Finite Group Databasebeta

← #29602256

Subgroups — #29602256

Order 15616.

2574 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic61C211yesmaximal, characteristic
2C2×#16839665C211yesmaximal, characteristic
3C2×#16839631C211yesmaximal, characteristic, complemented
4C2×#16839620C214yesmaximal, complemented
5C2×C4×#11758196C211yesmaximal, characteristic, complemented
6#21444062C214yesmaximal, complemented
7#21444049C211yesmaximal, characteristic, complemented
8#21444048C211yesmaximal, characteristic, complemented
9C2×C61×#390C211yesmaximal, characteristic
10C24×Dic61C2212yes—
11C2×D4×Dic61—24no—
12C2×D4×Dic61—24no—
13C2×D4×Dic61C418yescomplemented
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×#11758196C2212yes—
19C22×C22⋊Dic61C2214yes—
20C22×C22⋊Dic61C2212yes—
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839665—24no—
24#16839631—24no—
25#16839620—216no—
26C4×#11758196—24no—
27C2×#14519684C2214yes—
28C2×#14519684C2214yes—
29C2×#14519684C2212yes—
30C23×D4×C61C2211yescharacteristic
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2214yes—
33C22×C42×C61C2211yescharacteristic
34C61×#390—24no—
35C23×Dic61—24no—
36C23×Dic61—24no—
37C23×Dic61—28no—
38C23×Dic61—28no—
39C23×Dic61C2×C418yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45D4×Dic61—232no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61C2×C414yes—
52C2×#11758196—24no—
53C2×#11758196—24no—
54C2×#11758196—28no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196—44no—
58C2×#11758196D418yescomplemented
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61—24no—
61C2×C22⋊Dic61—28no—
62C2×C22⋊Dic61—28no—
63C2×C22⋊Dic61—44no—
64C2×C22⋊Dic61C2×C418yes—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2×C414yes—
69C2×C4×Dic61D414yes—
70C2×C4×Dic61D414yes—
71#14519684—28no—
72#14519684—24no—
73#14519684—28no—
74C25×C61C2312yes—
75C22×D4×C61—22no—
76C22×D4×C61—22no—
77C22×D4×C61—22no—
78C22×D4×C61—22no—
79C22×D4×C61C2×C414yes—
80C22×D4×C61D414yes—
81C22×D4×C61Q814yes—
82C23×C4×C61C2312yes—
83C23×C4×C61C2312yes—
84C23×C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C22⋊C4×C61—28no—
89C2×C22⋊C4×C61—28no—
90C2×C22⋊C4×C61—44no—
91C2×C42×C61—22no—
92C2×C42×C61—22no—
93C2×C42×C61—22no—
94C24×C61—24no—
95C24×C61—22no—
96C24×C61—22no—
97C24×C61—24no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—24no—
101C24×C61—24no—
102C24×C61C22×C414yes—
103C24×C61C2×D414yes—
104C24×C61C2×Q814yes—
105C24×C61C2411yescharacteristic
106C2×D4×C61—28no—
107C2×D4×C61—28no—
108C2×D4×C61—28no—
109C2×D4×C61—28no—
110C2×D4×C61—28no—
111C2×D4×C61—28no—
112C2×D4×C61C4⋊C4116yescomplemented
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D414yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×Q812yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—216no—
145C22×Dic61—232no—
146C22×Dic61—24no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—24no—
151C22×Dic61—416no—
152C22×Dic61—48no—
153C22×Dic61—48no—
154C22×Dic61C22×C414yes—
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C412yes—
157C22×Dic61C2×D414yes—
158C22×Dic61C2×D412yes—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D414yes—
163C22×Dic61#4114yes—
164C4⋊C4×C61—44no—
165C22⋊C4×C61—416no—
166C42×C61—44no—
167C22⋊Dic61—216no—
168C22⋊Dic61—416no—
169C4⋊Dic61—28no—
170C4⋊Dic61—44no—
171#11758196—216no—
172#11758196—48no—
173#11758196—48no—
174C4×Dic61—44no—
175C4×Dic61C22⋊C4116yescomplemented
176C23×C61—24no—
177C23×C61—24no—
178C23×C61—28no—
179C23×C61—28no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—28no—
185C23×C61—28no—
186C23×C61—28no—
187C23×C61—28no—
188C23×C61—44no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—44no—
194C23×C61—44no—
195C23×C61—44no—
196C23×C61—44no—
197C23×C61—44no—
198C23×C61C2×C4⋊C418yes—
199C23×C61C23×C411yescharacteristic
200C23×C61C22×D411yescharacteristic
201C23×C61C22×D411yescharacteristic
202C23×C61C22×D411yescharacteristic
203C23×C61C22×D412yes—
