Finite Group Databasebeta

← #10812480

Subgroups — #10812480

Order 768.

1038 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#84088C211yesmaximal, characteristic
2C2×#5232C211yesmaximal, characteristic
3#73878C211yesmaximal, characteristic, complemented
4C3×#1788C211yesmaximal, characteristic
5#70256C211yesmaximal, characteristic, complemented
6#70254C211yesmaximal, characteristic, complemented
7#70242C211yesmaximal, characteristic, complemented
8#70241C211yesmaximal, characteristic, complemented
9#70169C211yesmaximal, characteristic, complemented
10#70167C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69949C211yesmaximal, characteristic, complemented
14#69925C211yesmaximal, characteristic, complemented
15#69513C211yesmaximal, characteristic, complemented
16#22582—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×#877—21no—
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5232C418yescomplemented
30#5229C2211yescharacteristic
31#5229C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5220—21no—
35#5220—21no—
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5212C2211yescharacteristic
39#5211—21no—
40C4×#217C2211yescharacteristic
41C2×C3⋊#95—21no—
42C2×#815C2211yescharacteristic
43C2×C8×Dic3C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4943—21no—
51#4654—21no—
52#4654—21no—
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#417—31no—
62#2344—31no—
63#2339—31no—
64#1935—31no—
65#1933—31no—
66#1928—31no—
67#1927—31no—
68#1864—31no—
69#1862—31no—
70#1824—31no—
71#1809—31no—
72#1788D311yescharacteristic
73#1762—31no—
74C4×#102—31no—
75#1753—31no—
76C2×C3×#41—22no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41—41no—
80C2×C3×#41—41no—
81C2×C3×#41C2×C414yes—
82C22×C3×Q8C2311yescharacteristic
83C22×C3×D4—21no—
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4—21no—
86C23×C3×C4C2311yescharacteristic
87C23×Dic3—21no—
88C23×Dic3—21no—
89C23×Dic3—21no—
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100D4×Dic3—22no—
101D4×Dic3—22no—
102D4×Dic3—41no—
103D4×Dic3—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3C2×C414yescomplemented
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×#208—41no—
118C2×#208—41no—
119C2×#208C2×C414yescomplemented
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#877—44no—
123C3×#103—41no—
124C3×#103—41no—
125C3×#102—22no—
126C3×#102—22no—
127C3×C22⋊C8—22no—
128C3×#95—21no—
129#835C2×C414yescomplemented
130#835C2×C414yescomplemented
131#832C2×C414yes—
132#832C2×C414yes—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95D411yescharacteristic
143C3⋊#95D411yescharacteristic
144#815—41no—
145C8×Dic3—41no—
146#808—41no—
147#807—41no—
148C22×#41C2×D311yescharacteristic
149C22×M16—31no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×#117—61no—
153C22×C4⋊C4—31no—
154C2×#107—31no—
155C2×#106—31no—
156#417—38no—
157C2×#103—31no—
158C2×#103—31no—
159C2×#103C2×D311yescharacteristic
160C2×#103C2×D311yescharacteristic
161C2×#102—31no—
162C2×#102—31no—
163C2×#102C2×D311yescharacteristic
164C2×#102C2×D311yescharacteristic
165C2×C22⋊C8C2×D311yescharacteristic
166C2×#97—31no—
167C2×C4×C8—31no—
168#391—31no—
169#391—31no—
170#389—31no—
171#389—31no—
172#387—61no—
173#387—61no—
174#386—61no—
175#386—61no—
176#381—61no—
177#381—61no—
178#380—31no—
179#380C2×D311yescharacteristic
180C4×C4⋊C4—31no—
181C4×C4⋊C4—31no—
182C4×C22⋊C4—31no—
183C2×#95—61no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192C24×C3—21no—
193C24×C3C2×D411yescharacteristic
194C3×#41—28no—
195C3×#41—44no—
196C3×#41—44no—
197C3×#41—44no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C22×C412yes—
204C2×C3×D4—24no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4C22×C412yes—
211C2×C3×D4#4112yes—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—24no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—42no—
222C22×C3×C4—42no—
223C22×C3×C4C22×C412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4#4112yes—
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—24no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—42no—
233C22×Dic3—42no—
234C22×Dic3—42no—
235C22×Dic3—42no—
236C22×Dic3—42no—
237C22×Dic3—42no—
238C22×Dic3—81no—
239C22×Dic3—81no—
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D411yescharacteristic
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C22⋊C4—42no—
255#217—22no—
256#217—41no—
257#217—42no—
258#217—42no—
259#217—81no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3C22×C412yes—
268C4⋊Dic3C22×C412yes—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277#208—44no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C22×C412yes—
283C2×C3⋊C8C22×C412yes—
284C2×#41—22no—
285C2×#41—22no—
286C2×#41—22no—
287C2×#41—41no—
288C2×#41—41no—
289C2×#41C4×D314yes—
290C22×Q8C22×D311yescharacteristic
291C22×D4—21no—
292C22×D4C22×D311yescharacteristic
293C23×C4—21no—
294C23×C4—61no—
295C23×C4—61no—
296C23×C4—61no—
297C23×C4C22×D311yescharacteristic
298C2×M16—34no—
299C2×M16—121no—
300C2×M16—121no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—121no—
308C4×D4—121no—
309#117—124no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—62no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—61no—
