Finite Group Databasebeta

← #10772000

Subgroups — #10772000

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86163C211yesmaximal, characteristic
2C3×#3252C211yesmaximal, characteristic, complemented
3#83349C211yesmaximal, characteristic, complemented
4#83347C211yesmaximal, characteristic, complemented
5#71751C211yesmaximal, characteristic, complemented
6#71163C211yesmaximal, characteristic, complemented
7#71159C211yesmaximal, characteristic, complemented
8#71147C211yesmaximal, characteristic, complemented
9#71143C211yesmaximal, characteristic, complemented
10#71065C211yesmaximal, characteristic, complemented
11#71060C211yesmaximal, characteristic, complemented
12#71049C211yesmaximal, characteristic, complemented
13#71044C211yesmaximal, characteristic, complemented
14#70503C211yesmaximal, characteristic, complemented
15#70497C211yesmaximal, characteristic, complemented
16#18595—31nomaximal, Sylow
17C3×#533C2211yescharacteristic, complemented
18C3×#529C2211yescharacteristic
19#5705—22no—
20#5691—21no—
21D3×#124—21no—
22#5657—22no—
23#5633—21no—
24#5612—21no—
25#5585C2211yescharacteristic
26Q8×D12C2211yescharacteristic
27#5563—21no—
28#5550—21no—
29D4×D12C2211yescharacteristic
30#5542C2211yescharacteristic
31#5534—21no—
32#5521—21no—
33#5477C2211yescharacteristic
34C4×#235C2211yescharacteristic
35C3×#479—21no—
36C3×#477C2211yescharacteristic, complemented
37C3×#476C2211yescharacteristic
38C3×#470—21no—
39C3×#468—21no—
40C3×#464—21no—
41C3×#462C2211yescharacteristic, complemented
42C3×#461C2211yescharacteristic
43C3×#423C2211yescharacteristic, complemented
44#5094C2211yescharacteristic, complemented
45#5091C2211yescharacteristic, complemented
46#5088C2211yescharacteristic
47#5087C2211yescharacteristic
48#5085C2211yescharacteristic
49#5084C2211yescharacteristic
50#5079C2211yescharacteristic, complemented
51#5078C2211yescharacteristic, complemented
52#5066—21no—
53#5063—21no—
54#5049—22no—
55#5026—22no—
56#5002—21no—
57#4998—21no—
58C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
59C4×#208C2211yescharacteristic, complemented
60#4535C2211yescharacteristic, complemented
61#4535C2211yescharacteristic, complemented
62D4⋊(C3⋊C8)C2211yescharacteristic
63D4⋊(C3⋊C8)C2211yescharacteristic
64#4485C2211yescharacteristic
65#4485C2211yescharacteristic
66#4484C2211yescharacteristic, complemented
67#4484C2211yescharacteristic, complemented
68#4483C2211yescharacteristic
69#4483C2211yescharacteristic
70#4482C2211yescharacteristic, complemented
71#4482C2211yescharacteristic, complemented
72#3514—31no—
73#3252D311yescharacteristic, complemented
74#3182—31no—
75#3076—31no—
76#1650—31no—
77#1649—31no—
78#1640—31no—
79#1639—31no—
