Finite Group Databasebeta

← #29665239

Subgroups — #29665239

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#21456947C211yesmaximal, characteristic, complemented
2C2×#16840049C211yesmaximal, characteristic, complemented
3#21455285C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445231C214yesmaximal, complemented
6#21445197C214yesmaximal, complemented
7#21444591C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic, complemented
9C61×#1548C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11#16840312C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840158C2212yescomplemented
14#16840110C2212yescomplemented
15#16840080—22no—
16#16840049—22no—
17#16840049—22no—
18#16840049C2214yes—
19#16839978—22no—
20#16839968C2212yes—
21#16839958C2212yes—
22C8×D244—22no—
23#16839827C2214yes—
24#16839815C2211yescharacteristic
25C2×#14519708—21no—
26C2×C4⋊#92418C2211yescharacteristic
27#16839764—21no—
28C2×#14519697—21no—
29C22⋊C8×D61—22no—
30#16839512C2214yescomplemented
31#16839507C2212yescomplemented
32#16839505C2212yescomplemented
33C2×C4⋊Q8×C61C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C61×#333C2214yescomplemented
37C2×#11758222—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40#14519805—44no—
41C2×#11758190—22no—
42C2×D488—22no—
43C22×#92418—21no—
44C2×C8×D61—24no—
45#14519777C2311yescharacteristic
46#14519774C2311yescharacteristic
47#14519765—21no—
48#14519751—21no—
49#14519751—21no—
50C4×D244—21no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244—21no—
54C4×D244D412yes—
55C4×D244C2312yes—
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758198—21no—
58#14519708—24no—
59#14519708—24no—
60C4⋊#92418—21no—
61C4⋊#92418D412yes—
62C4⋊#92418C2312yes—
63#14519702—22no—
64#14519699—22no—
65#14519697—24no—
66#14519697—24no—
67C22⋊#92418D412yes—
68#14519689—24no—
69C4×#92418C2312yes—
70C22×Q8×C61—21no—
71C2×M16×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61C2312yes—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×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—
85C2×Q8×C61—22no—
86C2×Q8×C61—41no—
87C2×Q8×C61—82no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C2×C22⋊D61—41no—
93C22×Dic61—21no—
94#11758222—42no—
95#11758222—42no—
96C2×D244—21no—
97C2×D244—22no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244SD1614yes—
102C2×D244C2×D412yes—
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—41no—
106C2×C4×D61—82no—
107C2×Dic122—21no—
108M16×C61—42no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758198—22no—
120#11758198—22no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61C2×D412yes—
127C2×#92418—22no—
128C2×#92418—42no—
129C2×#92418C2×D414yes—
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—
139Q8×C61—82no—
140Q8×C61—82no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—82no—
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61C2×SD1612yes—
149C2×C4×C61#13512yes—
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C22⋊D61—82no—
154C2×Dic61—22no—
155C2×Dic61—41no—
156C2×Dic61—81no—
157D244—22no—
158D244—24no—
159D244—41no—
160D244—41no—
161D244—42no—
162D244—81no—
163D244C2×SD1614yes—
164C4×D61—44no—
165C4×D61—82no—
166C4×D61—82no—
167Dic122—41no—
168Dic122—41no—
169C8×C61—42no—
170C8×C61—42no—
171#92418—24no—
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—82no—
189C4×C61#46512yes—
190C4×C61C22×SD1611yescharacteristic
191C4×C61C2×#13511yescharacteristic
192C4×C61#57712yes—
193Dic61—42no—
194Dic61—81no—
195#3227—611no—
196C2×#476—611no—
197#3036—611no—
198#2972—611no—
199#1677—614no—
200#1671—614no—
201#1570—611no—
202#1548D6111yescharacteristic, complemented
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⋊Q8C2×D6111yescharacteristic
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⋊C8—611no—
230#417—1221no—
231C2×#102—1221no—
232C2×C22⋊C8—1222no—
233#334—614no—
234#333C2×D6114yescomplemented
235#332—612no—
236#331—612no—
237C61#1939111yescharacteristic, complemented, Sylow
238C2×#41—1221no—
239C22×Q8—21no—
240C22×D4—611no—
241C23×C4—1221no—
242AGL(1,ℤ/8)—2444no—
243C2×SD16—1222no—
244C2×D8—1222no—
245C2×M16—21no—
246C22×C8—1224no—
247C22×C8—1221no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—611no—
252C4⋊Q8C22×D6112yes—
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—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—611no—
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—1224no—
278#102—1224no—
279#102—1222no—
280C22⋊C8—21no—
281C22⋊C8—612no—
282C22⋊C8—1224no—
283C4×C8—21no—
284C4×C8—612no—
285C24—1221no—
286#41—2442no—
287#41—2442no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—1221no—
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—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—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—614no—
315C2×C8—1222no—
316C2×C8—4884no—
317C2×C8—2442no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—612no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C22⋊C4—1222no—
327C22⋊C4—1222no—
328C22⋊C4—2441no—
329C22⋊C4—2441no—
330C42#1175822211yescharacteristic
331C42D4×D6111yescharacteristic
332C42D4×D6111yescharacteristic
333C42C23×D6111yescharacteristic
334C23—1224no—
335C23—1222no—
336C23—4881no—
337C23—4881no—
338C23—2441no—
339C23#1451975011yescharacteristic
340Q8—82no—
341Q8—82no—
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—42no—
355C2×C4—82no—
356C2×C4—1222no—
357C2×C4—4882no—
358C2×C4—4882no—
359C2×C4—4881no—
360C2×C4—2441no—
361C2×C4—2444no—
362C2×C4#1451975011yescharacteristic
363C2×C4#1451975011yescharacteristic
364C2×C4#1451975011yescharacteristic
365C2×C4SD16×D6112yes—
366C2×C4#1451980012yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C2×D4×D6111yescharacteristic
369C2×C4C2×#1175822211yescharacteristic
370C8—42no—
371C8—42no—
372C8—1224no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—1224no—
377C22—4884no—
378C22—4881no—
379C22—4881no—
380C22—4881no—
381C22—2442no—
382C22—2442no—
383C22#1683999412yes—
384C22C2×#1451975011yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—4881no—
390C4—2442no—
391C4#1683999412yes—
392C4C2×SD16×D6111yescharacteristic
393C4C2×#1451980011yescharacteristic
394C4#1684057112yes—
395C2—22no—
396C2—9761no—
397C2—2444no—
398C2#2145533011yescharacteristic
399C2C2×#1683999411yescharacteristic
400C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radical#21455285C2
hypercenter#1548D61
hypocenterC61#19391
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #21456947C2semidirect
2. C2×#16840049C2semidirect
4. #21454842C2semidirect
5. #21445231C2semidirect
6. #21445197C2semidirect
7. #21444591C2semidirect
8. #21444466C2semidirect
9. C61×#1548C2semidirect
13. #16840158C22semidirect
14. #16840110C22semidirect
30. #16839512C22semidirect
31. #16839507C22semidirect
32. #16839505C22semidirect
36. C61×#333C22semidirect
202. #1548D61semidirect
234. #333C2×D61semidirect
237. C61#19391semidirect

Maximal subgroups

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

subgroupindex
#214569472
C2×#168400492
#214552852
#214548422
#214452312
#214451972
#214445912
#214444662
C61×#15482
#1939161