Finite Group Databasebeta

← #30617313

Subgroups — #30617313

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊Q8×#6923C211yesmaximal, characteristic, complemented
2C4×D4×#6923C211yesmaximal, characteristic
3#21461998C211yesmaximal, characteristic, complemented
4#21461931C211yesmaximal, characteristic, complemented
5#21461883C212yesmaximal, complemented
6#21461784C212yesmaximal
7#21461687C211yesmaximal, characteristic
8#21461634C212yesmaximal, complemented
9#21461606C211yesmaximal, characteristic, complemented
10#21461590C212yesmaximal
11#474×D61C211yesmaximal, characteristic
12C2×Q8×#6923—21no—
13C2×D4×#6923C2211yescharacteristic
14C2×D4×#6923C2211yescharacteristic
15C22×C4×#6923—21no—
16C2×#14519845C2212yes—
17D61⋊(C4⋊C8)C2211yescharacteristic
18C2×#14519839C2212yes—
19C2×#14519831C2212yes—
20(C2×#6923)⋊Q8—21no—
21#16840713—21no—
22#16840713C2212yes—
23#16840708C2211yescharacteristic
24C4⋊C4×#6923—21no—
25C4⋊C4×#6923C2212yes—
26C4⋊C4×#6923C2211yescharacteristic
27C4⋊C4×#6923C2211yescharacteristic
28C4×C4⋊#6923C2211yescharacteristic
29C22⋊C4×#6923—21no—
30C22⋊C4×#6923—21no—
31C4×C22⋊#6923—21no—
32C42×#6923C2211yescharacteristic
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36#16840635C2212yes—
37(C2×#6923)⋊C8C2212yes—
38C4⋊Q8×D61C2211yescharacteristic
39C4×D4×D61C2211yescharacteristic
40#16840043C411yescharacteristic, complemented
41#16840042C411yescharacteristic, complemented
42#16840041C412yescomplemented
43#16840040C412yescomplemented
44#16840039C411yescharacteristic, complemented
45#107×D61C2212yes—
46C4⋊C8×D61C2211yescharacteristic
47#102×D61C2212yes—
48C61×#474C411yescharacteristic, complemented
49C23×#6923—21no—
50C23×#6923—21no—
51C2×D61⋊C8C2312yes—
52Q8×#6923—44no—
53D4×#6923—22no—
54D4×#6923—22no—
55D4×#6923—22no—
56D4×#6923Q814yescomplemented
57C2×C4⋊#6923—22no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—21no—
60C2×C4⋊#6923C2312yes—
61C2×C4⋊#6923C2311yescharacteristic
62C2×C4⋊#6923C2311yescharacteristic
63C2×C22⋊#6923—21no—
64C2×C22⋊#6923—21no—
65C2×C4×#6923—21no—
66C2×C4×#6923—21no—
67C2×C4×#6923—22no—
68C2×C4×#6923—21no—
69C2×C4×#6923—44no—
70C2×C4×#6923D411yescharacteristic
71C2×C4×#6923D411yescharacteristic
72C2×C4×#6923D411yescharacteristic
73C2×C4×#6923D411yescharacteristic
74C2×C4×#6923C2311yescharacteristic
75C2×C4×#6923C2311yescharacteristic
76#14519845—24no—
77#14519844—21no—
78C4⋊#92420—21no—
79#14519839—24no—
80#14519831—24no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83(C2×#6923)⋊C4—21no—
84(C2×#6923)⋊C4—21no—
85(C2×#6923)⋊C4—21no—
86C2×Q8×D61—21no—
87C2×D4×D61C2311yescharacteristic
88C22×C4×D61—21no—
89C2×C8×D61C2312yes—
90#14519777C2×C411yescharacteristic, complemented
91#14519776—21no—
92#14519776C2×C412yescomplemented
93#14519775—21no—
94C4×D244C2×C411yescharacteristic
95D4×Dic61C2×C411yescharacteristic
96#14519739C2×C411yescharacteristic
97C4×C22⋊D61—21no—
98#14519737—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61C2312yes—
101C4⋊C4×D61C2311yescharacteristic
102C22⋊C4×D61—21no—
103C42×D61C2311yescharacteristic
104#14519711C2×C412yescomplemented
105#14519710C2×C412yescomplemented
106#14519710C2×C412yescomplemented
107C4⋊#92418C2×C411yescharacteristic, complemented
108#14519706C2×C412yescomplemented
109#14519699C2×C412yes—
110#14519698C2×C412yes—
111#14519697C2×C412yes—
112C4⋊Q8×C61C2×C411yescharacteristic, complemented
