Finite Group Databasebeta

← #30603520

Subgroups — #30603520

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462300C211yesmaximal, characteristic
2C2×#16840641C211yesmaximal, characteristic
3#21461508C211yesmaximal, characteristic
4#21461507C211yesmaximal, characteristic
5C2×#16840633C211yesmaximal, characteristic
6#21460997C211yesmaximal, characteristic
7#339×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2211yescharacteristic
9C2×#14519856C2211yescharacteristic
10C22×C4⋊#6923C2211yescharacteristic
11C22×C4⋊#6923C2211yescharacteristic
12#16840713—21no—
13#16840713—21no—
14#16840709—22no—
15#16840708—21no—
16#16840706—21no—
17C2×(C2×#6923)⋊C4—22no—
18#16840641—22no—
19#16840636—21no—
20#16840636—21no—
21#16840635—21no—
22#16840635—21no—
23#16840633—22no—
24C2×M16×D61C2211yescharacteristic
25#117×D61C2211yescharacteristic
26C2×C42×D61C2211yescharacteristic
27#16839523C411yescharacteristic, complemented
28#16839522C411yescharacteristic, complemented
29#16839521C411yescharacteristic, complemented
30C61×#339C411yescharacteristic, complemented
31C23×#6923—21no—
32C23×#6923D412yes—
33C2×#11758213—21no—
34D61⋊M16—21no—
35D61⋊M16—21no—
36D61⋊M16—21no—
37D61⋊M16D412yes—
38C2×#11758217—21no—
39C2×D61⋊C8—21no—
40#14519856—21no—
41#14519856—21no—
42#14519856—21no—
43#14519856—21no—
44C2×C4⋊#6923—21no—
45C2×C4⋊#6923—21no—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923—21no—
48C2×C4⋊#6923—21no—
49C2×C4⋊#6923—21no—
50C2×C4⋊#6923—21no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923—22no—
56C2×C4⋊#6923—22no—
57C2×C4⋊#6923D411yescharacteristic
58C2×C4⋊#6923D411yescharacteristic
59C2×C4⋊#6923Q812yes—
60C2×C22⋊#6923—21no—
61C2×C4×#6923—21no—
62(C2×#6923)⋊C4—22no—
63(C2×#6923)⋊C4—22no—
64(C2×#6923)⋊C4—44no—
65C22×C4×D61—21no—
66C22×C4×D61C2311yescharacteristic
67C2×#11758194C2×C411yescharacteristic
68C2×#11758189—21no—
69M16×D61—21no—
70M16×D61—21no—
71M16×D61—21no—
72M16×D61—21no—
73C2×C8×D61—21no—
74#14519736C2×C411yescharacteristic
75#14519735Q811yescharacteristic
76#14519735Q811yescharacteristic
77#14519734—21no—
78#14519733—21no—
79C4⋊C4×D61D411yescharacteristic
80C4⋊C4×D61D411yescharacteristic
81C22⋊C4×D61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2×C411yescharacteristic
84C42×D61—21no—
85C42×D61—21no—
86C42×D61—21no—
87C42×D61—21no—
88C42×D61—21no—
89C2×M16×C61C2×C411yescharacteristic
90#117×C61C2×C411yescharacteristic
91C2×C42×C61C2×C411yescharacteristic
92C23×D61C2×D411yescharacteristic
93C22×#6923—21no—
94C22×#6923—22no—
95C22×#6923—21no—
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—42no—
100C22×#6923—42no—
101C22×#6923—42no—
102C22×#6923—42no—
103C22×#6923—41no—
104C22×#6923—41no—
105C22×C4×C61—21no—
106C22×C4×C61C22×C411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61#4111yescharacteristic
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×C4×D61C2×D411yescharacteristic
120C2×C4×D61C2×Q811yescharacteristic
121C2×C4×D61C2×Q811yescharacteristic
122C2×C4×D61#4111yescharacteristic
123C2×C4×D61#4111yescharacteristic
124C2×C4×D61#4111yescharacteristic
125C2×C4×D61#4111yescharacteristic
126C2×C4×D61#4111yescharacteristic
127C2×C4×D61#4111yescharacteristic
128C2×C4×D61#4111yescharacteristic
129C22⋊#6923—41no—
130C22⋊#6923—41no—
131#11758217—22no—
132#11758217—41no—
133C2×#92420—41no—
134C4⋊#6923—21no—
135C4⋊#6923—21no—
136C4⋊#6923—22no—
137C4⋊#6923—22no—
138C4⋊#6923—21no—
139C4⋊#6923—21no—
140C4⋊#6923—41no—
141C4⋊#6923—41no—
142C4⋊#6923—41no—
143C4⋊#6923—41no—
144C4⋊#6923—41no—
145C4⋊#6923—41no—
146C4⋊#6923—82no—
147C4⋊#6923—82no—
148C4×#6923—41no—
149C4×#6923—41no—
150#11758213—22no—
151#11758213—41no—
152D61⋊C8—22no—
153D61⋊C8—41no—
154M16×C61—21no—
155M16×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61C4⋊C411yescharacteristic
