66C22⋊#92418D412yes—
67#14519689—24no—
68C4×#92418C2312yes—
69C2×#41×C61—21no—
70C22×C8×C61—21no—
71C4⋊Q8×C61C2311yescharacteristic
72#127×C61C2311yescharacteristic
73#124×C61—21no—
74#121×C61—21no—
75#121×C61—21no—
76C4×D4×C61—21no—
77C4×D4×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C23×D61—21no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—41no—
98C2×D244—21no—
99C2×D244—22no—
100C2×D244SD1614yes—
101C2×D244C2×D412yes—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×C4×D61—82no—
105C2×Dic122—22no—
106C2×Dic122—41no—
107C2×Dic122—82no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—42no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758201—44no—
120#11758199—44no—
121#11758198—22no—
122#11758198—22no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61—22no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4⋊Dic61C2×D412yes—
129#11758194—42no—
130C2×#92418—22no—
131C2×#92418C2×D414yes—
132C8×D61—84no—
133C23×C61—41no—
134C23×C61#12111yescharacteristic
135C22×D61—22no—
136C22×D61—24no—
137C22×D61—81no—
138C22×D61—81no—
139Q8×C61—41no—
140Q8×C61—41no—
141D4×C61—41no—
142D4×C61—41no—
143D4×C61—42no—
144D4×C61—81no—
145D4×C61—82no—
146C2×C4×C61—22no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—82no—
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61C2×SD1612yes—
154C2×C4×C61#13512yes—
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C2×#4111yescharacteristic
158C2×Dic61—22no—
159C2×Dic61—42no—
160C2×Dic61—81no—
161C2×Dic61—82no—
162D244—22no—
163D244—24no—
164D244C2×SD1614yes—
165C4×D61—44no—
166C4×D61—82no—
167Dic122—82no—
168Dic122—82no—
169C8×C61—42no—
170C8×C61—42no—
171#92418—24no—
172#92418—42no—
173#19391—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61#46512yes—
180C22×C61C2×#12111yescharacteristic
181C2×D61—24no—
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—84no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—81no—
189C4×C61#46512yes—
190C4×C61C22×SD1611yescharacteristic
191C4×C61C2×#13511yescharacteristic
192C4×C61#57712yes—
193Dic61—42no—
194Dic61—82no—
195#3227—611no—
196C2×#476—611no—
197#3036—611no—
198#2972—611no—
199#1677—614no—
200#1671—614no—
201#1570—611no—
202#1548—611no—
203#1540D6111yescharacteristic, complemented
204C2×C61—22no—
205C2×C61—161no—
206C2×C61#304211yescharacteristic
207C2×C61C2×#46511yescharacteristic
208C2×C61#306811yescharacteristic
209D61—44no—
210C2×AGL(1,ℤ/8)—1221no—
211#532C2×D6111yescharacteristic
212C2×C4⋊Q8—611no—
213C2×C4×D4—611no—
214#498—612no—
215#486—612no—
216#481—1222no—
217#476—614no—
218#476—1222no—
219#476—1222no—
220#463—1222no—
221#461—612no—
222#459—612no—
223C8×D4—614no—
224C8×D4—1222no—
225#431—611no—
226#431C2×D6111yescharacteristic
227C2×#106—1221no—
228#423—611no—
229C2×C4⋊C8C2×D6111yescharacteristic
230#417—1221no—
231C2×#102—1221no—
232C2×C22⋊C8—1222no—
233#334—614no—
234#333—614no—
235#332C2×D6112yescomplemented
236#331C2×D6112yescomplemented
237C61#1939111yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—1221no—
240C22×D4—611no—
241C23×C4—1221no—
242AGL(1,ℤ/8)—2444no—
243C2×SD16—1222no—
244C2×D8—1222no—
245C2×M16—1221no—
246C22×C8—21no—
247C22×C8—1224no—
248C4⋊Q8—612no—
249C4⋊Q8—1221no—
250C4⋊Q8—1221no—
251C4⋊Q8—1221no—
252C4⋊Q8C22×D6111yescharacteristic
253#127C22×D6111yescharacteristic