113C4×D4×C61C2×C411yescharacteristic
114#107×C61C2×C412yescomplemented
115C4⋊C8×C61C2×C411yescharacteristic, complemented
116#102×C61C2×C412yes—
117C23×D61—21no—
118C22×#6923—24no—
119C22×#6923—21no—
120C22×#6923—21no—
121C22×#6923—24no—
122C22×#6923—22no—
123C22×#6923—42no—
124C22×#6923—42no—
125C22×#6923—42no—
126C22×#6923—42no—
127C22×#6923C2×D411yescharacteristic
128C22×#6923C2×D411yescharacteristic
129C22×#6923C2×D411yescharacteristic
130C22×#6923C2×D411yescharacteristic
131C2×Q8×C61—21no—
132C2×D4×C61C22×C411yescharacteristic
133C22×C4×C61—21no—
134C2×C22⋊D61—21no—
135C22×Dic61—21no—
136Q8×D61—42no—
137Q8×D61—42no—
138D4×D61—21no—
139D4×D61—21no—
140D4×D61C2×Q812yes—
141D4×D61#4112yes—
142C2×D244C22×C411yescharacteristic
143C2×C4×D61—21no—
144C2×C4×D61—22no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2411yescharacteristic
152C2×Dic122—21no—
153C2×Dic122—22no—
154C22⋊#6923—24no—
155C22⋊#6923—24no—
156C2×#92420—22no—
157C4⋊#6923—21no—
158C4⋊#6923—21no—
159C4⋊#6923—22no—
160C4⋊#6923—22no—
161C4⋊#6923—42no—
162C4⋊#6923—44no—
163C4⋊#6923—42no—
164C4⋊#6923C2×Q812yes—
165C4⋊#6923#4112yes—
166C4×#6923—22no—
167C4×#6923—22no—
168C4×#6923—22no—
169C4×#6923—21no—
170C4×#6923—21no—
171C4×#6923—41no—
172C4×#6923—42no—
173C4×#6923—42no—
174C4×#6923—44no—
175C4×#6923—42no—
176C4×#6923C2×Q812yes—
177C4×#6923#4112yes—
178D61⋊C8—22no—
179D61⋊C8—22no—
180C2×C8×C61C22×C412yes—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C22×C412yes—
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—21no—
185C42×C61C22×C411yescharacteristic
186C22⋊Dic61—21no—
187#11758198—21no—
188#11758198—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C412yes—
191C4⋊Dic61C22×C411yescharacteristic
192#11758196—22no—
193#11758196—21no—
194#11758196—22no—
195C4×Dic61C22×C411yescharacteristic
196C4×Dic61#4111yescharacteristic
197C4×Dic61#4111yescharacteristic
198C2×#92418C22×C412yes—
199C8×D61—22no—
200C8×D61—22no—
201C23×C61—21no—
202C22×D61—21no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61C22×D411yescharacteristic
210C2×#6923—21no—
211C2×#6923—21no—
212C2×#6923—21no—
213C2×#6923—21no—
214C2×#6923—21no—
215C2×#6923—21no—
216C2×#6923—22no—
217C2×#6923—21no—
218C2×#6923—22no—
219C2×#6923—21no—
220C2×#6923—21no—
221C2×#6923—48no—
222C2×#6923—48no—
223C2×#6923—41no—
224C2×#6923—41no—
225C2×#6923—41no—
226C2×#6923—41no—
227C2×#6923—42no—
228C2×#6923—42no—
229C2×#6923—42no—
230C2×#6923C22⋊Q812yes—
231C2×#6923C22⋊Q812yes—
232C2×#6923#13712yes—
233Q8×C61—42no—
234D4×C61—21no—
235D4×C61C4×Q812yescomplemented
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—42no—
239C2×C4×C61C4×D411yescharacteristic
240C2×C4×C61C4×D411yescharacteristic
241C2×C4×C61C23×C411yescharacteristic
242C22⋊D61—22no—
243C22⋊D61—22no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—42no—
247C2×Dic61C4×D411yescharacteristic
248C2×Dic61C4×D411yescharacteristic
249C2×Dic61C2×#4111yescharacteristic
250D244—21no—
251D244C4×Q812yescomplemented
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—41no—
255C4×D61—42no—
256C4×D61—41no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61AGL(1,ℤ/8)12yes—
261C4×D61AGL(1,ℤ/8)12yes—
262C4×D61C22×Q811yescharacteristic
263C4×D61C2×#4111yescharacteristic
264C4×D61C2×#4111yescharacteristic
265C4×D61C2×#4111yescharacteristic
266Dic122—42no—
267Dic122—44no—
268#92420—42no—
269C8×C61—22no—
270#92418—22no—
