Finite Group Databasebeta

← #29602235

Subgroups — #29602235

Order 15616.

3526 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic61C211yesmaximal, characteristic
2C2×#16839623C211yesmaximal, characteristic, complemented
3C2×#16839623C211yesmaximal, characteristic, complemented
4C2×#16839620C211yesmaximal, characteristic, complemented
5C2×#16839618C211yesmaximal, characteristic
6C2×C22⋊C4×Dic61C211yesmaximal, characteristic, complemented
7#21444062C214yesmaximal, complemented
8#21444056C214yesmaximal, complemented
9C2×C61×#379C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C24×Dic61C2211yescharacteristic
12C22×#11758196C2212yescomplemented
13C22×C22⋊Dic61—24no—
14C22×C22⋊Dic61—24no—
15C22×C22⋊Dic61C418yescomplemented
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2214yes—
18C22×C22⋊Dic61C2212yes—
19C22×C22⋊Dic61C2212yes—
20C22×C22⋊Dic61C2211yescharacteristic
21C22×C4×Dic61C2212yescomplemented
22#16839623—24no—
23#16839623—24no—
24#16839620—24no—
25#16839618—24no—
26C22⋊C4×Dic61—24no—
27C2×#14519684C2214yescomplemented
28C2×#14519684C2214yescomplemented
29C2×#14519684C2212yescomplemented
30C2×#14519684C2212yescomplemented
31C26×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2214yes—
34C22×C22⋊C4×C61C2211yescharacteristic
35C61×#379—24no—
36C23×Dic61—22no—
37C23×Dic61—22no—
38C23×Dic61—22no—
39C23×Dic61—22no—
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—44no—
43C23×Dic61C2×C414yes—
44C23×Dic61C2×C414yes—
45C23×Dic61D414yes—
46C23×Dic61D414yes—
47C23×Dic61C2312yes—
48C23×Dic61C2312yes—
49C23×Dic61C2312yes—
50C23×Dic61C2311yescharacteristic
51C23×Dic61C2311yescharacteristic
52C2×#11758196—24no—
53C2×#11758196—44no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—22no—
56C2×C22⋊Dic61—232no—
57C2×C22⋊Dic61—24no—
58C2×C22⋊Dic61—24no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61—24no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—28no—
65C2×C22⋊Dic61—28no—
66C2×C22⋊Dic61—44no—
67C2×C22⋊Dic61C2×C418yes—
68C2×C22⋊Dic61C2×C418yes—
69C2×C22⋊Dic61D414yes—
70C2×C4×Dic61—24no—
71C2×C4×Dic61—44no—
72#14519684—28no—
73#14519684—24no—
74#14519684—24no—
75#14519684—28no—
76C25×C61—22no—
77C25×C61—22no—
78C25×C61—28no—
79C25×C61—22no—
80C25×C61—22no—
81C25×C61—22no—
82C25×C61—22no—
83C25×C61—44no—
84C25×C61C2×C414yes—
85C25×C61C2312yes—
86C25×C61C2311yescharacteristic
87C23×C4×C61C2312yes—
88C23×C4×C61C2312yes—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—28no—
95C2×C22⋊C4×C61—28no—
96C2×C22⋊C4×C61—44no—
97C2×C22⋊C4×C61D414yes—
98C24×C61—24no—
99C24×C61—24no—
100C24×C61—216no—
101C24×C61—216no—
102C24×C61—28no—
103C24×C61—216no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61—22no—
107C24×C61—22no—
108C24×C61—24no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—24no—
112C24×C61—24no—
113C24×C61—28no—
114C24×C61—28no—
115C24×C61—48no—
116C24×C61—416no—
117C24×C61—416no—
118C24×C61—416no—
119C24×C61—416no—
120C24×C61—48no—
121C24×C61—48no—
122C24×C61—44no—
123C24×C61—44no—
124C24×C61—44no—
125C24×C61—44no—
126C24×C61C22×C414yes—
127C24×C61C22×C412yes—
128C24×C61C2×D412yes—
129C24×C61C2×D412yes—
130C24×C61C2×D412yes—
131C24×C61C2×D412yes—
132C24×C61#4118yes—
133C24×C61#4112yes—
134C24×C61#4112yes—
135C24×C61C2411yescharacteristic