80#1638—31no—
81#1637—31no—
82#1630—31no—
83#1629—31no—
84#1547—31no—
85#1489—31no—
86#1483—31no—
87C2×D3×Q8—21no—
88C2×D3×D4—21no—
89C2×#235—21no—
90C2×#235D411yescharacteristic
91C22×D12—21no—
92C2×C3×Q16—21no—
93C2×C3×SD16—21no—
94C2×C3×M16—21no—
95C3×C4⋊Q8C2311yescharacteristic
96C3×C4⋊Q8C2311yescharacteristic
97C3×#126—21no—
98C3×#125—21no—
99C3×#125D411yescharacteristic
100C3×#124D411yescharacteristic
101C3×#123—21no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8D411yescharacteristic
105C3×#121D411yescharacteristic
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8C2311yescharacteristic
109C3×C4×D4—21no—
110C3×C4×D4C2311yescharacteristic
111C3×#117—21no—
112C3×#117—21no—
113C2×C3×C42C2311yescharacteristic
114#947—21no—
115C2×#215—22no—
116#938—22no—
117#937—22no—
118#936—21no—
119#930—21no—
120#930—21no—
121#930D412yes—
122C4×#67—21no—
123C4×#67—21no—
124#926—21no—
125#926D411yescharacteristic
126#924—21no—
127C2×#208—21no—
128C22×C3⋊C8—21no—
129#898—22no—
130#897—21no—
131#897—22no—
132#896—22no—
133#895—21no—
134#894—22no—
135#894—22no—
136#893—22no—
137#893—22no—
138#893—21no—
139#892—22no—
140#887—21no—
141#885—22no—
142#885—22no—
143#885—22no—
144#884—22no—
145#883—22no—
146#882—22no—
147D3×C22⋊C4—22no—
148D3×C22⋊C4—22no—
149D3×C22⋊C4—22no—
150#879—22no—
151#875—21no—
152#875—21no—
153#874C2311yescharacteristic
154C4×D12—21no—
155C4×D12—21no—
156C4×D12—21no—
157C4×D12D412yes—
158C4×D12D411yescharacteristic
159C4×D12C2311yescharacteristic
160#872—21no—
161C42×D3—21no—
162#869—21no—
163#869C2311yescharacteristic
164C4×Dic6D411yescharacteristic
165C4×Dic6C2311yescharacteristic
166C3×#107—21no—
167C3×#106—21no—
168C3×C4⋊C8—21no—
169C3×C4⋊C8C2311yescharacteristic
170C3×C4⋊C8C2311yescharacteristic
171C3×#103—21no—
172C3×#103—21no—
173C3×#103—21no—
174C3×#102—21no—
175#835—22no—
176#810—21no—
177#810—21no—
178#809—21no—
179#809—21no—
180#809—22no—
181#808—21no—
182#808—21no—
183#807—21no—
184#807—21no—
185#807—22no—
186C4⋊(C3⋊C8)—21no—
187C4⋊(C3⋊C8)C2311yescharacteristic
188C4⋊(C3⋊C8)C2311yescharacteristic
189#803—21no—
190C4×C3⋊C8C2311yescharacteristic
191C4×C3⋊C8C2311yescharacteristic
192#559—31no—
193#556—31no—
194#555—61no—
195#551—62no—
196D4×Q8—31no—
197#548—61no—
198#546—61no—
199#546—61no—
200D42—31no—
201#542—61no—
202#538—62no—
203#533C2×D311yescharacteristic, complemented
204#532—31no—
205#529—61no—
206#529C2×D311yescharacteristic
207C2×#124—61no—
208#518—61no—
209C4×#41—31no—
210#500—31no—
211#492—31no—
212#487—31no—
213#486—31no—
214#479—21no—
215#479—61no—
216#478—61no—
217#477—31no—
218#477C2×D311yescharacteristic, complemented
219#476—31no—
220#476C2×D311yescharacteristic
