Finite Group Databasebeta

← #472×#6923 #30617096

Subgroups — #472×#6923 #30617096

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1AGL(1,ℤ/8)×#6923C212yesmaximal, complemented
2C2×D8×#6923C211yesmaximal, characteristic, complemented
3#131×#6923C211yesmaximal, characteristic, complemented
4#21461980C211yesmaximal, characteristic, complemented
5#21461937C211yesmaximal, characteristic, complemented
6#21461905C212yesmaximal, complemented
7#21461858C212yesmaximal, complemented
8#21461708C211yesmaximal, characteristic, complemented
9#108×#6923C211yesmaximal, characteristic, complemented
10#100×#6923C212yesmaximal, complemented
11#472×D61C211yesmaximal, characteristic
12D61⋊#131C2211yescharacteristic
13#41×#6923C2211yescharacteristic
14C2×D4×#6923C2212yes—
15#16840803—22no—
16#16840801—22no—
17SD16×#6923—22no—
18D8×#6923—22no—
19D8×#6923—21no—
20D8×#6923—21no—
21D8×#6923—21no—
22D8×#6923—21no—
23D61⋊#108C2211yescharacteristic
24#16840773C2212yes—
25C2×#14519846C2211yescharacteristic
26#16840754C2212yes—
27#16840753C2212yes—
28C2×#14519839C2212yes—
29D61⋊#100C2212yes—
30#16840728—21no—
31#16840728C2211yescharacteristic
32#16840725—21no—
33#16840725C2211yescharacteristic
34M16×#6923—21no—
35M16×#6923C2212yes—
36M16×#6923C2211yescharacteristic
37C2×C8×#6923—21no—
38C2×C8×#6923C2211yescharacteristic
39#16840641C2212yes—
40#16840641C2212yes—
41#16840637C2212yes—
42AGL(1,ℤ/8)×D61C2212yes—
43C2×D8×D61C2211yescharacteristic
44#131×D61C2211yescharacteristic
45#16840033C411yescharacteristic, complemented
46#16840032C411yescharacteristic, complemented
47#16840031C412yescomplemented
48#16840030C412yescomplemented
49#16840029C411yescharacteristic, complemented
50#108×D61C2211yescharacteristic
51#100×D61C2212yes—
52C61×#472C411yescharacteristic, complemented
53C23×#6923—22no—
54#14519865—21no—
55#14519864—21no—
56#14519864—22no—
57D61⋊M16—21no—
58D61⋊M16C2312yes—
59D61⋊M16C2311yescharacteristic
60C2×D61⋊C8—21no—
61C2×D61⋊C8C2311yescharacteristic
62Q8×#6923D411yescharacteristic
63Q8×#6923D411yescharacteristic
64D4×#6923—21no—
65D4×#6923—21no—
66D4×#6923—22no—
67D4×#6923—22no—
68D4×#6923—22no—
69D4×#6923—22no—
70D4×#6923D411yescharacteristic
71D4×#6923D411yescharacteristic
72#14519856—21no—
73#14519856C2311yescharacteristic
74C2×C4⋊#6923C2312yes—
75C2×C22⋊#6923—22no—
76C2×C4×#6923—21no—
77C2×C4×#6923C2311yescharacteristic
78#14519848—21no—
79#14519847—21no—
80#14519846—22no—
81#14519846—21no—
82#14519846—21no—
83#14519845—22no—
84#14519840—22no—
85#14519839—22no—
86#14519839—22no—
87#14519839—22no—
88#14519839—22no—
89#14519839—22no—
90#14519837—24no—
91#14519831—21no—
92#14519831—22no—
93#14519831—21no—
94#14519831—21no—
95C8×#6923—21no—
96C8×#6923—22no—
97C8×#6923—21no—
98C8×#6923—21no—
99C8×#6923D411yescharacteristic
100C8×#6923D411yescharacteristic
101C8×#6923D411yescharacteristic
102C8×#6923D411yescharacteristic
103#41×D61C2311yescharacteristic
104C2×D4×D61C2312yes—
105#14519805C2×C412yescomplemented
106#14519804—22no—
107#14519803C2×C412yescomplemented
108#14519802C2×C412yescomplemented
