Finite Group Databasebeta

← #10824352

Subgroups — #10824352

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic, complemented
2C2×#4788C211yesmaximal, characteristic, complemented
3#76477C211yesmaximal, characteristic, complemented
4#76293C211yesmaximal, characteristic
5C3×#2055C211yesmaximal, characteristic, complemented
6#72056C211yesmaximal, characteristic
7#70483C211yesmaximal, characteristic, complemented
8Dic3×C4⋊C8C211yesmaximal, characteristic, complemented
9#70233C211yesmaximal, characteristic, complemented
10#70211C211yesmaximal, characteristic
11Dic3×#102C211yesmaximal, characteristic, complemented
12#69680C211yesmaximal, characteristic, complemented
13#69652C211yesmaximal, characteristic, complemented
14#68835C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#23678—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#67⋊C4—21no—
24C2×#67⋊C4—21no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#390C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34Dic3×SD16—24no—
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4957—21no—
46#4957—21no—
47#4939C2211yescharacteristic
48C42×Dic3C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4788—24no—
52#4654—21no—
53#4653—21no—
54#4649—21no—
55#4648—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×SD16—31no—
64C4×#127—31no—
65#2295—31no—
66#2084—31no—
67#2072—31no—
68#2065—31no—
69#2065—31no—
70#2055—31no—
71#2055D311yescharacteristic, complemented
72#2041—31no—
73#2014—31no—
74C4×C4⋊C8—31no—
75C4×#102—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C23×Dic3—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95Q8×Dic3—22no—
96Q8×Dic3D414yescomplemented
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—22no—
100#940—22no—
101#940—22no—
102D4×Dic3—22no—
103D4×Dic3—22no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#893—22no—
126#893—22no—
127#67⋊C4—44no—
128#67⋊C4—44no—
129C3×C4⋊C8—22no—
130C3×#103—22no—
131C3×#102—22no—
132C3×#95—21no—
133C3×#95—21no—
134#835—22no—
135#832—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#821—22no—
143#821—22no—
144#817—22no—
145C8×Dic3—22no—
146C8×Dic3—22no—
147#809—22no—
148#808—22no—
149C4⋊(C3⋊C8)—22no—
150C22×SD16—31no—
151C22×SD16C2×D311yescharacteristic
152C2×#127—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—61no—
157C2×C4×D4—61no—
158C4×SD16—64no—
159C4×SD16—64no—
160C2×#106—31no—
161C2×#106—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102C2×D311yescharacteristic
172C2×C4×C8—31no—
173C2×C4×C8—31no—
174#391—31no—
175#391C2×D311yescharacteristic
176#390—31no—
177#390—61no—
178#390—61no—
179#390C2×D311yescharacteristic
180#389—31no—
181#389—31no—
182#387—61no—
183#387—61no—
184#384—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—61no—
187#376—61no—
188C43—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#336—31no—
194#336—31no—
195C24×C3—21no—
196C23×D3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4#4112yes—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C2×#67—24no—
216C2×#67—24no—
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—42no—
220C2×#67—42no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3—42no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12#4112yes—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C3×SD16—28no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8#4112yes—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C42—21no—
254C3×C42—21no—
255#217—42no—
256#215—28no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260C4⋊Dic3—21no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D412yes—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—24no—
274C4×Dic3—22no—
275C4×Dic3—22no—
276C4×Dic3—41no—
277C4×Dic3—42no—
278C4×Dic3SD1614yescomplemented
279C4×Dic3C2×D412yes—
280C4×Dic3#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8#4112yes—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C22×D4—61no—
290C22×D4—61no—
291C22×D4C22×D311yescharacteristic
292C23×C4—61no—
293C23×C4—61no—
294C2×SD16—22no—
295C2×SD16—22no—
296C2×SD16—22no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—62no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304#127—62no—
305#127—62no—
306C4×Q8—34no—
307C4×Q8—62no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4—62no—
311C4×D4—62no—
312C4×D4—124no—
313C4×D4—124no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#106—62no—
340#106—62no—
341C4⋊C8—22no—
342C4⋊C8—62no—
343#103—22no—
344#103—62no—
345#102—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351C4×C8—62no—
352C4×C8—62no—
353#95—21no—
354#95—21no—
355#95—61no—
356#95—61no—
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×Q8—22no—