221#475—31no—
222#474—31no—
223#470—21no—
224#468—21no—
225#468—62no—
226#464—21no—
227#462C2×D311yescharacteristic, complemented
228#461C2×D311yescharacteristic
229#440—62no—
230#427—61no—
231#423C2×D311yescharacteristic, complemented
232C2×C4⋊C8—31no—
233#416—61no—
234C4×M16—31no—
235#332—31no—
236#332—31no—
237#331—31no—
238#331—31no—
239Q8⋊C8—31no—
240Q8⋊C8—31no—
241Q8⋊C8—31no—
242Q8⋊C8—31no—
243D4⋊C8—31no—
244D4⋊C8—31no—
245D4⋊C8—31no—
246D4⋊C8—31no—
247C23×D3—22no—
248C2×C3×Q8—21no—
249C2×C3×Q8—21no—
250C2×C3×Q8C2×D411yescharacteristic
251C2×C3×D4—21no—
252C2×C3×D4C2×D411yescharacteristic
253C22×C3×C4—21no—
254C22×C3×C4C2×D411yescharacteristic
255C22×C3×C4C2×D411yescharacteristic
256C22×C3×C4C2×D411yescharacteristic
257C2×#67—21no—
258C2×#67—22no—
259C2×#67—22no—
260D3×Q8—81no—
261D3×Q8—81no—
262D3×D4—81no—
263D3×D4—81no—
264#235—22no—
265#235—41no—
266#235—81no—
267#235—81no—
268C2×D12—22no—
269C2×D12—21no—
270C2×D12—21no—
271C2×D12—42no—
272C2×D12—42no—
273C2×D12C2×D411yescharacteristic
274C2×D12C2×D412yes—
275C2×C4×D3—21no—
276C2×C4×D3—21no—
277C2×C4×D3—22no—
278C2×C4×D3—22no—
279C2×C4×D3—21no—
280C2×C4×D3—22no—
281C2×Dic6—21no—
282C2×Dic6—22no—
283C2×Dic6C2×D411yescharacteristic
284C3×Q16—81no—
285C3×SD16—81no—
286C3×M16—81no—
287C2×C3×C8—21no—
288C2×C3×C8—21no—
289C3×C4⋊C4—21no—
290C3×C4⋊C4—21no—
291C3×C4⋊C4—21no—
292C3×C4⋊C4—21no—
293C3×C4⋊C4—21no—
294C3×C4⋊C4—21no—
295C3×C4⋊C4—21no—
296C3×C4⋊C4—21no—
297C3×C4⋊C4—41no—
298C3×C4⋊C4—41no—
299C3×C4⋊C4C2×D411yescharacteristic
300C3×C4⋊C4C2×D411yescharacteristic
301C3×C22⋊C4—21no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—21no—
304C3×C22⋊C4—21no—
305C3×C22⋊C4—21no—
306C3×C22⋊C4—41no—
307C3×C42—21no—
308C3×C42—41no—
309C3×C42C2×D411yescharacteristic
310C3×C42C2×D411yescharacteristic
311C3×C42C2×D411yescharacteristic
312C3×C42C2411yescharacteristic
313#217—22no—
314#217—22no—
315#217—21no—
316#215—44no—
317(C2×D3)⋊C4—22no—
318(C2×D3)⋊C4—22no—
319(C2×D3)⋊C4—22no—
320(C2×D3)⋊C4—22no—
321(C2×D3)⋊C4—22no—
322(C2×D3)⋊C4—21no—
323(C2×D3)⋊C4—22no—
324(C2×D3)⋊C4—22no—
325(C2×D3)⋊C4—21no—
326C4⋊Dic3—22no—
327C4⋊Dic3—21no—
328C4⋊Dic3C2×D411yescharacteristic
329C4⋊Dic3C2×D411yescharacteristic
330C4⋊Dic3C2×D412yes—
331Dic3⋊C4—22no—
332Dic3⋊C4—21no—
333Dic3⋊C4—21no—
334C4×Dic3—22no—
335C4×Dic3—21no—
336C4×Dic3—21no—
337C4×Dic3—21no—
338#208—42no—
339C2×C3⋊C8—21no—
340C2×C3⋊C8—21no—
341C2×C3⋊C8—21no—
342C2×C3⋊C8—21no—
343C2×C3⋊C8—42no—
344C2×#41—31no—
345C2×#41—61no—
346C22×Q8—61no—
347C22×D4—61no—
348C22×D4—61no—
349C2×Q16—21no—
350C2×SD16—21no—
351C2×SD16—62no—
352C2×M16—21no—
353C2×M16—61no—
354C22×C8—61no—
355C4⋊Q8—31no—
356C4⋊Q8—61no—
357C4⋊Q8C22×D311yescharacteristic
