Finite Group Databasebeta

← #11869662

Subgroups — #11869662

Order 1152.

1005 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596098C211yesmaximal, characteristic
2C2×#63849C211yesmaximal, characteristic, complemented
3C3×#5044C211yesmaximal, characteristic, complemented
4C3×#5043C211yesmaximal, characteristic, complemented
5#10591561C212yesmaximal, complemented
6#10591560C212yesmaximal, complemented
7#10591152C211yesmaximal, characteristic, complemented
8#10591133C211yesmaximal, characteristic, complemented
9#10591125C211yesmaximal, characteristic, complemented
10#10590574C212yesmaximal, complemented
11#10590451C212yesmaximal, complemented
12#81467—31nomaximal
13#81426—31nomaximal
14C2×#3832C2211yescharacteristic
15C22×C4×C3⋊D3C2211yescharacteristic
16C32×C22⋊Q8C2211yescharacteristic
17#64155C2211yescharacteristic
18#64139C2211yescharacteristic
19#64104C2211yescharacteristic
20#64103—21no—
21#64102—21no—
22#64101C2211yescharacteristic
23#64091—21no—
24#64090—21no—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×#3722—21no—
28C2×Dic32—21no—
29#63849—21no—
30#63849—21no—
31#63849C2212yes—
32#63849C2212yes—
33C2×#3710C2212yes—
34C3×#835C2212yescomplemented
35C3×#835C2212yescomplemented
36C3×#830C2211yescharacteristic
37C3×#830C2211yescharacteristic
38C3×#808C2212yescomplemented
39C3×#807C2212yescomplemented
40#63575C2212yescomplemented
41#63573C2212yescomplemented
42#63564C2212yescomplemented
43#63564C2212yescomplemented
44#63557C2212yes—
45#5627—31no—
46#5627—31no—
47#5627—61no—
48C2×#892—31no—
49C2×#892—31no—
50C3×#484—31no—
51C3×#483—31no—
52#5044D311yescharacteristic, complemented
53#5043D311yescharacteristic, complemented
54#4968—31no—
55#4967—31no—
56#4885—32no—
57#4880—32no—
58#4866—32no—
59#4860—32no—
60#4815—32no—
61#4814—32no—
62#4804—32no—
63#4804—32no—
64#4736—31no—
65#4735—31no—
66D3×C22⋊C8—31no—
67D3×C22⋊C8—31no—
68C23×C3⋊D3D411yescharacteristic
69C2×C32×Q8C2311yescharacteristic
70C22×C32×C4C2311yescharacteristic
71C2×#638—21no—
72C22×C3⋊Dic3D411yescharacteristic
73#3832—24no—
74C2×C4⋊(C3⋊D3)—21no—
75C2×C4⋊(C3⋊D3)—21no—
76C2×C4⋊(C3⋊D3)C2311yescharacteristic
77C2×C4×C3⋊D3—24no—
78C2×C4×C3⋊D3—21no—
79C2×C4×C3⋊D3D411yescharacteristic
80C2×C4×C3⋊D3D411yescharacteristic
81C2×C4×C3⋊D3C2311yescharacteristic
82C2×C4×C3⋊D3C2311yescharacteristic
83C22×C3×Dic3—21no—
84C22×C3×Dic3—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2311yescharacteristic
87C32×C22⋊C4—21no—
88#3752—21no—
89#3752—21no—
90#3752—21no—
91#3752—21no—
92C4⋊(C3⋊Dic3)C2311yescharacteristic
93#3750—21no—
94C4×C3⋊Dic3—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3C2312yes—
98C3×C4⋊Dic3C2312yes—
99C2×C3×C3⋊C8C2312yes—
100C2×C3×C3⋊C8C2312yes—
101#3722—22no—
102#3722—22no—
103Dic32—22no—
104Dic32—22no—
105#3710—24no—
106#3091—91noSylow
107C2×#239—31no—
