Finite Group Databasebeta

← #10845245

Subgroups — #10845245

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82344C211yesmaximal, characteristic, complemented
2C2×#4787C211yesmaximal, characteristic
3C2×#4765C211yesmaximal, characteristic, complemented
4#74334C211yesmaximal, characteristic, complemented
5C3×#1802C211yesmaximal, characteristic, complemented
6#71613C211yesmaximal, characteristic
7#70482C211yesmaximal, characteristic, complemented
8#70452C211yesmaximal, characteristic, complemented
9#70212C211yesmaximal, characteristic
10#70165C211yesmaximal, characteristic, complemented
11#69589C211yesmaximal, characteristic, complemented
12#68909C211yesmaximal, characteristic, complemented
13#68904C211yesmaximal, characteristic, complemented
14#68799C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#25334—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#896—21no—
22C2×#896—21no—
23C2×#893C2211yescharacteristic
24C2×#888C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C3×#384C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5210—21no—
33C4×#217—21no—
34C2×#821C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4993—21no—
40#4993—21no—
41#4988C2211yescharacteristic
42#4975C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47#4945—21no—
48#4944—21no—
49#4939C2211yescharacteristic
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52#4787—24no—
53#4765—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#463—31no—
64#2542—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2055—31no—
69#2054—31no—
70#1987—31no—
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1803—31no—
75#1802D311yescharacteristic, complemented
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×D4—22no—
86C3×C4×D4—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98#936—22no—
99#936D414yes—
100C2×#213—22no—
101C2×#213—22no—
102C2×#213—41no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×#204—22no—
123C2×#204—22no—
124C2×#204—41no—
125#896—44no—
126#896—44no—
127#893—22no—
128#893—22no—
129#888—22no—
130#888—22no—
131#887—22no—
132#887—41no—
133C3×C4⋊C8—41no—
134C3×#102—22no—
135C3×#102—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#821—22no—
143#816—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—41no—
147#810—22no—
148#810—22no—
149#809—22no—
150C22×SD16—31no—
151C22×D8—31no—
152C2×C4⋊Q8—31no—
153C2×#124—31no—
154C2×C22⋊Q8—61no—
155C2×C22⋊Q8—61no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158C2×C4×D4C2×D311yescharacteristic
159#489—64no—
160#463—64no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97—31no—
173#399—61no—
174#399—61no—
175#391—31no—
176#390—31no—
177#390—31no—
178#390—31no—
179#388—61no—
180#388—61no—
181#385—61no—
182#384—31no—
183#384C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193#336—31no—
194#336—31no—
195C24×C3—21no—
196C23×D3—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12#4112yes—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—42no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6C2×D412yes—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4#4112yes—
251C3×C22⋊C4—42no—
252C3×C42—21no—
253C3×C42—21no—
254#217—24no—
255#217—24no—
256#217—42no—
257#217—81no—
258#213—44no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268Dic3⋊C4—22no—
269Dic3⋊C4—22no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—42no—
277C4×Dic3—81no—
278C4×Dic3C2×D412yes—
279C4×Dic3#4112yes—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286#204—44no—
287C22×Q8—31no—
288C22×Q8—61no—
289C22×D4—31no—
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C23×C4—61no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—121no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305#124—34no—
306#124—62no—
307C22⋊Q8—124no—
308C22⋊Q8—124no—
309C4×Q8—62no—
310C4×Q8—121no—
311C4×D4—22no—
312C4×D4—41no—
313C4×D4—62no—
314C4×D4—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—61no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343C4⋊C8—41no—
344C4⋊C8—62no—
345C4⋊C8—121no—
346#103—62no—
347#103—62no—
348#103—62no—
349#102—22no—
350#102—22no—
351#102—62no—
352#102—62no—
353#102—62no—
354#97—62no—
355#95—31no—
356#95—31no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3C22×D411yescharacteristic