109#14519801—22no—
110SD16×D61—22no—
111C2×D488C2×C411yescharacteristic, complemented
112C2×#11758199C2×C412yescomplemented
113D8×D61—22no—
114D8×D61—21no—
115D8×D61—21no—
116D8×D61—21no—
117D8×D61—21no—
118#14519785—21no—
119#14519785C2×C411yescharacteristic, complemented
120#14519784C2×C411yescharacteristic, complemented
121#14519783—21no—
122#14519783C2×C411yescharacteristic, complemented
123M16×D61—21no—
124M16×D61C2311yescharacteristic
125M16×D61C2312yes—
126C2×C8×D61—21no—
127C2×C8×D61C2311yescharacteristic
128#14519713C2×C411yescharacteristic, complemented
129#14519712C2×C412yescomplemented
130#14519694C2×C412yescomplemented
131#14519693C2×C412yescomplemented
132#14519693C2×C412yescomplemented
133AGL(1,ℤ/8)×C61C2×C412yescomplemented
134C2×D8×C61C2×C411yescharacteristic, complemented
135#131×C61C2×C411yescharacteristic, complemented
136#108×C61C2×C411yescharacteristic, complemented
137#100×C61C2×C412yescomplemented
138C23×D61—22no—
139C22×#6923—21no—
140C22×#6923—21no—
141C22×#6923—22no—
142C22×#6923—42no—
143C22×#6923—42no—
144C22×#6923—84no—
145C22×#6923#4111yescharacteristic
146C22×#6923#4111yescharacteristic
147#41×C61C22×C411yescharacteristic
148C2×D4×C61C22×C412yes—
149C2×C22⋊D61—22no—
150#11758226#4111yescharacteristic
151Q8×D61C2×D411yescharacteristic
152#11758224—21no—
153#11758224#4111yescharacteristic
154D4×D61—21no—
155D4×D61—22no—
156D4×D61—22no—
157D4×D61—22no—
158D4×D61—22no—
159D4×D61C2×D411yescharacteristic
160#11758222—21no—
161#11758222C22×C411yescharacteristic
162C2×D244C22×C412yes—
163C2×C4×D61—21no—
164C2×C4×D61C2411yescharacteristic
165C22⋊#6923—21no—
166C22⋊#6923—21no—
167C22⋊#6923—21no—
168C22⋊#6923—42no—
169C22⋊#6923—42no—
170#11758217—22no—
171#11758217—22no—
172#11758217#4112yes—
173C2×#92420—22no—
174C2×#92420—21no—
175C4⋊#6923—21no—
176C4⋊#6923—21no—
177C4⋊#6923—22no—
178C4⋊#6923—22no—
179C4⋊#6923C2×D411yescharacteristic
180C4⋊#6923C2×D411yescharacteristic
181C4×#6923—21no—
182C4×#6923—21no—
183C4×#6923—21no—
184C4×#6923C2×D411yescharacteristic
185C4×#6923C2×D411yescharacteristic
186C4×#6923C2×D411yescharacteristic
187C4×#6923C2×D411yescharacteristic
188#11758213—22no—
189#11758213—21no—
190#11758213—22no—
191D61⋊C8—22no—
192D61⋊C8—21no—
193D61⋊C8—21no—
194D61⋊C8—22no—
195D61⋊C8C2×D411yescharacteristic
196D61⋊C8C2×D411yescharacteristic
197SD16×C61—22no—
198D8×C61—22no—
199D8×C61—21no—
200D8×C61—21no—
201M16×C61—21no—
202M16×C61C22×C411yescharacteristic
203M16×C61C22×C412yes—
204C2×C8×C61—21no—
205C2×C8×C61C22×C411yescharacteristic
206#11758201—22no—
207#11758200—22no—
208#11758199—22no—
209#11758199—22no—
210#11758199—22no—
211#11758199—22no—
212#11758194—21no—
213#11758194C22×C411yescharacteristic
214#11758194C22×C412yes—
215C2×#92418—21no—
216C2×#92418C22×C411yescharacteristic
217D488—22no—
218D488—21no—
219D488—21no—
220#11758190—22no—
221#11758189—21no—
222#11758189—22no—
223#11758189—21no—
224#11758189—21no—
225C8×D61—21no—
226C8×D61—22no—
227C8×D61—21no—
228C8×D61—21no—
229C8×D61C2×D411yescharacteristic
