Finite Group Databasebeta

← #29601969

Subgroups — #29601969

Order 15616.

4046 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic61C211yesmaximal, characteristic
2C2×#16839635C212yesmaximal, complemented
3C2×#16839620C212yesmaximal, complemented
4C2×#16839618C211yesmaximal, characteristic
5#21444062C214yesmaximal, complemented
6#21444042C212yesmaximal, complemented
7#21444041C212yesmaximal, complemented
8C23×C22⋊C4×C61C211yesmaximal, characteristic
9C24×Dic61C2212yes—
10C22×#11758196C2214yescomplemented
11C22×C22⋊Dic61—28no—
12C22×C22⋊Dic61C418yescomplemented
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61C2211yescharacteristic
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2212yes—
18C22×C22⋊Dic61C2212yes—
19C22×C22⋊Dic61C2211yescharacteristic
20#16839635—28no—
21#16839620—28no—
22#16839618—24no—
23C2×#14519684C2214yescomplemented
24C2×#14519684C2214yescomplemented
25C2×#14519684C2214yescomplemented
26C26×C61C2211yescharacteristic
27C24×C4×C61C2212yes—
28C22×C22⋊C4×C61—24no—
29C22×C22⋊C4×C61—28no—
30C22×C22⋊C4×C61C2212yes—
31C22×C22⋊C4×C61C2212yes—
32C23×Dic61—24no—
33C23×Dic61—24no—
34C23×Dic61—28no—
35C23×Dic61—44no—
36C23×Dic61C2×C418yescomplemented
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C2×#11758196—28no—
42C2×#11758196—28no—
43C2×#11758196—44no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—28no—
46C2×C22⋊Dic61—232no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—22no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—22no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—44no—
56C2×C22⋊Dic61C2×C414yes—
57C2×C22⋊Dic61C2×C414yes—
58C2×C22⋊Dic61C2×C418yescomplemented
59C2×C22⋊Dic61D418yes—
60#14519684—28no—
61#14519684—28no—
62#14519684—44no—
63C25×C61—24no—
64C25×C61—24no—
65C25×C61—24no—
66C25×C61—24no—
67C25×C61—44no—
68C25×C61C2×C414yes—
69C25×C61D414yes—
70C25×C61Q814yes—
71C25×C61C2311yescharacteristic
72C25×C61C2311yescharacteristic
73C25×C61C2311yescharacteristic
74C23×C4×C61—24no—
75C23×C4×C61—28no—
76C23×C4×C61—24no—
77C23×C4×C61—24no—
78C23×C4×C61—44no—
79C23×C4×C61C2314yes—
80C2×C22⋊C4×C61—232no—
81C2×C22⋊C4×C61—24no—
82C2×C22⋊C4×C61—24no—
83C2×C22⋊C4×C61—24no—
84C2×C22⋊C4×C61—28no—
85C2×C22⋊C4×C61D418yes—
86C24×C61—22no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—24no—
91C24×C61—24no—
92C24×C61—28no—
93C24×C61—28no—
94C24×C61—216no—
95C24×C61—28no—
96C24×C61—216no—
97C24×C61—216no—
98C24×C61—216no—
99C24×C61—28no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—24no—
105C24×C61—48no—
106C24×C61—48no—
107C24×C61—48no—
108C24×C61—416no—
109C24×C61—416no—
110C24×C61—416no—
111C24×C61—416no—
112C24×C61C4⋊C4116yescomplemented
113C24×C61C22×C412yes—
114C24×C61C22×C412yes—
115C24×C61C22×C412yes—
116C24×C61C2×D412yes—
117C24×C61C2×D414yes—
118C24×C61C2×D412yes—
119C24×C61C2×D412yes—
120C24×C61C2×D414yes—
121C24×C61C2×Q812yes—
122C24×C61C2×Q812yes—
123C24×C61C2×Q812yes—
124C24×C61#4114yes—
125C24×C61#4114yes—
126C24×C61C2411yescharacteristic
127C22×C4×C61—216no—
128C22×C4×C61—24no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—24no—
133C22×C4×C61—24no—
134C22×C4×C61—416no—
135C22×C4×C61—416no—
136C22×C4×C61—48no—
137C22×C4×C61—48no—
138C22×C4×C61C2×D418yes—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—24no—
142C22×Dic61—216no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—24no—
147C22×Dic61—216no—
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—24no—
158C22×Dic61—48no—
159C22×Dic61—416no—
160C22×Dic61—416no—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C414yes—
163C22×Dic61C2×D418yes—
164C22⋊C4×C61—216no—
165C22⋊C4×C61—216no—
166C22⋊Dic61—28no—
167C22⋊Dic61—216no—
168C22⋊Dic61—48no—
169C22⋊Dic61—44no—
170C22⋊Dic61—48no—
171C22⋊Dic61—44no—
172C22⋊Dic61C22⋊C4116yescomplemented
173#11758196—416no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—24no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—28no—
189C23×C61—28no—
190C23×C61—28no—
191C23×C61—28no—
192C23×C61—28no—
193C23×C61—28no—
194C23×C61—28no—
195C23×C61—28no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—24no—
