Finite Group Databasebeta

← #10858063

Subgroups — #10858063

Order 768.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#428C211yesmaximal, characteristic, complemented
2#82244C211yesmaximal, characteristic
3#82172C211yesmaximal, characteristic, complemented
4C2×#4891C211yesmaximal, characteristic, complemented
5#71995C211yesmaximal, characteristic, complemented
6#71990C211yesmaximal, characteristic
7#71576C211yesmaximal, characteristic, complemented
8#71560C211yesmaximal, characteristic
9#70638C211yesmaximal, characteristic, complemented
10#70616C211yesmaximal, characteristic, complemented
11#70609C211yesmaximal, characteristic, complemented
12#70586C211yesmaximal, characteristic, complemented
13#70585C211yesmaximal, characteristic, complemented
14#70568C211yesmaximal, characteristic, complemented
15#70566C211yesmaximal, characteristic, complemented
16#26363—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C3×#428—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28#5212—21no—
29#5212C2211yescharacteristic
30#5210—21no—
31C4×#217C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4988C2211yescharacteristic
38#4986—21no—
39#4984C2211yescharacteristic
40#4982—21no—
41#4977C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4957—21no—
50#4956—21no—
51#4940—21no—
52C4×(C2×D3)⋊C4—21no—
53#4891—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#428D311yescharacteristic, complemented
64#2442—31no—
65#2363—31no—
66#2092—31no—
67#2088—31no—
68#2065—31no—
69#2059—31no—
70#1949—31no—
71#1918—31no—
72#1895—31no—
73#1874—31no—
74#1862—31no—
75#1854—31no—
76#1851—31no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×#67—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#117—22no—
88C3×#117—41no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—41no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×#217—21no—
99C2×#217D411yescharacteristic
100C2×#217D411yescharacteristic
101#930—22no—
102#930Q814yes—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#893—22no—
121#893—41no—
122#890—22no—
123#890—41no—
124C3×#107—22no—
125C3×#107D414yescomplemented
126C3×#106—22no—
127C3×#106—22no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#830—22no—
138#821—22no—
139#821—22no—
140Dic3⋊C8—22no—
141Dic3⋊C8—22no—
142#809—22no—
143#809—41no—
144#808—22no—
145#807—41no—
146C22×M16C2×D311yescharacteristic
147C2×#125—31no—
148C2×#121—31no—
149C2×C4×D4—31no—
150C2×#117C2×D311yescharacteristic
151C22×C4⋊C4C2×D311yescharacteristic
152#478—64no—
153#428—24no—
154C2×#107—31no—
155C2×#107C2×D311yescharacteristic
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×#106C2×D311yescharacteristic
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C22⋊C8—31no—
167#400—61no—
168#398—61no—
169#390—31no—
170#390—61no—
171#389—31no—
172#389—31no—
173#389—31no—
174#388—61no—
175#386—61no—
176#385—61no—
177#382—31no—
178#380—31no—
179#380—61no—
180C4×C4⋊C4—31no—
181C4×C22⋊C4—31no—
182C4×C22⋊C4—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336—31no—
190#336—31no—
191C24×C3C2×D411yescharacteristic
192C23×D3—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×Q812yes—
206C22×C3×C4#4112yes—
207C22×C3×C4C2411yescharacteristic
208C2×#67—24no—
209C2×#67—81no—
210C2×#67—81no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12C2×Q812yes—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—81no—
227C3×M16—28no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C2×D412yes—
233C2×C3×C8#4112yes—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C42—42no—
245#217—41no—
246#217—41no—
247#217—42no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—81no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3—81no—
259C4⋊Dic3C2×Q812yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—81no—
262Dic3⋊C4—81no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—81no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C22×D4—31no—
275C22×D4—61no—
276C23×C4—21no—
277C23×C4—61no—
278C23×C4C22×D311yescharacteristic
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286#125—62no—
287#125—121no—
288#121—34no—
289#121—62no—
290C4×D4—62no—
291C4×D4—121no—
292#117—22no—
293#117—41no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—21no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321#107—22no—
322#107—121no—
323#107#6714yescomplemented
324#106—22no—
325#106—22no—
326#106—62no—