368C22×D3—22no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—41no—
378C3×D4—41no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—84no—
394C2×C3×C4AGL(1,ℤ/8)12yes—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3#12112yes—
420C2×Dic3#12112yes—
421C2×Dic3AGL(1,ℤ/8)12yes—
422D12—22no—
423D12—22no—
424D12—41no—
425D12—41no—
426C4×D3—84no—
427Dic6—22no—
428Dic6—22no—
429Dic6—41no—
430Dic6—41no—
431Dic6—84no—
432C3×C8—41no—
433C3×C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—21no—
437C24—61no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8—122no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—32no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4D3×D412yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—81no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479D8—124no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4#23512yes—
510C22⋊C4—42no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C42—21no—
521C42—21no—
522C42—32no—
523C42—32no—
524C42—61no—
525C42—61no—
526C42—122no—
527C42—241no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3C2×#12111yescharacteristic
536C22×C3#54611yescharacteristic
537C22×C3#54811yescharacteristic
538C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
539C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
540C22×C3#57711yescharacteristic
541C22×C3#57811yescharacteristic
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C3×C4—22no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—81no—
553C3×C4—81no—
554C3×C4#52911yescharacteristic
555C3×C4#54211yescharacteristic
556C3×C4#54611yescharacteristic
557C3×C4#54711yescharacteristic
558Dic3—22no—
559Dic3—22no—
560Dic3—22no—
561Dic3—42no—
562Dic3—41no—
563Dic3—81no—
564Dic3—81no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—62no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23C2×D3×D411yescharacteristic
580Q8—62no—
581Q8—62no—
582Q8—121no—
583Q8—121no—
584Q8—244no—
585D4—22no—
586D4—22no—
587D4—41no—
588D4—41no—
589D4—62no—
590D4—62no—
591D4—121no—
592D4—121no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—62no—
620C2×C4—84no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—244no—
630C2×C4#91112yes—
631C2×C4C2×#23511yescharacteristic
632C2×C4C2×D3×D411yescharacteristic
633C2×C4#100411yescharacteristic
634C2×C4#100711yescharacteristic
635C2×C4D3×#4111yescharacteristic
636C2×C4D3×#4111yescharacteristic
637C8—41no—
638C8—41no—
639C8—121no—
640C8—121no—
641C2×C3—82no—
642C2×C3#242711yescharacteristic
643C2×C3#303811yescharacteristic
644C2×C3#304611yescharacteristic
645C2×C3#330611yescharacteristic
646C2×C3#330811yescharacteristic
647C2×C3#336711yescharacteristic
648C2×C3#337311yescharacteristic
649D3—82no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—122no—
658C22—122no—
659C22—122no—
660C22—241no—
661C22—241no—
662C22—241no—
663C22—241no—
664C22#549011yescharacteristic
665C22D3×#12111yescharacteristic
666C22#563311yescharacteristic
667C22C2×#91111yescharacteristic
668C22#576211yescharacteristic
669C22D3×AGL(1,ℤ/8)11yescharacteristic
670C22#577911yescharacteristic
671C4—22no—
672C4—41no—
673C4—42no—
674C4—62no—
675C4—62no—
676C4—62no—
677C4—81no—
678C4—81no—
679C4—122no—
680C4—121no—
681C4—241no—
682C4—241no—
683C4#550711yescharacteristic
684C4#552811yescharacteristic
685C4#561011yescharacteristic
686C4#564011yescharacteristic
687C3#2533411yescharacteristic, complemented, Sylow
688C2—82no—
689C2—242no—
690C2#7389511yescharacteristic
691C2#7876111yescharacteristic
692C2#7889611yescharacteristic
693C2#8033311yescharacteristic
694C2#8043111yescharacteristic
695C2#8099311yescharacteristic
696C2#8105211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1802D3
hypocenterC3#25334
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. #82344C2semidirect
3. C2×#4765C2semidirect
4. #74334C2semidirect
5. C3×#1802C2semidirect
7. #70482C2semidirect
8. #70452C2semidirect
10. #70165C2semidirect
11. #69589C2semidirect
12. #68909C2semidirect
13. #68904C2semidirect
14. #68799C2semidirect
15. #68782C2semidirect
75. #1802D3semidirect
687. C3#25334semidirect

Maximal subgroups

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

subgroupindex
#823442
C2×#47872
C2×#47652
#743342
C3×#18022
#716132
#704822
#704522
#702122
#701652
#695892
#689092
#689042
#687992
#687822
#253343