199C23×C61—24no—
200C23×C61—22no—
201C23×C61—24no—
202C23×C61—24no—
203C23×C61—24no—
204C23×C61—22no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—232no—
208C23×C61—232no—
209C23×C61—416no—
210C23×C61—432no—
211C23×C61—432no—
212C23×C61—432no—
213C23×C61—432no—
214C23×C61—464no—
215C23×C61—44no—
216C23×C61—44no—
217C23×C61—44no—
218C23×C61—44no—
219C23×C61C2×C22⋊C414yes—
220C23×C61C2×C4⋊C414yes—
221C23×C61C2×C4⋊C414yes—
222C23×C61C2×C4⋊C414yes—
223C23×C61#11714yes—
224C23×C61C4×D418yes—
225C23×C61#12014yes—
226C23×C61#12118yes—
227C23×C61C22⋊Q818yes—
228C23×C61C22⋊Q818yes—
229C23×C61#12314yes—
230C23×C61C23×C411yescharacteristic
231C23×C61C22×D412yes—
232C23×C61C22×D411yescharacteristic
233C23×C61C22×D412yes—
234C23×C61C22×Q811yescharacteristic
235C23×C61C2×#4112yes—
236C23×C61C2×#4112yes—
237C23×C61H(5,𝔽2)14yes—
238C2×C4×C61—28no—
239C2×C4×C61—28no—
240C2×C4×C61—28no—
241C2×C4×C61—432no—
242C2×C4×C61—44no—
243C2×C4×C61—44no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—28no—
249C2×Dic61—28no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—432no—
254C2×Dic61—44no—
255C2×Dic61—44no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—44no—
263C2×Dic61—44no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61C2×C22⋊C418yescomplemented
267#13877—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—28no—
271C22×C61—28no—
272C22×C61—28no—
273C22×C61—28no—
274C22×C61—28no—
275C22×C61—28no—
276C22×C61—28no—
277C22×C61—28no—
278C22×C61—216no—
279C22×C61—216no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—24no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—464no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—48no—
291C22×C61#38018yescomplemented
292C22×C61#38518yescomplemented
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61C22×C4⋊C411yescharacteristic
295C22×C61C2×#11711yescharacteristic
296C22×C61C2×C4×D412yes—
297C22×C61#51812yes—
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×#12112yes—
300C22×C61C2×C22⋊Q812yes—
301C22×C61C2×C22⋊Q812yes—
302C22×C61C2×#12311yescharacteristic
303C22×C61#53412yes—
304C22×C61#54312yes—
305C22×C61D4214yes—
306C22×C61#54614yes—
307C22×C61#54614yes—
308C22×C61#55114yes—
309C4×C61—48no—
310Dic61—28no—
311Dic61—44no—
312Dic61—82no—
313Dic61—82no—
314C23×C22⋊C4—611no—
315C23×C22⋊C4D6111yescharacteristic
316C2×#385—612no—
317C2×#380—612no—
318C2×#379—611no—
319#1439—614no—
320C22⋊#95—612no—
321C22⋊#95—612no—
322C2×C61—28no—
323C2×C61—28no—
324C2×C61—44no—
325C2×C61C2×#38011yescharacteristic
326C2×C61C2×#38511yescharacteristic
327C2×C61#231511yescharacteristic
328C2×C61D4×C22⋊C412yes—
329C2×C61#234212yes—
330C2×C61#241012yes—
331C2×C61#241212yes—
332C2×C61#243714yes—
333C26C2×D6111yescharacteristic
334C24×C4—612no—
335C24×C4C2×D6112yes—
336C22×C4⋊C4—614no—
337C22×C22⋊C4—24no—
338C22×C22⋊C4—28no—
339C22×C22⋊C4—611no—
340C22×C22⋊C4—618no—
341C22×C22⋊C4—611no—
342C22×C22⋊C4—612no—
343C22×C22⋊C4—611no—
344C22×C22⋊C4—612no—
345C22×C22⋊C4—611no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4—1228no—
348C22×C22⋊C4C2×D6112yes—
349C22×C22⋊C4C2×D6112yes—
350#385—1228no—
351#380—1228no—
352#379—1224no—
353C2×#95—614no—
354C2×#95—614no—
355C2×#95—614no—
356C61#1387711yescharacteristic, complemented, Sylow
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25Dic12214yes—
363C25C4×D6114yes—
364C25C22⋊D6114yes—
365C25C22×D6111yescharacteristic
366C25C22×D6111yescharacteristic
367C25C22×D6111yescharacteristic
368C23×C4—24no—
369C23×C4—28no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—44no—
373C23×C4—612no—
374C23×C4—612no—
375C23×C4—618no—
376C23×C4—612no—
377C23×C4—612no—
378C23×C4—1228no—
379C23×C4—1224no—
380C23×C4—1224no—
381C23×C4—2444no—
382C23×C4C22×D6114yes—
383C2×C4⋊C4—1228no—
384C2×C4⋊C4—1228no—
385C2×C4⋊C4—2444no—
386C2×C22⋊C4—232no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—614no—
392C2×C22⋊C4—618no—
393C2×C22⋊C4—614no—
394C2×C22⋊C4—618no—
395C2×C22⋊C4—12232no—
396C2×C22⋊C4—1228no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—2444no—
407C2×C22⋊C4C22⋊D6118yes—
408#95—1228no—
409#95—1228no—