358C4⋊Q8C22×D311yescharacteristic
359#127—31no—
360#126—21no—
361#126—62no—
362#126—62no—
363#125—21no—
364#125#6711yescharacteristic
365#124—61no—
366#124—62no—
367#124—62no—
368#124—62no—
369#124—61no—
370#124—61no—
371#124—61no—
372#124#6711yescharacteristic
373#123—21no—
374#123—62no—
375#123—62no—
376#123—62no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—62no—
380C22⋊Q8—62no—
381C22⋊Q8—61no—
382C22⋊Q8—61no—
383C22⋊Q8#6711yescharacteristic
384#121—32no—
385#121—62no—
386#121—61no—
387#121—62no—
388#121—61no—
389#121—61no—
390#121#6711yescharacteristic
391#120—62no—
392#120—62no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—31no—
396C4×Q8—31no—
397C4×Q8—61no—
398C4×Q8C22×D311yescharacteristic
399C4×D4—21no—
400C4×D4—31no—
401C4×D4—32no—
402C4×D4—31no—
403C4×D4—61no—
404C4×D4—61no—
405C4×D4—62no—
406C4×D4—62no—
407C4×D4—61no—
408C4×D4—61no—
409C4×D4—61no—
410C4×D4—61no—
411C4×D4C22×D311yescharacteristic
412#117—21no—
413#117—21no—
414#117—31no—
415#117—62no—
416#117—61no—
417#117—61no—
418C2×C22⋊C4—62no—
419C2×C22⋊C4—62no—
420C2×C22⋊C4—62no—
421C2×C42—61no—
422C2×C42C22×D311yescharacteristic
423#107—21no—
424#107—61no—
425#107—61no—
426#107—62no—
427#106—21no—
428#106—61no—
429#106—61no—
430C4⋊C8—21no—
431C4⋊C8—31no—
432C4⋊C8—31no—
433C4⋊C8—61no—
434C4⋊C8C22×D311yescharacteristic
435C4⋊C8C22×D311yescharacteristic
436#103—21no—
437#103—21no—
438#103—21no—
439#103—61no—
440#103—61no—
441#103—62no—
442#102—21no—
443#102—61no—
444#102—61no—
445#102—62no—
446#97—61no—
447C4×C8—31no—
448C4×C8—31no—
449C23×C3—21no—
450C23×C3#12011yescharacteristic
451C22×D3—21no—
452C22×D3—22no—
453C22×D3—44no—
454C22×D3—82no—
455C22×D3—82no—
456C3×Q8—21no—
457C3×Q8—21no—
458C3×Q8—42no—
459C3×Q8—41no—
460C3×Q8—41no—
461C3×D4—21no—
462C3×D4—21no—
463C3×D4—81no—
464C2×C3×C4—21no—
465C2×C3×C4—21no—
466C2×C3×C4—21no—
467C2×C3×C4—21no—
468C2×C3×C4—21no—
469C2×C3×C4—21no—
470C2×C3×C4—21no—
471C2×C3×C4—41no—
472C2×C3×C4—41no—
473C2×C3×C4—41no—
474C2×C3×C4—41no—
475C2×C3×C4—41no—
476C2×C3×C4#12011yescharacteristic
477C2×C3×C4#12011yescharacteristic
478C2×C3×C4#12011yescharacteristic
479C2×C3×C4C22×D411yescharacteristic
480C2×C3×C4C22×D411yescharacteristic
481C2×C3×C4C22×D411yescharacteristic
482#67—42no—
483#67—82no—
484#67—82no—
485C2×Dic3—21no—
486C2×Dic3—21no—
487C2×Dic3—21no—
488C2×Dic3—22no—
489D12—21no—
490D12—42no—
491D12—44no—
492D12—41no—
493D12—41no—
494D12—41no—
495D12—41no—
496D12#13712yes—
497C4×D3—42no—
498C4×D3—41no—
499C4×D3—41no—
500C4×D3—81no—
501C4×D3—82no—
502C4×D3—82no—
503C4×D3—82no—
504Dic6—21no—
505Dic6—21no—
506Dic6—41no—
507Dic6—41no—
508Dic6—82no—
509C3×C8—41no—
510C3×C8—41no—
511C3⋊C8—22no—
