Finite Group Databasebeta

← #29663820

Subgroups — #29663820

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic, complemented
2#21455919C211yesmaximal, characteristic, complemented
3#21455911C211yesmaximal, characteristic
4C2×C8×D244C211yesmaximal, characteristic, complemented
5#21445271C212yesmaximal, complemented
6#21445257C212yesmaximal, complemented
7#21445195C212yesmaximal, complemented
8#21445161C212yesmaximal, complemented
9#21444520C211yesmaximal, characteristic, complemented
10#21444515C211yesmaximal, characteristic, complemented
11C61×#1568C211yesmaximal, characteristic, complemented
12C22×C8×D61—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16840160C2211yescharacteristic, complemented
17#16840135C2211yescharacteristic, complemented
18#16840113C2211yescharacteristic, complemented
19#16840109C2211yescharacteristic, complemented
20#16840098—21no—
21#16840081—21no—
22#16840053C2212yes—
23#16840043C2212yes—
24#16839991—21no—
25#16839991—21no—
26#16839968C2212yes—
27#16839958C2212yes—
28#16839915—21no—
29#16839898—21no—
30#16839875—22no—
31#16839864—22no—
32C4×#11758190—21no—
33C4×#11758190—21no—
34C8×D244—21no—
35C8×D244—21no—
36C8×D244C2212yes—
37C8×D244C2212yes—
38C2×C4⋊#92418C2211yescharacteristic
39C2×#14519689—21no—
40#16839510C2212yescomplemented
41#16839509C2212yescomplemented
42#16839507C2212yescomplemented
43#16839505C2212yescomplemented
44C61×#432C2211yescharacteristic
45C61×#432C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C61×#335C2212yescomplemented
48C61×#334C2212yescomplemented
49C2×#11758222—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758190—22no—
53C2×#11758190—21no—
54C2×#11758190—21no—
55C2×D488—22no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58C2×C8×D61—22no—
59C2×C8×D61—44no—
60#14519777C2311yescharacteristic
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519769—21no—
65#14519765—21no—
66#14519756—21no—
67#14519756—21no—
68#14519751—21no—
69#14519751—21no—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×D244—21no—
73C4×D244—21no—
74C4×D244—21no—
75C4×D244—21no—
76C4×D244D412yes—
77C4×D244C2312yes—
78#14519736—21no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×#11758198—21no—
81#14519711—22no—
82#14519709—21no—
83#14519709—21no—
84#14519709—22no—
85C4⋊#92418—21no—
86C4⋊#92418C2312yes—
87#14519702—21no—
88#14519702—21no—
89#14519702—22no—
90#14519699—22no—
91#14519699—22no—
92#14519699—21no—
93#14519699—21no—
94#14519699—22no—
95#14519689—22no—
96#14519689—22no—
97C22×C8×C61D411yescharacteristic
98C22×C8×C61D411yescharacteristic
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61C2312yes—
101C4⋊C8×C61C2312yes—
102C22⋊C8×C61—21no—
103C22⋊C8×C61—21no—
104#97×C61—21no—
105#97×C61—21no—
106C4×C8×C61D411yescharacteristic
107C4×C8×C61D411yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C23×D61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—41no—
115C22×Dic61—21no—
116#11758222—82no—
117C2×D244—21no—
118C2×D244—24no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244C2×D412yes—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—41no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143C22⋊Dic61—41no—
144#11758198—22no—
145#11758198—22no—
146#11758198—41no—
147#11758198—41no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61C2×D412yes—
156#11758196—41no—
157#11758196—41no—
158C4×Dic61—41no—
