Finite Group Databasebeta

← #29601839

Subgroups — #29601839

Order 15616.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839698C211yesmaximal, characteristic
2C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839613C212yesmaximal, complemented
4C2×C22⋊C4×Dic61C211yesmaximal, characteristic
5C2×C22⋊C4×Dic61C212yesmaximal
6#21444086C211yesmaximal, characteristic, complemented
7#21444085C212yesmaximal, complemented
8#21444082C211yesmaximal, characteristic, complemented
9#21444062C212yesmaximal
10#95×Dic61C211yesmaximal, characteristic, complemented
11C61×#1439C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758196C2212yes—
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19C22×C4×Dic61C2212yes—
20C22×C4×Dic61C2212yes—
21C22×C4×Dic61C2211yescharacteristic
22#16839698C418yescomplemented
23C4⋊C4×Dic61—24no—
24#16839613—28no—
25C22⋊C4×Dic61—28no—
26C22⋊C4×Dic61—24no—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2211yescharacteristic
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2212yes—
37C22×C22⋊C4×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C23×Dic61—24no—
41C23×Dic61—22no—
42C23×Dic61—24no—
43C23×Dic61—22no—
44C23×Dic61Q814yes—
45C23×Dic61C2312yes—
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2312yes—
50C23×Dic61C2311yescharacteristic
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61C2×C414yescomplemented
54C2×#11758196—24no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196C2×C418yescomplemented
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—24no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—24no—
65C2×C22⋊Dic61C2×C418yes—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—24no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—24no—
71C2×C4×Dic61—24no—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—24no—
74C2×C4×Dic61D418yes—
75C2×C4×Dic61D414yes—
76#14519684—24no—
77#14519684—22no—
78#14519684—24no—
79#14519684—22no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684C2×C414yescomplemented
84C25×C61C2311yescharacteristic
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2312yes—
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—24no—
91C2×C22⋊C4×C61—24no—
92C2×C22⋊C4×C61—24no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—24no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61C2×C414yes—
97#95×C61—24no—
98#95×C61—22no—
99C24×C61—22no—
100C24×C61—21no—
101C24×C61—22no—
102C24×C61—21no—
103C24×C61—21no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61—21no—
107C24×C61C22×C412yes—
108C24×C61C2×Q812yes—
109C24×C61#4112yes—
110C24×C61C2411yescharacteristic
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
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—22no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D414yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4114yes—
134C22×C4×C61#4112yes—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—28no—
158C22×Dic61—216no—
159C22×Dic61—48no—
160C22×Dic61—44no—
161C22×Dic61—44no—
162C22×Dic61C22×C414yes—
163C22×Dic61C22×C414yes—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×D414yes—
166C22×Dic61C2×D414yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×Q812yes—
169C22×Dic61C2×Q812yes—
170C22×Dic61#4112yes—
171C22×Dic61#4114yes—
172C22×Dic61#4114yes—
173C22×Dic61#4112yes—
174C4⋊C4×C61—28no—
175C22⋊C4×C61—216no—
176C22⋊C4×C61—28no—
177C22⋊Dic61—28no—
178C22⋊Dic61—216no—
179C4⋊Dic61—28no—
180#11758196—216no—
181#11758196—216no—
182C4×Dic61—216no—
183C4×Dic61—28no—
184C4×Dic61—44no—
185C4×Dic61—48no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—22no—
193C23×C61—24no—
194C23×C61—24no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—41no—
206C23×C61—42no—
207C23×C61—41no—
208C23×C61C4×Q814yes—
209C23×C61C23×C411yescharacteristic
210C23×C61C22×D412yes—
211C23×C61C22×D411yescharacteristic
