Finite Group Databasebeta

← #29715238

Subgroups — #29715238

Order 15616.

1494 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455279C212yesmaximal, complemented
2C2×#16839930C212yesmaximal, complemented
3#21448541C211yesmaximal, characteristic, complemented
4#21447045C212yesmaximal, complemented
5#21446861C212yesmaximal, complemented
6#21446652C212yesmaximal, complemented
7#21445811C211yesmaximal, characteristic, complemented
8#21445773C212yesmaximal, complemented
9C61×#3496C211yesmaximal, characteristic, complemented
10C2×#14519802C2214yescomplemented
11C2×#14519773C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic, complemented
13#16839974—24no—
14#16839964—22no—
15#16839954—22no—
16#16839930—28no—
17#16839786C2212yescomplemented
18C2×#14519704C2214yescomplemented
19C2×#14519704C2214yescomplemented
20#16839763C2214yescomplemented
21C2×#14519694C2212yes—
22#16839665C2211yescharacteristic
23#16839663C2211yescharacteristic, complemented
24#16839641—22no—
25#16839638—22no—
26#16839627C2211yescharacteristic, complemented
27#16839608C2211yescharacteristic, complemented
28#16839531C2212yescomplemented
29C2×H(5,𝔽2)×C61C2212yes—
30C61×#547—22no—
31C61×#546—24no—
32D42×C61—22no—
33C61×#537C2212yes—
34C61×#534—22no—
35C61×#532C2212yes—
36C2×#121×C61—22no—
37C2×C4×D4×C61C2211yescharacteristic
38C22×D244C2311yescharacteristic
39C22×C22⋊D61—22no—
40C22×C4×D61—21no—
41#14519802—28no—
42#14519802—44no—
43C2×#11758201—24no—
44C2×#11758199—24no—
45C2×#11758194C2314yes—
46#14519773—22no—
47#14519773D414yescomplemented
48C4×D244—22no—
49C4×D244—22no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C22⋊Dic61—22no—
52C2×#11758198—21no—
53C2×#11758198—22no—
54C2×C4×Dic61C2311yescharacteristic
55C2×C4×Dic61C2311yescharacteristic
56C4⋊#92418—22no—
57C4⋊#92418—22no—
58#14519704—28no—
59#14519704—44no—
60#14519701—24no—
61#14519698—24no—
62#14519694—24no—
63#14519684—21no—
64#14519684—21no—
65#14519684—22no—
66H(5,𝔽2)×C61—24no—
67H(5,𝔽2)×C61—24no—
68H(5,𝔽2)×C61—44no—
69C2×#41×C61—24no—
70C2×#41×C61C2314yes—
71C22×D4×C61—24no—
72C22×D4×C61—22no—
73C22×D4×C61—24no—
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61D412yes—
76C4⋊Q8×C61D412yescomplemented
77#127×C61D412yescomplemented
78#124×C61D414yescomplemented
79#123×C61—28no—
80C22⋊Q8×C61—24no—
81#121×C61—24no—
82#121×C61—24no—
83#121×C61—24no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—24no—
87#120×C61—28no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61—24no—
95C2×C22⋊C4×C61D412yes—
96C2×C42×C61C2311yescharacteristic
97C24×C61—24no—
98C24×C61C2×D412yes—
99C23×D61—21no—
100#41×C61—24no—
101#41×C61—24no—
102#41×C61—24no—
103#41×C61—24no—
104#41×C61—48no—
105#41×C61—48no—
106C2×Q8×C61C2×D414yes—
107C2×D4×C61—28no—
108C2×D4×C61—24no—
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—48no—
115C2×D4×C61—48no—
116C2×D4×C61—48no—
117C2×D4×C61—48no—
118C2×D4×C61—84no—
119C2×D4×C61C2×D414yes—
120C2×D4×C61C2×D412yes—
121C2×D4×C61#4112yes—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61—24no—
125C22×C4×C61—22no—
126C22×C4×C61—82no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—28no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61—82no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244C2×D412yescomplemented
141C2×C4×D61—42no—
142C2×C4×D61—82no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C22⋊C4×C61—28no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—24no—
151C42×C61C2×D412yes—
152C42×C61C2×D412yes—
153C22⋊Dic61—82no—
154#11758201—48no—
155#11758199—48no—
