Finite Group Databasebeta

← #29602008

Subgroups — #29602008

Order 15616.

2222 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758196C211yesmaximal, characteristic
2C2×#16839646C212yesmaximal, complemented
3C2×#16839620C211yesmaximal, characteristic
4#21444085C212yesmaximal, complemented
5#21444070C212yesmaximal, complemented
6#21444052C212yesmaximal, complemented
7#21444042C211yesmaximal, characteristic
8#21444041C211yesmaximal, characteristic
9#21444041C211yesmaximal, characteristic
10#21444041C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C22×#11758196—28no—
15C22×#11758196C418yescomplemented
16C22×#11758196C2212yes—
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×#11758196C2211yescharacteristic
20C22×#11758196C2211yescharacteristic
21C22×C22⋊Dic61C2211yescharacteristic
22C22×C22⋊Dic61C2211yescharacteristic
23#16839646—28no—
24#16839620—24no—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C24×C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2211yescharacteristic
37C22×C42×C61C2212yes—
38C4×C22⋊C4×C61—24no—
39C2×#95×C61C2212yes—
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—24no—
43C23×Dic61—24no—
44C23×Dic61—22no—
45C23×Dic61—22no—
46C23×Dic61—24no—
47C23×Dic61—42no—
48C23×Dic61C2×C414yes—
49C23×Dic61C2×C414yes—
50C23×Dic61C2312yes—
51C23×Dic61C2312yes—
52C23×Dic61C2311yescharacteristic
53C23×Dic61C2311yescharacteristic
54C23×Dic61C2311yescharacteristic
55C23×Dic61C2311yescharacteristic
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196—232no—
60C2×#11758196—24no—
61C2×#11758196—24no—
62C2×#11758196—42no—
63C2×#11758196C2×C414yes—
64C2×#11758196C2×C414yes—
65C2×#11758196C2×C414yes—
66C2×#11758196C2×C414yes—
67C2×#11758196D418yes—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74#14519684—24no—
75#14519684—24no—
76#14519684—24no—
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—42no—
82#14519684—42no—
83C25×C61C2311yescharacteristic
84C23×C4×C61—22no—
85C23×C4×C61—22no—
86C23×C4×C61—24no—
87C23×C4×C61—42no—
88C23×C4×C61C2×C414yes—
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2312yes—
91C23×C4×C61C2312yes—
92C23×C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61D414yes—
98C2×C22⋊C4×C61Q814yes—
99C2×C42×C61—24no—
100C2×C42×C61—24no—
101C2×C42×C61—42no—
102#95×C61—24no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61—22no—
107C24×C61C22×C412yes—
108C24×C61C2×D412yes—
109C24×C61C2×Q812yes—
110C24×C61#4112yes—
111C24×C61#4112yes—
112C24×C61#4112yes—
113C24×C61#4112yes—
114C24×C61C2411yescharacteristic
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—28no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—28no—
136C22×C4×C61—44no—
137C22×C4×C61—48no—
138C22×C4×C61—48no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2×Q814yes—
144C22×Dic61—216no—
145C22×Dic61—216no—
146C22×Dic61—28no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61—28no—
157C22×Dic61—28no—
158C22×Dic61—28no—
159C22×Dic61—21no—
160C22×Dic61—22no—
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61—28no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61—22no—
175C22×Dic61—22no—
176C22×Dic61—22no—
177C22×Dic61—22no—
178C22×Dic61—44no—
179C22×Dic61—48no—
180C22×Dic61—48no—
181C22×Dic61C22×C412yes—
182C22×Dic61C22×C412yes—
183C22×Dic61C22×C412yes—
184C22×Dic61C22×C412yes—
185C22×Dic61C2×D414yes—
186C22×Dic61C2×D414yes—
187C22×Dic61#4114yes—
188C22⋊C4×C61—28no—
189C22⋊C4×C61—28no—
190C42×C61—48no—
191C22⋊Dic61—44no—
192C22⋊Dic61—44no—
193#11758196—28no—
194#11758196—28no—
195#11758196—28no—
196#11758196—48no—
197#11758196C22⋊C4116yescomplemented
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—24no—
203C23×C61—24no—
204C23×C61—24no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—24no—
209C23×C61—24no—
210C23×C61—24no—
211C23×C61—24no—
212C23×C61—24no—
213C23×C61—24no—
214C23×C61#11714yes—
215C23×C61C4×D414yes—
216C23×C61C4×Q814yes—
217C23×C61C22⋊Q814yes—
218C23×C61#12314yes—
219C23×C61#12514yes—
220C23×C61#12614yes—
