Finite Group Databasebeta

← #29665241

Subgroups — #29665241

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#21456949C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3#21455285C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445229C214yesmaximal, complemented
6#21445195C214yesmaximal, complemented
7#21444501C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C61×#1570C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840160C2212yescomplemented
15#16840109C2212yescomplemented
16#16840081—22no—
17#16840053—22no—
18#16840053—22no—
19#16840053C2214yes—
20#16839978—22no—
21#16839968C2212yes—
22#16839958C2212yes—
23C8×D244C2214yes—
24#16839827—22no—
25#16839815C2211yescharacteristic
26C2×#14519709—21no—
27#16839764—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839510C2214yescomplemented
31#16839509C2212yescomplemented
32#16839505C2212yescomplemented
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#334C2214yescomplemented
37C2×#11758222—21no—
38C22×Dic122—21no—
39C22×D244D411yescharacteristic
40C22×C4×D61—21no—
41#14519805—44no—
42C2×#11758190—22no—
43C2×D488—22no—
44C2×C8×D61—24no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777C2311yescharacteristic
49#14519777C2312yes—
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244—21no—
57C4×D244—21no—
58C4×D244D412yes—
59C4×D244C2312yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×#11758198—21no—
64#14519709—24no—
65#14519709—24no—
66C4⋊#92418C2312yes—
67#14519702—22no—
68#14519699—24no—
69#14519699—24no—
70#14519699—22no—
71C22⋊#92418—21no—
72#14519689—24no—
73C4×#92418—21no—
74C2×M16×C61—21no—
75C22×C8×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—21no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C2×C22⋊D61—41no—
89C22×Dic61—21no—
90C22×Dic61—41no—
91#11758222—42no—
92#11758222—42no—
93C2×D244—21no—
94C2×D244—22no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244SD1614yes—
99C2×D244C2×D412yes—
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—41no—
103C2×C4×D61—82no—
104C2×Dic122—21no—
105C2×Dic122—22no—
106C2×Dic122—41no—
107C2×Dic122—82no—
108M16×C61—42no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—42no—
112C2×C8×C61C2×D414yes—
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61#4111yescharacteristic
116C42×C61C2411yescharacteristic
117#11758198—22no—
118#11758198—22no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—22no—
123C4⋊Dic61—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4⋊Dic61C2×D412yes—
129C2×#92418—22no—
130D488—44no—
131#11758190—44no—
132C8×D61—84no—
133C23×C61#12111yescharacteristic
134C22×D61—22no—
135C22×D61—24no—
136C22×D61—41no—
137C22×D61—81no—
138C22×D61—81no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61C2×SD1612yes—
145C2×C4×C61#13512yes—
146C2×C4×C61C22×D411yescharacteristic
147C2×C4×C61C22×D411yescharacteristic
148C2×C4×C61C2×#4111yescharacteristic
149C22⋊D61—82no—
150C2×Dic61—22no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—81no—
154C2×Dic61—82no—
155D244—22no—
156D244—24no—
157D244—41no—
158D244—41no—
159D244—42no—
160D244—81no—
161D244C2×SD1614yes—
162C4×D61—44no—
163C4×D61—82no—
164C4×D61—82no—
165Dic122—41no—
166Dic122—41no—
167Dic122—82no—
168Dic122—82no—
169C8×C61—24no—
170C8×C61—42no—
171C8×C61—42no—
172#92418—42no—
173#19391—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61#46512yes—
177C22×C61C2×#12111yescharacteristic
178C2×D61—24no—
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—84no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61#46512yes—
189C4×C61C22×SD1611yescharacteristic
190C4×C61C2×#13511yescharacteristic
191C4×C61#57712yes—
192Dic61—42no—
193Dic61—81no—
194Dic61—82no—
195#3227—611no—
196C2×#476—611no—
197#3036—611no—
198#2972—611no—
199#1677—614no—
200#1671—614no—
201#1570D6111yescharacteristic, complemented
202#1548—611no—
203#1540—611no—
204C2×C61—22no—
205C2×C61#304211yescharacteristic
206C2×C61C2×#46511yescharacteristic
207C2×C61#306811yescharacteristic
208D61—44no—
209D61—161no—
210C2×AGL(1,ℤ/8)—1221no—
211#532—611no—