156#11758198—41no—
157#11758198—41no—
158#11758198—82no—
159C4⋊Dic61—22no—
160C4×Dic61—21no—
161C4×Dic61—22no—
162C4×Dic61C2×D412yescomplemented
163#11758194—24no—
164#11758194—24no—
165C2×#92418—24no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—24no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—48no—
172C23×C61—84no—
173C23×C61—84no—
174C23×C61#12114yes—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—41no—
180C22×D61—41no—
181Q8×C61—24no—
182Q8×C61—24no—
183D4×C61—24no—
184D4×C61—24no—
185D4×C61—24no—
186D4×C61—24no—
187D4×C61—44no—
188D4×C61—416no—
189D4×C61—44no—
190D4×C61—44no—
191D4×C61—82no—
192D4×C61—84no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—44no—
198C2×C4×C61—48no—
199C2×C4×C61—44no—
200C2×C4×C61—44no—
201C2×C4×C61—42no—
202C2×C4×C61—82no—
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61#12012yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D412yes—
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61H(5,𝔽2)12yes—
210C22⋊D61—88no—
211C2×Dic61—22no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219D244—24no—
220D244—42no—
221C4×D61—84no—
222#92418—44no—
223#24226—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—24no—
227C22×C61—44no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—44no—
231C22×C61—44no—
232C22×C61—44no—
233C22×C61—82no—
234C22×C61—168no—
235C22×C61C2×#12011yescharacteristic
236C22×C61C2×#12111yescharacteristic
237C22×C61#53511yescharacteristic
238C2×D61—44no—
239C2×D61—42no—
240C2×D61—82no—
241C2×D61—81no—
242C2×D61—81no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—22no—
246C4×C61—44no—
247C4×C61—42no—
248C4×C61D4212yes—
249C4×C61#54612yes—
250Dic61—42no—
251Dic61—81no—
252Dic61—81no—
253#3496D6111yescharacteristic, complemented
254#3036—612no—
255C2×#453—612no—
256#2328—611no—
257#2040—612no—
258#2006—612no—
259#1963—612no—
260#1800—611no—
261#1792—612no—
262C2×C61—21no—
263C2×C61—21no—
264C2×C61—44no—
265C2×C61—84no—
266C2×C61#240811yescharacteristic
267C2×C61#301911yescharacteristic
268C2×C61#302111yescharacteristic
269D61—82no—
270C2×H(5,𝔽2)C2×D6112yes—
271C2×AGL(1,ℤ/8)—614no—
272#547—22no—
273#546—24no—
274D42—22no—
275#537C2×D6112yes—
276#534—22no—
277#532C2×D6112yes—
278C2×#127—611no—
279C2×#121—22no—
280C2×C4×D4—611no—
281C2×C4×D4C2×D6111yescharacteristic
282#463—1224no—
283#461—1222no—
284#459—1222no—
285#453—1228no—
286#423—612no—
287C2×#104—614no—
288C2×#104—614no—
289#417—614no—
290C2×#100—612no—
291#390—611no—
292#390—611no—
293#386—1222no—
294#386—1222no—
295#382—611no—
296#376—611no—
297#343—612no—
298C61#2422611yescharacteristic, complemented, Sylow
299H(5,𝔽2)—24no—
300H(5,𝔽2)—24no—
301H(5,𝔽2)—44no—
302C2×#41—24no—
303C2×#41C22×D6114yes—
304C22×D4—24no—
305C22×D4—22no—
306C22×D4—24no—
307C22×D4—611no—
308C22×D4—1222no—
309C22×D4C22×D6111yescharacteristic
310C23×C4—1221no—
311C23×C4C22⋊D6112yes—
312AGL(1,ℤ/8)—1228no—
313AGL(1,ℤ/8)—2444no—
314C2×SD16—1224no—
315C2×D8—1224no—
316C2×M16—614no—
317C4⋊Q8C22⋊D6112yescomplemented
318#127—614no—
319#127—1222no—
320#127C22⋊D6112yescomplemented
321#124C22⋊D6114yescomplemented
322#123—28no—
323C22⋊Q8—24no—
324#121—24no—
325#121—24no—
326#121—24no—
327#121—22no—
328#121—24no—
329#121—22no—
330#120—28no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—1222no—
336C4×D4—1222no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C22⋊C4—24no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4C22⋊D6112yes—
345C2×C42—611no—
346C2×C42—611no—
347C2×C42C22×D6111yescharacteristic
348C4⋊C8—1222no—