230C8×D61C2×D411yescharacteristic
231C8×D61#4111yescharacteristic
232C8×D61#4111yescharacteristic
233C23×C61—22no—
234C22×D61—21no—
235C22×D61—22no—
236C22×D61—42no—
237C22×D61—42no—
238C22×D61—82no—
239C22×D61—82no—
240C22×D61C2×#4111yescharacteristic
241C2×#6923—22no—
242C2×#6923—21no—
243C2×#6923—21no—
244C2×#6923—41no—
245C2×#6923—42no—
246C2×#6923—42no—
247C2×#6923—42no—
248C2×#6923—84no—
249C2×#6923—84no—
250C2×#6923—82no—
251C2×#6923#12111yescharacteristic
252C2×#6923#12111yescharacteristic
253C2×#6923#12111yescharacteristic
254C2×#6923#12111yescharacteristic
255Q8×C61C4×D411yescharacteristic
256D4×C61—21no—
257D4×C61—22no—
258D4×C61—22no—
259D4×C61C4×D411yescharacteristic
260C2×C4×C61—21no—
261C2×C4×C61C23×C411yescharacteristic
262C22⋊D61—21no—
263C22⋊D61—21no—
264C22⋊D61—21no—
265C22⋊D61—42no—
266C22⋊D61—42no—
267C2×Dic61—21no—
268C2×Dic61C2×#4111yescharacteristic
269D244—21no—
270D244—22no—
271D244—22no—
272D244C4×D411yescharacteristic
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—21no—
276C4×D61C22×D411yescharacteristic
277C4×D61C22×D411yescharacteristic
278C4×D61C2×#4111yescharacteristic
279C4×D61C2×#4111yescharacteristic
280Dic122—21no—
281Dic122C4×D411yescharacteristic
282#92420—21no—
283#92420—22no—
284#92420—21no—
285#92420—42no—
286C8×C61—21no—
287C8×C61—22no—
288C8×C61C4×D411yescharacteristic
289C8×C61C4×D411yescharacteristic
290#92418—21no—
291#92418—22no—
292#92418C4×D411yescharacteristic
293#92418C4×D411yescharacteristic
294C4×#472—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—42no—
297C22×C61—82no—
298C22×C61C4×#4111yescharacteristic
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—42no—
305C2×D61—84no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61—82no—
309C2×D61C2×#12111yescharacteristic
310C2×D61#52911yescharacteristic
311#6923—21no—
312#6923—21no—
313#6923—21no—
314#6923—41no—
315#6923—41no—
316#6923—82no—
317#6923#47212yesdirect product factor
318C4×C61—21no—
319C4×C61C2×C4×D411yescharacteristic
320C4×C61C2×C4×D411yescharacteristic
321Dic61—21no—
322Dic61#52511yescharacteristic
323Dic61#53211yescharacteristic
324C2×#472—611no—
325C4×AGL(1,ℤ/8)—612no—
326C2×C4×D8—611no—
327C4×#131—611no—
328#1978—611no—
329#1952—611no—
330#1931—612no—
331#1907—612no—
332#1816—611no—
333C4×#108—611no—
334C4×#100—612no—
335C2×C61—21no—
336C2×C61—41no—
337C2×C61—82no—
338C2×C61C4×#12111yescharacteristic
339D61—21no—
340D61—41no—
341D61—82no—
342D61C2×#47211yescharacteristic
343D61#306711yescharacteristic
344C2×AGL(1,ℤ/8)—612no—
345C22×D8—611no—
346C2×#131—611no—
347C2×#131—611no—
348C4×#41—611no—
349C2×C4×D4—612no—
350#472—612no—
351#472—611no—
352#472—611no—
353#472—611no—
354#472—612no—
355#472#692311yescharacteristic, direct product factor
356#442—1222no—
357#440—1222no—
358C4×SD16—1222no—
359C4×D8—1221no—
360C4×D8—1221no—
361C4×D8—1221no—
362C4×D8—1221no—
363C4×D8—1222no—
364C2×#108—611no—
365C2×#108—611no—
366#428—612no—
367C2×#107—611no—
368#418—612no—
369#417—612no—
370C2×#102—612no—
371C2×#100—612no—
372C2×#100—612no—
373#405—611no—
374#405—611no—