512C3⋊C8—41no—
513C3⋊C8—41no—
514C3⋊C8—41no—
515C24—62no—
516#41—62no—
517#41—121no—
518#41—241no—
519#41—241no—
520C2×Q8—21no—
521C2×Q8—21no—
522C2×Q8—31no—
523C2×Q8—62no—
524C2×Q8—61no—
525C2×Q8—241no—
526C2×Q8—241no—
527C2×Q8C2×#6711yescharacteristic
528C2×D4—21no—
529C2×D4—31no—
530C2×D4—32no—
531C2×D4—62no—
532C2×D4—61no—
533C2×D4—62no—
534C2×D4—62no—
535C2×D4—61no—
536C2×D4—61no—
537C2×D4—122no—
538C2×D4—122no—
539C2×D4—241no—
540C2×D4—241no—
541C2×D4C2×#6711yescharacteristic
542C22×C4—21no—
543C22×C4—62no—
544C22×C4—62no—
545C22×C4—61no—
546C22×C4—61no—
547C22×C4—61no—
548C22×C4—62no—
549C22×C4D3×D411yescharacteristic
550C22×C4D3×D411yescharacteristic
551C22×C4C2×#6711yescharacteristic
552Q16—81no—
553SD16—81no—
554SD16—124no—
555M16—81no—
556M16—122no—
557C2×C8—21no—
558C2×C8—21no—
559C2×C8—61no—
560C2×C8—61no—
561C2×C8—61no—
562C2×C8—61no—
563C2×C8—122no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—32no—
573C4⋊C4—31no—
574C4⋊C4—31no—
575C4⋊C4—41no—
576C4⋊C4—41no—
577C4⋊C4—62no—
578C4⋊C4—62no—
579C4⋊C4—61no—
580C4⋊C4—61no—
581C4⋊C4—61no—
582C4⋊C4C2×#6711yescharacteristic
583C4⋊C4C2×#6711yescharacteristic
584C22⋊C4—21no—
585C22⋊C4—21no—
586C22⋊C4—21no—
587C22⋊C4—21no—
588C22⋊C4—21no—
589C22⋊C4—41no—
590C22⋊C4—62no—
591C22⋊C4—62no—
592C22⋊C4—62no—
593C22⋊C4—62no—
594C22⋊C4—62no—
595C22⋊C4—62no—
596C22⋊C4—62no—
597C22⋊C4—61no—
598C22⋊C4—61no—
599C22⋊C4—62no—
600C22⋊C4—62no—
601C22⋊C4—61no—
602C42—21no—
603C42—41no—
604C42—62no—
605C42—61no—
606C42—61no—
607C42—61no—
608C42D3×D411yescharacteristic
609C42D3×D411yescharacteristic
610C42C2×#6711yescharacteristic
611C42C23×D311yescharacteristic
612C22×C3—21no—
613C22×C3—21no—
614C22×C3—21no—
615C22×C3—41no—
616C22×C3—41no—
617C22×C3—41no—
618C22×C3C2×#12011yescharacteristic
619C2×D3—22no—
620C2×D3—41no—
621C2×D3—41no—
622C2×D3—42no—
623C2×D3—42no—
624C2×D3—42no—
625C2×D3—164no—
626C3×C4—21no—
627C3×C4—21no—
628C3×C4—21no—
629C3×C4—21no—
630C3×C4—41no—
631C3×C4—41no—
632C3×C4—41no—
633C3×C4—81no—
634C3×C4—81no—
635C3×C4C2×#13711yescharacteristic
636C3×C4#57511yescharacteristic
637C3×C4#57611yescharacteristic
638C3×C4#57711yescharacteristic
639Dic3—41no—
640Dic3—42no—
641Dic3—41no—
642Dic3—41no—
643C23—21no—
644C23—61no—
645C23—62no—
646C23—124no—
647C23—242no—
648C23—242no—
649C23#93711yescharacteristic
650Q8—21no—
651Q8—21no—
652Q8—42no—
653Q8—41no—
654Q8—41no—
655Q8—61no—
656Q8—61no—
657Q8—121no—
658Q8—121no—
659Q8—242no—
660D4—21no—
661D4—21no—
662D4—32no—
663D4—61no—
664D4—81no—
665D4—122no—
666D4—124no—
667D4—122no—
668D4—121no—
669D4—121no—
670D4—121no—
671D4—121no—
672D4—242no—
673D4—242no—
674C2×C4—21no—