376C3×Q8—22no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—41no—
382C3×D4—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4C2×SD1612yes—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404#67—48no—
405#67—48no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—84no—
422C2×Dic3#12112yes—
423C2×Dic3#12112yes—
424C2×Dic3C2×SD1612yes—
425C2×Dic3C2×SD1612yes—
426C2×Dic3#13512yes—
427C2×Dic3AGL(1,ℤ/8)12yes—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433C3×C8—22no—
434C3×C8—41no—
435C3⋊C8—22no—
436C3⋊C8—41no—
437C24—21no—
438C24—61no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8D3×D412yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—32no—
451C2×D4—64no—
452C2×D4—64no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—122no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—122no—
462C2×D4#23712yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482SD16—28no—
483SD16—68no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8#23712yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—32no—
501C4⋊C4—64no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4—122no—
510C4⋊C4—122no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C42—21no—
518C42—21no—
519C42—34no—
520C42—32no—
521C42—32no—
522C42—62no—
523C42—62no—
524C42—64no—
525C42—61no—
526C42—61no—
527C42—121no—
528C42—122no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3C2×#12111yescharacteristic
537C22×C3#52511yescharacteristic
538C22×C3#53211yescharacteristic
539C22×C3C22×SD1611yescharacteristic
540C22×C3C2×#13511yescharacteristic
541C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
542C22×C3#57511yescharacteristic
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—41no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4#52511yescharacteristic
556C3×C4#52511yescharacteristic
557C3×C4#52911yescharacteristic
558C3×C4#52911yescharacteristic
559Dic3—22no—
560Dic3—22no—
561Dic3—22no—
562Dic3—42no—
563Dic3—81no—
564Dic3—81no—
565Dic3#46014yes—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—62no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23C2×D3×D411yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—41no—
584Q8—41no—
585D4—22no—
586D4—22no—
587D4—41no—
588D4—41no—
589D4—62no—
590D4—62no—
591D4—128no—
592D4—128no—
593D4—121no—
594D4—121no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—32no—
604C2×C4—32no—
605C2×C4—32no—
606C2×C4—32no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—62no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—244no—
631C2×C4—244no—
632C2×C4D3×SD1612yes—
633C2×C4C2×D3×D411yescharacteristic
634C2×C4C2×#23711yescharacteristic
635C2×C4C2×#23711yescharacteristic
636C2×C4D3×#4111yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4D3×#4111yescharacteristic
639C8—22no—
640C8—41no—
641C8—62no—
642C8—121no—
643C2×C3—82no—
644C2×C3#239211yescharacteristic
645C2×C3C2×#46011yescharacteristic
646C2×C3#303511yescharacteristic
647C2×C3#310111yescharacteristic
648C2×C3#310511yescharacteristic
649C2×C3#316711yescharacteristic
650C2×C3#317111yescharacteristic
651D3—82no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—241no—
663C22—241no—
664C22—241no—
665C22—241no—
666C22#559611yescharacteristic
667C22D3×#12111yescharacteristic
668C22#560611yescharacteristic
669C22C2×D3×SD1611yescharacteristic
670C22D3×#13511yescharacteristic
671C22D3×AGL(1,ℤ/8)11yescharacteristic
672C22#577711yescharacteristic
673C4—22no—
674C4—34no—
675C4—42no—
676C4—41no—
677C4—62no—
678C4—62no—
679C4—62no—
680C4—81no—
681C4—81no—
682C4—122no—
683C4—241no—
684C4—241no—
685C4#558611yescharacteristic
686C4#564911yescharacteristic
687C4#565611yescharacteristic
688C4#568011yescharacteristic
689C3#2367811yescharacteristic, complemented, Sylow
690C2—82no—
691C2—242no—
692C2#7689011yescharacteristic
693C2D3×#46011yescharacteristic
694C2#8040811yescharacteristic
695C2#8050511yescharacteristic
696C2#8059711yescharacteristic
697C2#8063211yescharacteristic
698C2#8078411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2055C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2055D3
hypocenterC3#23678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×SD16C2semidirect
2. C2×#4788C2semidirect
3. #76477C2semidirect
5. C3×#2055C2semidirect
7. #70483C2semidirect
8. Dic3×C4⋊C8C2semidirect
9. #70233C2semidirect
11. Dic3×#102C2semidirect
12. #69680C2semidirect
13. #69652C2semidirect
14. #68835C2semidirect
15. #68780C2semidirect
71. #2055D3semidirect
96. Q8×Dic3D4semidirect
278. C4×Dic3SD16semidirect
689. C3#23678semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
C2×#47882
#764772
#762932
C3×#20552
#720562
#704832
Dic3×C4⋊C82
#702332
#702112
Dic3×#1022
#696802
#696522
#688352
#687802
#236783