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61C2×D414yes—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—28no—
149C22×Dic61—28no—
150C22×Dic61—28no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—24no—
165C22×Dic61—24no—
166C22×Dic61—48no—
167C22×Dic61—48no—
168C22×Dic61—48no—
169C22×Dic61—44no—
170C22×Dic61—44no—
171C22×Dic61—44no—
172C22×Dic61—44no—
173C22×Dic61C22⋊C4116yescomplemented
174C22×Dic61C22×C414yes—
175C22×Dic61C22×C412yes—
176C22×Dic61C22×C412yes—
177C22×Dic61C2×D414yes—
178C22×Dic61C2×D412yes—
179C22×Dic61C2×D412yes—
180C22×Dic61C2×D412yes—
181C22×Dic61C2×D412yes—
182C22⋊C4×C61—28no—
183C22⋊C4×C61—416no—
184C22⋊C4×C61—44no—
185C22⋊Dic61—28no—
186C22⋊Dic61—216no—
187C22⋊Dic61—216no—
188C22⋊Dic61—416no—
189C22⋊Dic61—44no—
190#11758196—48no—
191C4×Dic61—48no—
192C23×C61—24no—
193C23×C61—24no—
194C23×C61—24no—
195C23×C61—24no—
196C23×C61—24no—
197C23×C61—24no—
198C23×C61—28no—
199C23×C61—28no—
200C23×C61—28no—
201C23×C61—28no—
202C23×C61—232no—
203C23×C61—24no—
204C23×C61—24no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—24no—
209C23×C61—28no—
210C23×C61—28no—
211C23×C61—28no—
212C23×C61—28no—
213C23×C61—28no—
214C23×C61—28no—
215C23×C61—432no—
216C23×C61—432no—
217C23×C61—432no—
218C23×C61—416no—
219C23×C61—432no—
220C23×C61—432no—
221C23×C61—432no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61—44no—
227C23×C61—44no—
228C23×C61—44no—
229C23×C61—44no—
230C23×C61—44no—
231C23×C61—44no—
232C23×C61—416no—
233C23×C61C2×C22⋊C414yes—
234C23×C61#11718yes—
235C23×C61C4×D414yes—
236C23×C61C4×D414yes—
237C23×C61#12114yes—
238C23×C61#12114yes—
239C23×C61#12414yes—
240C23×C61#12714yes—
241C23×C61C23×C411yescharacteristic
242C23×C61C22×D411yescharacteristic
243C23×C61C22×D411yescharacteristic
244C23×C61C22×D411yescharacteristic
245C23×C61C22×D411yescharacteristic
246C23×C61C2×#4114yes—
247C23×C61C2×#4111yescharacteristic
248C23×C61C2×#4111yescharacteristic
249C23×C61H(5,𝔽2)14yes—
250C2×C4×C61—24no—
251C2×C4×C61—28no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—44no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—44no—
259C2×C4×C61—44no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—28no—
271C2×Dic61—416no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—44no—
279C2×Dic61—416no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61—44no—
284C2×Dic61—44no—
285C2×Dic61C2×C22⋊C414yes—
286C2×Dic61C2×C22⋊C414yes—
287#13903—611nomaximal, Sylow
288C22×C61—28no—
289C22×C61—24no—
290C22×C61—28no—
291C22×C61—28no—
292C22×C61—28no—
293C22×C61—28no—
294C22×C61—24no—
295C22×C61—28no—
296C22×C61—28no—
297C22×C61—28no—
298C22×C61—28no—
299C22×C61—464no—
300C22×C61—48no—
301C22×C61—44no—
302C22×C61—44no—
303C22×C61—44no—
304C22×C61—48no—
305C22×C61—48no—
306C22×C61—48no—
307C22×C61—44no—
308C22×C61#39018yes—
309C22×C61C22×C22⋊C411yescharacteristic
310C22×C61C2×#11712yes—
311C22×C61C2×C4×D411yescharacteristic
312C22×C61C2×C4×D411yescharacteristic
313C22×C61#51812yes—
314C22×C61C2×#12111yescharacteristic
315C22×C61C2×#12111yescharacteristic
316C22×C61#52512yes—
317C22×C61#52512yes—
318C22×C61C2×#12411yescharacteristic
319C22×C61C2×#12711yescharacteristic
320C22×C61#53412yes—
321C22×C61#53412yes—
322C22×C61#54614yes—
323C22×C61#54614yes—
324C22×C61#55114yes—