375#405—1221no—
376#405—1221no—
377C4×M16—611no—
378C4×M16—612no—
379C4×M16—1221no—
380C2×C4×C8—611no—
381C2×C4×C8—1221no—
382#343—612no—
383#343—612no—
384#341—612no—
385C61C4×#47211yescharacteristic, complemented, Sylow
386C2×#41—611no—
387C22×D4—612no—
388C23×C4—1222no—
389AGL(1,ℤ/8)—612no—
390AGL(1,ℤ/8)—612no—
391AGL(1,ℤ/8)—612no—
392AGL(1,ℤ/8)—1222no—
393AGL(1,ℤ/8)—1222no—
394AGL(1,ℤ/8)C2×#692312yescomplemented
395C2×SD16—1222no—
396C2×D8—612no—
397C2×D8—611no—
398C2×D8—1221no—
399C2×D8—1221no—
400C2×D8—1221no—
401C2×D8—1221no—
402C2×D8—1222no—
403C2×D8C2×#692311yescharacteristic, complemented
404#131—611no—
405#131—611no—
406#131—611no—
407#131—1221no—
408#131—1221no—
409#131—1222no—
410#131—1221no—
411#131—1221no—
412#131C2×#692311yescharacteristic, complemented
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—612no—
416C2×M16—612no—
417C2×M16—1221no—
418C2×M16—1221no—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8—1221no—
422C22×C8—1221no—
423C4×Q8—611no—
424C4×Q8—611no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1221no—
432C4×D4—1221no—
433#117—611no—
434#117—1221no—
435C2×C4⋊C4—612no—
436C2×C22⋊C4—1222no—
437C2×C42—611no—
438C2×C42—1221no—
439#108—612no—
440#108—611no—
441#108—1221no—
442#108—1221no—
443#108C2×#692311yescharacteristic, complemented
444#107—1222no—
445#107—1221no—
446#107—1221no—
447#106—1222no—
448#103—1222no—
449#102—1222no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#100—612no—
455#100—612no—
456#100—612no—
457#100—1224no—
458#100C2×#692312yescomplemented
459#97—1221no—
460#97—1222no—
461#97—1221no—
462#97—1221no—
463C4×C8—611no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8—611no—
467C4×C8—1221no—
468C4×C8—1222no—
469C4×C8—1221no—
470C4×C8—1221no—
471C24—1222no—
472#41—611no—
473#41—611no—
474#41—611no—
475#41—1221no—
476#41—1221no—
477#41C22×#692311yescharacteristic
478C2×Q8—611no—
479C2×D4—611no—
480C2×D4—612no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4C22×#692312yes—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—4884no—
496C22×C4—2442no—
497C22×C4—2442no—
498SD16—22no—
499SD16—1222no—
500SD16—1222no—
501SD16—1222no—
502D8—22no—
503D8—21no—
504D8—21no—
505D8—1222no—
506D8—1222no—
507D8—1222no—
508D8—1222no—
509D8—1222no—
510D8—1221no—
511D8—1221no—
512M16—21no—
513M16—612no—
514M16—611no—
515M16—612no—
516M16—1221no—
517M16—1221no—
518M16—1222no—
519M16—1221no—
520M16—1221no—
521M16—1222no—
522M16—1222no—
523M16—1221no—
524M16—1222no—
525M16—1222no—
526M16C22×#692311yescharacteristic
527M16C22×#692312yes—
528C2×C8—21no—
529C2×C8—611no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—611no—
535C2×C8—611no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1222no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1222no—
544C2×C8—1222no—
545C2×C8—1221no—
546C2×C8—1222no—
547C2×C8C22×#692311yescharacteristic
548C4⋊C4—611no—
549C4⋊C4—611no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C22⋊C4—1221no—
555C22⋊C4—1221no—
556C22⋊C4—1221no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C23—22no—
