Finite Group Databasebeta

← #30602919

Subgroups — #30602919

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic
2#21462257C211yesmaximal, characteristic
3#21461507C211yesmaximal, characteristic
4C2×#16840634C211yesmaximal, characteristic
5C2×#16840634C211yesmaximal, characteristic
6#21460996C211yesmaximal, characteristic
7#343×D61C211yesmaximal, characteristic
8C2×#14519856C2211yescharacteristic
9C2×#14519856C2211yescharacteristic
10C22×C4⋊#6923C2211yescharacteristic
11C22×C4×#6923C2211yescharacteristic
12#16840773—22no—
13#16840773C414yescomplemented
14C2×#14519846—21no—
15C2×#14519845—21no—
16C4⋊C4×#6923—21no—
17C4⋊C4×#6923—21no—
18C4⋊C4×#6923—21no—
19C4⋊C4×#6923—21no—
20C4×C22⋊#6923—22no—
21C2×(C2×#6923)⋊C4—22no—
22#16840636—21no—
23#16840636—21no—
24#16840634—22no—
25#16840634C414yescomplemented
26C2×M16×D61C2211yescharacteristic
27C2×M16×D61C2211yescharacteristic
28C2×C4⋊C4×D61C2211yescharacteristic
29#16839531C411yescharacteristic, complemented
30#16839530C411yescharacteristic, complemented
31#16839530C411yescharacteristic, complemented
32C61×#343C411yescharacteristic, complemented
33C23×#6923—21no—
34C23×#6923—21no—
35#14519856—21no—
36#14519856—21no—
37#14519856—21no—
38#14519856—21no—
39#14519856C2×C412yes—
40#14519856C2×C412yes—
41#14519856D412yes—
42#14519856Q812yes—
43C2×C4⋊#6923—21no—
44C2×C4⋊#6923—21no—
45C2×C4⋊#6923—21no—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923—21no—
48C2×C4⋊#6923C2×C412yes—
49C2×C4⋊#6923D411yescharacteristic
50C2×C4⋊#6923D411yescharacteristic
51C2×C4⋊#6923D411yescharacteristic
52C2×C4⋊#6923D411yescharacteristic
53C2×C22⋊#6923—21no—
54C2×C22⋊#6923—21no—
55C2×C4×#6923—21no—
56C2×C4×#6923—21no—
57C2×C4×#6923—21no—
58C2×C4×#6923—21no—
59C2×C4×#6923—21no—
60C2×C4×#6923—21no—
61C2×C4×#6923—21no—
62C2×C4×#6923—22no—
63C2×C4×#6923—22no—
64C2×C4×#6923C2×C412yes—
65#14519846—24no—
66#14519845—24no—
67(C2×#6923)⋊C4—22no—
68(C2×#6923)⋊C4—22no—
69(C2×#6923)⋊C4—44no—
70C22×C4×D61D412yes—
71C22×C4×D61C2311yescharacteristic
72C2×#11758194C2×C411yescharacteristic
73C2×#11758194C2×C411yescharacteristic
74C2×#11758189—21no—
75C2×#11758189—21no—
76M16×D61—21no—
77M16×D61—21no—
78M16×D61—21no—
79M16×D61—21no—
80M16×D61—21no—
81M16×D61—21no—
82M16×D61C2×C412yes—
83M16×D61C2×C412yes—
84C2×C8×D61—21no—
85C2×C8×D61—21no—
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×#11758196Q812yes—
88C4⋊C4×D61—21no—
89C4⋊C4×D61—21no—
90C4⋊C4×D61—21no—
91C4⋊C4×D61—21no—
92C2×M16×C61C2×C411yescharacteristic
93C2×M16×C61C2×C411yescharacteristic
94C2×C4⋊C4×C61C2×C411yescharacteristic
95C23×D61C2×D411yescharacteristic
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—21no—
100C22×#6923—21no—
101C22×#6923—21no—
102C22×#6923—21no—
103C22×#6923—22no—
104C22×#6923—41no—
105C22×#6923—41no—
106C22×#6923—41no—
107C22×#6923—41no—
108C22×#6923—42no—
109C22×#6923—42no—
110C22×#6923—41no—
111C22×#6923—41no—
112C22×C4×C61C4⋊C412yes—
113C22×C4×C61C22×C411yescharacteristic
114C22×Dic61C4⋊C412yes—
115C22×Dic61C2×Q811yescharacteristic
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61C22×C411yescharacteristic
121C2×C4×D61C22×C411yescharacteristic
122C2×C4×D61C2×D411yescharacteristic
123C2×C4×D61C2×D411yescharacteristic
124C2×C4×D61C2×D411yescharacteristic
125C2×C4×D61C2×Q811yescharacteristic
126C2×C4×D61#4111yescharacteristic
127C2×C4×D61#4111yescharacteristic
128C2×C4×D61#4111yescharacteristic
129C2×C4×D61#4111yescharacteristic
130C2×C4×D61#4111yescharacteristic
131C2×C4×D61#4111yescharacteristic
132C22⋊#6923—22no—
133C22⋊#6923—22no—
134C22⋊#6923—22no—
135C22⋊#6923—22no—
136C4⋊#6923—22no—
137C4⋊#6923—22no—
138C4⋊#6923—21no—
139C4⋊#6923—21no—
140C4⋊#6923—21no—
141C4⋊#6923—21no—
142C4⋊#6923—41no—
143C4⋊#6923—41no—
144C4⋊#6923C22⋊C412yes—
145C4⋊#6923C22⋊C412yes—
146C4⋊#6923C4⋊C412yes—
147C4⋊#6923C4⋊C412yes—
148C4×#6923—22no—
149C4×#6923—22no—
150C4×#6923—22no—
151C4×#6923—22no—
152C4×#6923—22no—
153C4×#6923—22no—
154C4×#6923—41no—
155C4×#6923—41no—
156C4×#6923—82no—