325C22×C61#55114yes—
326C4×C61—44no—
327C4×C61—82no—
328Dic61—24no—
329Dic61—24no—
330Dic61—44no—
331Dic61—44no—
332Dic61—82no—
333C23×C22⋊C4—611no—
334C2×#381—611no—
335C2×#381—611no—
336C2×#380—611no—
337C2×#379—611no—
338C2×#379D6111yescharacteristic
339C2×C4×C22⋊C4—611no—
340#1439—614no—
341#1437—614no—
342C2×C61—28no—
343C2×C61—48no—
344C2×C61#228011yescharacteristic
345C2×C61C2×#39011yescharacteristic
346C2×C61#230811yescharacteristic
347C2×C61#234212yes—
348C2×C61#234212yes—
349C2×C61#253412yes—
350C2×C61#253412yes—
351C2×C61#253912yes—
352C2×C61#256112yes—
353C26C2×D6111yescharacteristic
354C24×C4—611no—
355C24×C4—611no—
356C22×C4⋊C4—612no—
357C22×C22⋊C4—618no—
358C22×C22⋊C4—612no—
359C22×C22⋊C4—612no—
360C22×C22⋊C4—611no—
361C22×C22⋊C4—611no—
362C22×C22⋊C4—614no—
363C22×C22⋊C4—1224no—
364C22×C22⋊C4—1224no—
365C22×C22⋊C4C2×D6111yescharacteristic
366C22×C22⋊C4C2×D6114yes—
367C22×C22⋊C4C2×D6111yescharacteristic
368C22×C42—612no—
369#381—1224no—
370#381—1224no—
371#380—1224no—
372#379—24no—
373#379—1224no—
374C4×C22⋊C4—1224no—
375C2×#95—612no—
376C2×#95—614no—
377C2×#95—614no—
378C2×#95—612no—
379C61#1390311yescharacteristic, complemented, Sylow
380C25—22no—
381C25—22no—
382C25—28no—
383C25—22no—
384C25—22no—
385C25—22no—
386C25—22no—
387C25—44no—
388C25C4×D6114yes—
389C25C22×D6112yes—
390C25C22×D6111yescharacteristic
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—614no—
394C23×C4—614no—
395C23×C4—614no—
396C23×C4—614no—
397C23×C4—612no—
398C23×C4—612no—
399C23×C4—612no—
400C23×C4—1222no—
401C23×C4—1222no—
402C23×C4—1222no—
403C23×C4—1222no—
404C23×C4—1222no—
405C23×C4—1222no—
406C23×C4—2444no—
407C23×C4C22×D6112yes—
408C23×C4C22×D6112yes—
409C2×C4⋊C4—1224no—
410C2×C4⋊C4—2444no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—28no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—28no—
418C2×C22⋊C4—44no—
419C2×C22⋊C4—618no—
420C2×C22⋊C4—618no—
421C2×C22⋊C4—614no—
422C2×C22⋊C4—1224no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1224no—
426C2×C22⋊C4—12232no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1228no—
430C2×C22⋊C4—1222no—
431C2×C22⋊C4—1222no—
432C2×C22⋊C4—1228no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—2444no—
435C2×C22⋊C4C22⋊D6114yes—
436C2×C42—1224no—
437C2×C42—2444no—
438#95—1224no—
439#95—1224no—
440#95—1228no—
441#95—1228no—
442C24—24no—
443C24—24no—
444C24—216no—
445C24—216no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—24no—
451C24—22no—
452C24—22no—
453C24—24no—
454C24—24no—
455C24—28no—
456C24—216no—
457C24—28no—
458C24—28no—
459C24—48no—
460C24—416no—
461C24—416no—
462C24—416no—
463C24—416no—
464C24—48no—
465C24—48no—
466C24—44no—
467C24—44no—
468C24—44no—
469C24—44no—
470C24C2×C4×D6114yes—
471C24C2×C4×D6112yes—
472C24#1175822212yes—
473C24D4×D6112yes—
474C24D4×D6112yes—
475C24D4×D6112yes—
476C24#1175822418yes—
477C24#1175822412yes—
478C24C2×C22⋊D6112yes—
479C24C23×D6111yescharacteristic
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—6116no—
496C22×C4—614no—
497C22×C4—614no—
498C22×C4—1224no—
499C22×C4—1224no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1224no—
510C22×C4—1228no—
511C22×C4—1228no—
512C22×C4—1228no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1224no—
517C22×C4—1224no—
518C22×C4—2448no—