108C2×#239—31no—
109C2×#239—61no—
110C22×C4×D3—31no—
111C22×C4×D3—31no—
112C22×C4×D3—61no—
113C3×C22⋊Q8—21no—
114C3×C22⋊Q8C2×D311yescharacteristic
115C3×C22⋊Q8C2×D311yescharacteristic
116C2×C3×C4⋊C4—31no—
117C2×C3×C4⋊C4—31no—
118#946—31no—
119#946—31no—
120#946—61no—
121#930—31no—
122#930—31no—
123#930—61no—
124C2×(C2×D3)⋊C4—61no—
125C2×(C2×D3)⋊C4—61no—
126C2×C4⋊Dic3C2×D311yescharacteristic
127C2×C4⋊Dic3C2×D311yescharacteristic
128C2×C4×Dic3—61no—
129C2×C4×Dic3—61no—
130C2×D3×C8—32no—
131C2×D3×C8—32no—
132#895—31no—
133#895—31no—
134#895—61no—
135#894—61no—
136#894—61no—
137#894—62no—
138#893—61no—
139#893—61no—
140#893—62no—
141#892—31no—
142#892—31no—
143#892—32no—
144#892—32no—
145#892—32no—
146#892—32no—
147#892—61no—
148#892—61no—
149#892—61no—
150#892—61no—
151#892—61no—
152#882—61no—
153#882—61no—
154#882—62no—
155#67⋊C4—61no—
156#67⋊C4—61no—
157#67⋊C4—62no—
158C3×#107—32no—
159C3×#106—32no—
160C3×#103—32no—
161C3×#103—32no—
162C3×C22⋊C8—31no—
163C3×C22⋊C8—31no—
164#835C2×D312yescomplemented
165#835C2×D312yescomplemented
166#830C2×D311yescharacteristic
167#830C2×D311yescharacteristic
168#821—32no—
169#821—32no—
170(C2×D3)⋊C8—32no—
171(C2×D3)⋊C8—32no—
172#818—32no—
173#817—32no—
174#809—32no—
175#809—32no—
176#808—32no—
177#808C2×D312yescomplemented
178#807—32no—
179#807C2×D312yescomplemented
180C23×C32C2×D411yescharacteristic
181C22×C3⋊D3—21no—
182C22×C3⋊D3—22no—
183C22×C3⋊D3—22no—
184C22×C3⋊D3—41no—
185C22×C3⋊D3C2×D411yescharacteristic
186C22×C3⋊D3C2×D411yescharacteristic
187C32×Q8—22no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—22no—
191C2×C32×C4C2×D411yescharacteristic
192C2×C32×C4C2411yescharacteristic
193#638—42no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—22no—
196C2×C3⋊Dic3C2×D411yescharacteristic
197C2×C3⋊Dic3C2×D411yescharacteristic
198C4⋊(C3⋊D3)—22no—
199C4⋊(C3⋊D3)—42no—
200C4⋊(C3⋊D3)—42no—
201C4×C3⋊D3—22no—
202C4×C3⋊D3—22no—
203C4×C3⋊D3—22no—
204C4×C3⋊D3—42no—
205C4×C3⋊D3#4114yes—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—22no—
208C2×C3×Dic3—42no—
209C2×C3×Dic3—42no—
210C3×C3⋊C8—22no—
211C3×C3⋊C8—22no—
212#525—91no—
213C2×#117—91no—
214#485—92no—
215#484—31no—
216#483—31no—
217#482—92no—
218#481—91no—
219#480—91no—
220#427—92no—
221#416—92no—
222C2×C22⋊C8—91no—
223C23×D3—31no—
224C23×D3—31no—
225C23×D3—61no—
226C2×C3×Q8—21no—
227C2×C3×Q8C22×D311yescharacteristic
228C2×C3×Q8C22×D311yescharacteristic
229C22×C3×C4—21no—
230C22×C3×C4—61no—
231C22×C3×C4—61no—
232C22×C3×C4C22×D311yescharacteristic
233C22×C3×C4C22×D311yescharacteristic
234C2×#67—62no—
235C2×#67—61no—
236C2×#67—61no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—31no—
240C22×Dic3—31no—
241C22×Dic3—61no—
242#239—64no—
243#239—64no—
244#239—124no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—61no—
248C2×D12—61no—
