Finite Group Databasebeta

← #29689102

Subgroups — #29689102

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840275C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic, complemented
3C2×#16839619C211yesmaximal, characteristic, complemented
4#21446256C212yesmaximal, complemented
5#21446249C212yesmaximal, complemented
6#21446136C211yesmaximal, characteristic, complemented
7#21446134C211yesmaximal, characteristic, complemented
8#21446127C211yesmaximal, characteristic, complemented
9#21445912C212yesmaximal, complemented
10#21445758C212yesmaximal, complemented
11C61×#1872C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758189C2212yes—
15#16840275—24no—
16#16840275C418yescomplemented
17C2×#14519762—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×#14519747—21no—
22C2×#14519739—21no—
23C2×#14519737—21no—
24C2×#14519735C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26C22×#11758198—21no—
27C2×#14519701C2212yescomplemented
28C2×#14519699C2212yescomplemented
29C2×#14519697C2212yescomplemented
30#16839724—24no—
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2212yes—
33#16839619—21no—
34#16839619—21no—
35#16839619C2212yes—
36#16839619C2212yes—
37C2×#14519684—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C2×#11758226—22no—
48C2×#11758226—42no—
49C2×#11758226—42no—
50C2×#11758226C2×C414yescomplemented
51C22×D244—21no—
52C22×D244—21no—
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C23×Dic61D411yescharacteristic
57C22×C4×D61—21no—
58C22×C4×D61—42no—
59C22×C4×D61—42no—
60C22×C4×D61C2×C414yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D414yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758189—24no—
67C2×#11758189—42no—
68C2×#11758189—42no—
69C22×#92418C2312yes—
70#14519762—28no—
71#14519753—22no—
72#14519753C2×C414yescomplemented
73#14519752—22no—
74#14519752C2×C414yescomplemented
75#14519751—22no—
76#14519751C2×C414yes—
77#14519747—28no—
78#14519739—28no—
79#14519737—28no—
80#14519735—22no—
81#14519735C2×C414yescomplemented
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—42no—
84C2×C4⋊Dic61C2312yes—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C2×#11758198—44no—
94C2×C4×Dic61—21no—
95#14519701—42no—
96#14519699—42no—
97#14519697—42no—
98C22⋊#92418—22no—
99#14519689—24no—
100#14519684—22no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2×C414yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—42no—
111C2×C4⋊C4×C61C2312yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114#103×C61—42no—
115C22⋊C8×C61—22no—
116C24×C61C2×D411yescharacteristic
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—21no—
120C23×D61—41no—
121C23×D61—41no—
122C23×D61—41no—
123C23×D61—41no—
124C23×D61C22⋊C414yes—
125C23×D61C2×D411yescharacteristic
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61C22×C412yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—24no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—41no—
149C22×Dic61—41no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61C22⋊C414yes—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156#11758226—48no—
157#11758226—48no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—24no—
161C2×D244—24no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—42no—
165C2×D244—42no—
166C2×D244—42no—
167C2×D244C22×C412yes—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—24no—
173C2×C4×D61—48no—
174C2×C4×D61—48no—
175C2×C4×D61—41no—
176C2×C4×D61—41no—
177C2×C4×D61—41no—
178C2×C4×D61—41no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×C4×D61C22⋊C414yes—
182C2×C4×D61C22⋊C414yes—
183C2×C4×D61C22×C412yes—
184C2×C4×D61C22×C412yes—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2×D412yes—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—42no—
195C4⋊C4×C61C22×C412yes—
196C22⋊C4×C61—24no—
197#11758198—24no—
198#11758198—24no—
199#11758198—24no—
200#11758198—24no—
201#11758198—44no—
202#11758198—44no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61—42no—
207C4⋊Dic61C22×C412yes—
208#11758196—24no—
209C4×Dic61—24no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213#11758189—48no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61C2×C22⋊C412yes—
219C23×C61AGL(1,ℤ/8)12yes—
220C23×C61#13712yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—28no—
235C22×D61—22no—
236C22×D61—44no—
237C22×D61—44no—
238C22×D61—44no—
239C22×D61—44no—
240C22×D61—44no—
241C22×D61—44no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61—41no—
