Finite Group Databasebeta

← #29680895

Subgroups — #29680895

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840245C211yesmaximal, characteristic
2#21454409C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic
4#21448442C211yesmaximal, characteristic, complemented
5#21446054C214yesmaximal, complemented
6#21446047C214yesmaximal, complemented
7C2×#16839530C211yesmaximal, characteristic, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C2×C61×#421C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11#16840245—22no—
12#16840245—22no—
13#16840245—22no—
14#16840245C2214yes—
15C2×Q8×Dic61—21no—
16C2×D4×Dic61—21no—
17C2×D4×Dic61—22no—
18C2×#14519736—21no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61—21no—
22C22×C4⋊Dic61C2211yescharacteristic
23#16839804—21no—
24#16839804—21no—
25#16839804C2212yescomplemented
26#16839804C2212yescomplemented
27#16839800—22no—
28#16839771—21no—
29#16839767—21no—
30#16839763C2212yes—
31#16839763C2212yes—
32#16839758—22no—
33#16839661—22no—
34#16839637—22no—
35#16839632C2212yescomplemented
36#16839628C2212yescomplemented
37#16839530C2214yescomplemented
38#16839522C2214yescomplemented
39C22×#41×C61C2211yescharacteristic
40C22×M16×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#421C418yescomplemented
43C2×#104×C61C2214yes—
44C23×Dic61—22no—
45C23×Dic61D412yes—
46C2×#11758194—21no—
47C2×#11758194—22no—
48C2×#11758194—22no—
49C2×#11758194C2312yes—
50C22×#92418—21no—
51Q8×Dic61—24no—
52Q8×Dic61—24no—
53D4×Dic61—28no—
54D4×Dic61—24no—
55D4×Dic61—24no—
56D4×Dic61—28no—
57#14519736—24no—
58#14519736—21no—
59#14519736—21no—
60#14519736—24no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736D412yes—
66#14519736C2312yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—24no—
74C2×C4⋊Dic61—24no—
75C2×C4⋊Dic61D412yes—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2312yes—
81C2×C22⋊Dic61—22no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519711—22no—
88#14519711—22no—
89#14519709—22no—
90#14519709—22no—
91#14519701—22no—
92#14519701—22no—
93#14519698—22no—
94#14519698—22no—
95#14519684—24no—
96C2×#41×C61—21no—
97C2×#41×C61—21no—
98C2×#41×C61—22no—
99C2×#41×C61—22no—
100C2×#41×C61C2×C414yes—
101C2×#41×C61D412yes—
102C2×#41×C61Q814yes—
103C2×#41×C61C2312yes—
104C22×Q8×C61D411yescharacteristic
105C22×D4×C61—21no—
106C22×D4×C61D411yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C2×M16×C61—21no—
110C2×M16×C61—22no—
111C2×M16×C61C2×C414yescomplemented
112C2×M16×C61C2312yes—
113C22×C8×C61—21no—
114#117×C61—22no—
115#117×C61C2×C414yescomplemented
116C2×C4⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61C2312yes—
119#104×C61C2×C4116yescomplemented
120C24×C61—21no—
121C24×C61C2×D411yescharacteristic
122#41×C61—24no—
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—24no—
127#41×C61—22no—
128#41×C61—22no—
129#41×C61C4⋊C418yescomplemented
130#41×C61C22×C414yes—
131#41×C61C2×Q814yes—
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61C4⋊C414yescomplemented
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—24no—
137C2×D4×C61—22no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61C4⋊C414yescomplemented
143C2×D4×C61C2×D412yes—
144C22×C4×C61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—24no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C22×C412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×Q812yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—22no—
162C22×Dic61—21no—
163C22×Dic61—28no—
164C22×Dic61—22no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—82no—
168C22×Dic61—84no—
169C22×Dic61C2×D414yes—
170M16×C61—24no—
171M16×C61—22no—
172M16×C61C22×C414yescomplemented
173C2×C8×C61—24no—
174C2×C8×C61—22no—
175C4⋊C4×C61—24no—
176C22⋊C4×C61—24no—
177C42×C61—22no—
178C42×C61C22×C414yescomplemented
179C22⋊Dic61—28no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—41no—
183C22⋊Dic61—41no—
184C4⋊Dic61—28no—
185C4⋊Dic61—22no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—84no—
196C4⋊Dic61C2×D412yes—
197C4⋊Dic61C2×D412yes—