159C2×#92418—22no—
160C2×#92418—42no—
161D488—44no—
162#11758190—42no—
163#11758190—44no—
164#11758190—42no—
165C8×D61—44no—
166C8×D61—44no—
167C23×C61#12111yescharacteristic
168C22×D61—24no—
169C22×D61—22no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—41no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C22⋊D61—82no—
184C2×Dic61—22no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189D244—24no—
190D244—22no—
191D244—42no—
192D244—41no—
193D244—41no—
194D244—81no—
195D244#13514yes—
196C4×D61—44no—
197C4×D61—42no—
198C4×D61—42no—
199C4×D61—82no—
200Dic122—42no—
201Dic122—41no—
202Dic122—41no—
203Dic122—81no—
204C8×C61—21no—
205C8×C61—21no—
206C8×C61—21no—
207C8×C61—42no—
208C8×C61—42no—
209C8×C61AGL(1,ℤ/8)12yes—
210#92418—42no—
211#19257—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61#47112yes—
215C22×C61C2×#12111yescharacteristic
216C2×D61—24no—
217C2×D61—48no—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61C2×#13512yes—
228C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
229C4×C61#57511yescharacteristic
230Dic61—42no—
231Dic61—81no—
232Dic61—81no—
233Dic61—81no—
234#3167—611no—
235#3110—611no—
236#3109—611no—
237C2×C8×D4—611no—
238#1683—612no—
239#1681—612no—
240#1671—612no—
241#1665—612no—
242#1568D6111yescharacteristic, complemented
243#1552—611no—
244#1551—611no—
245C2×C61—22no—
246C2×C61#303611yescharacteristic
247C2×C61#305011yescharacteristic
248C2×C61C2×#47111yescharacteristic
249D61—44no—
250D61—161no—
251C23×C8—1221no—
252#533—611no—
253#532—611no—
254C2×C4×D4—611no—
255#498—611no—
256#492—611no—
257#487—611no—
258#486—611no—
259#484—1221no—
260#481—1221no—
261#476—612no—
262#474—612no—
263#465—1221no—
264#465—1221no—
265#461—612no—
266#459—612no—
267#450—1221no—
268#447—1221no—
269#442—1222no—
270#440—1222no—
271C4×SD16—1221no—
272C4×SD16—1221no—
273C8×D4—612no—
274C8×D4—612no—
275C8×D4—1221no—
276C8×D4—1221no—
277#432C2×D6111yescharacteristic
278#432C2×D6111yescharacteristic
279C2×C4⋊C8—611no—
280C2×C22⋊C8—1221no—
281C2×C4×C8C2×D6111yescharacteristic
282#335C2×D6112yescomplemented
283#334C2×D6112yescomplemented
284#333—612no—
285#332—612no—
286#332—612no—
287#331—612no—
288C61#1925711yescharacteristic, complemented, Sylow
289C2×#41—1221no—
290C22×D4—611no—
291C23×C4—1221no—
292C2×SD16—1222no—
293C2×SD16—1221no—
294C2×SD16—1221no—
295C2×D8—1222no—
296C22×C8—1221no—
297C22×C8—1222no—
298C22×C8—1222no—
299C22×C8—2444no—
300C22×C8D24411yescharacteristic
301C22×C8D24411yescharacteristic
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305#127—611no—
306#125—1221no—
307#124—1221no—
308C22⋊Q8—1221no—
309C22⋊Q8—1221no—
310#121—1221no—
311#121—1221no—
312C4×Q8—1221no—
313C4×Q8—1221no—
314C4×D4—612no—
315C4×D4—612no—
316C4×D4—1221no—
317C4×D4—1221no—
318C4×D4—1221no—
319C4×D4—1221no—
320#117—1221no—
321C2×C4⋊C4—611no—
322C2×C22⋊C4—1221no—
323C2×C42C22×D6111yescharacteristic
324#107—1222no—
325#106—1222no—
326#106—1221no—
327#106—1221no—
328C4⋊C8—612no—
329C4⋊C8—1221no—
330C4⋊C8C22×D6112yes—
331C4⋊C8C22×D6112yes—
332#103—1222no—
333#103—1221no—
334#103—1221no—
335#102—1222no—
336#102—1222no—
337#102—1222no—
338#102—1221no—
339#102—1221no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—1222no—
343C22⋊C8—1222no—
344#97—21no—
345#97—21no—
346C4×C8D24411yescharacteristic
347C4×C8D24411yescharacteristic
348C4×C8C22×D6111yescharacteristic