271#21613—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61C2×C4×D411yescharacteristic
276C2×D61—22no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61C2×C22⋊Q811yescharacteristic
284C2×D61#52511yescharacteristic
285C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
286C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
287C2×D61C2×#13711yescharacteristic
288C2×D61#57511yescharacteristic
289#6923—22no—
290#6923—22no—
291#6923—22no—
292#6923—42no—
293#6923—42no—
294#6923—41no—
295#6923—82no—
296C4×C61—21no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61#44212yes—
300C4×C61C2×C4×Q811yescharacteristic
301C4×C61C4×#4111yescharacteristic
302Dic61—21no—
303Dic61—42no—
304Dic61—41no—
305Dic61#44212yes—
306Dic61#52911yescharacteristic
307Dic61#53311yescharacteristic
308C2×#474—611no—
309C4×C4⋊Q8—611no—
310C42×D4—611no—
311#1988—611no—
312#1948—611no—
313#1921—612no—
314#1864—612no—
315#1802—611no—
316#1776—612no—
317#1767—611no—
318#1762—612no—
319C2×C61—42no—
320C2×C61C4×C22⋊Q811yescharacteristic
321C2×C61C2×#44211yescharacteristic
322C2×C61C4×#13711yescharacteristic
323D61—42no—
324D61#307711yescharacteristic
325D61#307911yescharacteristic
326D61#311011yescharacteristic
327D61#311511yescharacteristic
328C2×C4⋊Q8—611no—
329C2×C4×Q8—1221no—
330C2×C4×D4—611no—
331C2×C4×D4—611no—
332C2×C4×D4—611no—
333C22×C42—1221no—
334#474—611no—
335#474—611no—
336#474—612no—
337#474—611no—
338#474—612no—
339#474#692311yescharacteristic, complemented
340C2×#107—612no—
341C2×#106—612no—
342C2×C4⋊C8—611no—
343C2×C4⋊C8—611no—
344C2×#102—612no—
345C2×#102—612no—
346C2×#97—612no—
347#391—1221no—
348#389—612no—
349#389—1221no—
350#384—611no—
351C4×C4⋊C4—612no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—1221no—
356C4×C22⋊C4—1221no—
357C4×C22⋊C4—1221no—
358C4×C22⋊C4—1221no—
359C43—611no—
360#340—612no—
361#340—612no—
362#340—612no—
363#340—612no—
364#336—612no—
365C61#2161311yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C22×D4—611no—
368C23×C4—1221no—
369C23×C4—1221no—
370C23×C4—1221no—
371C22×C8—612no—
372C22×C8—612no—
373C4⋊Q8—612no—
374C4⋊Q8—611no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377C4⋊Q8C2×#692311yescharacteristic, complemented
378C4×Q8—2444no—
379C4×D4—614no—
380C4×D4—611no—
381C4×D4—611no—
382C4×D4—611no—
383C4×D4—1221no—
384C4×D4—1221no—
385C4×D4—1222no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4C2×#692311yescharacteristic
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1222no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42—2444no—
413#107—612no—
414#107—612no—
415#107—612no—
416#107C2×#692312yescomplemented
417#106—1224no—
418C4⋊C8—612no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8C2×#692311yescharacteristic, complemented
423#102—612no—
424#102—612no—
425#102—612no—
426#102—1224no—
427#102C2×#692312yes—
428#97—1224no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1222no—
433#95—1222no—
434C24—1221no—
435C2×Q8—21no—
436C2×Q8—1222no—
437C2×Q8—1221no—
438C2×Q8—2442no—
439C2×Q8—2442no—
440C2×D4—612no—
441C2×D4—612no—
442C2×D4—611no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4C22×#692311yescharacteristic
447C22×C4—21no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1222no—
458C22×C4—1221no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1224no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C2×C8—612no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8C22×#692312yes—