330C2×C42—61no—
331#107—121no—
332#106—121no—
333#103—34no—
334#103—34no—
335#103—41no—
336#103—41no—
337#102—22no—
338#102—22no—
339#102—34no—
340#102—34no—
341C22⋊C8—22no—
342#97—121no—
343C4×C8—121no—
344#95—21no—
345#95—31no—
346#95—31no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3C4×D412yes—
370C23×C3C22×D411yescharacteristic
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—48no—
379C3×D4#11714yescomplemented
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—48no—
388C2×C3×C4—41no—
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×C4C4×D412yes—
397C2×C3×C4C23×C411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—48no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—84no—
428C2×Dic3—84no—
429C2×Dic3AGL(1,ℤ/8)12yes—
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—22no—
434C24—21no—
435C24D3×D411yescharacteristic
436#41—28no—
437#41—44no—
438#41—44no—
439#41—44no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8C2×C4×D312yes—
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4C2×C4×D312yes—
453C2×D4#23712yes—
454C2×D4#23712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—64no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4—122no—
480C22×C4—241no—
481C22×C4—241no—
482C22×C4C2×C4×D312yes—
483C22×C4D3×D411yescharacteristic
484C22×C4#23712yes—
485C22×C4C23×D311yescharacteristic
486M16—124no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—32no—
494C2×C8—32no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—32no—
504C4⋊C4—32no—
505C4⋊C4—64no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C22⋊C4—42no—
515C22⋊C4—62no—
516C22⋊C4—121no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—122no—
526C22×C3—22no—
527C22×C3—24no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—41no—
540C22×C3C2×C4×D411yescharacteristic
541C22×C3#54611yescharacteristic
542C22×C3#54611yescharacteristic
543C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57711yescharacteristic
546C22×C3#57811yescharacteristic
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—81no—
550C3×C4—81no—
551C3×C4C2×#11711yescharacteristic
552C3×C4#51811yescharacteristic
553C3×C4#55111yescharacteristic
554C3×C4#55111yescharacteristic
555Dic3—22no—
556Dic3—41no—
557Dic3—42no—
558Dic3—42no—
559Dic3—41no—
560Dic3—41no—
561Dic3—81no—
562Dic3—81no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—21no—
571C23—21no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23#67⋊C412yes—
577C23C2×D3×D411yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—41no—
581Q8—41no—
582D4—22no—
583D4—22no—
584D4—22no—
585D4—48no—
586D4#87714yescomplemented
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—32no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—128no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—244no—
629C2×C4—244no—
630C2×C4#67⋊C412yes—
631C2×C4C22×C4×D311yescharacteristic
632C2×C4C2×#23711yescharacteristic
633C2×C4C2×#23711yescharacteristic
634C2×C4C2×#23711yescharacteristic
635C2×C4#100411yescharacteristic
636C2×C4#100411yescharacteristic
637C8—41no—
638C8—41no—
639C8—62no—
640C8—62no—
641C2×C3—42no—
642C2×C3—44no—
643C2×C3#234211yescharacteristic
644C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
645C2×C3#294711yescharacteristic
646C2×C3#330611yescharacteristic
647C2×C3#331511yescharacteristic
648C2×C3#332311yescharacteristic
649C2×C3#333011yescharacteristic
650C22—22no—
651C22—22no—
652C22—24no—
653C22—22no—
654C22—42no—
655C22—41no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22C2×#67⋊C411yescharacteristic
665C22#559111yescharacteristic
666C22#559111yescharacteristic
667C22#577011yescharacteristic
668C22#577111yescharacteristic
669C22D3×AGL(1,ℤ/8)11yescharacteristic
670C22#577711yescharacteristic
671C4—22no—
672C4—42no—
673C4—62no—
674C4—81no—
675C4—81no—
676C4—122no—
677C4—122no—
678C4—121no—
679C4—121no—
680C4—121no—
681C4—241no—
682C4—241no—
683C4C2×#87711yescharacteristic
684C4#548611yescharacteristic
685C4#558711yescharacteristic
686C4#558711yescharacteristic
687C3#2258211yescharacteristic, complemented, Sylow
688C2—42no—
689C2—44no—
690C2#7382211yescharacteristic
691C2#7882111yescharacteristic
692C2#7886611yescharacteristic
693C2#8010611yescharacteristic
694C2#8011611yescharacteristic
695C2#8025411yescharacteristic
696C2#8025811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
cosocleC22×C3×C4C24
hypercenter#1788D3
hypocenterC3#22582
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
3. #73878C2semidirect
5. #70256C2semidirect
6. #70254C2semidirect
7. #70242C2semidirect
8. #70241C2semidirect
9. #70169C2semidirect
10. #70167C2semidirect
11. #70152C2semidirect
12. Dic3×#102C2semidirect
13. #69949C2semidirect
14. #69925C2semidirect
15. #69513C2semidirect
29. #5232C4semidirect
106. C2×C4⋊Dic3C2×C4semidirect
119. C2×#208C2×C4semidirect
129. #835C2×C4semidirect
130. #835C2×C4semidirect
379. C3×D4#117semidirect
586. D4#877semidirect
687. C3#22582semidirect

Maximal subgroups

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

subgroupindex
#840882
C2×#52322
#738782
C3×#17882
#702562
#702542
#702422
#702412
#701692
#701672
#701522
Dic3×#1022
#699492
#699252
#695132
#225823