254#124—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—612no—
260C4×D4—612no—
261C4×D4—1221no—
262C4×D4—1221no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C42C22×D6111yescharacteristic
268#106—1224no—
269#106—1224no—
270C4⋊C8—21no—
271C4⋊C8—612no—
272C4⋊C8—612no—
273C4⋊C8—1221no—
274C4⋊C8C22×D6112yes—
275C4⋊C8C22×D6112yes—
276#103—1222no—
277#102—1222no—
278#102—1224no—
279#102—1224no—
280C22⋊C8—21no—
281C22⋊C8—612no—
282C22⋊C8—1224no—
283C4×C8—21no—
284C4×C8—612no—
285C24—1221no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—1222no—
290C2×Q8—4882no—
291C2×Q8—2441no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—612no—
297C2×D4—614no—
298C2×D4—1221no—
299C2×D4—1222no—
300C22×C4—41no—
301C22×C4—1221no—
302C22×C4—1222no—
303C22×C4—1222no—
304C22×C4—4882no—
305C22×C4—2441no—
306C22×C4#1175822211yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309SD16—2444no—
310D8—2444no—
311M16—2442no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—614no—
316C2×C8—1222no—
317C2×C8—4884no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—612no—
321C4⋊C4—1221no—
322C4⋊C4—1222no—
323C4⋊C4—1222no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1222no—
329C22⋊C4—1222no—
330C42#1175822211yescharacteristic
331C42D4×D6111yescharacteristic
332C42D4×D6111yescharacteristic
333C42C23×D6111yescharacteristic
334C23—41no—
335C23—1224no—
336C23—1222no—
337C23—4881no—
338C23—4881no—
339C23#1451975011yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—4882no—
343Q8—4882no—
344D4—41no—
345D4—41no—
346D4—42no—
347D4—81no—
348D4—82no—
349D4—614no—
350D4—1224no—
351D4—1222no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—82no—
356C2×C4—1222no—
357C2×C4—4882no—
358C2×C4—4882no—
359C2×C4—4881no—
360C2×C4—2442no—
361C2×C4—2444no—
362C2×C4#1451975011yescharacteristic
363C2×C4#1451975011yescharacteristic
364C2×C4#1451975011yescharacteristic
365C2×C4SD16×D6112yes—
366C2×C4#1451979612yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C2×D4×D6111yescharacteristic
369C2×C4C2×#1175822211yescharacteristic
370C8—42no—
371C8—42no—
372C8—1224no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—1224no—
380C22—4884no—
381C22—2442no—
382C22—2442no—
383C22#1683999012yes—
384C22C2×#1451975011yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—4882no—
390C4—2442no—
391C4#1683999012yes—
392C4C2×SD16×D6111yescharacteristic
393C4C2×#1451979611yescharacteristic
394C4#1684056912yes—
395C2—22no—
396C2—161no—
397C2—2444no—
398C2#2145533411yescharacteristic
399C2C2×#1683999011yescharacteristic
400C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#1540D61
hypocenterC61#19391
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #21456944C2semidirect
2. C2×#16840053C2semidirect
4. #21454842C2semidirect
5. #21445227C2semidirect
6. #21445193C2semidirect
7. #21444589C2semidirect
8. #21444499C2semidirect
9. C61×#1540C2semidirect
13. #16840158C22semidirect
14. #16840108C22semidirect
30. #16839512C22semidirect
31. #16839510C22semidirect
35. C61×#332C22semidirect
36. C61×#331C22semidirect
203. #1540D61semidirect
235. #332C2×D61semidirect
236. #331C2×D61semidirect
237. C61#19391semidirect

Maximal subgroups

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

subgroupindex
#214569442
C2×#168400532
#214552792
#214548422
#214452272
#214451932
#214445892
#214444992
C61×#15402
#1939161