212C23×C61C22×Q811yescharacteristic
213C23×C61C2×#4112yes—
214C23×C61C2×#4111yescharacteristic
215C23×C61C2×#4111yescharacteristic
216C23×C61C2×#4112yes—
217C23×C61C2×#4111yescharacteristic
218C23×C61C2×#4111yescharacteristic
219C23×C61C2×#4112yes—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61C4×D414yes—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—24no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—42no—
261C2×Dic61—41no—
262C2×Dic61—42no—
263C2×Dic61—41no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—416no—
273C2×Dic61C4×D418yes—
274C2×Dic61C22⋊Q814yes—
275C2×Dic61C22⋊Q818yes—
276C2×Dic61#12318yes—
277C2×Dic61#12314yes—
278C2×Dic61#12714yes—
279C4×#399—611nomaximal, Sylow
280C22×C61—24no—
281C22×C61—28no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—44no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—44no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—44no—
291C22×C61C2×C4×D412yes—
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C2×C4×Q811yescharacteristic
294C22×C61C4×#4112yes—
295C22×C61C4×#4111yescharacteristic
296C22×C61C2×C22⋊Q811yescharacteristic
297C22×C61C2×C22⋊Q812yes—
298C22×C61C2×#12312yes—
299C22×C61C2×#12311yescharacteristic
300C22×C61#52512yes—
301C22×C61#52511yescharacteristic
302C22×C61#52511yescharacteristic
303C22×C61#52512yes—
304C22×C61#52912yes—
305C22×C61#52912yes—
306C22×C61#52911yescharacteristic
307C22×C61#52912yes—
308C22×C61#52912yes—
309C22×C61C2×#12711yescharacteristic
310C22×C61#53212yes—
311C22×C61#53211yescharacteristic
312C22×C61#53312yes—
313C22×C61#53311yescharacteristic
314C4×C61—24no—
315C4×C61—42no—
316C4×C61—44no—
317C4×C61—44no—
318Dic61—28no—
319Dic61—44no—
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323Dic61#39918yescomplemented
324C2×#399—611no—
325C2×C4×C4⋊C4—612no—
326C2×C4×C4⋊C4—611no—
327C2×C4×C22⋊C4—612no—
328C2×C4×C22⋊C4—611no—
329#1445—612no—
330#1445—611no—
331#1444—611no—
332#1439—612no—
333#1439D6111yescharacteristic
334C4×#95—611no—
335C2×C61—44no—
336C2×C61C4×#12311yescharacteristic
337C2×C61C4×#12312yes—
338C2×C61C4×C22⋊Q811yescharacteristic
339C2×C61C4×C22⋊Q812yes—
340C2×C61C4×#12711yescharacteristic
341C2×C61C2×#39911yescharacteristic
342C2×C61#239512yes—
343C2×C61#239511yescharacteristic
344C2×C61#239611yescharacteristic
345C2×C61#239612yes—
346C2×C61#239711yescharacteristic
347C24×C4—611no—
348C22×C4⋊C4—612no—
349C22×C4⋊C4—611no—
350C22×C4⋊C4—612no—
351C22×C4⋊C4C2×D6111yescharacteristic
352C22×C22⋊C4—612no—
353C22×C22⋊C4—611no—
354C22×C22⋊C4C2×D6112yes—
355C22×C22⋊C4C2×D6111yescharacteristic
356C22×C42—612no—
357C22×C42—611no—
358C22×C42—611no—
359C22×C42—612no—
360#399—618no—
361C4×C4⋊C4—1228no—
362C4×C4⋊C4—1224no—
363C4×C22⋊C4—1228no—
364C4×C22⋊C4—1224no—
365C2×#95—611no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—611no—
371C2×#95—611no—
372C2×#95—612no—
373C2×#95C2×D6111yescharacteristic
374C2×#95C2×D6112yes—
375C61C4×#39911yescharacteristic, complemented, Sylow
376C25C22×D6111yescharacteristic
377C23×C4—611no—
378C23×C4—612no—
379C23×C4—614no—
380C23×C4—612no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—1224no—
385C23×C4—1222no—
386C23×C4—1224no—
387C23×C4—1222no—
388C23×C4C22×D6111yescharacteristic
389C23×C4C22×D6112yes—
390C23×C4C22×D6112yes—
391C23×C4C22×D6111yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—618no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1224no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1224no—
402C2×C4⋊C4—1222no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—618no—
409C2×C22⋊C4—1224no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1224no—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4C4×D6114yes—
415C2×C42—618no—
416C2×C42—614no—
417C2×C42—1224no—
418C2×C42—1224no—
419C2×C42—1224no—
420C2×C42—1222no—
421C2×C42—1224no—
422C2×C42—1224no—
423C2×C42—1224no—