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#423C2×D6111yescharacteristic
229C2×C4⋊C8C2×D6111yescharacteristic
230#417—1221no—
231C2×#102—1221no—
232C2×C22⋊C8—1222no—
233#334C2×D6114yescomplemented
234#333—614no—
235#332—612no—
236#331—612no—
237C61#1939111yescharacteristic, complemented, Sylow
238C2×#41—1221no—
239C22×Q8—1221no—
240C22×D4—611no—
241C23×C4—1221no—
242AGL(1,ℤ/8)—2444no—
243C2×SD16—1222no—
244C2×D8—1222no—
245C2×M16—21no—
246C22×C8—21no—
247C22×C8—1224no—
248C4⋊Q8—612no—
249C4⋊Q8—611no—
250C4⋊Q8—1221no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253#127—611no—
254#124—1221no—
255#121—1221no—
256#121—1221no—
257C4×D4—612no—
258C4×D4—612no—
259C4×D4—1221no—
260C4×D4—1221no—
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—21no—
272C4⋊C8—612no—
273C4⋊C8D24412yes—
274C4⋊C8C22×D6112yes—
275C4⋊C8C22×D6112yes—
276#103—1222no—
277#102—1222no—
278#102—1224no—
279#102—1224no—
280C22⋊C8—1221no—
281C22⋊C8—1224no—
282C22⋊C8D24412yes—
283C4×C8—1221no—
284C4×C8C22×D6112yes—
285C24—1221no—
286#41—2442no—
287#41—2442no—
288C2×Q8—1221no—
289C2×Q8—1222no—
290C2×Q8—4882no—
291C2×Q8—2441no—
292C2×D4—612no—
293C2×D4—614no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—1222no—
299C2×D4—2441no—
300C22×C4—1221no—
301C22×C4—1222no—
302C22×C4—1222no—
303C22×C4—4882no—
304C22×C4—2441no—
305C22×C4—2441no—
306C22×C4C2×D24411yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4#1175822611yescharacteristic
309SD16—2444no—
310D8—2444no—
311M16—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—1222no—
316C2×C8—4884no—
317C2×C8C2×D24414yes—
318C4⋊C4—612no—
319C4⋊C4—1221no—
320C4⋊C4—1222no—
321C4⋊C4—1222no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C22⋊C4—1222no—
327C22⋊C4—1222no—
328C22⋊C4—2441no—
329C22⋊C4—2441no—
330C42C2×D24411yescharacteristic
331C42D4×D6111yescharacteristic
332C42#1175822611yescharacteristic
333C42C23×D6111yescharacteristic
334C23—1224no—
335C23—1222no—
336C23—4881no—
337C23—4881no—
338C23—2441no—
339C23#1451975211yescharacteristic
340Q8—4882no—
341Q8—4882no—
342Q8—2441no—
343Q8—2441no—
344D4—614no—
345D4—1224no—
346D4—1222no—
347D4—4881no—
348D4—4882no—
349D4—2441no—
350D4—2441no—
351D4—2442no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—1222no—
355C2×C4—4882no—
356C2×C4—4882no—
357C2×C4—4882no—
358C2×C4—4881no—
359C2×C4—2441no—
360C2×C4—2442no—
361C2×C4—2444no—
362C2×C4#1451975211yescharacteristic
363C2×C4#1451975211yescharacteristic
364C2×C4#1451975211yescharacteristic
365C2×C4SD16×D6112yes—
366C2×C4#1451979812yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C22×D24411yescharacteristic
369C2×C4C2×#1175822611yescharacteristic
370C8—24no—
371C8—42no—
372C8—42no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—1224no—
377C22—4884no—
378C22—4881no—
379C22—4881no—
380C22—4881no—
381C22—2442no—
382C22—2442no—
383C22#1683999212yes—
384C22C2×#1451975211yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—4881no—
389C4—4882no—
390C4—2442no—
391C4#1683999212yes—
392C4C2×SD16×D6111yescharacteristic
393C4C2×#1451979811yescharacteristic
394C4#1684057412yes—
395C2—22no—
396C2—9761no—
397C2—2444no—
398C2#2145533211yescharacteristic
399C2C2×#1683999211yescharacteristic
400C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1570C2
socleC22×C61C2×#121
prime radical#21455285C2
hypercenter#1570D61
hypocenterC61#19391
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
1. #21456949C2semidirect
2. C2×#16840053C2semidirect
4. #21454842C2semidirect
5. #21445229C2semidirect
6. #21445195C2semidirect
7. #21444501C2semidirect
8. #21444469C2semidirect
9. C61×#1570C2semidirect
14. #16840160C22semidirect
15. #16840109C22semidirect
30. #16839510C22semidirect
31. #16839509C22semidirect
32. #16839505C22semidirect
36. C61×#334C22semidirect
201. #1570D61semidirect
233. #334C2×D61semidirect
237. C61#19391semidirect

Maximal subgroups

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

subgroupindex
#214569492
C2×#168400532
#214552852
#214548422
#214452292
#214451952
#214445012
#214444692
C61×#15702
#1939161