Finite Group Databasebeta

← #29707448

Subgroups — #29707448

Order 15616.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic, complemented
2C2×#16839778C211yesmaximal, characteristic
3C2×#16839778C211yesmaximal, characteristic
4C2×#16839735C211yesmaximal, characteristic
5#21446994C218yesmaximal, complemented
6#21445530C211yesmaximal, characteristic, complemented
7C2×#16839519C211yesmaximal, characteristic, complemented
8C2×C61×#344C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10#16840393—22no—
11#16840390—22no—
12#16840390—22no—
13#16840388—22no—
14#16840319—22no—
15#16840315—22no—
16#16840315C2214yescomplemented
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839778—22no—
21#16839778—22no—
22#16839778—22no—
23#16839778C2214yes—
24C2×#14519703—22no—
25C2×#14519703C2214yes—
26#16839735—22no—
27#16839735C2214yes—
28C2×#14519691—22no—
29#16839523C2214yescomplemented
30#16839519C2214yescomplemented
31C22×M16×C61C2211yescharacteristic
32C22×M16×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C61×#344C2214yescomplemented
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×#11758222—21no—
45C2×#11758222Q812yescomplemented
46C2×#11758222C2312yes—
47C22×Dic122Q811yescharacteristic, complemented
48C22×D244Q811yescharacteristic, complemented
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51#14519777—24no—
52#14519776—24no—
53#14519776—24no—
54#14519769—24no—
55#14519768—24no—
56#14519768—24no—
57#14519756—28no—
58#14519754—28no—
59#14519754—28no—
60C4×Dic122—28no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736D412yes—
69#14519736D412yes—
70#14519736D412yes—
71#14519736C2312yes—
72#14519736C2312yes—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D412yes—
81C2×C4×Dic61C2312yes—
82#14519703—28no—
83#14519703—28no—
84#14519691—28no—
85C23×C4×C61C2311yescharacteristic
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61—21no—
96C2×M16×C61C2312yes—
97C2×M16×C61C2312yes—
98C22×C8×C61—21no—
99C22×C8×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61D412yes—
103#117×C61C2312yes—
104C2×C4⋊C4×C61—21no—
105C2×C22⋊C4×C61D412yes—
106C2×C42×C61—21no—
107C24×C61C2×D411yescharacteristic
108C23×D61—21no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×Q811yescharacteristic
116C22×C4×C61C2×Q811yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—41no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—41no—
143#11758222—41no—
144#11758222—41no—
145#11758222—41no—
146#11758222—41no—
147#11758222—41no—
148#11758222—41no—
149#11758222—41no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C2×Q812yes—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—28no—
168C2×Dic122C2×Q812yes—
169M16×C61—22no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—41no—
174M16×C61—41no—
175M16×C61—41no—
176M16×C61—41no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—41no—
180C2×C8×C61—41no—
181C2×C8×C61—41no—
182C2×C8×C61—41no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61C2×D414yes—
187C42×C61—22no—
188C42×C61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—41no—
195C4⋊Dic61—28no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—41no—
202#11758196—28no—
203#11758196—28no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—41no—
208C4×Dic61C2×D414yes—
209C4×Dic61C2×D414yes—
210C23×C61—81no—
211C23×C61#12011yescharacteristic
212C23×C61#12011yescharacteristic
213C23×C61C22⋊Q811yescharacteristic
214C23×C61C22⋊Q811yescharacteristic
215C23×C61#12411yescharacteristic
216C23×C61#12411yescharacteristic
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—82no—
222C2×C4×C61—24no—
223C2×C4×C61—82no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61C22⋊Q811yescharacteristic
227C2×C4×C61C22⋊Q811yescharacteristic
228C2×C4×C61C22⋊Q811yescharacteristic
229C2×C4×C61C22⋊Q811yescharacteristic
230C2×C4×C61C22⋊Q811yescharacteristic
231C2×C4×C61C22⋊Q811yescharacteristic
232C2×C4×C61C22⋊Q811yescharacteristic
233C2×C4×C61C22⋊Q811yescharacteristic
234C2×C4×C61C22⋊Q811yescharacteristic
