Finite Group Databasebeta

← #29682601

Subgroups — #29682601

Order 15616.

1000 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839778C211yesmaximal, characteristic
2C2×#16839778C211yesmaximal, characteristic
3C2×#16839744C211yesmaximal, characteristic
4#21454237C211yesmaximal, characteristic, complemented
5#21446487C218yesmaximal, complemented
6C2×#16839533C211yesmaximal, characteristic, complemented
7#21445530C211yesmaximal, characteristic, complemented
8C2×C61×#430C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13#16839778—22no—
14#16839778—22no—
15#16839778—22no—
16#16839778C2214yes—
17C2×#14519703—22no—
18C2×#14519703C2214yes—
19#16839744C414yescomplemented
20#16839744C2214yes—
21C2×#14519695C412yescomplemented
22C2×#14519693C412yescomplemented
23#16839720—22no—
24#16839717—22no—
25M16×Dic61C2214yescomplemented
26#16839533C2214yescomplemented
27#16839523C2214yescomplemented
28C22×M16×C61C2211yescharacteristic
29C22×M16×C61C2211yescharacteristic
30C22×M16×C61C2211yescharacteristic
31C61×#430—22no—
32C61×#430C2214yescomplemented
33C2×#108×C61—22no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222C2×C412yes—
45C2×#11758222C2312yes—
46C22×Dic122C2×C411yescharacteristic
47C22×D244C2×C411yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194C2×C412yes—
54C2×#11758194C2×C412yes—
55C2×#11758194C2312yes—
56C22×#92418—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—21no—
61#14519736—21no—
62#14519736D412yes—
63#14519736D412yes—
64#14519736C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2312yes—
71#14519703—28no—
72#14519703—28no—
73#14519695C2×C418yescomplemented
74#14519693C2×C418yescomplemented
75#14519687—24no—
76C8×Dic61—24no—
77C23×C4×C61C2311yescharacteristic
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61C2×C412yes—
90C2×M16×C61C2×C412yes—
91C2×M16×C61D412yes—
92C2×M16×C61D412yes—
93C2×M16×C61C2312yes—
94C2×M16×C61C2312yes—
95C2×M16×C61C2312yes—
96C22×C8×C61—21no—
97C22×C8×C61—21no—
98C22×C8×C61—21no—
99#108×C61—28no—
100C24×C61#4111yescharacteristic
101C23×D61—21no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—22no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—41no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140#11758222—41no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C22×C412yes—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122C22×C412yes—
159M16×C61—22no—
160M16×C61—22no—
161M16×C61—22no—
162M16×C61—22no—
163M16×C61—22no—
164M16×C61—22no—
165M16×C61—22no—
166M16×C61—22no—
167M16×C61—41no—
168M16×C61—41no—
169M16×C61C22×C414yes—
170M16×C61C2×D414yes—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—41no—
179C2×C8×C61—41no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—41no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—41no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—41no—
191C4×Dic61C2×D414yes—
192#11758194—22no—
193#11758194—22no—
194#11758194—22no—
195#11758194C22×C414yes—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C23×C61—81no—
200C23×C61#11711yescharacteristic
201C23×C61#11711yescharacteristic
202C23×C61#12311yescharacteristic
203C23×C61#12311yescharacteristic
204C23×C61#12411yescharacteristic
205C23×C61#12411yescharacteristic
206C23×C61C2×#4111yescharacteristic
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—82no—
211C2×C4×C61#11711yescharacteristic
212C2×C4×C61#11711yescharacteristic
213C2×C4×C61C4×D411yescharacteristic
214C2×C4×C61C4×D411yescharacteristic
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C4×D411yescharacteristic
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12311yescharacteristic
226C2×C4×C61#12311yescharacteristic
227C2×C4×C61#12411yescharacteristic
228C2×C4×C61#12411yescharacteristic