157C4×#6923—82no—
158M16×C61—21no—
159M16×C61—21no—
160M16×C61—21no—
161M16×C61C4212yescomplemented
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168#11758196—22no—
169#11758196—22no—
170#11758194—21no—
171#11758194—21no—
172#11758194—21no—
173#11758194C4212yescomplemented
174C2×#92418—21no—
175C2×#92418—21no—
176#11758189—22no—
177#11758189—22no—
178#11758189—41no—
179#11758189—41no—
180C8×D61—22no—
181C8×D61—22no—
182C8×D61—41no—
183C8×D61—41no—
184C23×C61C2×C4⋊C411yescharacteristic
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61C4×D411yescharacteristic
190C22×D61C4×D411yescharacteristic
191C22×D61#12011yescharacteristic
192C22×D61#12111yescharacteristic
193C22×D61C22⋊Q811yescharacteristic
194C22×D61#12311yescharacteristic
195C2×#6923—24no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—41no—
207C2×#6923—41no—
208C2×#6923—41no—
209C2×#6923—41no—
210C2×#6923—41no—
211C2×#6923—41no—
212C2×#6923—41no—
213C2×#6923—41no—
214C2×#6923—44no—
215C2×#6923—42no—
216C2×#6923—82no—
217C2×#6923—81no—
218C2×#6923—81no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61C2×C4211yescharacteristic
222C2×C4×C61#11711yescharacteristic
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×Q811yescharacteristic
227C2×Dic61—22no—
228C2×Dic61—42no—
229C2×Dic61C4×Q811yescharacteristic
230C2×Dic61C4×Q811yescharacteristic
231C2×Dic61C22⋊Q811yescharacteristic
232C2×Dic61C22⋊Q811yescharacteristic
233C2×Dic61#12511yescharacteristic
234C2×Dic61C4⋊Q811yescharacteristic
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—82no—
238C4×D61C2×C22⋊C411yescharacteristic
239C4×D61C2×C4⋊C411yescharacteristic
240C4×D61#11711yescharacteristic
241C4×D61#11711yescharacteristic
242C4×D61C4×D411yescharacteristic
243C4×D61C4×D411yescharacteristic
244C4×D61C4×D411yescharacteristic
245C4×D61C4×Q811yescharacteristic
246C4×D61#12111yescharacteristic
247C4×D61C22⋊Q811yescharacteristic
248C4×D61#12311yescharacteristic
249C4×D61#12311yescharacteristic
250C4×D61#12411yescharacteristic
251C4×D61#12511yescharacteristic
252C4×D61#12611yescharacteristic
253C4×D61#12611yescharacteristic
254C8×C61—22no—
255C8×C61—41no—
256#92418—22no—
257#92418—41no—
258#10899—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61C4×C4⋊C411yescharacteristic
262C22×C61#38511yescharacteristic
263C22×C61#38911yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
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#38511yescharacteristic
275C2×D61#38611yescharacteristic
276C2×D61#38711yescharacteristic
277C2×D61#38811yescharacteristic
278C2×D61#45311yescharacteristic
279C2×D61#45411yescharacteristic
280C2×D61#45511yescharacteristic
281C2×D61#45611yescharacteristic
282#6923—44no—
283#6923—42no—
284#6923—42no—
285#6923—42no—
286#6923—42no—
287#6923—82no—
288C4×C61—42no—
289C4×C61C4×C22⋊C411yescharacteristic
290C4×C61C4×C4⋊C411yescharacteristic
291C4×C61#38711yescharacteristic
292C4×C61#38811yescharacteristic
293Dic61—42no—
294Dic61#38711yescharacteristic
295Dic61#38711yescharacteristic
296Dic61#38911yescharacteristic
297Dic61#39111yescharacteristic
298C2×#428—611no—
299#2274—611no—
300C2×#343—611no—
301#1736—611no—
302C2×#339—611no—
303C2×#339—611no—
304C4⋊#95—611no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61#143611yescharacteristic
308C2×C61#185611yescharacteristic
309C2×C61#185711yescharacteristic
310D61—21no—
311D61—21no—
312D61#185911yescharacteristic
313D61#186011yescharacteristic
314D61#188211yescharacteristic
315D61#188311yescharacteristic
316C22×M16—611no—
317C22×M16—611no—
318C2×#117—611no—
319C2×#117—611no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4—611no—
322C22×C42—611no—
323#428—614no—
324#428—1222no—
325C2×#107—1221no—
326C2×#106—1221no—
327C4×C4⋊C4—1221no—
328C4×C4⋊C4—1221no—
329C4×C4⋊C4—1221no—
330C4×C4⋊C4—1221no—
331C4×C22⋊C4—1222no—
332C2×#95—1222no—
333#343—611no—
334#343—611no—
335#343—611no—
336#343#692311yescharacteristic, complemented
337#340—1221no—
338#340—1221no—
339#339—614no—
340#339—1222no—
