Finite Group Databasebeta

← #29692215

Subgroups — #29692215

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840052C211yesmaximal, characteristic
2#21448567C211yesmaximal, characteristic, complemented
3#21446771C211yesmaximal, characteristic, complemented
4#21446587C211yesmaximal, characteristic, complemented
5#21446559C211yesmaximal, characteristic, complemented
6#21446544C211yesmaximal, characteristic, complemented
7#21446519C211yesmaximal, characteristic
8#21446505C211yesmaximal, characteristic
9#21446443C211yesmaximal, characteristic
10#21446436C211yesmaximal, characteristic
11#21446401C211yesmaximal, characteristic, complemented
12#21446394C211yesmaximal, characteristic, complemented
13#21445829C211yesmaximal, characteristic
14#21445676C211yesmaximal, characteristic, complemented
15C61×#2319C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17#16840052C418yescomplemented
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839666—21no—
31#16839666—21no—
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839645—21no—
35#16839645—21no—
36#16839628C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C61×#390C2211yescharacteristic
51C61×#390C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#388—21no—
55C61×#388—21no—
56C61×#381—21no—
57C61×#380—21no—
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#376C2211yescharacteristic
60C22×Dic122—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777C2×C414yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76#14519710—41no—
77#14519708—41no—
78#14519708C2×C414yes—
79#14519708C2×C414yes—
80C4⋊#92418—22no—
81C4⋊#92418C2×C414yes—
82#14519698—41no—
83#14519698—41no—
84#14519698C2×C414yes—
85#14519698C2×C414yes—
86#14519688—41no—
87C4×#92418—41no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C4×D4×C61—22no—
97C4×D4×C61C2×C414yes—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61D411yescharacteristic
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119C24×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61C22×C412yes—
126C22×C4×C61—21no—
127C22×C4×C61—24no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—81no—
131C22×C4×C61—81no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C2×Dic122—24no—
144C2×Dic122—81no—
145C2×Dic122—81no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—81no—
152C4⋊C4×C61—81no—
153C4⋊C4×C61C22×C412yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—81no—
156C22⋊C4×C61—81no—
157C22⋊C4×C61—81no—
158C22⋊C4×C61—81no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—81no—
164C42×C61C22×C412yes—
165C42×C61#4112yes—
166C4⋊Dic61—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C22×C412yes—
174C4⋊Dic61C22×C412yes—
175#11758196—81no—
176#11758196—81no—
177C4×Dic61—81no—
178C4×Dic61—81no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418C22×C412yes—
190C2×#92418C22×C412yes—
191C23×C61—22no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61C22×D411yescharacteristic
199D4×C61—22no—
200D4×C61—22no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—84no—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C4×D412yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249Dic122—84no—
250#92418—22no—
251#92418—22no—
252#92418—41no—
253#92418—41no—
254#22123—611nomaximal, Sylow
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C2×C4×D411yescharacteristic
263C22×C61#53611yescharacteristic
264C22×C61#53711yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57711yescharacteristic
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61#51911yescharacteristic
279C4×C61#52011yescharacteristic
280C4×C61#54111yescharacteristic
281C4×C61#54211yescharacteristic
282Dic61—42no—
283Dic61—42no—
284Dic61—81no—
285Dic61—81no—
286C2×#476—611no—
287#2331—611no—
288#2319D6111yescharacteristic
289#1988—611no—
290#1949—611no—
291#1944—611no—
292#1941—611no—
293#1936—611no—
294#1934—611no—
295#1926—611no—
296#1925—611no—
297#1920—611no—
298#1919—611no—
299#1804—611no—
300#1768—611no—
301C2×C61—82no—
302C2×C61#232311yescharacteristic
303C2×C61#294611yescharacteristic
304C2×C61#294911yescharacteristic
305C2×C61#324111yescharacteristic