249C2×D12—61no—
250C2×D12—61no—
251C2×D12—61no—
252C2×D12—62no—
253C2×D12—62no—
254C2×C4×D3—31no—
255C2×C4×D3—31no—
256C2×C4×D3—31no—
257C2×C4×D3—31no—
258C2×C4×D3—31no—
259C2×C4×D3—31no—
260C2×C4×D3—31no—
261C2×C4×D3—31no—
262C2×C4×D3—61no—
263C2×C4×D3—61no—
264C2×C4×D3—61no—
265C2×C4×D3—61no—
266C2×C4×D3—64no—
267C2×C4×D3—64no—
268C2×C4×D3—61no—
269C2×C4×D3—61no—
270C2×C4×D3—62no—
271C2×C4×D3—124no—
272C2×C3×C8—32no—
273C2×C3×C8—32no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—21no—
277C3×C4⋊C4—21no—
278C3×C4⋊C4—32no—
279C3×C4⋊C4—32no—
280C3×C4⋊C4—61no—
281C3×C4⋊C4—61no—
282C3×C4⋊C4C22×D311yescharacteristic
283C3×C4⋊C4C22×D311yescharacteristic
284C3×C22⋊C4—22no—
285C3×C22⋊C4—21no—
286C3×C22⋊C4—21no—
287(C2×D3)⋊C4—62no—
288(C2×D3)⋊C4—62no—
289(C2×D3)⋊C4—62no—
290(C2×D3)⋊C4—62no—
291(C2×D3)⋊C4—61no—
292(C2×D3)⋊C4—61no—
293(C2×D3)⋊C4—61no—
294(C2×D3)⋊C4—61no—
295(C2×D3)⋊C4—61no—
296(C2×D3)⋊C4—61no—
297(C2×D3)⋊C4—62no—
298(C2×D3)⋊C4—62no—
299(C2×D3)⋊C4—62no—
300(C2×D3)⋊C4—62no—
301(C2×D3)⋊C4—61no—
302(C2×D3)⋊C4—61no—
303C4⋊Dic3—21no—
304C4⋊Dic3—21no—
305C4⋊Dic3—31no—
306C4⋊Dic3—31no—
307C4⋊Dic3—61no—
308C4⋊Dic3C22×D312yes—
309C4⋊Dic3C22×D312yes—
310Dic3⋊C4—61no—
311Dic3⋊C4—61no—
312Dic3⋊C4—62no—
313C4×Dic3—62no—
314C4×Dic3—62no—
315C4×Dic3—62no—
316C4×Dic3—62no—
317C4×Dic3—61no—
318C4×Dic3—61no—
319C4×Dic3—62no—
320C2×C3⋊C8C22×D312yes—
321C2×C3⋊C8C22×D312yes—
322D3×C8—64no—
323D3×C8—64no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32#13512yes—
327C22×C32C22×D411yescharacteristic
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—21no—
331C2×C3⋊D3—21no—
332C2×C3⋊D3—21no—
333C2×C3⋊D3—44no—
334C2×C3⋊D3—41no—
335C2×C3⋊D3—41no—
336C2×C3⋊D3—41no—
337C2×C3⋊D3#12312yes—
338C32×C4—21no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4C2×#4112yes—
342C3⋊Dic3—21no—
343C3⋊Dic3—41no—
344C3⋊Dic3#12312yes—
345C3×Dic3—42no—
346C3×Dic3—42no—
347C2×#41—91no—
348C23×C4—91no—
349C22×C8—92no—
350#123—181no—
351C22⋊Q8—91no—
352C22⋊Q8D3211yescharacteristic
353#121—91no—
354#121—91no—
355#120—181no—
356C4×D4—181no—
357C4×D4—181no—
358#117—92no—
359#117—92no—
360#117—91no—
361#117—181no—
362#117—181no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—31no—
365C2×C22⋊C4—181no—
366C2×C42—181no—
367#107—32no—
368#107—92no—
369#106—32no—
370#106—92no—
371#103—32no—
372#103—32no—
373#102—92no—
374#102—92no—
375C22⋊C8—31no—
376C22⋊C8—31no—
377C22⋊C8—92no—
378C23×C3—21no—
379C23×C3D3×D411yescharacteristic
380C23×C3D3×D411yescharacteristic
381C22×D3—31no—
382C22×D3—31no—
383C22×D3—31no—
384C22×D3—31no—
385C22×D3—61no—
386C22×D3—61no—
387C22×D3—62no—
388C22×D3—62no—
389C22×D3—62no—
390C22×D3—62no—
391C22×D3—62no—
392C22×D3—61no—
393C22×D3—61no—
394C22×D3—122no—
395C22×D3—122no—
396C22×D3—121no—
397C22×D3—121no—
398C22×D3—122no—
399C3×Q8—22no—
400C3×Q8—22no—