198C4×Dic61—28no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—22no—
209C23×C61—24no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61C4×D412yes—
215C23×C61#12011yescharacteristic
216C23×C61#12011yescharacteristic
217C23×C61C22⋊Q812yes—
218C23×C61C22×D411yescharacteristic
219Q8×C61—22no—
220Q8×C61—24no—
221Q8×C61C2×C4⋊C414yes—
222D4×C61—24no—
223D4×C61—24no—
224D4×C61—28no—
225D4×C61—22no—
226D4×C61—24no—
227D4×C61C2×C4⋊C414yes—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—28no—
233C2×C4×C61—24no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C4⋊C412yes—
237C2×C4×C61C2×C4⋊C412yes—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C22⋊Q812yes—
244C2×C4×C61C22⋊Q812yes—
245C2×C4×C61C22⋊Q812yes—
246C2×C4×C61#12312yes—
247C2×C4×C61#12312yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×Q811yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—42no—
258C2×Dic61—44no—
259C2×Dic61—42no—
260C2×Dic61—44no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—82no—
266C2×Dic61—88no—
267C8×C61—24no—
268#92418—42no—
269#20864—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—24no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—48no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61#38512yes—
283C22×C61#38512yes—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61C2×#12011yescharacteristic
286C22×C61C2×C22⋊Q811yescharacteristic
287C4×C61—24no—
288C4×C61—44no—
289C4×C61#38512yes—
290C4×C61#38512yes—
291C4×C61C22×C4⋊C411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×C22⋊Q811yescharacteristic
294C4×C61C2×#12311yescharacteristic
295Dic61—44no—
296Dic61—42no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
300C2×#520—611no—
301#2911—611no—
302C2×#421D6111yescharacteristic
303#2896—611no—
304#2317—611no—
305#1860—614no—
306#1859—614no—
307C2×#343—611no—
308#1733—611no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61—44no—
314C2×C61C2×#38511yescharacteristic
315C2×C61#301911yescharacteristic
316C2×C61#302011yescharacteristic
317C22×#41C2×D6111yescharacteristic
318C22×M16—611no—
319C22×M16C2×D6111yescharacteristic
320#520—614no—
321#520—1222no—
322#520—1222no—
323#520—1222no—
324C2×C4×Q8—1221no—
325C2×C4×D4—1222no—
326C2×C4×D4—1221no—
327C2×#117—611no—
328C2×#117—611no—
329C2×#117—1221no—
330C2×#117C2×D6111yescharacteristic
331C22×C4⋊C4—611no—
332C22×C4⋊C4—1221no—
333#428—612no—
334#428—612no—
335#428—1221no—
336#428—1221no—
337#427—1222no—
338#421Dic6118yescomplemented
339C2×#104C2×D6114yes—
340#419—1221no—
341#418—1221no—
342#417—612no—
343#417—612no—
344#416—1222no—
345#389—1222no—
346#385—1222no—
347#384—612no—
348#382—612no—
349#343—614no—
350#339—614no—
351C61#2086411yescharacteristic, complemented, Sylow
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41Dic12214yes—
357C2×#41D24412yes—
358C2×#41C2×Dic6114yes—
359C2×#41C22×D6112yes—
360C22×Q8D24411yescharacteristic
361C22×D4—21no—
362C22×D4D24411yescharacteristic
363C23×C4—21no—
364C23×C4—612no—
365C23×C4—1222no—
366C23×C4C22×D6111yescharacteristic
367C2×M16—21no—
368C2×M16—22no—
369C2×M16—612no—
370C2×M16—1221no—
371C2×M16—1222no—
372C2×M16—1222no—
373C2×M16C2×Dic6114yescomplemented
374C2×M16C22×D6112yes—
375C22×C8—21no—
376C22×C8—1221no—
377C4×Q8—1224no—
378C4×Q8—1224no—
379C4×D4—1228no—
380C4×D4—1228no—
381C4×D4—1224no—
382C4×D4—1224no—
383#117—22no—
384#117—612no—
385#117—612no—
386#117—1224no—
387#117—1224no—
388#117—1221no—
389#117—1221no—
390#117—1221no—
391#117—1221no—
392#117—1221no—
393#117—1221no—
394#117C2×Dic6114yescomplemented
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1224no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1224no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—1222no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42C22×D6112yes—
418#107—1222no—
419#107—1222no—
420#106—1222no—
421#106—1222no—
422#104C2×Dic61116yescomplemented
423#103—1222no—
424#103—1222no—
425#102—1222no—
426#102—1222no—
427#95—1224no—
428C24—21no—
429C24D4×D6111yescharacteristic
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41—24no—
437#41C4⋊Dic6118yescomplemented
