Finite Group Databasebeta

← #29654430

Subgroups — #29654430

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459583C211yesmaximal, characteristic
2#21456621C212yesmaximal, complemented
3#21444821C212yesmaximal, complemented
4#21444804C212yesmaximal, complemented
5#21444787C212yesmaximal, complemented
6#21444770C212yesmaximal, complemented
7#21444347C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#2638C211yesmaximal, characteristic, complemented
10C2×#14519822—21no—
11#16840416—22no—
12#16840407—24no—
13D4×D244—22no—
14#16840343C2212yes—
15#16840324—21no—
16#16840324C2212yes—
17#16840312—21no—
18C2×#14519751—21no—
19C2×#14519751C2212yes—
20#16840095—22no—
21#16840075C2214yescomplemented
22#16840010—24no—
23#16839912—22no—
24#16839901C2214yescomplemented
25#16839726C2212yescomplemented
26#16839495C2214yescomplemented
27#16839495C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yescomplemented
30C2×C22⋊Q8×C61C2211yescharacteristic, complemented
31C2×C4×D4×C61C2211yescharacteristic
32C61×#398C2211yescharacteristic, complemented
33C61×#397—21no—
34C61×#397C2211yescharacteristic, complemented
35C61×#395C2211yescharacteristic, complemented
36C61×#394—21no—
37C61×#393C2211yescharacteristic, complemented
38C61×#387—21no—
39C61×#385—21no—
40C61×#381C2211yescharacteristic, complemented
41#14519822—44no—
42#14519822—44no—
43C2×#11758222—22no—
44C2×#11758224—24no—
45C22×D244—21no—
46C22×D244D412yes—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C2×D4×D61—24no—
50C2×#11758201—24no—
51C2×#11758194—22no—
52#14519777—21no—
53#14519774—21no—
54#14519765—22no—
55#14519761—24no—
56#14519761—24no—
57#14519757—24no—
58#14519752—24no—
59#14519751—22no—
60#14519751—22no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751—22no—
65#14519751D414yes—
66#14519751C2314yes—
67#14519747—24no—
68#14519747—24no—
69C4×D244—22no—
70C4×D244—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D412yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75#14519710—24no—
76#14519701—24no—
77#14519697—24no—
78C22⋊#92418—22no—
79C22⋊#92418C2314yes—
80C22×Q8×C61C2311yescharacteristic
81C22×D4×C61D411yescharacteristic
82C23×C4×C61D411yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61C2311yescharacteristic
99C2×C42×C61D411yescharacteristic
100#95×C61—21no—
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—21no—
105#95×C61C2311yescharacteristic
106#95×C61C2311yescharacteristic
107C24×C61C2×D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—81no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—81no—
123C22×C4×C61—81no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61—44no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140#11758224—416no—
141D4×D61—416no—
142#11758222—44no—
143#11758222—44no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—82no—
149C2×D244—82no—
150C2×D244C2×D414yes—
151C2×C4×D61—24no—
152C2×Dic122—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61—41no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—81no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—42no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—81no—
164C22⋊C4×C61—81no—
165C42×C61—21no—
166C42×C61—21no—
167#11758201—84no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61C2×D414yes—
176#11758194—82no—
177C2×#92418—24no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61#12012yes—
187C23×C61AGL(1,ℤ/8)12yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—24no—
190C22×D61—44no—
191C22×D61—44no—
192C22×D61—82no—
193C22×D61—82no—
194C22×D61—82no—
195C22×D61—82no—
196Q8×C61—44no—
197D4×C61—28no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—81no—
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61AGL(1,ℤ/8)12yes—
225C2×C4×C61C22×D412yes—
226C22⋊D61—48no—
227C22⋊D61—48no—
228C22⋊D61—44no—
229C22⋊D61—44no—
230C2×Dic61—24no—
231C2×Dic61—44no—
232C2×Dic61—44no—
233D244—44no—
234D244—44no—
235D244—44no—
236D244—82no—
237D244—82no—
238C4×D61—48no—
239Dic122—44no—
240#92418—44no—
241#18351—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C2×#12011yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C2×D61—44no—
258C2×D61—44no—
259C2×D61—44no—
260C2×D61—164no—
261C2×D61—164no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61—81no—
272Dic61—44no—
273#3496—611no—
274#3193—612no—
275#2638D6111yescharacteristic, complemented
276#1610—612no—
277#1607—612no—
278#1604—612no—
279#1601—612no—
280#1510—612no—
281#1465—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—42no—
285C2×C61—42no—
286C2×C61#300311yescharacteristic
287C2×C61#302211yescharacteristic
288C2×C61#302611yescharacteristic
289D61—84no—
290C2×H(5,𝔽2)—1221no—
291#547—1222no—
292#546—1224no—
293D42—1222no—
294#537—612no—
295#534—612no—
296#534—1221no—
297#532—1221no—
298C2×C22⋊Q8C2×D6111yescharacteristic, complemented
299C2×#121—612no—
300C2×#121—1221no—