158C4⋊C4×C61C4⋊C411yescharacteristic
159C22⋊C4×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C22⋊Dic61—21no—
164#11758198—21no—
165#11758198—21no—
166C4⋊Dic61C4⋊C411yescharacteristic
167C4⋊Dic61C4⋊C411yescharacteristic
168#11758196—21no—
169#11758196—21no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C4×Dic61—41no—
176C4×Dic61—41no—
177#11758194—21no—
178#11758194—21no—
179C2×#92418—21no—
180#11758189—41no—
181#11758189—41no—
182C8×D61—41no—
183C8×D61—41no—
184C23×C61C4×D411yescharacteristic
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61#12011yescharacteristic
190C22×D61#12111yescharacteristic
191C22×D61C22⋊Q811yescharacteristic
192C22×D61C22⋊Q811yescharacteristic
193C22×D61#12311yescharacteristic
194C22×D61#12411yescharacteristic
195C2×#6923—44no—
196C2×#6923—44no—
197C2×#6923—42no—
198C2×#6923—41no—
199C2×#6923—41no—
200C2×#6923—42no—
201C2×#6923—41no—
202C2×#6923—42no—
203C2×#6923—41no—
204C2×#6923—41no—
205C2×#6923—42no—
206C2×#6923—41no—
207C2×#6923—82no—
208C2×#6923—81no—
209C2×#6923—81no—
210C2×#6923—82no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61C2×C4⋊C411yescharacteristic
217C2×C4×C61#11711yescharacteristic
218C2×C4×C61#11711yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×Q811yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61C22⋊Q811yescharacteristic
228C2×Dic61#12311yescharacteristic
229C2×Dic61#12311yescharacteristic
230C2×Dic61#12511yescharacteristic
231C2×Dic61#12611yescharacteristic
232C2×Dic61#12611yescharacteristic
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61#12111yescharacteristic
242C4×D61#12111yescharacteristic
243C4×D61C22⋊Q811yescharacteristic
244C4×D61C22⋊Q811yescharacteristic
245C4×D61#12311yescharacteristic
246C4×D61#12311yescharacteristic
247C4×D61#12411yescharacteristic
248C4×D61#12411yescharacteristic
249C4×D61#12411yescharacteristic
250C4×D61#12511yescharacteristic
251C4×D61#12511yescharacteristic
252C4×D61#12511yescharacteristic
253C4×D61#12511yescharacteristic
254C4×D61#12611yescharacteristic
255C4×D61#12611yescharacteristic
256C4×D61C4⋊Q811yescharacteristic
257#92420—42no—
258C8×C61—41no—
259#92418—41no—
260#11276—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61#38511yescharacteristic
264C22×C61#38711yescharacteristic
265C22×C61#38811yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61#39311yescharacteristic
277C2×D61#39411yescharacteristic
278C2×D61#39811yescharacteristic
279C2×D61#40011yescharacteristic
280C2×D61#45311yescharacteristic
281C2×D61#45411yescharacteristic
282C2×D61#45511yescharacteristic
283C2×D61#45611yescharacteristic
284#6923—82no—
285#6923—81no—
286#6923—82no—
287#6923—81no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61#38211yescharacteristic
292C4×C61#38711yescharacteristic
293C4×C61#38811yescharacteristic
294C4×C61#38911yescharacteristic
295Dic61—42no—
296Dic61—41no—
297Dic61—41no—
298Dic61#39711yescharacteristic
299Dic61#39811yescharacteristic
300Dic61#40011yescharacteristic
301Dic61#40011yescharacteristic
302#2317—611no—
303C2×#343—611no—
304#1736—611no—
305#1736—611no—
306C2×#339—611no—
307C2×#339—611no—
308C4⋊#95—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61#144211yescharacteristic
312C2×C61#185911yescharacteristic
313C2×C61#186011yescharacteristic
314D61—21no—
315D61—21no—
316D61#202711yescharacteristic
317D61#202811yescharacteristic
318D61#203711yescharacteristic
319D61#203811yescharacteristic
320C22×M16—611no—
321C22×M16—611no—
322C2×#117—611no—
323C2×#117—611no—
324C22×C4⋊C4—611no—
325C22×C4⋊C4—611no—
326C22×C42—611no—
327#389—1221no—
328#389—1221no—
329#385—1222no—
330#384—1221no—
331#382—1221no—
332C2×#95—1222no—
333#343—1222no—
334#340—1221no—
335#340—1221no—
336#340—1221no—
337#340—1221no—
338#339—611no—
