Finite Group Databasebeta

← #29683033

Subgroups — #29683033

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448297C211yesmaximal, characteristic, complemented
2C4×D4×Dic61C211yesmaximal, characteristic
3#21446462C211yesmaximal, characteristic, complemented
4#21446455C211yesmaximal, characteristic, complemented
5#21446413C211yesmaximal, characteristic, complemented
6#21446392C211yesmaximal, characteristic, complemented
7#21446155C211yesmaximal, characteristic
8#21446155C211yesmaximal, characteristic
9#21446082C211yesmaximal, characteristic
10#21446068C211yesmaximal, characteristic
11#21445716C211yesmaximal, characteristic, complemented
12#21445649C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic
14#21445567C211yesmaximal, characteristic, complemented
15C2×C61×#440C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C22×C4×Dic61—21no—
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26#16839669C2211yescharacteristic
27#16839669C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839648—21no—
31#16839648—21no—
32C4⋊C4×Dic61C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34C4×C4⋊Dic61—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C22⋊C4×Dic61—21no—
38C22⋊C4×Dic61—21no—
39C4×C22⋊Dic61—21no—
40#16839608—21no—
41C42×Dic61C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839516C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#440C418yescomplemented
56C2×#107×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67D4×Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—44no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519709—22no—
89#14519701—22no—
90#14519701—22no—
91#14519698—22no—
92#14519698—22no—
93#14519688—22no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104C22×Q8×C61C2311yescharacteristic
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C2×SD16×C61—22no—
108C2×SD16×C61—22no—
109C2×SD16×C61C2×C414yes—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×Q8×C61—22no—
113C4×Q8×C61C2×C414yescomplemented
114C4×D4×C61—22no—
115C4×D4×C61C2×C414yes—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61C2×C414yescomplemented
123#103×C61C2×C414yescomplemented
124#102×C61C2×C414yes—
125#97×C61C2×C414yescomplemented
126C24×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C22×C412yes—
137C2×D4×C61#4112yes—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—24no—
149C22×Dic61—24no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162SD16×C61—28no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C22×C412yes—
168C2×C8×C61C22×C412yes—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22×C412yes—
173C4⋊C4×C61C22×C412yes—
174C22⋊C4×C61—24no—
175C42×C61—24no—
176C42×C61C22×C412yes—
177C42×C61#4112yes—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61#4112yes—
189C4⋊Dic61#4112yes—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61—41no—
197C4×Dic61—41no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C22×D411yescharacteristic
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—41no—
219Q8×C61—41no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—22no—
223D4×C61#11714yes—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—48no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—48no—
259C2×Dic61—48no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61AGL(1,ℤ/8)12yes—
271C2×Dic61#13712yes—
272C8×C61—22no—
273C8×C61—22no—
274#92418—41no—
275#92418—41no—
276#21121—611nomaximal, Sylow
277C22×C61—24no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61#52511yescharacteristic
286C22×C61#52511yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61C2×#11711yescharacteristic
297C4×C61C4×#4111yescharacteristic
298C4×C61#52911yescharacteristic
299C4×C61#52911yescharacteristic
300Dic61—22no—
301Dic61—22no—
302Dic61—42no—
303Dic61—41no—
304Dic61—42no—
305Dic61—41no—
306Dic61—81no—
307Dic61—81no—
308C2×#440D6111yescharacteristic
309#2295—611no—
310C42×D4—611no—
311#1928—611no—
312#1927—611no—
313#1921—611no—
314#1918—611no—
315#1877—611no—
316#1877—611no—
317#1864—611no—
318#1862—611no—
319#1776—611no—
320#1763—611no—
321#1762—611no—
322#1745—611no—
323C2×C61—44no—
324C2×C61#229011yescharacteristic
325C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C61C4×#13711yescharacteristic
327C2×C61#311011yescharacteristic
328C2×C61#311111yescharacteristic
329C2×C61#311411yescharacteristic
330C2×C61#311611yescharacteristic
331C22×SD16C2×D6111yescharacteristic