204C23×C61C22×Q811yescharacteristic
205C23×C61C2×#4111yescharacteristic
206C23×C61C2×#4111yescharacteristic
207C23×C61C2×#4112yes—
208C23×C61H(5,𝔽2)12yes—
209C23×C612−1+412yes—
210D4×C61—232no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61C2×C22⋊C414yes—
224C2×C4×C61C2×C4⋊C414yes—
225C2×C4×C61#11714yes—
226C2×C4×C61#12114yes—
227C2×C4×C61#12114yes—
228C2×C4×C61C22⋊Q814yes—
229C2×C4×C61C22⋊Q814yes—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—28no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—42no—
240C2×Dic61—44no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—432no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—44no—
251C2×Dic61—44no—
252C2×Dic61C2×C22⋊C414yes—
253C2×Dic61C2×C22⋊C414yes—
254C2×Dic61C4×D418yes—
255C2×Dic61#12118yes—
256C2×Dic61#12414yes—
257C2×Dic61#12714yes—
258#13906—611nomaximal, Sylow
259C22×C61—216no—
260C22×C61—28no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—48no—
264C22×C61—48no—
265C22×C61—48no—
266C22×C61—44no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61C22×C22⋊C411yescharacteristic
271C22×C61C22×C4⋊C411yescharacteristic
272C22×C61C2×#11711yescharacteristic
273C22×C61C2×C4×D412yes—
274C22×C61#52012yes—
275C22×C61C2×#12112yes—
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12111yescharacteristic
278C22×C61C2×C22⋊Q811yescharacteristic
279C22×C61C2×C22⋊Q811yescharacteristic
280C22×C61C2×#12411yescharacteristic
281C22×C61C2×#12711yescharacteristic
282C22×C61#53511yescharacteristic
283C22×C61#53611yescharacteristic
284C22×C61#53712yes—
285C22×C61D4212yes—
286C22×C61#54712yes—
287C22×C61#54712yes—
288C22×C61D4×Q812yes—
289C22×C61#55012yes—
290C22×C61#55412yes—
291C22×C61#55512yes—
292C22×C61#55612yes—
293C4×C61—44no—
294C4×C61—82no—
295C4×C61#38018yescomplemented
296Dic61—24no—
297Dic61—44no—
298Dic61—44no—
299Dic61—82no—
300Dic61#39018yescomplemented
301C22×C4×D4—611no—
302C2×#390—611no—
303C2×#390D6111yescharacteristic
304C2×#384—611no—
305C2×#380—614no—
306C2×C4×C4⋊C4—611no—
307#1439—614no—
308C4⋊#95—611no—
309C4⋊#95—611no—
310C2×C61—48no—
311C2×C61C2×#38011yescharacteristic
312C2×C61C2×#39011yescharacteristic
313C2×C61#231111yescharacteristic
314C2×C61D4×C4⋊C412yes—
315C2×C61#235412yes—
316C2×C61#253712yes—
317C2×C61#253812yes—
318C2×C61#254111yescharacteristic
319C2×C61#254211yescharacteristic
320C2×C61#254711yescharacteristic
321C2×C61#256311yescharacteristic
322C23×D4C2×D6111yescharacteristic
323C24×C4—612no—
324C2×C4×D4—618no—
325C2×C4×D4—1224no—
326C2×C4×D4—1224no—
327C22×C4⋊C4—612no—
328C22×C4⋊C4—611no—
329C22×C4⋊C4—612no—
330C22×C4⋊C4—611no—
331C22×C4⋊C4—612no—
332C22×C4⋊C4C2×D6111yescharacteristic
333C22×C22⋊C4—612no—
334C22×C22⋊C4—614no—
335C22×C22⋊C4C2×D6114yes—
336C22×C42—611no—
337C22×C42—611no—
338C22×C42C2×D6111yescharacteristic
339#390—24no—
340#390—1224no—
341#384—1224no—
342#380—12216no—
343C4×C4⋊C4—1224no—
344C2×#95—614no—
345C2×#95—614no—
346C2×#95—612no—
347C61#1390611yescharacteristic, complemented, Sylow
348C25C22×D6112yes—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4Dic12214yes—
354C22×D4C4×D6114yes—
355C22×D4C22⋊D6114yes—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—618no—
359C23×C4—612no—
360C23×C4—612no—
361C23×C4—612no—
362C23×C4—1224no—
363C23×C4—1224no—
364C23×C4—1228no—
365C23×C4—1228no—
366C23×C4C22×D6112yes—
367C23×C4C22×D6112yes—
368C23×C4C22×D6111yescharacteristic
369C4×D4—12232no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—618no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1228no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1224no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—2444no—
386C2×C22⋊C4—28no—
387C2×C22⋊C4—28no—
388C2×C22⋊C4—44no—
389C2×C22⋊C4—618no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1228no—
393C2×C22⋊C4—1228no—
394C2×C22⋊C4—2444no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—614no—
399C2×C42—614no—
400C2×C42—614no—
401C2×C42—1222no—
402C2×C42—1222no—
403C2×C42—1222no—