675C2×C4—21no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—21no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—61no—
687C2×C4—61no—
688C2×C4—61no—
689C2×C4—62no—
690C2×C4—122no—
691C2×C4—121no—
692C2×C4—121no—
693C2×C4—242no—
694C2×C4—241no—
695C2×C4—242no—
696C2×C4—242no—
697C2×C4#93711yescharacteristic
698C2×C4#93711yescharacteristic
699C2×C4#93711yescharacteristic
700C2×C4C2×D3×D411yescharacteristic
701C2×C4C2×D3×D411yescharacteristic
702C2×C4C22×#6711yescharacteristic
703C8—41no—
704C8—41no—
705C8—62no—
706C8—121no—
707C8—121no—
708C8—121no—
709C2×C3—41no—
710C2×C3—81no—
711C2×C3#300711yescharacteristic
712C2×C3#301211yescharacteristic
713C2×C3#301811yescharacteristic
714D3—42no—
715D3—82no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—62no—
723C22—121no—
724C22—121no—
725C22—122no—
726C22—122no—
727C22—122no—
728C22—484no—
729C22C2×#93711yescharacteristic
730C4—21no—
731C4—21no—
732C4—21no—
733C4—21no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—81no—
738C4—81no—
739C4—121no—
740C4—121no—
741C4—121no—
742C4—122no—
743C4D3×#13711yescharacteristic
744C4#577711yescharacteristic
745C4#583511yescharacteristic
746C4#583711yescharacteristic
747C3#1859511yescharacteristic, complemented, Sylow
748C2—41no—
749C2—81no—
750C2—122no—
751C2—242no—
752C2#8342411yescharacteristic
753C2#8398111yescharacteristic
754C2#8400411yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3252C2
socleC22×C3C2×#120
prime radical#86163C2
hypercenter#3252D3
hypocenterC3#18595
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3252

Complements

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

NH ≅ G/Ntype
2. C3×#3252C2semidirect
3. #83349C2semidirect
4. #83347C2semidirect
5. #71751C2semidirect
6. #71163C2semidirect
7. #71159C2semidirect
8. #71147C2semidirect
9. #71143C2semidirect
10. #71065C2semidirect
11. #71060C2semidirect
12. #71049C2semidirect
13. #71044C2semidirect
14. #70503C2semidirect
15. #70497C2semidirect
17. C3×#533C22semidirect
36. C3×#477C22semidirect
41. C3×#462C22semidirect
43. C3×#423C22semidirect
44. #5094C22semidirect
45. #5091C22semidirect
50. #5079C22semidirect
51. #5078C22semidirect
58. C2×C4⋊(C3⋊C8)C22semidirect
59. C4×#208C22semidirect
60. #4535C22semidirect
61. #4535C22semidirect
66. #4484C22semidirect
67. #4484C22semidirect
70. #4482C22semidirect
71. #4482C22semidirect
73. #3252D3semidirect
203. #533C2×D3semidirect
218. #477C2×D3semidirect
227. #462C2×D3semidirect
231. #423C2×D3semidirect
747. C3#18595semidirect

Maximal subgroups

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

subgroupindex
#861632
C3×#32522
#833492
#833472
#717512
#711632
#711592
#711472
#711432
#710652
#710602
#710492
#710442
#705032
#704972
#185953