301C2×C4×D4C2×D6111yescharacteristic
302#484—1222no—
303#480—614no—
304#468—1224no—
305#450—1222no—
306#448—614no—
307#407—612no—
308#398C2×D6111yescharacteristic, complemented
309#397—21no—
310#397C2×D6111yescharacteristic, complemented
311#395C2×D6111yescharacteristic, complemented
312#394—21no—
313#393C2×D6111yescharacteristic, complemented
314#387—21no—
315#385—21no—
316#381C2×D6111yescharacteristic, complemented
317#327—614no—
318#327—614no—
319#324—612no—
320#323—612no—
321C61#1835111yescharacteristic, complemented, Sylow
322H(5,𝔽2)—2444no—
323H(5,𝔽2)—2444no—
324C2×#41—1224no—
325C2×#41—1222no—
326C22×Q8C22×D6111yescharacteristic
327C22×D4—612no—
328C22×D4—1221no—
329C22×D4—1224no—
330C22×D4—1222no—
331C22×D4—1222no—
332C22×D4C22⋊D6111yescharacteristic
333C23×C4C22⋊D6111yescharacteristic
334C23×C4C22×D6111yescharacteristic
335C2×SD16—1224no—
336C2×M16—1222no—
337C4⋊Q8—1221no—
338#127—1221no—
339#124—1222no—
340#123—1224no—
341#123—1224no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—1224no—
345#121—614no—
346#121—614no—
347#121—1224no—
348#121—1222no—
349#121—1222no—
350#121—1222no—
351#121—1222no—
352#121—1222no—
353#121—1222no—
354#120—1224no—
355#120—1224no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—1222no—
359C4×D4—1222no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4C22⋊D6111yescharacteristic
374C2×C22⋊C4C22×D6111yescharacteristic
375C2×C42C22⋊D6111yescharacteristic
376#107—1224no—
377#103—1224no—
378#102—1224no—
379C22⋊C8—614no—
380C22⋊C8—1222no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95C22×D6111yescharacteristic
387#95C22×D6111yescharacteristic
388C24—1222no—
389C24—1222no—
390C24C2×C22⋊D6111yescharacteristic
391C24C2×C22⋊D6111yescharacteristic
392#41—2444no—
393#41—24416no—
394#41—2444no—
395C2×Q8—22no—
396C2×Q8—81no—
397C2×Q8—1222no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—614no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—1224no—
406C2×D4—1224no—
407C2×D4—1224no—
408C2×D4—1222no—
409C2×D4—4882no—
410C2×D4—4882no—
411C2×D4—24416no—
412C2×D4—2444no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4—2444no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—1224no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4D4×D6112yes—
428C22×C4D4×D6112yes—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C23×D6111yescharacteristic
434SD16—4884no—
435M16—4882no—
436C2×C8—1224no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—81no—
443C4⋊C4—614no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—1224no—
454C22⋊C4—1224no—
455C22⋊C4—1224no—
456C22⋊C4—1224no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—1224no—
468C23—4882no—
469C23—4882no—
470C23—4882no—
471C23—4882no—
472C23—2444no—
473C23—2444no—
474C23#1451974512yes—
475C23#1451980212yes—
476C23C22×C22⋊D6111yescharacteristic
477Q8—44no—
478Q8—2444no—
479D4—28no—
480D4—4882no—
481D4—4882no—
482D4—2444no—
483D4—2444no—
484D4—2444no—
485D4—2448no—
486D4—2448no—
487D4—2444no—
488D4—2444no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—81no—
513C2×C4—1224no—
514C2×C4—2448no—
515C2×C4—2444no—
516C2×C4—2444no—
517C2×C4#1451974511yescharacteristic
518C2×C4#1451974511yescharacteristic
519C2×C4#1451980212yes—
520C2×C4C2×D4×D6112yes—
521C8—2444no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—9764no—
535C22—9764no—
536C22—2444no—
537C22—2444no—
538C22—2444no—
539C22C2×#1451974511yescharacteristic
540C22C2×#1451980211yescharacteristic
541C22C2×#1451980211yescharacteristic
542C4—22no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—2444no—
553C2—21no—
554C2—21no—
555C2—42no—
556C2—42no—
557C2—4884no—
558C2#2145504511yescharacteristic
559C2#2145519711yescharacteristic
560C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2638C2
socleC22×C61C2×#120
prime radical#21459583C2
hypercenter#2638D61
hypocenterC61#18351
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2638

Complements

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

NH ≅ G/Ntype
2. #21456621C2semidirect
3. #21444821C2semidirect
4. #21444804C2semidirect
5. #21444787C2semidirect
6. #21444770C2semidirect
7. #21444347C2semidirect
8. #21444157C2semidirect
9. C61×#2638C2semidirect
21. #16840075C22semidirect
24. #16839901C22semidirect
25. #16839726C22semidirect
26. #16839495C22semidirect
27. #16839495C22semidirect
28. #16839488C22semidirect
29. #16839486C22semidirect
30. C2×C22⋊Q8×C61C22semidirect
32. C61×#398C22semidirect
34. C61×#397C22semidirect
35. C61×#395C22semidirect
37. C61×#393C22semidirect
40. C61×#381C22semidirect
275. #2638D61semidirect
298. C2×C22⋊Q8C2×D61semidirect
308. #398C2×D61semidirect
310. #397C2×D61semidirect
311. #395C2×D61semidirect
313. #393C2×D61semidirect
316. #381C2×D61semidirect
321. C61#18351semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214566212
#214448212
#214448042
#214447872
#214447702
#214443472
#214441572
C61×#26382
#1835161