478C4⋊C4—21no—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—612no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—2442no—
492C4⋊C4—2444no—
493C4⋊C4—2442no—
494C4⋊C4C22×#692312yes—
495C4⋊C4C22×#692311yescharacteristic
496C22⋊C4—21no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1224no—
500C22⋊C4—1224no—
501C22⋊C4—1221no—
502C42—612no—
503C42—612no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—1222no—
508C42—1221no—
509C42—1221no—
510C42—1222no—
511C42—1222no—
512C42—2442no—
513C42—2441no—
514C42—2442no—
515C42—2442no—
516C42—2444no—
517C42C22×#692311yescharacteristic
518C23—21no—
519C23—611no—
520C23—1222no—
521C23—1222no—
522C23—1221no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23—2441no—
527Q8—42no—
528Q8—2444no—
529Q8—2442no—
530D4—21no—
531D4—612no—
532D4—1222no—
533D4—1222no—
534D4—1221no—
535D4Q8×#692312yescomplemented
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—42no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—611no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—2442no—
567C2×C4—2441no—
568C2×C4—2442no—
569C2×C4—2441no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2448no—
573C2×C4—2448no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2442no—
582C2×C4D4×#692311yescharacteristic
583C2×C4D4×#692311yescharacteristic
584C2×C4C23×#692311yescharacteristic
585C8—22no—
586C8—1222no—
587C8—2442no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—611no—
592C22—611no—
593C22—611no—
594C22—611no—
595C22—611no—
596C22—611no—
597C22—1222no—
598C22—2442no—
599C22—2442no—
600C22—2442no—
601C22—2442no—
602C22—2441no—
603C22—2441no—
604C22C2×D4×#692311yescharacteristic
605C4—21no—
606C4—41no—
607C4—42no—
608C4—611no—
609C4—611no—
610C4—612no—
611C4—1222no—
612C4—1222no—
613C4—1222no—
614C4—1221no—
615C4—4882no—
616C4—2441no—
617C4—2442no—
618C4—2442no—
619C4—2441no—
620C4—2442no—
621C4#1684080312yes—
622C4C2×Q8×#692311yescharacteristic
623C4#41×#692311yescharacteristic
624C2—42no—
625C2—611no—
626C2—611no—
627C2—611no—
628C2—611no—
629C2—2442no—
630C2C22⋊Q8×#692311yescharacteristic
631C2C2×#1684080311yescharacteristic
632C2#137×#692311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#474C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#474#6923
hypocenterC61#21613
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×#6923C2semidirect
3. #21461998C2semidirect
4. #21461931C2semidirect
5. #21461883C2semidirect
8. #21461634C2semidirect
9. #21461606C2semidirect
40. #16840043C4semidirect
41. #16840042C4semidirect
42. #16840041C4semidirect
43. #16840040C4semidirect
44. #16840039C4semidirect
48. C61×#474C4semidirect
56. D4×#6923Q8semidirect
90. #14519777C2×C4semidirect
92. #14519776C2×C4semidirect
104. #14519711C2×C4semidirect
105. #14519710C2×C4semidirect
106. #14519710C2×C4semidirect
107. C4⋊#92418C2×C4semidirect
108. #14519706C2×C4semidirect
112. C4⋊Q8×C61C2×C4semidirect
114. #107×C61C2×C4semidirect
115. C4⋊C8×C61C2×C4semidirect
235. D4×C61C4×Q8semidirect
251. D244C4×Q8semidirect
339. #474#6923semidirect
365. C61#21613semidirect
377. C4⋊Q8C2×#6923semidirect
416. #107C2×#6923semidirect
422. C4⋊C8C2×#6923semidirect
535. D4Q8×#6923semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×#69232
C4×D4×#69232
#214619982
#214619312
#214618832
#214617842
#214616872
#214616342
#214616062
#214615902
#474×D612
#2161361