339#339—611no—
340#339—611no—
341#339—1222no—
342#339#692311yescharacteristic, complemented
343C61#1127611yescharacteristic, complemented, Sylow
344C23×C4—611no—
345C23×C4—612no—
346C23×C4—1221no—
347C23×C4—1221no—
348C2×M16—612no—
349C2×M16—611no—
350C2×M16—1221no—
351C2×M16—1221no—
352C2×M16—1221no—
353C2×M16—1221no—
354C2×M16—1221no—
355C2×M16—1221no—
356C2×M16—1221no—
357C2×M16—1221no—
358C2×M16—1221no—
359C2×M16—1221no—
360C2×M16C2×#692311yescharacteristic
361C22×C8—1221no—
362C22×C8—1221no—
363#117—611no—
364#117—611no—
365#117—611no—
366#117—1221no—
367#117—1221no—
368#117—1221no—
369#117—1221no—
370#117—1221no—
371#117—1221no—
372#117C2×#692311yescharacteristic
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C42—611no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42C2×#692311yescharacteristic
402#95—1222no—
403#95—1222no—
404#95—2444no—
405C24—611no—
406C22×C4—21no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—2441no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4C22×#692311yescharacteristic
442M16—21no—
443M16—21no—
444M16—1222no—
445M16—1222no—
446M16—1221no—
447M16—1221no—
448M16—2441no—
449M16—2441no—
450M16—2441no—
451M16—2441no—
452C2×C8—21no—
453C2×C8—1222no—
454C2×C8—1221no—
455C2×C8—2441no—
456C2×C8—2441no—
457C2×C8—2441no—
458C2×C8—2441no—
459C4⋊C4—611no—
460C4⋊C4—611no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—4882no—
470C4⋊C4—4882no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4C4⋊#692311yescharacteristic
478C4⋊C4C4⋊#692311yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—1221no—
481C22⋊C4—1221no—
482C22⋊C4—1221no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—2441no—
494C42—2441no—
495C42—2441no—
496C42—2441no—
497C23—611no—
498C23—611no—
499C23—611no—
500C23—611no—
501C23—611no—
502C23—611no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23D4×#692311yescharacteristic
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—611no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—4881no—
539C2×C4—4881no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2444no—
561C2×C4—2444no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4C2×C4⋊#692311yescharacteristic
565C2×C4#1451985611yescharacteristic
566C2×C4#1451985611yescharacteristic
567C2×C4D4×#692311yescharacteristic
568C2×C4D4×#692311yescharacteristic
569C2×C4Q8×#692311yescharacteristic
570C8—41no—
571C8—2442no—
572C8—2441no—
573C22—21no—
574C22—21no—
575C22—611no—
576C22—611no—
577C22—611no—
578C22—611no—
579C22—611no—
580C22—611no—
581C22—611no—
582C22—611no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22#1684070911yescharacteristic
594C22#1684071111yescharacteristic
595C22#1684071211yescharacteristic
596C4—42no—
597C4—41no—
598C4—41no—
599C4—611no—
600C4—611no—
601C4—611no—
602C4—611no—
603C4—4882no—
604C4—4882no—
605C4—4881no—
606C4—4881no—
607C4—2442no—
608C4—2441no—
609C4—2441no—
610C4#1684070611yescharacteristic
611C4#1684071111yescharacteristic
612C4#1684071211yescharacteristic
613C4#1684071311yescharacteristic
614C2—21no—
615C2—21no—
616C2—611no—
617C2—611no—
618C2—611no—
619C2—611no—
620C2—1221no—
621C2—1221no—
622C2#2146100511yescharacteristic
623C2#2146177811yescharacteristic
624C2#2146177911yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#339C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#339#6923
hypocenterC61#11276
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
27. #16839523C4semidirect
28. #16839522C4semidirect
29. #16839521C4semidirect
30. C61×#339C4semidirect
342. #339#6923semidirect
343. C61#11276semidirect

Maximal subgroups

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

subgroupindex
#214623002
C2×#168406412
#214615082
#214615072
C2×#168406332
#214609972
#339×D612
#1127661