221C23×C61C23×C411yescharacteristic
222C23×C61C22×D412yes—
223C23×C61C22×D411yescharacteristic
224C23×C61C22×Q811yescharacteristic
225C23×C61C2×#4111yescharacteristic
226C23×C61C2×#4111yescharacteristic
227C23×C61C2×#4112yes—
228C23×C61C2×#4111yescharacteristic
229C23×C61C2×#4111yescharacteristic
230C23×C61H(5,𝔽2)12yes—
231C23×C612−1+412yes—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—416no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61C2×C22⋊C414yes—
257C2×Dic61—232no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—416no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—42no—
289C2×Dic61—42no—
290C2×Dic61C2×C22⋊C414yes—
291C2×Dic61C2×C22⋊C414yes—
292C2×Dic61#11714yes—
293#13882—611nomaximal, Sylow
294C22×C61—24no—
295C22×C61—24no—
296C22×C61—24no—
297C22×C61—24no—
298C22×C61—24no—
299C22×C61—28no—
300C22×C61—28no—
301C22×C61—28no—
302C22×C61—28no—
303C22×C61—24no—
304C22×C61#38718yescomplemented
305C22×C61C22×C22⋊C411yescharacteristic
306C22×C61C2×#11711yescharacteristic
307C22×C61C2×#11711yescharacteristic
308C22×C61C2×C4×D411yescharacteristic
309C22×C61C2×C4×Q811yescharacteristic
310C22×C61#51811yescharacteristic
311C22×C61#51911yescharacteristic
312C22×C61C2×C22⋊Q811yescharacteristic
313C22×C61C2×#12311yescharacteristic
314C22×C61#52512yes—
315C22×C61#52512yes—
316C22×C61C2×#12511yescharacteristic
317C22×C61C2×#12611yescharacteristic
318C22×C61#53912yes—
319C22×C61#53912yes—
320C22×C61D4212yes—
321C22×C61#54612yes—
322C22×C61#54712yes—
323C22×C61#54812yes—
324C22×C61D4×Q812yes—
325C22×C61#55212yes—
326C22×C61#55312yes—
327C22×C61#55412yes—
328C4×C61—24no—
329C4×C61—42no—
330C4×C61—44no—
331C4×C61—44no—
332Dic61—24no—
333Dic61—24no—
334Dic61—24no—
335Dic61—42no—
336Dic61—82no—
337Dic61—82no—
338C23×C4⋊C4—611no—
339C2×#387—612no—
340C2×#380—611no—
341C2×C4×C22⋊C4D6111yescharacteristic
342#1445—612no—
343#1441—612no—
344#1436—612no—
345C22⋊#95—611no—
346C22⋊#95—611no—
347C22⋊#95—611no—
348C22⋊#95—611no—
349C2×C61—28no—
350C2×C61#228011yescharacteristic
351C2×C61C2×#38711yescharacteristic
352C2×C61#231311yescharacteristic
353C2×C61D4×C22⋊C411yescharacteristic
354C2×C61Q8×C22⋊C411yescharacteristic
355C2×C61#234411yescharacteristic
356C2×C61#234511yescharacteristic
357C2×C61#244312yes—
358C2×C61#244912yes—
359C2×C61#247312yes—
360C2×C61#248112yes—
361C24×C4—611no—
362C24×C4—611no—
363C24×C4C2×D6111yescharacteristic
364C22×C4⋊C4—611no—
365C22×C4⋊C4—618no—
366C22×C4⋊C4—611no—
367C22×C4⋊C4—611no—
368C22×C4⋊C4—611no—
369C22×C4⋊C4—612no—
370C22×C4⋊C4—1228no—
371C22×C22⋊C4—611no—
372C22×C22⋊C4—611no—
373C22×C22⋊C4C2×D6111yescharacteristic
374C22×C22⋊C4C2×D6111yescharacteristic
375C22×C42C2×D6112yes—
376#387—1228no—
377#380—1224no—
378C4×C22⋊C4—24no—
379C2×#95—612no—
380C2×#95—612no—
381C2×#95—612no—
382C2×#95—612no—
383C2×#95—612no—
384C2×#95—612no—
385C2×#95—612no—
386C2×#95—612no—
387C2×#95—612no—
388C2×#95C2×D6112yes—
389C61#1388211yescharacteristic, complemented, Sylow
390C25C22×D6111yescharacteristic
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—611no—
398C23×C4—614no—
399C23×C4—611no—
400C23×C4—614no—
401C23×C4—612no—
402C23×C4—612no—
403C23×C4—1224no—
404C23×C4—1224no—
405C23×C4—1224no—
406C23×C4—1222no—
407C23×C4—1222no—
408C23×C4—1222no—
409C23×C4—1222no—
410C23×C4—2442no—
411C23×C4C4×D6114yes—
412C23×C4C22×D6112yes—
413C23×C4C22×D6111yescharacteristic
414C23×C4C22×D6112yes—
415C23×C4C22×D6111yescharacteristic
416C2×C4⋊C4—614no—
417C2×C4⋊C4—618no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4—614no—
420C2×C4⋊C4—614no—
421C2×C4⋊C4—12232no—
422C2×C4⋊C4—1224no—
423C2×C4⋊C4—1224no—
424C2×C4⋊C4—1224no—
425C2×C4⋊C4—1224no—
426C2×C4⋊C4—1224no—
427C2×C4⋊C4—2442no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—1222no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1222no—
437C2×C22⋊C4—1222no—
438C2×C22⋊C4Dic12214yes—
439C2×C22⋊C4C22⋊D6114yes—
440C2×C42—24no—
441C2×C42—24no—
442C2×C42—42no—
443#95—24no—
444#95—1224no—