229C2×C4×C61#12611yescharacteristic
230C2×C4×C61#12611yescharacteristic
231C2×C4×C61#12611yescharacteristic
232C2×C4×C61#12611yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C22⋊D61—42no—
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C2×Dic61—22no—
244C2×Dic61—24no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—82no—
249C2×Dic61—82no—
250D244—22no—
251D244—22no—
252D244—41no—
253D244—41no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—42no—
258Dic122—22no—
259Dic122—22no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—24no—
263C8×C61—24no—
264C8×C61—42no—
265#92418—24no—
266#21080—611nomaximal, Sylow
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61#38811yescharacteristic
275C22×C61#38811yescharacteristic
276C22×C61#38811yescharacteristic
277C22×C61#38811yescharacteristic
278C22×C61C2×#11711yescharacteristic
279C22×C61C2×#12311yescharacteristic
280C22×C61C2×#12411yescharacteristic
281C2×D61—42no—
282C2×D61—82no—
283C2×D61—82no—
284C2×D61—82no—
285C4×C61#38811yescharacteristic
286C4×C61#38811yescharacteristic
287C4×C61#38811yescharacteristic
288C4×C61#38811yescharacteristic
289C4×C61C2×C4×D411yescharacteristic
290C4×C61C2×C4×D411yescharacteristic
291C4×C61C2×#12111yescharacteristic
292C4×C61C2×#12611yescharacteristic
293Dic61—44no—
294Dic61—42no—
295Dic61—82no—
296C2×#430D6111yescharacteristic, complemented
297C2×#421—611no—
298C2×#421—611no—
299C2×#413—611no—
300#2873—611no—
301#1932—618no—
302C2×#344—611no—
303#1733—611no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61C2×#38811yescharacteristic
312D61—82no—
313C22×#41—611no—
314C22×M16—611no—
315C22×M16C2×D6111yescharacteristic
316C22×M16C2×D6111yescharacteristic
317C22×M16C2×D6111yescharacteristic
318C2×#117—611no—
319C2×#117—611no—
320#430—22no—
321#430C2×D6114yescomplemented
322C2×#108—22no—
323#421—614no—
324#421—1222no—
325#421—1222no—
326#421—1222no—
327C2×#104—614no—
328C2×#104—1222no—
329#413—614no—
330#413—614no—
331C2×#101—612no—
332C2×#100—612no—
333#405—1222no—
334#405—1222no—
335C4×M16—614no—
336#344—614no—
337#339—614no—
338C61#2108011yescharacteristic, complemented, Sylow
339C2×#41—612no—
340C2×#41—612no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C2×#41—1221no—
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1221no—
350C2×#41—1221no—
351C22×Q8—611no—
352C22×D4—611no—
353C22×D4—1221no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—612no—
368C2×M16—612no—
369C2×M16—612no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16C4×D6112yes—
374C2×M16C4×D6112yes—
375C2×M16C22⋊D6112yes—
376C2×M16C22⋊D6112yes—
377C2×M16C22×D6112yes—
378C2×M16C22×D6112yes—
379C2×M16C22×D6112yes—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—21no—
383C22×C8—1221no—
384#117—612no—
385#117—612no—
386#117—612no—
387#117—1221no—
388#117—1221no—
389#117—1221no—
390#117—1221no—
391#117—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C42—612no—
397C2×C42—1221no—
398#108—28no—
399#104—1228no—
400#104—1228no—
401#101—618no—
402#100—618no—
403#97—1224no—
404C4×C8—1224no—
405C24—1221no—
406C24#1175822611yescharacteristic
407#41—1222no—
408#41—1222no—
409#41—1222no—
410#41—1222no—
411#41—1222no—
412#41—1222no—
413#41—1222no—
414#41—1222no—
415#41—2441no—
416#41—2441no—
417#41—2441no—
418#41—2441no—
419#41—2441no—
420#41—2441no—
421#41—2441no—
422#41—2441no—
423C2×Q8—612no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×D4—612no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4—2441no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4—2441no—
450C22×C4C2×C4×D6111yescharacteristic
451C22×C4C2×C4×D6111yescharacteristic
452C22×C4#1175822211yescharacteristic
453C22×C4#1175822211yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822411yescharacteristic