567C23—611no—
568C23—1221no—
569C23—1222no—
570C23—4882no—
571C23—4882no—
572C23—2442no—
573C23—2442no—
574Q8—611no—
575Q8—1221no—
576Q8D4×#692311yescharacteristic
577D4—21no—
578D4—22no—
579D4—22no—
580D4—611no—
581D4—1221no—
582D4—1221no—
583D4—1221no—
584D4—1221no—
585D4—1222no—
586D4—1222no—
587D4—2442no—
588D4—2442no—
589D4D4×#692311yescharacteristic
590C2×C4—21no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—4884no—
608C2×C4—4884no—
609C2×C4—4882no—
610C2×C4—2442no—
611C2×C4—2441no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4C23×#692311yescharacteristic
615C8—21no—
616C8—22no—
617C8—611no—
618C8—611no—
619C8—1221no—
620C8—1221no—
621C8—1222no—
622C8—1221no—
623C8—1222no—
624C8—2442no—
625C8D4×#692311yescharacteristic
626C8D4×#692311yescharacteristic
627C22—21no—
628C22—42no—
629C22—82no—
630C22—611no—
631C22—611no—
632C22—1221no—
633C22—1221no—
634C22—1221no—
635C22—4882no—
636C22—4882no—
637C22—4882no—
638C22—4884no—
639C22—2441no—
640C22—2441no—
641C22—2442no—
642C22#41×#692311yescharacteristic
643C4—21no—
644C4—612no—
645C4—611no—
646C4—611no—
647C4—1221no—
648C4—1221no—
649C4—1221no—
650C4—1221no—
651C4—4882no—
652C4—2441no—
653C4—2441no—
654C4C2×D4×#692311yescharacteristic
655C4C2×D4×#692311yescharacteristic
656C2—21no—
657C2—41no—
658C2—82no—
659C2—611no—
660C2—611no—
661C2—1221no—
662C2—4882no—
663C2—2441no—
664C2#121×#692311yescharacteristic
subgroupG/H
centerC2#121×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#472C4
socleC2×C61C4×#121
prime radical#472×D61C2
cosocleC2×C4×D61C24
hypercenter#472#6923
hypocenterC61C4×#472
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. AGL(1,ℤ/8)×#6923C2semidirect
2. C2×D8×#6923C2semidirect
3. #131×#6923C2semidirect
4. #21461980C2semidirect
5. #21461937C2semidirect
6. #21461905C2semidirect
7. #21461858C2semidirect
8. #21461708C2semidirect
9. #108×#6923C2semidirect
10. #100×#6923C2semidirect
45. #16840033C4semidirect
46. #16840032C4semidirect
47. #16840031C4semidirect
48. #16840030C4semidirect
49. #16840029C4semidirect
52. C61×#472C4semidirect
105. #14519805C2×C4semidirect
107. #14519803C2×C4semidirect
108. #14519802C2×C4semidirect
111. C2×D488C2×C4semidirect
112. C2×#11758199C2×C4semidirect
119. #14519785C2×C4semidirect
120. #14519784C2×C4semidirect
122. #14519783C2×C4semidirect
128. #14519713C2×C4semidirect
129. #14519712C2×C4semidirect
130. #14519694C2×C4semidirect
131. #14519693C2×C4semidirect
132. #14519693C2×C4semidirect
133. AGL(1,ℤ/8)×C61C2×C4semidirect
134. C2×D8×C61C2×C4semidirect
135. #131×C61C2×C4semidirect
136. #108×C61C2×C4semidirect
137. #100×C61C2×C4semidirect
317. #6923#472direct (N × H)
355. #472#6923direct (N × H)
385. C61C4×#472semidirect
394. AGL(1,ℤ/8)C2×#6923semidirect
403. C2×D8C2×#6923semidirect
412. #131C2×#6923semidirect
443. #108C2×#6923semidirect
458. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
AGL(1,ℤ/8)×#69232
C2×D8×#69232
#131×#69232
#214619802
#214619372
#214619052
#214618582
#214617082
#108×#69232
#100×#69232
#472×D612
C4×#47261