401C3×Q8—42no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—42no—
413C2×C3×C4—62no—
414C2×C3×C4—62no—
415C2×C3×C4—122no—
416C2×C3×C4—122no—
417C2×C3×C4D3×D411yescharacteristic
418C2×C3×C4D3×D411yescharacteristic
419C2×C3×C4C23×D311yescharacteristic
420C2×C3×C4C23×D311yescharacteristic
421#67—124no—
422#67—122no—
423#67—122no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—31no—
427C2×Dic3—31no—
428C2×Dic3—31no—
429C2×Dic3—31no—
430C2×Dic3—42no—
431C2×Dic3—42no—
432C2×Dic3—61no—
433C2×Dic3—61no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—61no—
438C2×Dic3—61no—
439C2×Dic3—122no—
440D12—62no—
441D12—62no—
442D12—122no—
443D12—122no—
444D12—122no—
445D12—124no—
446D12—122no—
447D12—122no—
448D12—124no—
449C4×D3—34no—
450C4×D3—34no—
451C4×D3—64no—
452C4×D3—62no—
453C4×D3—62no—
454C4×D3—62no—
455C4×D3—62no—
456C4×D3—62no—
457C4×D3—62no—
458C4×D3—122no—
459C4×D3—122no—
460C4×D3—122no—
461C4×D3—122no—
462C4×D3—122no—
463C4×D3—124no—
464C3×C8—62no—
465C3×C8—62no—
466C3⋊C8—22no—
467C3⋊C8—22no—
468C2×C32—22no—
469C2×C32C2×#12311yescharacteristic
470C2×C32C2×#13511yescharacteristic
471C2×C32#57511yescharacteristic
472C3⋊D3—22no—
473C3⋊D3—41no—
474C3⋊D3—81no—
475C24—91no—
476#41—184no—
477C2×Q8C2×D3211yescharacteristic
478C2×D4—91no—
479C2×D4—181no—
480C2×D4—181no—
481C2×D4—181no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—91no—
485C22×C4—91no—
486C22×C4—91no—
487C22×C4—91no—
488C22×C4—91no—
489C22×C4—184no—
490C22×C4—181no—
491C22×C4C2×D3211yescharacteristic
492C2×C8—32no—
493C2×C8—32no—
494C2×C8—184no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—32no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—91no—
501C4⋊C4—181no—
502C4⋊C4C2×D3211yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—181no—
505C22⋊C4—181no—
506C22⋊C4—181no—
507C22⋊C4—182no—
508C22⋊C4—182no—
509C22⋊C4—181no—
510C42—181no—
511C42—182no—
512C42—182no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—22no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3#91612yes—
522C22×C3#91912yes—
523C22×C3C2×D3×D411yescharacteristic
524C22×C3C2×D3×D411yescharacteristic
525C2×D3—32no—
526C2×D3—32no—
527C2×D3—62no—
528C2×D3—61no—
529C2×D3—61no—
530C2×D3—61no—
531C2×D3—61no—
532C2×D3—62no—
533C2×D3—61no—
534C2×D3—61no—
535C2×D3—61no—
536C2×D3—61no—
537C2×D3—61no—
538C2×D3—61no—
539C2×D3—121no—
540C2×D3—121no—
541C2×D3—121no—
542C2×D3—121no—
543C2×D3—124no—
544C2×D3—124no—
545C2×D3—128no—
546C2×D3—122no—
547C2×D3—122no—
548C2×D3—122no—
549C2×D3—121no—
550C2×D3—121no—
551C2×D3—121no—
552C2×D3—121no—
553C2×D3—121no—
554C2×D3—121no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—22no—
558C3×C4—41no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—41no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4—122no—
566C3×C4—122no—
567C3×C4D3×#4112yes—