457C22×C4#1175822411yescharacteristic
458C22×C4#1175822611yescharacteristic
459C22×C4#1175822611yescharacteristic
460C22×C4#1175822611yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464M16—22no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—41no—
473M16—41no—
474M16—614no—
475M16—1222no—
476M16—1222no—
477M16—1222no—
478M16C2×C4×D6114yes—
479M16C2×C22⋊D6114yes—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—2441no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—2441no—
500C42—614no—
501C42—1222no—
502C42—1222no—
503C42—2441no—
504C23—81no—
505C23—1222no—
506C23—4882no—
507C23—2441no—
508C23—2441no—
509C23#1451973511yescharacteristic
510C23#1451973511yescharacteristic
511C23#1451976211yescharacteristic
512C23#1451976211yescharacteristic
513C23#1451976611yescharacteristic
514C23#1451976611yescharacteristic
515C23C2×#1175822611yescharacteristic
516Q8—1222no—
517Q8—1222no—
518Q8—2441no—
519Q8—2441no—
520D4—1222no—
521D4—1222no—
522D4—2442no—
523D4—2442no—
524D4—2442no—
525D4—2442no—
526D4—2441no—
527D4—2441no—
528C2×C4—1222no—
529C2×C4—1224no—
530C2×C4—4882no—
531C2×C4—4882no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4#1451973511yescharacteristic
540C2×C4#1451973511yescharacteristic
541C2×C4C4×C22⋊D6111yescharacteristic
542C2×C4C4×C22⋊D6111yescharacteristic
543C2×C4C4×C22⋊D6111yescharacteristic
544C2×C4C4×C22⋊D6111yescharacteristic
545C2×C4#1451973911yescharacteristic
546C2×C4#1451973911yescharacteristic
547C2×C4#1451973911yescharacteristic
548C2×C4#1451973911yescharacteristic
549C2×C4#1451974911yescharacteristic
550C2×C4#1451974911yescharacteristic
551C2×C4#1451974911yescharacteristic
552C2×C4#1451974911yescharacteristic
553C2×C4#1451976211yescharacteristic
554C2×C4#1451976211yescharacteristic
555C2×C4#1451976611yescharacteristic
556C2×C4#1451976611yescharacteristic
557C2×C4#1451977211yescharacteristic
558C2×C4#1451977211yescharacteristic
559C2×C4#1451977211yescharacteristic
560C2×C4#1451977211yescharacteristic
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4C2×#1175822611yescharacteristic
567C8—24no—
568C8—24no—
569C8—42no—
570C8—1224no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22—2442no—
582C22#1683965211yescharacteristic
583C22#1683965211yescharacteristic
584C22#1683965211yescharacteristic
585C22#1683965211yescharacteristic
586C22C2×#1451973511yescharacteristic
587C22C2×#1451976211yescharacteristic
588C22C2×#1451976611yescharacteristic
589C4—4882no—
590C4—2444no—
591C4—2442no—
592C4#1683965211yescharacteristic
593C4#1683965211yescharacteristic
594C4#1683965211yescharacteristic
595C4#1683965211yescharacteristic
596C4C2×C4×C22⋊D6111yescharacteristic
597C4C2×#1451973911yescharacteristic
598C4C2×#1451974911yescharacteristic
599C4C2×#1451977211yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—4882no—
608C2C2×#1683965211yescharacteristic
subgroupG/H
centerC4C2×#14519749
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC2×C61C2×#388
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#430D61
hypocenterC61#21080
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
4. #21454237C2semidirect
5. #21446487C2semidirect
6. C2×#16839533C2semidirect
7. #21445530C2semidirect
8. C2×C61×#430C2semidirect
19. #16839744C4semidirect
21. C2×#14519695C4semidirect
22. C2×#14519693C4semidirect
25. M16×Dic61C22semidirect
26. #16839533C22semidirect
27. #16839523C22semidirect
32. C61×#430C22semidirect
73. #14519695C2×C4semidirect
74. #14519693C2×C4semidirect
296. C2×#430D61semidirect
321. #430C2×D61semidirect
338. C61#21080semidirect

Maximal subgroups

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

subgroupindex
C2×#168397782
C2×#168397442
#214542372
#214464872
C2×#168395332
#214455302
C2×C61×#4302
#2108061