235C2×C4×C61C22⋊Q811yescharacteristic
236C2×C4×C61#12411yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12411yescharacteristic
239C2×C4×C61#12411yescharacteristic
240C2×C4×C61#12411yescharacteristic
241C2×C4×C61#12411yescharacteristic
242C2×C4×C61#12411yescharacteristic
243C2×C4×C61#12411yescharacteristic
244C2×C4×C61#12411yescharacteristic
245C2×C4×C61#12411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×Q811yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—42no—
253C22⋊D61—42no—
254C22⋊D61—42no—
255C22⋊D61—42no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—42no—
262C2×Dic61—82no—
263C2×Dic61—82no—
264C2×Dic61—82no—
265D244—22no—
266D244—22no—
267D244—41no—
268D244—41no—
269C4×D61—42no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—42no—
273Dic122—22no—
274Dic122—22no—
275Dic122—41no—
276Dic122—41no—
277Dic122—88no—
278C8×C61—42no—
279C8×C61—42no—
280#23472—611nomaximal, Sylow
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61#39311yescharacteristic
289C22×C61#39311yescharacteristic
290C22×C61#39311yescharacteristic
291C22×C61#39311yescharacteristic
292C22×C61C2×#12011yescharacteristic
293C22×C61C2×C22⋊Q811yescharacteristic
294C22×C61C2×#12411yescharacteristic
295C2×D61—42no—
296C2×D61—82no—
297C2×D61—82no—
298C2×D61—82no—
299C4×C61—44no—
300C4×C61#39311yescharacteristic
301C4×C61#39311yescharacteristic
302C4×C61#39311yescharacteristic
303C4×C61#39311yescharacteristic
304C4×C61C2×C22⋊Q811yescharacteristic
305C4×C61C2×C22⋊Q811yescharacteristic
306C4×C61C2×#12411yescharacteristic
307C4×C61C2×#12411yescharacteristic
308Dic61—44no—
309Dic61—44no—
310Dic61—42no—
311Dic61—82no—
312#3458—611no—
313C2×#421—611no—
314C2×#421—611no—
315C2×#410—611no—
316#2031—618no—
317C2×#344D6111yescharacteristic, complemented
318#1733—611no—
319C2×H(3,ℤ/4)—611no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61C2×#39311yescharacteristic
328D61—82no—
329C22×#41—611no—
330C22×M16C2×D6111yescharacteristic
331C22×M16C2×D6111yescharacteristic
332#544—1222no—
333#544—1222no—
334#544—1222no—
335#543—1222no—
336#533—614no—
337#533—1222no—
338#533—1222no—
339C2×#117—611no—
340C2×#117—611no—
341C2×#117—611no—
342C2×#117C2×D6111yescharacteristic
343#421—614no—
344#421—1222no—
345#421—1222no—
346#421—1222no—
347C2×#104—614no—
348C2×#104—1222no—
349#410—614no—
350#410—1222no—
351C2×#99—1222no—
352#344C2×D6114yescomplemented
353#339—614no—
354H(3,ℤ/4)—614no—
355C61#2347211yescharacteristic, complemented, Sylow
356C2×#41—612no—
357C2×#41—612no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C22×Q8—611no—
369C22×D4—611no—
370C22×D4—1221no—
371C23×C4—1221no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16C22×D6112yes—
384C2×M16C22×D6112yes—
385C22×C8—21no—
386C22×C8—21no—
387C4⋊Q8—1224no—
388C4⋊Q8—1224no—
389C4⋊Q8—1224no—
390#125—1224no—
391#125—1224no—
392#125—1224no—
393C22⋊Q8—1228no—
394C22⋊Q8—1228no—
395C22⋊Q8—1228no—
396C4×Q8—1228no—
397#117—21no—
398#117—21no—
399#117—612no—
400#117—612no—
401#117—612no—
402#117—612no—
403#117—612no—
404#117—1221no—
405#117—1221no—
406#117—1221no—
407#117—1221no—
408#117—1221no—
409#117—1221no—
410#117—1221no—
411#117D24412yes—
412#117C22×D6112yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4D24412yes—
421C2×C42—21no—
422C2×C42—612no—
423C2×C42—612no—
424C2×C42—1221no—
425#104—1228no—
426#104—1228no—
427#99—1228no—
428C24—1221no—
429C24C2×D24411yescharacteristic
430#41—1222no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—1222no—
437#41—1222no—
438#41—2441no—
439#41—2441no—
440#41—2441no—
441#41—2441no—
442#41—2441no—
443#41—2441no—
444#41—2441no—
445#41—2441no—
446C2×Q8—612no—
447C2×Q8—1228no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×D4—612no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C22×C4—22no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—2441no—
472C22×C4—2441no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4—2441no—
476C22×C4C2×D24411yescharacteristic
477C22×C4#1175822211yescharacteristic
478C22×C4#1175822211yescharacteristic
