Finite Group Databasebeta

← #397×Dic61 #29601834

Subgroups — #397×Dic61 #29601834

Order 15616.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic61C212yesmaximal
3C2×C22⋊C4×Dic61C211yesmaximal, characteristic
4#21444087C211yesmaximal, characteristic, complemented
5#21444081C212yesmaximal, complemented
6#21444076C211yesmaximal, characteristic, complemented
7#21444068C212yesmaximal
8#21444064C211yesmaximal, characteristic
9#95×Dic61C212yesmaximal, complemented
10#95×Dic61C211yesmaximal, characteristic, complemented
11C2×C61×#397C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2211yescharacteristic
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C4⋊C4×Dic61—24no—
21C22⋊C4×Dic61—24no—
22C22⋊C4×Dic61—28no—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C22⋊C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2212yes—
36C61×#397C418yescomplemented
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C23×Dic61—22no—
40C23×Dic61—22no—
41C23×Dic61—24no—
42C23×Dic61—24no—
43C23×Dic61D414yes—
44C23×Dic61C2311yescharacteristic
45C23×Dic61C2311yescharacteristic
46C23×Dic61C2312yes—
47C23×Dic61C2312yes—
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—24no—
58C2×C4×Dic61—24no—
59C2×C4×Dic61—24no—
60C2×C4×Dic61—24no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61—24no—
63C2×C4×Dic61—24no—
64C2×C4×Dic61—24no—
65C2×C4×Dic61—24no—
66C2×C4×Dic61—24no—
67C2×C4×Dic61D414yes—
68#14519684—22no—
69#14519684—22no—
70#14519684—22no—
71#14519684—24no—
72#14519684—24no—
73#14519684—24no—
74#14519684—24no—
75#14519684—24no—
76#14519684—24no—
77#14519684—24no—
78C25×C61C2311yescharacteristic
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2312yes—
82C23×C4×C61C2311yescharacteristic
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61C2×C414yescomplemented
86C2×C22⋊C4×C61—24no—
87C2×C22⋊C4×C61—24no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61C2×C418yes—
91C2×C22⋊C4×C61C2×C414yes—
92#95×C61C2×C418yescomplemented
93#95×C61C2×C414yescomplemented
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—21no—
102C24×C61C22×C412yes—
103C24×C61C2×D412yes—
104C24×C61#4112yes—
105C24×C61C2411yescharacteristic
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61C22×C414yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C414yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—28no—
155C22×Dic61—216no—
156C22×Dic61—44no—
157C22×Dic61—44no—
158C22×Dic61—48no—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D412yes—
163C22×Dic61#4114yes—
164C22×Dic61#4112yes—
165C22×Dic61#4114yes—
166C22×Dic61#4112yes—
167C22×Dic61#4114yes—
168C22×Dic61#4114yes—
169C4⋊C4×C61—28no—
170C22⋊C4×C61—216no—
171C22⋊C4×C61—28no—
172C22⋊Dic61—28no—
173C22⋊Dic61—216no—
174C4⋊Dic61—28no—
175C4×Dic61—28no—
176C4×Dic61—48no—
177C4×Dic61—48no—
178C4×Dic61—44no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—22no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—41no—
201C23×C61C4×D414yes—
202C23×C61C23×C411yescharacteristic
203C23×C61C22×D411yescharacteristic
204C23×C61C22×D411yescharacteristic
205C23×C61C2×#4111yescharacteristic
206C23×C61C2×#4111yescharacteristic
207C23×C61C2×#4111yescharacteristic
208C23×C61C2×#4112yes—
209C23×C61C2×#4112yes—
210C23×C61C2×#4112yes—
211C23×C61C2×#4111yescharacteristic