568C3×C4D3×#4112yes—
569Dic3—32no—
570Dic3—32no—
571Dic3—42no—
572Dic3—42no—
573Dic3—62no—
574Dic3—61no—
575Dic3—61no—
576Dic3—121no—
577Dic3—122no—
578Dic3—121no—
579Dic3—121no—
580C32#309111yescharacteristic, complemented, Sylow
581C23—91no—
582C23—91no—
583C23—182no—
584C23—182no—
585C23—181no—
586C23—361no—
587C23#381111yescharacteristic
588Q8—22no—
589D4—182no—
590D4—362no—
591D4—362no—
592D4—362no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—91no—
599C2×C4—94no—
600C2×C4—91no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—182no—
604C2×C4—182no—
605C2×C4—182no—
606C2×C4—182no—
607C2×C4—181no—
608C2×C4—362no—
609C2×C4#381111yescharacteristic
610C2×C4C22×D3211yescharacteristic
611C8—62no—
612C8—62no—
613C2×C3—22no—
614C2×C3—22no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—42no—
619C2×C3D3×#12311yescharacteristic
620C2×C3D3×#12311yescharacteristic
621C2×C3C2×#91611yescharacteristic
622C2×C3C2×#91911yescharacteristic
623C2×C3#577311yescharacteristic
624C2×C3#577711yescharacteristic
625D3—62no—
626D3—62no—
627D3—122no—
628D3—122no—
629D3—121no—
630D3—121no—
631D3—242no—
632D3—241no—
633D3—241no—
634C22—21no—
635C22—21no—
636C22—92no—
637C22—181no—
638C22—181no—
639C22—181no—
640C22—181no—
641C22—181no—
642C22—364no—
643C22—361no—
644C22—361no—
645C22—361no—
646C22#6395012yes—
647C22C2×#381111yescharacteristic
648C4—21no—
649C4—41no—
650C4—41no—
651C4—92no—
652C4—122no—
653C4—122no—
654C4—181no—
655C4—361no—
656C4#6431212yes—
657C3—21no—
658C3#8142611yescharacteristic, complemented
659C3#8146711yescharacteristic, complemented
660C2—22no—
661C2—182no—
662C2—361no—
663C2—721no—
664C2#1059516911yescharacteristic
665C2C2×#6395011yescharacteristic
666C2#1059547111yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C22⋊Q8C22
socleC22×C32C22×D4
prime radical#10596098C2
hypercenterC22⋊Q8D32
hypocenterC32#3091
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#63849C2semidirect
3. C3×#5044C2semidirect
4. C3×#5043C2semidirect
5. #10591561C2semidirect
6. #10591560C2semidirect
7. #10591152C2semidirect
8. #10591133C2semidirect
9. #10591125C2semidirect
10. #10590574C2semidirect
11. #10590451C2semidirect
34. C3×#835C22semidirect
35. C3×#835C22semidirect
38. C3×#808C22semidirect
39. C3×#807C22semidirect
40. #63575C22semidirect
41. #63573C22semidirect
42. #63564C22semidirect
43. #63564C22semidirect
52. #5044D3semidirect
53. #5043D3semidirect
164. #835C2×D3semidirect
165. #835C2×D3semidirect
177. #808C2×D3semidirect
179. #807C2×D3semidirect
580. C32#3091semidirect
658. C3#81426semidirect
659. C3#81467semidirect

Maximal subgroups

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

subgroupindex
#105960982
C2×#638492
C3×#50442
C3×#50432
#105915612
#105915602
#105911522
#105911332
#105911252
#105905742
#105904512
#814673
#814263