246C22×D61—41no—
247C22×D61—41no—
248C22×D61C2×C22⋊C412yes—
249C22×D61C2×C22⋊C412yes—
250Q8×C61—24no—
251Q8×C61—42no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—24no—
255C2×C4×C61—24no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—84no—
266C2×C4×C61C2×C22⋊C412yes—
267C2×C4×C61C23×C411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61H(5,𝔽2)12yes—
270C2×C4×C61H(5,𝔽2)12yes—
271C22⋊D61—48no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—22no—
277C2×Dic61—44no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61—84no—
284C2×Dic61C2×C22⋊C412yes—
285C2×Dic61C2×C22⋊C412yes—
286D244—24no—
287D244—48no—
288D244—48no—
289D244—42no—
290C4×D61—24no—
291C4×D61—24no—
292C4×D61—24no—
293C4×D61—24no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—48no—
297C8×C61—42no—
298#92418—42no—
299#21769—611nomaximal, Sylow
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61#41714yes—
307C22×C61C22×C22⋊C411yescharacteristic
308C22×C61#53411yescharacteristic
309C22×C61#53411yescharacteristic
310C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C61C2×#13711yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57511yescharacteristic
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—416no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—41no—
329C2×D61—42no—
330C2×D61—42no—
331C2×D61—42no—
332C2×D61—81no—
333C2×D61—81no—
334C2×D61—81no—
335C2×D61—81no—
336C4×C61—22no—
337C4×C61—42no—
338C4×C61—42no—
339C4×C61—82no—
340C4×C61#51812yes—
341C4×C61#53412yes—
342Dic61—22no—
343Dic61—22no—
344Dic61—42no—
345Dic61—82no—
346C2×#525—611no—
347C2×#407—611no—
348C2×#380—611no—
349#1895—612no—
350#1894—612no—
351#1874—611no—
352#1874—611no—
353#1872—611no—
354#1872D6111yescharacteristic, complemented
355#1825—612no—
356#1788—612no—
357C2×C61—24no—
358C2×C61#230811yescharacteristic
359C2×C61C2×#41711yescharacteristic
360C2×C61#289611yescharacteristic
361C2×C61#319411yescharacteristic
362C2×C61#319411yescharacteristic
363C2×C61#319611yescharacteristic
364C2×C61#319611yescharacteristic
365D61—42no—
366D61—42no—
367D61—82no—
368C22×#41—611no—
369C24×C4—611no—
370C22×M16—612no—
371#525—618no—
372#525—1224no—
373C2×#123—1221no—
374C2×C22⋊Q8—611no—
375C2×C22⋊Q8C2×D6111yescharacteristic
376C2×#121—611no—
377C2×#121—611no—
378C2×#120—1221no—
379C2×C4×D4—1221no—
380C2×C4×D4—1221no—
381C2×#117—611no—
382C22×C4⋊C4—611no—
383C22×C4⋊C4C2×D6111yescharacteristic
384C22×C22⋊C4—1221no—
385C2×#103—612no—
386C2×#103C2×D6112yescomplemented
387C2×#102—612no—
388C2×#102—612no—
389#407—1224no—
390C2×C22⋊C8—611no—
391C2×C22⋊C8—612no—
392C2×C22⋊C8C2×D6111yescharacteristic
393#380—612no—
394#380—612no—
395#380—1221no—
396#380—1221no—
397C2×#95—1221no—
398#340—612no—
399#340—612no—
400#340—612no—
401#340C2×D6112yescomplemented
402C61#2176911yescharacteristic, complemented, Sylow
403C25—611no—
404C2×#41—614no—
405C2×#41—1222no—
406C2×#41—2442no—
407C2×#41—2442no—
408C22×Q8C22×D6111yescharacteristic
409C22×D4—611no—
410C22×D4—1221no—
411C22×D4—1221no—
412C22×D4—1221no—
413C23×C4—21no—
414C23×C4—611no—
415C23×C4—611no—
416C23×C4—611no—
417C23×C4—614no—
418C23×C4—611no—
419C23×C4—611no—
420C23×C4—614no—
421C23×C4—1221no—
422C23×C4—1221no—
423C23×C4—2442no—
424C23×C4—2442no—
425C23×C4C22×D6111yescharacteristic
426C2×M16—1224no—
427C2×M16—2442no—
428C2×M16—2442no—
429C22×C8—612no—
430C22×C8C22×D6112yes—
431#123—1228no—
432C22⋊Q8—22no—
433C22⋊Q8—614no—
434C22⋊Q8—1222no—
435C22⋊Q8C4×D6114yes—
436#121—614no—
437#121—614no—
438#121—1222no—
439#121—1222no—
440#120—1228no—
441C4×D4—1228no—
442C4×D4—1228no—
443#117—614no—
444#117—1222no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—42no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—2442no—
453C2×C4⋊C4C22×D6112yes—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4—2444no—
463C2×C42—1221no—
464#103—42no—
465#103—2442no—
466#102—2442no—
467#102—2442no—
468C22⋊C8—22no—
469C22⋊C8—1224no—
470C22⋊C8—1222no—
471#95—1222no—
472#95—1222no—
473C24—611no—
474C24—611no—
475C24—614no—
476C24—1222no—
477C24—1222no—
478C24—1221no—
479C24—2441no—
480C24—2441no—
481C24—2441no—
482C24—2441no—
483C24D4×D6111yescharacteristic
484#41—2448no—
485#41—2448no—
486C2×Q8—21no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8C2×C4×D6112yes—
490C2×D4—612no—
491C2×D4—1224no—
492C2×D4—1224no—
493C2×D4—1224no—
494C2×D4—1221no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—2442no—
498C2×D4—2442no—
499C2×D4—2442no—
500C2×D4—2442no—
501C2×D4—2442no—