327C4⋊C8—62no—
328C4⋊C8—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#102—121no—
333C22⋊C8—62no—
334#95—31no—
335#95—31no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3C22⋊Q812yes—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—84no—
377C2×C3×C4C22⋊Q812yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×Q811yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C42−1+411yescharacteristic
384#67—84no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406D12—22no—
407D12—22no—
408D12—41no—
409D12—41no—
410C4×D3—84no—
411C3×C8—22no—
412C3×C8—22no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C24—61no—
416C24D3×D411yescharacteristic
417C2×D4—32no—
418C2×D4—64no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—241no—
425C2×D4—241no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—81no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—241no—
447C22×C4#23512yes—
448C22×C4D3×D411yescharacteristic
449C22×C4D3×Q812yes—
450C22×C4C23×D311yescharacteristic
451M16—28no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8#23512yes—
463C2×C8C2×#6712yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4C2×#6712yes—
483C4⋊C4C2×#6712yes—
484C22⋊C4—42no—
485C22⋊C4—64no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—121no—
489C22⋊C4—121no—
490C22⋊C4—122no—
491C22⋊C4—241no—
492C42—42no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—61no—
497C42—241no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3C2×C22⋊Q811yescharacteristic
506C22×C3D4211yescharacteristic
507C22×C3#54711yescharacteristic
508C22×C3#57511yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57611yescharacteristic
511C22×C3#57711yescharacteristic
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C3×C4—22no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4#52511yescharacteristic
525C3×C4#53911yescharacteristic
526C3×C4D4×Q811yescharacteristic
527C3×C4#55411yescharacteristic
528Dic3—42no—
529Dic3—41no—
530Dic3—41no—
531Dic3—41no—
532Dic3—41no—
533Dic3—81no—
534Dic3—81no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—62no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23#89612yes—
549C23C2×D3×D411yescharacteristic
550D4—62no—
551D4—62no—
552D4—121no—
553D4—121no—
554D4—244no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—84no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—244no—
596C2×C4#89612yes—
597C2×C4C2×#23511yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4#100411yescharacteristic
600C2×C4C2×D3×Q811yescharacteristic
601C2×C4#100711yescharacteristic
602C2×C4C22×#6711yescharacteristic
603C8—22no—
604C8—22no—
605C8—121no—
606C8—121no—
607C2×C3—42no—
608C2×C3#244911yescharacteristic
609C2×C3#308011yescharacteristic
610C2×C3#308111yescharacteristic
611C2×C3#327111yescharacteristic
612C2×C3#327311yescharacteristic
613C2×C3#334811yescharacteristic
614C2×C3#335511yescharacteristic
615D3—82no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22#550711yescharacteristic
631C22C2×#89611yescharacteristic
632C22#577311yescharacteristic
633C22#577411yescharacteristic
634C22#577711yescharacteristic
635C22#577811yescharacteristic
636C22D4×#6711yescharacteristic
637C4—22no—
638C4—42no—
639C4—42no—
640C4—81no—
641C4—81no—
642C4—121no—
643C4—121no—
644C4—121no—
645C4—122no—
646C4—121no—
647C4—241no—
648C4—241no—
649C4#551111yescharacteristic
650C4#551511yescharacteristic
651C4#579111yescharacteristic
652C4Q8×#6711yescharacteristic
653C3#2636311yescharacteristic, complemented, Sylow
654C2—42no—
655C2—242no—
656C2#7922811yescharacteristic
657C2#7923611yescharacteristic
658C2#7924111yescharacteristic
659C2#7924211yescharacteristic
660C2#8217311yescharacteristic
661C2#8396111yescharacteristic
662C2#8399011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#428D3
hypocenterC3#26363
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×C3×#428C2semidirect
3. #82172C2semidirect
4. C2×#4891C2semidirect
5. #71995C2semidirect
7. #71576C2semidirect
9. #70638C2semidirect
10. #70616C2semidirect
11. #70609C2semidirect
12. #70586C2semidirect
13. #70585C2semidirect
14. #70568C2semidirect
15. #70566C2semidirect
63. C2×#428D3semidirect
125. C3×#107D4semidirect
323. #107#67semidirect
653. C3#26363semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4282
#822442
#821722
C2×#48912
#719952
#719902
#715762
#715602
#706382
#706162
#706092
#705862
#705852
#705682
#705662
#263633