349C4⋊C8—1222no—
350#104—1228no—
351#104—2444no—
352#103—1224no—
353#102—1224no—
354#100—1224no—
355#95—1221no—
356#95—1221no—
357#95—1222no—
358C24—24no—
359C24—1221no—
360C24C2×C22⋊D6112yes—
361#41—24no—
362#41—24no—
363#41—24no—
364#41—24no—
365#41—48no—
366#41—48no—
367C2×Q8C2×C22⋊D6114yes—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—84no—
380C2×D4—612no—
381C2×D4—1228no—
382C2×D4—1222no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—4882no—
386C2×D4—4882no—
387C2×D4D4×D6112yes—
388C2×D4#1175822412yes—
389C2×D4C2×C22⋊D6114yes—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—82no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—4882no—
399C22×C4—2442no—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404SD16—2448no—
405D8—2448no—
406M16—1224no—
407M16—1224no—
408C2×C8—1224no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—1222no—
414C22⋊C4—28no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—4882no—
419C22⋊C4—4882no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C42—612no—
423C42—1221no—
424C42—1222no—
425C42C2×C22⋊D6112yes—
426C42C2×C22⋊D6112yes—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—42no—
431C23—42no—
432C23—48no—
433C23—84no—
434C23—84no—
435C23—1222no—
436C23—2442no—
437C23—2442no—
438C23—2441no—
439C23—2441no—
440C23#1451974814yes—
441C23C22×C22⋊D6111yescharacteristic
442Q8—24no—
443Q8—24no—
444D4—24no—
445D4—24no—
446D4—24no—
447D4—24no—
448D4—44no—
449D4—416no—
450D4—44no—
451D4—44no—
452D4—82no—
453D4—84no—
454D4—1224no—
455D4—4888no—
456D4—2442no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—1222no—
468C2×C4—4884no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451974611yescharacteristic
477C2×C4#1451974612yes—
478C2×C4#1451974611yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C22×C22⋊D6112yes—
481C2×C4C2×#1175822411yescharacteristic
482C2×C4#1451982212yes—
483C8—2444no—
484C22—21no—
485C22—21no—
486C22—24no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—82no—
494C22—168no—
495C22—4882no—
496C22—4881no—
497C22—4881no—
498C22—2444no—
499C22—2442no—
500C22C2×#1451974611yescharacteristic
501C22C2×#1451974811yescharacteristic
502C22#1684032911yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—42no—
507C4—44no—
508C4—4881no—
509C4—4881no—
510C4—2442no—
511C4D4×C22⋊D6112yes—
512C4#1684040612yes—
513C2—21no—
514C2—21no—
515C2—44no—
516C2—84no—
517C2—4882no—
518C2#2144916911yescharacteristic
519C2#2145517611yescharacteristic
520C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#24226
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21455279C2semidirect
2. C2×#16839930C2semidirect
3. #21448541C2semidirect
4. #21447045C2semidirect
5. #21446861C2semidirect
6. #21446652C2semidirect
7. #21445811C2semidirect
8. #21445773C2semidirect
9. C61×#3496C2semidirect
10. C2×#14519802C22semidirect
12. C2×C4×D244C22semidirect
17. #16839786C22semidirect
18. C2×#14519704C22semidirect
19. C2×#14519704C22semidirect
20. #16839763C22semidirect
23. #16839663C22semidirect
26. #16839627C22semidirect
27. #16839608C22semidirect
28. #16839531C22semidirect
47. #14519773D4semidirect
76. C4⋊Q8×C61D4semidirect
77. #127×C61D4semidirect
78. #124×C61D4semidirect
140. C2×D244C2×D4semidirect
162. C4×Dic61C2×D4semidirect
253. #3496D61semidirect
298. C61#24226semidirect
317. C4⋊Q8C22⋊D61semidirect
320. #127C22⋊D61semidirect
321. #124C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214552792
C2×#168399302
#214485412
#214470452
#214468612
#214466522
#214458112
#214457732
C61×#34962
#2422661