479C22×C4#1175822211yescharacteristic
480C22×C4#1175822211yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4#1175822411yescharacteristic
486C22×C4#1175822411yescharacteristic
487C22×C4Q8×D6111yescharacteristic
488C22×C4Q8×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490M16—22no—
491M16—22no—
492M16—22no—
493M16—22no—
494M16—41no—
495M16—41no—
496M16—41no—
497M16—41no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—1228no—
507C4⋊C4—1228no—
508C4⋊C4—1228no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—2441no—
515C22⋊C4—22no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—2441no—
522C22⋊C4C2×D24414yes—
523C42—22no—
524C42—22no—
525C42—614no—
526C42—614no—
527C42—1222no—
528C42—1222no—
529C42—1222no—
530C42—2441no—
531C23—81no—
532C23—1222no—
533C23—4882no—
534C23—2441no—
535C23—2441no—
536C23#1451974711yescharacteristic
537C23#1451974711yescharacteristic
538C23#1451975711yescharacteristic
539C23#1451975711yescharacteristic
540C23#1451976511yescharacteristic
541C23#1451976511yescharacteristic
542C23C22×D24411yescharacteristic
543Q8—1222no—
544Q8—1222no—
545Q8—4888no—
546Q8—2441no—
547Q8—2441no—
548D4—1222no—
549D4—1222no—
550D4—2442no—
551D4—2442no—
552D4—2442no—
553D4—2442no—
554D4—2441no—
555D4—2441no—
556C2×C4—24no—
557C2×C4—82no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1222no—
561C2×C4—4882no—
562C2×C4—4882no—
563C2×C4—4882no—
564C2×C4—2442no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4#1451974711yescharacteristic
572C2×C4#1451974711yescharacteristic
573C2×C4#1451975511yescharacteristic
574C2×C4#1451975511yescharacteristic
575C2×C4#1451975511yescharacteristic
576C2×C4#1451975511yescharacteristic
577C2×C4#1451975511yescharacteristic
578C2×C4#1451975511yescharacteristic
579C2×C4#1451975511yescharacteristic
580C2×C4#1451975511yescharacteristic
581C2×C4#1451975711yescharacteristic
582C2×C4#1451975711yescharacteristic
583C2×C4#1451976311yescharacteristic
584C2×C4#1451976311yescharacteristic
585C2×C4#1451976311yescharacteristic
586C2×C4#1451976311yescharacteristic
587C2×C4#1451976311yescharacteristic
588C2×C4#1451976311yescharacteristic
589C2×C4#1451976311yescharacteristic
590C2×C4#1451976311yescharacteristic
591C2×C4#1451976511yescharacteristic
592C2×C4#1451976511yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C2×Q8×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C8—42no—
600C8—42no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—4882no—
609C22—4882no—
610C22—4882no—
611C22—2442no—
612C22#1683967511yescharacteristic
613C22#1683967511yescharacteristic
614C22#1683967511yescharacteristic
615C22#1683967511yescharacteristic
616C22C2×#1451974711yescharacteristic
617C22C2×#1451975711yescharacteristic
618C22C2×#1451976511yescharacteristic
619C4—44no—
620C4—4882no—
621C4—2444no—
622C4—2444no—
623C4—2442no—
624C4#1683967511yescharacteristic
625C4#1683967511yescharacteristic
626C4#1683967511yescharacteristic
627C4#1683967511yescharacteristic
628C4C2×#1451975511yescharacteristic
629C4C2×#1451975511yescharacteristic
630C4C2×#1451976311yescharacteristic
631C4C2×#1451976311yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—4882no—
640C2C2×#1683967511yescharacteristic
subgroupG/H
centerC4C2×#14519763
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#344C2
socleC2×C61C2×#393
prime radicalC22×#11758222C22
hypercenterC2×#344D61
hypocenterC61#23472
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
1. #21459224C2semidirect
5. #21446994C2semidirect
6. #21445530C2semidirect
7. C2×#16839519C2semidirect
8. C2×C61×#344C2semidirect
16. #16840315C22semidirect
29. #16839523C22semidirect
30. #16839519C22semidirect
34. C61×#344C22semidirect
45. C2×#11758222Q8semidirect
47. C22×Dic122Q8semidirect
48. C22×D244Q8semidirect
317. C2×#344D61semidirect
352. #344C2×D61semidirect
355. C61#23472semidirect

Maximal subgroups

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

subgroupindex
#214592242
C2×#168397782
C2×#168397352
#214469942
#214455302
C2×#168395192
C2×C61×#3442
#2347261