410#95—2444no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—24no—
422C24—28no—
423C24—28no—
424C24—216no—
425C24—28no—
426C24—216no—
427C24—216no—
428C24—216no—
429C24—28no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24#11758196116yescomplemented
438C24C2×Dic12212yes—
439C24C2×Dic12212yes—
440C24C2×Dic12212yes—
441C24C2×C4×D6112yes—
442C24C2×C4×D6112yes—
443C24C2×C4×D6112yes—
444C24#1175822214yes—
445C24D4×D6114yes—
446C24#1175822414yes—
447C24C2×C22⋊D6112yes—
448C24C2×C22⋊D6112yes—
449C24C2×C22⋊D6112yes—
450C24C2×C22⋊D6114yes—
451C24C23×D6111yescharacteristic
452C22×C4—216no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—614no—
464C22×C4—618no—
465C22×C4—614no—
466C22×C4—12216no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1224no—
474C22×C4—12216no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1224no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—2448no—
486C22×C4—24416no—
487C22×C4—24416no—
488C22×C4C2×C22⋊D6118yes—
489C4⋊C4—24416no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—6116no—
493C22⋊C4—12216no—
494C22⋊C4—1228no—
495C22⋊C4—2448no—
496C22⋊C4—2444no—
497C22⋊C4—2448no—
498C22⋊C4—2444no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—232no—
532C23—24no—
533C23—232no—
534C23—416no—
535C23—464no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C22⋊C4×D6114yes—
545C23C2×#1175819614yes—
546C23C2×#1175819614yes—
547C23C2×#1175819614yes—
548C23#1451973314yes—
549C23C4×C22⋊D6118yes—
550C23#1451974614yes—
551C23#1451974918yes—
552C23#1451975418yes—
553C23#1451975618yes—
554C23#1451976014yes—
555C23C22×C4×D6111yescharacteristic
556C23C2×D4×D6112yes—
557C23C22×C22⋊D6111yescharacteristic
558C23C22×C22⋊D6112yes—
559C23C22×Dic12211yescharacteristic
560C23C2×#1175822412yes—
561C23C2×#1175822212yes—
562C23#1451982214yes—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—432no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—618no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1228no—
575C2×C4—1228no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—24432no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2444no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2444no—
592C22—216no—
593C22—216no—
594C22—24no—
595C22—28no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22#1683962018yescomplemented
616C22#1683963518yescomplemented
617C22C2×C22⋊C4×D6111yescharacteristic
618C22C22×#1175819611yescharacteristic
619C22C2×#1451973311yescharacteristic
620C22C2×C4×C22⋊D6112yes—
621C22#1684023512yes—
622C22C2×#1451974611yescharacteristic
623C22C2×#1451974912yes—
624C22C2×#1451975412yes—
625C22C2×#1451975612yes—
626C22C2×#1451976011yescharacteristic
627C22#1684032112yes—
628C22#1684038812yes—
629C22D4×C22⋊D6114yes—
630C22#1684040214yes—
631C22#1684040614yes—
632C22#1684043814yes—
633C4—48no—
634C4—1228no—
635C4—4882no—
636C4—4882no—
637C4—2444no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×#1683962011yescharacteristic
642C2C2×#1683963511yescharacteristic
643C2#2144843111yescharacteristic
644C2C22⋊C4×C22⋊D6112yes—
645C2#2144866412yes—
646C2#2144919312yes—
647C2#2144921512yes—
648C2#2144948014yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC23×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC26×C61C22
hypercenterC23×C22⋊C4D61
hypocenterC61#13877
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839635C2semidirect
3. C2×#16839620C2semidirect
5. #21444062C2semidirect
6. #21444042C2semidirect
7. #21444041C2semidirect
10. C22×#11758196C22semidirect
12. C22×C22⋊Dic61C4semidirect
23. C2×#14519684C22semidirect
24. C2×#14519684C22semidirect
25. C2×#14519684C22semidirect
36. C23×Dic61C2×C4semidirect
58. C2×C22⋊Dic61C2×C4semidirect
112. C24×C61C4⋊C4semidirect
172. C22⋊Dic61C22⋊C4semidirect
266. C2×Dic61C2×C22⋊C4semidirect
291. C22×C61#380semidirect
292. C22×C61#385semidirect
356. C61#13877semidirect
437. C24#11758196semidirect
615. C22#16839620semidirect
616. C22#16839635semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic612
C2×#168396352
C2×#168396202
C2×#168396182
#214440622
#214440422
#214440412
C23×C22⋊C4×C612
#1387761