212C23×C61C2×#4112yes—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61C4×D414yes—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—41no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—416no—
267C2×Dic61#12114yes—
268C2×Dic61#12314yes—
269C2×Dic61#12318yes—
270C2×Dic61#12414yes—
271C2×Dic61#12618yes—
272C4×#397—611nomaximal, Sylow
273C22×C61—24no—
274C22×C61—28no—
275C22×C61—44no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—44no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C4×#4111yescharacteristic
287C22×C61C4×#4112yes—
288C22×C61C4×#4112yes—
289C22×C61C2×#12111yescharacteristic
290C22×C61C2×#12311yescharacteristic
291C22×C61C2×#12312yes—
292C22×C61#52512yes—
293C22×C61#52511yescharacteristic
294C22×C61#52511yescharacteristic
295C22×C61C2×#12411yescharacteristic
296C22×C61C2×#12612yes—
297C22×C61#52912yes—
298C22×C61#52912yes—
299C22×C61#52912yes—
300C22×C61#52912yes—
301C22×C61#52911yescharacteristic
302C22×C61#52912yes—
303C22×C61#52912yes—
304C22×C61#52912yes—
305C22×C61#53211yescharacteristic
306C22×C61#53211yescharacteristic
307C4×C61—24no—
308C4×C61—44no—
309C4×C61—44no—
310C4×C61—42no—
311Dic61—42no—
312Dic61—44no—
313Dic61—44no—
314Dic61—42no—
315Dic61—42no—
316Dic61#39718yesdirect product factor
317C2×#397D6111yescharacteristic
318C2×C4×C4⋊C4—611no—
319C2×C4×C22⋊C4—612no—
320C2×C4×C22⋊C4—611no—
321#1445—611no—
322#1443—612no—
323#1442—611no—
324#1440—612no—
325#1439—611no—
326C4×#95—612no—
327C4×#95—611no—
328C2×C61—44no—
329C2×C61C4×#12111yescharacteristic
330C2×C61C4×#12311yescharacteristic
331C2×C61C4×#12312yes—
332C2×C61C4×#12411yescharacteristic
333C2×C61C4×#12612yes—
334C2×C61C2×#39711yescharacteristic
335C2×C61#239211yescharacteristic
336C2×C61#239512yes—
337C2×C61#239511yescharacteristic
338C2×C61#239811yescharacteristic
339C2×C61#240112yes—
340C24×C4—611no—
341C22×C4⋊C4—611no—
342C22×C4⋊C4C2×D6111yescharacteristic
343C22×C22⋊C4—612no—
344C22×C22⋊C4—611no—
345C22×C22⋊C4C2×D6111yescharacteristic
346C22×C22⋊C4C2×D6112yes—
347C22×C42—611no—
348C22×C42—611no—
349C22×C42—612no—
350C22×C42—612no—
351#397Dic6118yesdirect product factor
352C4×C4⋊C4—1224no—
353C4×C22⋊C4—1224no—
354C4×C22⋊C4—1228no—
355C2×#95—612no—
356C2×#95—611no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—611no—
363C2×#95—611no—
364C2×#95—612no—
365C2×#95C2×D6111yescharacteristic
366C2×#95C2×D6112yes—
367C61C4×#39711yescharacteristic, complemented, Sylow
368C25C22×D6111yescharacteristic
369C23×C4—611no—
370C23×C4—614no—
371C23×C4—612no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—612no—
375C23×C4—611no—
376C23×C4—1222no—
377C23×C4—1224no—
378C23×C4—1224no—
379C23×C4—1222no—
380C23×C4C22×D6112yes—
381C23×C4C22×D6111yescharacteristic
382C23×C4C22×D6112yes—
383C23×C4C22×D6111yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4C2×Dic6114yescomplemented
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1224no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4C2×Dic6118yes—
400C2×C22⋊C4C2×Dic6114yes—
401C2×C42—614no—
402C2×C42—1224no—
403C2×C42—1222no—
404C2×C42—1224no—
405C2×C42—1224no—
406C2×C42—1224no—
407C2×C42—1224no—
408C2×C42—1224no—
409C2×C42—1224no—
410C2×C42—1224no—
411#95—1224no—
412#95—1224no—
413#95—1222no—
414#95—1224no—
415#95—1224no—
416#95—1224no—
417#95—1224no—
418#95—1224no—
419#95—1222no—
420#95—1222no—
421#95C2×Dic6118yescomplemented
422#95C2×Dic6114yescomplemented
423C24—21no—