519C22×C4—2448no—
520C22×C4—2448no—
521C22×C4—2444no—
522C22×C4—2444no—
523C22×C4—2444no—
524C22×C4—2444no—
525C22×C4C2×C22⋊D6114yes—
526C4⋊C4—2448no—
527C22⋊C4—28no—
528C22⋊C4—416no—
529C22⋊C4—44no—
530C22⋊C4—12216no—
531C22⋊C4—12216no—
532C22⋊C4—1228no—
533C22⋊C4—24416no—
534C22⋊C4—2444no—
535C42—2448no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—232no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—28no—
559C23—432no—
560C23—432no—
561C23—416no—
562C23—432no—
563C23—432no—
564C23—432no—
565C23—432no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—416no—
575C23—44no—
576C23—44no—
577C23C22⋊C4×D6114yes—
578C23#1451973318yes—
579C23#1451973714yes—
580C23C4×C22⋊D6114yes—
581C23#1451974914yes—
582C23#1451975014yes—
583C23#1451976314yes—
584C23#1451977314yes—
585C23C22×C4×D6111yescharacteristic
586C23C2×D4×D6111yescharacteristic
587C23C2×D4×D6111yescharacteristic
588C23C2×D4×D6111yescharacteristic
589C23C22×C22⋊D6111yescharacteristic
590C23C2×#1175822414yes—
591C23C2×#1175822411yescharacteristic
592C23C2×#1175822211yescharacteristic
593C23#1451982214yes—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—614no—
605C2×C4—614no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—1224no—
616C2×C4—1228no—
617C2×C4—24416no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2444no—
622C2×C4—2444no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4—24416no—
630C2×C4—2444no—
631C22—24no—
632C22—24no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—464no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—48no—
648C22—44no—
649C22—48no—
650C22—48no—
651C22#1683966218yes—
652C22C2×C22⋊C4×D6111yescharacteristic
653C22C2×#1451973312yes—
654C22C2×#1451973711yescharacteristic
655C22C2×C4×C22⋊D6111yescharacteristic
656C22#1684023512yes—
657C22C2×#1451974911yescharacteristic
658C22C2×#1451975011yescharacteristic
659C22#1684026812yes—
660C22#1684026812yes—
661C22C2×#1451976311yescharacteristic
662C22C2×#1451977311yescharacteristic
663C22#1684032212yes—
664C22#1684032212yes—
665C22#1684039914yes—
666C22#1684040614yes—
667C22#1684043814yes—
668C22#1684044314yes—
669C4—44no—
670C4—82no—
671C4—1224no—
672C4—1224no—
673C4—4882no—
674C4—2444no—
675C4—2444no—
676C2—28no—
677C2—48no—
678C2#2144819711yescharacteristic
679C2C2×#1683966211yescharacteristic
680C2#2144839511yescharacteristic
681C2#2144866412yes—
682C2#2144867012yes—
683C2#2145070612yes—
684C2#2145071212yes—
685C2#2145074712yes—
686C2#2145097412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#379C2
socle, cosocleC24×C61C24
prime radicalC26×C61C22
hypercenterC2×#379D61
hypocenterC61#13903
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
2. C2×#16839623C2semidirect
3. C2×#16839623C2semidirect
4. C2×#16839620C2semidirect
6. C2×C22⋊C4×Dic61C2semidirect
7. #21444062C2semidirect
8. #21444056C2semidirect
12. C22×#11758196C22semidirect
15. C22×C22⋊Dic61C4semidirect
21. C22×C4×Dic61C22semidirect
27. C2×#14519684C22semidirect
28. C2×#14519684C22semidirect
29. C2×#14519684C22semidirect
30. C2×#14519684C22semidirect
173. C22×Dic61C22⋊C4semidirect
379. C61#13903semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic612
C2×#168396232
C2×#168396202
C2×#168396182
C2×C22⋊C4×Dic612
#214440622
#214440562
C2×C61×#3792
#1390361