438#41C2×Dic12214yes—
439#41C22×Dic6114yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8C4⋊Dic6114yescomplemented
443C2×Q8C2×D24412yes—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4C4⋊Dic6114yescomplemented
451C2×D4C2×D24412yes—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—614no—
459C22×C4—1228no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—4884no—
467C22×C4—4882no—
468C22×C4C2×Dic12212yes—
469C22×C4C2×D24411yescharacteristic
470C22×C4C2×D24411yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4#1175822412yes—
475C22×C4#1175822412yes—
476C22×C4C22×Dic6112yes—
477C22×C4C23×D6111yescharacteristic
478M16—22no—
479M16—24no—
480M16—1222no—
481M16—1222no—
482M16—2442no—
483M16C22×Dic6114yescomplemented
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—1222no—
487C2×C8—2442no—
488C4⋊C4—24no—
489C4⋊C4—612no—
490C4⋊C4—612no—
491C4⋊C4—1228no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1221no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—4884no—
503C22⋊C4—24no—
504C22⋊C4—1228no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—2441no—
508C22⋊C4—2441no—
509C42—22no—
510C42—1228no—
511C42—1222no—
512C42—1222no—
513C42—2441no—
514C42—2441no—
515C42C22×Dic6114yescomplemented
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23D4×Dic6112yes—
523C23#1451974711yescharacteristic
524C23#1451974711yescharacteristic
525C23#1451975412yes—
526C23C2×D4×D6111yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8C2×C4⋊Dic6114yes—
530D4—24no—
531D4—24no—
532D4—28no—
533D4—22no—
534D4—24no—
535D4C2×C4⋊Dic6114yes—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1224no—
547C2×C4—4882no—
548C2×C4—4888no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4C2×C4⋊Dic6112yes—
558C2×C4C2×C4⋊Dic6112yes—
559C2×C4D4×Dic6112yes—
560C2×C4D4×Dic6112yes—
561C2×C4D4×Dic6112yes—
562C2×C4#1451974711yescharacteristic
563C2×C4#1451974711yescharacteristic
564C2×C4#1451975412yes—
565C2×C4#1451975412yes—
566C2×C4#1451975412yes—
567C2×C4#1451976112yes—
568C2×C4#1451976112yes—
569C2×C4C23×Dic6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C22×D24411yescharacteristic
572C2×C4C22×Dic12211yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C8—24no—
576C8—2442no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#1683963712yes—
590C22#1683963712yes—
591C22C2×D4×Dic6111yescharacteristic
592C22C2×#1451974711yescharacteristic
593C22C2×#1451975411yescharacteristic
594C4—24no—
595C4—44no—
596C4—4881no—
597C4—4881no—
598C4—2442no—
599C4—2444no—
600C4—2442no—
601C4#1683963712yes—
602C4#1683963712yes—
603C4C22×C4⋊Dic6111yescharacteristic
604C4C2×D4×Dic6111yescharacteristic
605C4C2×#1451975411yescharacteristic
606C4C2×#1451976111yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—44no—
612C2C2×#1683963711yescharacteristic
613C2#2145517811yescharacteristic
614C2#2145518511yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#421C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#421D61
hypocenterC61#20864
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. #21454409C2semidirect
4. #21448442C2semidirect
5. #21446054C2semidirect
6. #21446047C2semidirect
7. C2×#16839530C2semidirect
8. #21445529C2semidirect
25. #16839804C22semidirect
26. #16839804C22semidirect
35. #16839632C22semidirect
36. #16839628C22semidirect
37. #16839530C22semidirect
38. #16839522C22semidirect
42. C61×#421C4semidirect
111. C2×M16×C61C2×C4semidirect
115. #117×C61C2×C4semidirect
119. #104×C61C2×C4semidirect
129. #41×C61C4⋊C4semidirect
134. C2×Q8×C61C4⋊C4semidirect
142. C2×D4×C61C4⋊C4semidirect
172. M16×C61C22×C4semidirect
178. C42×C61C22×C4semidirect
338. #421Dic61semidirect
351. C61#20864semidirect
373. C2×M16C2×Dic61semidirect
394. #117C2×Dic61semidirect
422. #104C2×Dic61semidirect
437. #41C4⋊Dic61semidirect
442. C2×Q8C4⋊Dic61semidirect
450. C2×D4C4⋊Dic61semidirect
483. M16C22×Dic61semidirect
515. C42C22×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402452
#214544092
#214543382
#214484422
#214460542
#214460472
C2×#168395302
#214455292
C2×C61×#4212
#2086461