424C2×C42—1224no—
425#95—24no—
426#95—22no—
427#95—614no—
428#95—1224no—
429#95—1222no—
430#95—1224no—
431#95—1224no—
432#95—1224no—
433#95—1222no—
434#95—1224no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24—22no—
443C24C2×C4×D6112yes—
444C24#1175822412yes—
445C24Q8×D6112yes—
446C24C23×D6111yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—614no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—614no—
469C22×C4—612no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—1228no—
478C22×C4—12216no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—2448no—
502C22×C4—2444no—
503C22×C4—2444no—
504C22×C4C2×C4×D6112yes—
505C22×C4C2×C4×D6112yes—
506C22×C4D4×D6114yes—
507C22×C4D4×D6112yes—
508C22×C4#1175822414yes—
509C22×C4#1175822412yes—
510C4⋊C4—28no—
511C4⋊C4—12216no—
512C4⋊C4—12216no—
513C4⋊C4—1228no—
514C22⋊C4—216no—
515C22⋊C4—28no—
516C22⋊C4—12216no—
517C22⋊C4—1228no—
518C42—1228no—
519C42—12216no—
520C42—2448no—
521C42—2444no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23#1451974214yes—
545C23C22×C4×D6111yescharacteristic
546C23C2×D4×D6112yes—
547C23C2×D4×D6111yescharacteristic
548C23C2×Q8×D6111yescharacteristic
549C23#41×D6112yes—
550C23#41×D6112yes—
551C23#41×D6111yescharacteristic
552C23#41×D6111yescharacteristic
553C23C2×#1175822411yescharacteristic
554C23C2×#1175822411yescharacteristic
555C23C2×#1175822412yes—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—614no—
580C2×C4—618no—
581C2×C4—618no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—618no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1224no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1224no—
600C2×C4—1222no—
601C2×C4—24416no—
602C2×C4—2442no—
603C2×C4—2441no—
604C2×C4—2442no—
605C2×C4—2441no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2441no—
609C2×C4—2442no—
610C2×C4—2441no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4#1451973714yes—
615C22—24no—
616C22—28no—
617C22—42no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22C4×D4×D6112yes—
627C22C2×#1451973711yescharacteristic
628C22C2×#1451974211yescharacteristic
629C22C4×#1175822412yes—
630C22#1684023011yescharacteristic
631C22C22⋊Q8×D6111yescharacteristic
632C22C22⋊Q8×D6112yes—
633C22#123×D6112yes—
634C22#123×D6111yescharacteristic
635C22#1684026812yes—
636C22#1684026811yescharacteristic
637C22#1684027211yescharacteristic
638C22#1684027212yes—
639C22#1684028912yes—
640C22#1684029112yes—
641C22#1684029311yescharacteristic
642C22#1684029312yes—
643C22#1684029512yes—
644C22#127×D6111yescharacteristic
645C22#1684030712yes—
646C22#1684031111yescharacteristic
647C22#1684031412yes—
648C22#1684031411yescharacteristic
649C4—24no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—618no—
654C4—1228no—
655C4—2442no—
656C4—2442no—
657C4—2442no—
658C4—2444no—
659C2—44no—
660C2#2144834411yescharacteristic
661C2#2144834612yes—
662C2#2144834711yescharacteristic
663C2#2144835112yes—
664C2#2144838311yescharacteristic
665C2#399×D6111yescharacteristic
666C2#2144907512yes—
667C2#2144908111yescharacteristic
668C2#2144908611yescharacteristic
669C2#2144909412yes—
670C2#2144909711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1439C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1439D61
hypocenterC61C4×#399
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic61C2semidirect
3. C2×#16839613C2semidirect
6. #21444086C2semidirect
7. #21444085C2semidirect
8. #21444082C2semidirect
10. #95×Dic61C2semidirect
22. #16839698C4semidirect
53. C2×C4⋊Dic61C2×C4semidirect
59. C2×#11758196C2×C4semidirect
83. #14519684C2×C4semidirect
323. Dic61#399semidirect
375. C61C4×#399semidirect

Maximal subgroups

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

subgroupindex
C2×#168396982
C2×C4⋊C4×Dic612
C2×#168396132
C2×C22⋊C4×Dic612
#214440862
#214440852
#214440822
#214440622
#95×Dic612
C61×#14392
C4×#39961