424C24—22no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24—22no—
429C24—22no—
430C24—21no—
431C24D4×D6112yes—
432C24#1175822412yes—
433C24C22×Dic6112yes—
434C24C23×D6111yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—614no—
457C22×C4—614no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—614no—
461C22×C4—612no—
462C22×C4—614no—
463C22×C4—1228no—
464C22×C4—12216no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4—2448no—
492C22×C4D4×D6112yes—
493C22×C4#1175822412yes—
494C22×C4C22×Dic6114yes—
495C22×C4C22×Dic6112yes—
496C22×C4C22×Dic6114yes—
497C22×C4C22×Dic6112yes—
498C4⋊C4—28no—
499C4⋊C4—1228no—
500C22⋊C4—216no—
501C22⋊C4—28no—
502C22⋊C4—12216no—
503C22⋊C4—1228no—
504C42—1228no—
505C42—2448no—
506C42—2448no—
507C42—2444no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23D4×Dic6114yes—
531C23C23×Dic6111yescharacteristic
532C23C2×D4×D6111yescharacteristic
533C23C2×D4×D6111yescharacteristic
534C23#41×D6111yescharacteristic
535C23#41×D6112yes—
536C23#41×D6112yes—
537C23#41×D6112yes—
538C23#41×D6112yes—
539C23#41×D6111yescharacteristic
540C23C2×#1175822411yescharacteristic
541C23C2×#1175822411yescharacteristic
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—614no—
566C2×C4—614no—
567C2×C4—618no—
568C2×C4—614no—
569C2×C4—618no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1222no—
583C2×C4—24416no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4D4×Dic6114yes—
601C22—24no—
602C22—28no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22C2×D4×Dic6111yescharacteristic
613C22C2×D4×Dic6111yescharacteristic
614C22#41×Dic6111yescharacteristic
615C22#41×Dic6112yes—
616C22#41×Dic6112yes—
617C22#121×D6111yescharacteristic
618C22#123×D6112yes—
619C22#123×D6111yescharacteristic
620C22#1684027012yes—
621C22#1684027011yescharacteristic
622C22#1684027211yescharacteristic
623C22#124×D6111yescharacteristic
624C22#126×D6112yes—
625C22#1684028912yes—
626C22#1684028912yes—
627C22#1684028912yes—
628C22#1684029112yes—
629C22#1684029311yescharacteristic
630C22#1684029512yes—
631C22#1684029512yes—
632C22#1684029512yes—
633C22#1684030711yescharacteristic
634C22#1684031111yescharacteristic
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—618no—
640C4—2442no—
641C4—2444no—
642C4—2444no—
643C4—2442no—
644C4—2442no—
645C2—44no—
646C2#121×Dic6111yescharacteristic
647C2#123×Dic6111yescharacteristic
648C2#123×Dic6112yes—
649C2#124×Dic6111yescharacteristic
650C2#126×Dic6112yes—
651C2#397×D6111yescharacteristic
652C2#2144905611yescharacteristic
653C2#2144907712yes—
654C2#2144908111yescharacteristic
655C2#2144910411yescharacteristic
656C2#2144911612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#397C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#397D61
hypocenterC61C4×#397
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×Dic61C2semidirect
4. #21444087C2semidirect
5. #21444081C2semidirect
6. #21444076C2semidirect
9. #95×Dic61C2semidirect
10. #95×Dic61C2semidirect
36. C61×#397C4semidirect
85. C2×C4⋊C4×C61C2×C4semidirect
92. #95×C61C2×C4semidirect
93. #95×C61C2×C4semidirect
316. Dic61#397direct (N × H)
351. #397Dic61direct (N × H)
367. C61C4×#397semidirect
388. C2×C4⋊C4C2×Dic61semidirect
421. #95C2×Dic61semidirect
422. #95C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic612
C2×C22⋊C4×Dic612
#214440872
#214440812
#214440762
#214440682
#214440642
#95×Dic612
C2×C61×#3972
C4×#39761