502C2×D4—2442no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—614no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—612no—
517C22×C4—612no—
518C22×C4—614no—
519C22×C4—614no—
520C22×C4—1224no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1221no—
526C22×C4—1222no—
527C22×C4—2448no—
528C22×C4—2448no—
529C22×C4—2441no—
530C22×C4—2441no—
531C22×C4—2441no—
532C22×C4—2441no—
533C22×C4—2442no—
534C22×C4—2442no—
535C22×C4—2441no—
536C22×C4—2441no—
537C22×C4—2442no—
538C22×C4—2442no—
539C22×C4—2442no—
540C22×C4C2×C4×D6112yes—
541C22×C4D4×D6112yes—
542C22×C4D4×D6111yescharacteristic
543C22×C4C23×D6111yescharacteristic
544M16—2448no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—1222no—
549C2×C8—1222no—
550C2×C8—1222no—
551C4⋊C4—21no—
552C4⋊C4—24no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—42no—
556C4⋊C4—612no—
557C4⋊C4—1224no—
558C4⋊C4—1221no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C4⋊C4—2442no—
562C4⋊C4C2×C4×D6112yes—
563C22⋊C4—24no—
564C22⋊C4—1224no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C22⋊C4—2444no—
569C22⋊C4—2444no—
570C42—1224no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—612no—
576C23—612no—
577C23—1228no—
578C23—1221no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23—1221no—
587C23—1221no—
588C23—1221no—
589C23—1221no—
590C23—1222no—
591C23—2444no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23—2444no—
596C23—2444no—
597C23—2441no—
598C23—2441no—
599C23—2441no—
600C23—2441no—
601C23—2441no—
602C23—2441no—
603C23C22⋊C4×D6112yes—
604C23#1451980412yes—
605C23#1451980912yes—
606C23C2×D4×D6111yescharacteristic
607Q8—24no—
608Q8—42no—
609D4—1224no—
610D4—2448no—
611D4—2448no—
612D4—2448no—
613D4—2442no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4—612no—
629C2×C4—612no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1224no—
635C2×C4—1224no—
636C2×C4—1224no—
637C2×C4—1224no—
638C2×C4—1222no—
639C2×C4—4884no—
640C2×C4—2444no—
641C2×C4—2444no—
642C2×C4—2448no—
643C2×C4—2444no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2442no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4C22⋊C4×D6112yes—
650C2×C4C22×C4×D6111yescharacteristic
651C2×C4C2×D4×D6111yescharacteristic
652C2×C4#1451982212yes—
653C2×C4#1451982312yes—
654C8—42no—
655C8—2442no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—1222no—
663C22—1222no—
664C22—1222no—
665C22—1222no—
666C22—1222no—
667C22—1222no—
668C22—4881no—
669C22—4881no—
670C22—4881no—
671C22—4881no—
672C22—24416no—
673C22—2442no—
674C22—2442no—
675C22—2442no—
676C22—2441no—
677C22—2441no—
678C22—2441no—
679C22—2441no—
680C22—2441no—
681C22—2441no—
682C22—2441no—
683C22—2441no—
684C22#1683976214yes—
685C22C2×C22⋊C4×D6111yescharacteristic
686C22#1684032211yescharacteristic
687C22#1684032311yescharacteristic
688C22C2×#1451980411yescharacteristic
689C22C2×#1451980911yescharacteristic
690C22#1684055811yescharacteristic
691C22#1684056111yescharacteristic
692C4—22no—
693C4—42no—
694C4—42no—
695C4—82no—
696C4—1222no—
697C4—1222no—
698C4—4882no—
699C4—2442no—
700C4#1684023412yes—
701C4#1684032012yes—
702C2—24no—
703C2—4882no—
704C2—2442no—
705C2—2442no—
706C2#2144839311yescharacteristic
707C2C2×#1683976211yescharacteristic
708C2#2145433511yescharacteristic
709C2#2145663611yescharacteristic
710C2#2145663811yescharacteristic
711C2#2145665211yescharacteristic
712C2#2145665411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC2×#16840275C2
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21769
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#16839724C2semidirect
3. C2×#16839619C2semidirect
4. #21446256C2semidirect
5. #21446249C2semidirect
6. #21446136C2semidirect
7. #21446134C2semidirect
8. #21446127C2semidirect
9. #21445912C2semidirect
10. #21445758C2semidirect
11. C61×#1872C2semidirect
16. #16840275C4semidirect
27. C2×#14519701C22semidirect
28. C2×#14519699C22semidirect
29. C2×#14519697C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#103×C61C22semidirect
45. C61×#340C22semidirect
50. C2×#11758226C2×C4semidirect
72. #14519753C2×C4semidirect
74. #14519752C2×C4semidirect
81. #14519735C2×C4semidirect
354. #1872D61semidirect
386. C2×#103C2×D61semidirect
401. #340C2×D61semidirect
402. C61#21769semidirect

Maximal subgroups

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

subgroupindex
C2×#168402752
C2×#168397242
C2×#168396192
#214462562
#214462492
#214461362
#214461342
#214461272
#214459122
#214457582
C61×#18722
#2176961