445#95—1224no—
446#95—1224no—
447#95—1224no—
448#95—1224no—
449#95—1224no—
450#95—1224no—
451#95—2442no—
452#95—2442no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×Dic12212yes—
458C24C2×C4×D6112yes—
459C24#1175822212yes—
460C24#1175822212yes—
461C24#1175822212yes—
462C24#1175822212yes—
463C24C2×C22⋊D6112yes—
464C24C23×D6111yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—614no—
493C22×C4—614no—
494C22×C4—614no—
495C22×C4—612no—
496C22×C4—1228no—
497C22×C4—1228no—
498C22×C4—1228no—
499C22×C4—12216no—
500C22×C4—12216no—
501C22×C4—1222no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—1228no—
511C22×C4—1222no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1222no—
515C22×C4—1228no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—2444no—
531C22×C4—2448no—
532C22×C4—2448no—
533C22×C4C2×Dic12214yes—
534C22×C4C2×C4×D6112yes—
535C22×C4C2×C4×D6112yes—
536C22×C4D4×D6114yes—
537C22×C4C2×C22⋊D6114yes—
538C4⋊C4—6116no—
539C4⋊C4—1228no—
540C4⋊C4—1228no—
541C4⋊C4—1228no—
542C4⋊C4—2448no—
543C22⋊C4—28no—
544C22⋊C4—28no—
545C22⋊C4—2444no—
546C22⋊C4—2444no—
547C42—48no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23#1451973414yes—
565C23C4×C22⋊D6114yes—
566C23C4×Dic12214yes—
567C23#1451975614yes—
568C23#1451976014yes—
569C23#1451976914yes—
570C23#1451977114yes—
571C23C22×C4×D6111yescharacteristic
572C23C2×D4×D6112yes—
573C23C22×C22⋊D6111yescharacteristic
574C23C22×Dic12211yescharacteristic
575C23#41×D6112yes—
576C23C2×#1175822211yescharacteristic
577C23C2×#1175822211yescharacteristic
578C23C2×#1175822211yescharacteristic
579C23C2×#1175822211yescharacteristic
580C23#1451982212yes—
581C23#1451982412yes—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—416no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—614no—
607C2×C4—614no—
608C2×C4—614no—
609C2×C4—12232no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1224no—
617C2×C4—1224no—
618C2×C4—1224no—
619C2×C4—1224no—
620C2×C4—1224no—
621C2×C4—1224no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—24416no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2442no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4C22⋊C4×D6114yes—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22#1683964618yescomplemented
654C22C2×C22⋊C4×D6111yescharacteristic
655C22#117×D6111yescharacteristic
656C22C2×#1451973411yescharacteristic
657C22C2×C4×C22⋊D6111yescharacteristic
658C22C2×C4×Dic12211yescharacteristic
659C22#1684023511yescharacteristic
660C22#1684023911yescharacteristic
661C22C2×#1451975611yescharacteristic
662C22C2×#1451976011yescharacteristic
663C22#1684027112yes—
664C22#1684027112yes—
665C22C2×#1451976911yescharacteristic
666C22C2×#1451977111yescharacteristic
667C22#1684035412yes—
668C22#1684035412yes—
669C22D4×C22⋊D6112yes—
670C22#1684040212yes—
671C22#1684041112yes—
672C22#1684042112yes—
673C22D4×Dic12212yes—
674C22#1684044512yes—
675C22#1684046112yes—
676C22#1684047112yes—
677C4—24no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—1224no—
682C4—1224no—
683C4—1224no—
684C4—4882no—
685C4—4882no—
686C4—2442no—
687C2—28no—
688C2#2144820011yescharacteristic
689C2C2×#1683964611yescharacteristic
690C2#2144841911yescharacteristic
691C2C22⋊C4×C22⋊D6111yescharacteristic
692C2C22⋊C4×Dic12211yescharacteristic
693C2#2144868911yescharacteristic
694C2#2144869711yescharacteristic
695C2#2144954812yes—
696C2#2144962212yes—
697C2#2144996012yes—
698C2#2145008012yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#13882
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839646C2semidirect
4. #21444085C2semidirect
5. #21444070C2semidirect
6. #21444052C2semidirect
15. C22×#11758196C4semidirect
197. #11758196C22⋊C4semidirect
304. C22×C61#387semidirect
389. C61#13882semidirect
653. C22#16839646semidirect

Maximal subgroups

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

subgroupindex
C23×#117581962
C2×#168396462
C2×#168396202
#214440852
#214440702
#214440522
#214440422
#214440412
C2×C4×C22⋊C4×C612
#1388261