341C61#1089911yescharacteristic, complemented, Sylow
342C23×C4—611no—
343C23×C4—612no—
344C23×C4—1221no—
345C23×C4—1221no—
346C2×M16—611no—
347C2×M16—612no—
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×M16C2×#692311yescharacteristic
359C2×M16C2×#692311yescharacteristic
360C22×C8—1221no—
361C22×C8—1221no—
362#117—612no—
363#117—612no—
364#117—612no—
365#117—612no—
366#117—1221no—
367#117—1221no—
368#117—1221no—
369#117—1221no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C2×#692311yescharacteristic
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C42—612no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1222no—
397C2×C42—1222no—
398C2×C42—1221no—
399#107—1224no—
400#106—1224no—
401#95—1222no—
402#95—1222no—
403#95—2444no—
404C24—611no—
405C22×C4—611no—
406C22×C4—611no—
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—612no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4—2441no—
432C22×C4—2441no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4—2441no—
438C22×C4—2441no—
439C22×C4C4⋊#692312yes—
440C22×C4C22×#692311yescharacteristic
441M16—21no—
442M16—21no—
443M16—21no—
444M16—612no—
445M16—1222no—
446M16—1222no—
447M16—1221no—
448M16—1221no—
449M16—1221no—
450M16—2441no—
451M16—2441no—
452M16C4×#692312yescomplemented
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—2441no—
460C2×C8—2441no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—612no—
464C4⋊C4—612no—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C22⋊C4—1222no—
480C22⋊C4—1222no—
481C22⋊C4—1222no—
482C22⋊C4—1222no—
483C42—1222no—
484C42—1222no—
485C42—1222no—
486C42—1222no—
487C42—1222no—
488C42—1222no—
489C42—4882no—
490C42—4882no—
491C42—2441no—
492C42—2441no—
493C23—611no—
494C23—611no—
495C23—611no—
496C23—611no—
497C23—611no—
498C23—611no—
499C23—1221no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23C2×C4⋊#692311yescharacteristic
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
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—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1224no—
533C2×C4—1222no—
534C2×C4—4882no—
535C2×C4—4881no—
536C2×C4—4881no—
537C2×C4—4882no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2444no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4C2×C4×#692311yescharacteristic
558C2×C4#1451985611yescharacteristic
559C2×C4D4×#692311yescharacteristic
560C2×C4D4×#692311yescharacteristic
561C2×C4D4×#692311yescharacteristic
562C2×C4Q8×#692311yescharacteristic
563C8—22no—
564C8—41no—
565C8—1222no—
566C8—2441no—
567C22—21no—
568C22—21no—
569C22—611no—
570C22—611no—
571C22—611no—
572C22—611no—
573C22—611no—
574C22—611no—
575C22—611no—
576C22—611no—
577C22—1221no—
578C22—1221no—
579C22—1221no—
580C22—1221no—
581C22—1221no—
582C22—1221no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22C4×C4⋊#692311yescharacteristic
588C22#1684070911yescharacteristic
589C22#1684071311yescharacteristic
590C4—42no—
591C4—611no—
592C4—611no—
593C4—611no—
594C4—611no—
595C4—4882no—
596C4—2442no—
597C4—2442no—
598C4—2442no—
599C4—2442no—
600C4—2444no—
601C4—2442no—
602C4C22⋊C4×#692311yescharacteristic
603C4C4⋊C4×#692311yescharacteristic
604C4#1684071111yescharacteristic
605C4#1684071211yescharacteristic
606C2—21no—
607C2—21no—
608C2—611no—
609C2—611no—
610C2—611no—
611C2—611no—
612C2—1221no—
613C2—1221no—
614C2#2146099911yescharacteristic
615C2#2146177411yescharacteristic
616C2#2146177511yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#343C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#343#6923
hypocenterC61#10899
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
13. #16840773C4semidirect
25. #16840634C4semidirect
29. #16839531C4semidirect
30. #16839530C4semidirect
31. #16839530C4semidirect
32. C61×#343C4semidirect
161. M16×C61C42semidirect
173. #11758194C42semidirect
336. #343#6923semidirect
341. C61#10899semidirect
452. M16C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
#214622572
#214615072
C2×#168406342
#214609962
#343×D612
#1089961