306C2×C61#324411yescharacteristic
307C2×C61#325711yescharacteristic
308C2×C61#326311yescharacteristic
309C2×C4⋊Q8—611no—
310C2×C4×D4C2×D6111yescharacteristic
311#476—618no—
312C2×#107—611no—
313C2×#106—611no—
314C2×#106—611no—
315C2×#106—611no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8—611no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#102—611no—
322C2×#97—611no—
323C2×C4×C8—611no—
324#391—1221no—
325#391—1221no—
326#390C2×D6111yescharacteristic
327#390C2×D6111yescharacteristic
328#389—611no—
329#389—611no—
330#389C2×D6111yescharacteristic
331#389C2×D6111yescharacteristic
332#388—21no—
333#388—21no—
334#387—1221no—
335#387—1221no—
336#382—611no—
337#381—21no—
338#380—21no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4C2×D6111yescharacteristic
342#376C2×D6111yescharacteristic
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#336—611no—
352#336—611no—
353C61#2212311yescharacteristic, complemented, Sylow
354C22×Q8—1221no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—611no—
361C4⋊Q8—614no—
362C4⋊Q8—1222no—
363C4×D4—22no—
364C4×D4C4×D6114yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C42—21no—
383C2×C42—1221no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386C2×C42C22×D6111yescharacteristic
387C2×C42C22×D6111yescharacteristic
388#107—2441no—
389#106—614no—
390#106—614no—
391#106—2441no—
392C4⋊C8—614no—
393C4⋊C8—1222no—
394#102—614no—
395#102—614no—
396#102—2441no—
397#102—2441no—
398#97—2441no—
399C4×C8—2441no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95C22⋊D6111yescharacteristic
410#95C22⋊D6111yescharacteristic
411#95C22⋊D6111yescharacteristic
412#95C22⋊D6111yescharacteristic
413C24—21no—
414C2×Q8—1224no—
415C2×Q8—4881no—
416C2×Q8—4881no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D6112yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—612no—
441C2×C8—612no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—612no—
460C4⋊C4—612no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1224no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—4881no—
469C4⋊C4—4881no—
470C4⋊C4C2×C4×D6112yes—
471C22⋊C4—24no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—4881no—
482C42—4881no—
483C42C2×C4×D6112yes—
484C42#1175822212yes—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C22×C22⋊D6111yescharacteristic
493Q8—4884no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—84no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C4×C22⋊D6112yes—
537C2×C4C4×C22⋊D6112yes—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C2×C4#1451982311yescharacteristic
541C2×C4#1451982411yescharacteristic
542C2×C4#1451982411yescharacteristic
543C2×C4#1451982411yescharacteristic
544C8—1222no—
545C8—1222no—
546C8—2441no—
547C8—2441no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×C4×C22⋊D6111yescharacteristic
556C22#1684033311yescharacteristic
557C22#1684034011yescharacteristic
558C22#1684056511yescharacteristic
559C22#1684057011yescharacteristic
560C22#1684057311yescharacteristic
561C22#1684057311yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—4881no—
572C4—4881no—
573C4—2442no—
574C4—2442no—
575C4#1684023911yescharacteristic
576C4#1684024211yescharacteristic
577C4#1684037411yescharacteristic
578C4#1684038311yescharacteristic
579C2—82no—
580C2#2144850111yescharacteristic
581C2#2145464711yescharacteristic
582C2#2145467611yescharacteristic
583C2#2145706611yescharacteristic
584C2#2145711111yescharacteristic
585C2#2145722411yescharacteristic
586C2#2145728411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2319C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#2319D61
hypocenterC61#22123
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #21448567C2semidirect
3. #21446771C2semidirect
4. #21446587C2semidirect
5. #21446559C2semidirect
6. #21446544C2semidirect
11. #21446401C2semidirect
12. #21446394C2semidirect
14. #21445676C2semidirect
17. #16840052C4semidirect
353. C61#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#168400522
#214485672
#214467712
#214465872
#214465592
#214465442
#214465192
#214465052
#214464432
#214464362
#214464012
#214463942
#214458292
#214456762
C61×#23192
#2212361