332C2×C4×Q8C2×D6111yescharacteristic
333C2×C4×D4—611no—
334C2×C4×D4—611no—
335C2×C4×D4C2×D6111yescharacteristic
336C22×C42—1221no—
337#440Dic6118yescomplemented
338C2×#107—611no—
339C2×#107C2×D6111yescharacteristic
340C2×#106—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103C2×D6111yescharacteristic
344C2×#102—611no—
345C2×#102—611no—
346C2×#102C2×D6111yescharacteristic
347C2×#97—611no—
348C2×#97C2×D6111yescharacteristic
349#391—611no—
350#391—611no—
351#389—611no—
352#389—611no—
353#387—1221no—
354#387—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C22⋊C4—1221no—
361C4×C22⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#376—1221no—
364C43—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#336—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375C61#2112111yescharacteristic, complemented, Sylow
376C22×Q8C22×D6111yescharacteristic
377C22×D4C22×D6111yescharacteristic
378C23×C4—21no—
379C23×C4—1221no—
380C23×C4—1221no—
381C2×SD16—22no—
382C2×SD16—22no—
383C2×SD16C2×Dic6114yes—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4×Q8—22no—
389C4×Q8C2×Dic6114yescomplemented
390C4×D4—22no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4C2×Dic6114yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C42—21no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42—1221no—
419C2×C42—2444no—
420C2×C42C22×D6111yescharacteristic
421#107—1222no—
422#107C2×Dic6114yescomplemented
423#106—1222no—
424#103—1222no—
425#103—1222no—
426#103C2×Dic6114yescomplemented
427#102—1222no—
428#102—1222no—
429#102C2×Dic6114yes—
430#97—1222no—
431#97C2×Dic6114yescomplemented
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C22×Dic6112yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4#1175822212yes—
453C2×D4#1175822212yes—
454C2×D4C22×Dic6112yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8C22×Dic6112yes—
490C2×C8C22×Dic6112yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4C22×Dic6112yes—
505C4⋊C4C22×Dic6112yes—
506C22⋊C4—24no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C42—24no—
510C42—1222no—
511C42—1222no—
512C42—1221no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—2441no—
517C42—2441no—
518C42—2442no—
519C42—2442no—
520C42—2442no—
521C42—2442no—
522C42#1175822412yes—
523C42C22×Dic6112yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D6111yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1451973614yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—2448no—
569C2×C4—2448no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4D4×Dic6112yes—
581C2×C4D4×Dic6112yes—
582C2×C4C23×Dic6111yescharacteristic
583C2×C4#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C2×C4C2×#1175822211yescharacteristic
588C8—22no—
589C8—22no—
590C8—2441no—
591C8—2441no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×D4×Dic6111yescharacteristic
600C22#1684027111yescharacteristic
601C22#1684027111yescharacteristic
602C22AGL(1,ℤ/8)×D6111yescharacteristic
603C22#137×D6111yescharacteristic
604C22#1684055811yescharacteristic
605C22#1684056111yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—1222no—
612C4—1222no—
613C4—4881no—
614C4—4881no—
615C4—2442no—
616C4—2441no—
617C4—2442no—
618C4—2441no—
619C4C2×#1451973611yescharacteristic
620C4#41×Dic6111yescharacteristic
621C4#1684029211yescharacteristic
622C4#1684029211yescharacteristic
623C2—44no—
624C2#2144826411yescharacteristic
625C2AGL(1,ℤ/8)×Dic6111yescharacteristic
626C2#137×Dic6111yescharacteristic
627C2#2145591611yescharacteristic
628C2#2145593211yescharacteristic
629C2#2145596111yescharacteristic
630C2#2145597711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#440D61
hypocenterC61#21121
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #21448297C2semidirect
3. #21446462C2semidirect
4. #21446455C2semidirect
5. #21446413C2semidirect
6. #21446392C2semidirect
11. #21445716C2semidirect
12. #21445649C2semidirect
14. #21445567C2semidirect
55. C61×#440C4semidirect
113. C4×Q8×C61C2×C4semidirect
122. #107×C61C2×C4semidirect
123. #103×C61C2×C4semidirect
125. #97×C61C2×C4semidirect
337. #440Dic61semidirect
375. C61#21121semidirect
389. C4×Q8C2×Dic61semidirect
422. #107C2×Dic61semidirect
426. #103C2×Dic61semidirect
431. #97C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214482972
C4×D4×Dic612
#214464622
#214464552
#214464132
#214463922
#214461552
#214460822
#214460682
#214457162
#214456492
#214456442
#214455672
C2×C61×#4402
#2112161