404#95—1224no—
405#95—1228no—
406#95—1228no—
407C24—24no—
408C24—22no—
409C24—22no—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—24no—
415C24C2×Dic12214yes—
416C24C2×C4×D6114yes—
417C24C2×C22⋊D6114yes—
418C24C23×D6111yescharacteristic
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4#11758196116yescomplemented
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—12216no—
448C22×C4—12232no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1224no—
466C22×C4—24416no—
467C22×C4—2448no—
468C22×C4—2448no—
469C22×C4C2×Dic12212yes—
470C22×C4C2×C4×D6112yes—
471C22×C4D4×D6114yes—
472C22×C4D4×D6112yes—
473C22×C4D4×D6112yes—
474C22×C4#1175822412yes—
475C22×C4#1175822412yes—
476C22×C4C2×C22⋊D6112yes—
477C4⋊C4—44no—
478C4⋊C4—12216no—
479C4⋊C4—1228no—
480C4⋊C4—2448no—
481C4⋊C4—2448no—
482C4⋊C4—2444no—
483C22⋊C4—416no—
484C22⋊C4—12216no—
485C22⋊C4—24416no—
486C42—44no—
487C42—6116no—
488C42—2444no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23C2×#1175819618yes—
512C23C22×C4×D6111yescharacteristic
513C23C2×D4×D6111yescharacteristic
514C23C2×D4×D6111yescharacteristic
515C23C2×D4×D6112yes—
516C23C22×C22⋊D6111yescharacteristic
517C23C22×Dic12211yescharacteristic
518C23#41×D6112yes—
519C23C2×#1175822411yescharacteristic
520C23C2×#1175822411yescharacteristic
521C23#1451982212yes—
522C23#1451982412yes—
523D4—232no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—614no—
537C2×C4—618no—
538C2×C4—618no—
539C2×C4—614no—
540C2×C4—614no—
541C2×C4—614no—
542C2×C4—1224no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1228no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—24432no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4—2444no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4C22⋊C4×D6114yes—
565C2×C4C2×#1175819614yes—
566C2×C4#1451973314yes—
567C2×C4#1451974914yes—
568C2×C4#1451974914yes—
569C2×C4#1451975414yes—
570C2×C4#1451975414yes—
571C22—216no—
572C22—28no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—48no—
577C22—44no—
578C22—44no—
579C22—48no—
580C22—44no—
581C22—48no—
582C22C2×C22⋊C4×D6111yescharacteristic
583C22C22×#1175819611yescharacteristic
584C22C2×#1451973311yescharacteristic
585C22C4×D4×D6112yes—
586C22#1684024112yes—
587C22#121×D6112yes—
588C22C2×#1451974911yescharacteristic
589C22C2×#1451974911yescharacteristic
590C22C2×#1451975411yescharacteristic
591C22C2×#1451975411yescharacteristic
592C22#124×D6111yescharacteristic
593C22#127×D6111yescharacteristic
594C22#1684032511yescharacteristic
595C22#1684033211yescharacteristic
596C22#1684034112yes—
597C22D4×C22⋊D6112yes—
598C22#1684041112yes—
599C22#1684041212yes—
600C22D4×Dic12212yes—
601C22#1684043212yes—
602C22#1684047112yes—
603C22#1684047612yes—
604C22#1684048112yes—
605C4—44no—
606C4—82no—
607C4—618no—
608C4—1224no—
609C4—4882no—
610C4—2444no—
611C4—2444no—
612C4#1683962018yescomplemented
613C2—48no—
614C2C2×#1683962011yescharacteristic
615C2#390×D6111yescharacteristic
616C2#2144840911yescharacteristic
617C2D4×#1175819612yes—
618C2#2144876712yes—
619C2#2145073312yes—
620C2#2145073912yes—
621C2#2145076311yescharacteristic
622C2#2145077011yescharacteristic
623C2#2145081211yescharacteristic
624C2#2145099011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#390C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC2×#390D61
hypocenterC61#13906
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
3. C2×#16839631C2semidirect
4. C2×#16839620C2semidirect
5. C2×C4×#11758196C2semidirect
6. #21444062C2semidirect
7. #21444049C2semidirect
8. #21444048C2semidirect
13. C2×D4×Dic61C4semidirect
58. C2×#11758196D4semidirect
112. C2×D4×C61C4⋊C4semidirect
175. C4×Dic61C22⋊C4semidirect
295. C4×C61#380semidirect
300. Dic61#390semidirect
347. C61#13906semidirect
425. C2×D4#11758196semidirect
612. C4#16839620semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic612
C2×#168396652
C2×#168396312
C2×#168396202
C2×C4×#117581962
#214440622
#214440492
#214440482
C2×C61×#3902
#1390661