349C4×C8C22×D6111yescharacteristic
350C24—1221no—
351#41—4882no—
352C2×Q8—1221no—
353C2×Q8—1221no—
354C2×Q8—1221no—
355C2×D4—612no—
356C2×D4—1224no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—2441no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4—1222no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2441no—
368C22×C4C2×D24411yescharacteristic
369C22×C4D4×D6111yescharacteristic
370C22×C4#1175822611yescharacteristic
371SD16—2442no—
372SD16—2444no—
373SD16—2442no—
374D8—2444no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—1222no—
380C2×C8—2444no—
381C2×C8—2442no—
382C2×C8—2444no—
383C2×C8C2×D24411yescharacteristic
384C2×C8C2×D24411yescharacteristic
385C2×C8C2×D24411yescharacteristic
386C2×C8C2×D24411yescharacteristic
387C4⋊C4—612no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C22⋊C4—1222no—
398C22⋊C4—1222no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C42—2441no—
403C42C2×D24411yescharacteristic
404C42D4×D6111yescharacteristic
405C42#1175822611yescharacteristic
406C42C23×D6111yescharacteristic
407C23—1224no—
408C23—1222no—
409C23—2442no—
410C23—2441no—
411C23—2442no—
412C23#1451975211yescharacteristic
413Q8—4881no—
414Q8—2442no—
415Q8—2441no—
416Q8—2441no—
417D4—614no—
418D4—1224no—
419D4—1222no—
420D4—4881no—
421D4—4882no—
422D4—2442no—
423D4—2441no—
424D4—2441no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—1222no—
430C2×C4—4882no—
431C2×C4—2444no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4#1451975211yescharacteristic
439C2×C4#1451975211yescharacteristic
440C2×C4#1451975211yescharacteristic
441C2×C4C2×D4×D6111yescharacteristic
442C2×C4C22×D24411yescharacteristic
443C2×C4C2×#1175822611yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8—42no—
448C8—42no—
449C8—2442no—
450C8#1451980512yes—
451C22—21no—
452C22—21no—
453C22—1224no—
454C22—4881no—
455C22—4881no—
456C22—4881no—
457C22—2448no—
458C22—2442no—
459C22—2442no—
460C22#1684002612yes—
461C22C2×#1451975211yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—4881no—
467C4—4881no—
468C4—4881no—
469C4—2442no—
470C4#135×D6112yes—
471C4C2×#1451980511yescharacteristic
472C4#1684056211yescharacteristic
473C2—22no—
474C2—9761no—
475C2—2444no—
476C2#2145528511yescharacteristic
477C2#2145540111yescharacteristic
478C2C2×#1684002611yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1568C2
socleC22×C61C2×#121
prime radical#21455911C2
hypercenter#1568D61
hypocenterC61#19257
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1568

Complements

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

NH ≅ G/Ntype
1. #21456378C2semidirect
2. #21455919C2semidirect
4. C2×C8×D244C2semidirect
5. #21445271C2semidirect
6. #21445257C2semidirect
7. #21445195C2semidirect
8. #21445161C2semidirect
9. #21444520C2semidirect
10. #21444515C2semidirect
11. C61×#1568C2semidirect
16. #16840160C22semidirect
17. #16840135C22semidirect
18. #16840113C22semidirect
19. #16840109C22semidirect
40. #16839510C22semidirect
41. #16839509C22semidirect
42. #16839507C22semidirect
43. #16839505C22semidirect
47. C61×#335C22semidirect
48. C61×#334C22semidirect
242. #1568D61semidirect
282. #335C2×D61semidirect
283. #334C2×D61semidirect
288. C61#19257semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214559192
#214559112
C2×C8×D2442
#214452712
#214452572
#214451952
#214451612
#214445202
#214445152
C61×#15682
#1925761