Finite Group Databasebeta

← #10824500

Subgroups — #10824500

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5164C211yesmaximal, characteristic, complemented
2C2×Dic3×SD16C211yesmaximal, characteristic, complemented
3#83172C211yesmaximal, characteristic
4#76483C211yesmaximal, characteristic, complemented
5C3×#2014C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic, complemented
7#72051C211yesmaximal, characteristic, complemented
8#71762C211yesmaximal, characteristic
9#70180C211yesmaximal, characteristic, complemented
10#70179C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69671C211yesmaximal, characteristic, complemented
14#68641C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#23690—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#391C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31#5220—21no—
32#5220—21no—
33#5220—21no—
34#5220—21no—
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5164—24no—
40Dic3×SD16—24no—
41C2×#818C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×C8×Dic3C2211yescharacteristic
47Dic3×C4⋊C4C2211yescharacteristic
48#4972C2211yescharacteristic
49#4936C2211yescharacteristic
50#4932C2211yescharacteristic
51#4928C2211yescharacteristic
52#4654—21no—
53#4653—21no—
54#4651—21no—
55#4650—21no—
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#440—31no—
63C2×C4×SD16—31no—
64#2328—31no—
65#2320—31no—
66#2085—31no—
67#2074—31no—
68#2067—31no—
69#2065—31no—
70#2055—31no—
71#2055—31no—
72#2041—31no—
73#2014D311yescharacteristic
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C23×Dic3—21no—
83C23×Dic3—21no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16—22no—
88C2×C3×SD16—22no—
89C2×C3×SD16—22no—
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C3×#127—22no—
93C3×#127—22no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102Q8×Dic3—22no—
103Q8×Dic3D414yescomplemented
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3—22no—
107D4×Dic3—22no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C3×#102—22no—
122C3×#102—22no—
123C3×#95—21no—
124C3×#95—21no—
125#835—22no—
126#835—22no—
127#832—22no—
128#832—22no—
129#832—22no—
130#832—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#818—22no—
143#817—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148C22×SD16C2×D311yescharacteristic
149C22×SD16C2×D311yescharacteristic
150C2×#127C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391C2×D311yescharacteristic
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#387—61no—
177#387—61no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#383—61no—
183#382—31no—
184#382—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—81no—
214C2×C3×D4—81no—
215C2×C3×D4—81no—
216C2×C3×D4—81no—
217C2×C3×D4#4112yes—
218C2×C3×D4#4112yes—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—42no—
229C22×Dic3—42no—
230C22×Dic3—81no—
231C22×Dic3—81no—
232C22×Dic3—81no—
233C22×Dic3—81no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C3×SD16—28no—
239C3×SD16—28no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8#4112yes—
245C2×C3×C8#4112yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C42—21no—
249C3×C42—21no—
250#217—42no—
251#217—42no—
252#217—81no—
253#217—81no—
254#217—81no—
255#217—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—24no—
270C4×Dic3—81no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22×D311yescharacteristic
283C22×D4C22×D311yescharacteristic
284C23×C4—61no—
285C23×C4—61no—
286C2×SD16—22no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296#127—22no—
297#127—22no—
298C4×Q8—34no—
299C4×Q8—62no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#106—62no—
325C4⋊C8—62no—
326C4⋊C8—62no—
327#103—62no—
328#103—62no—
329#102—22no—
330#102—22no—
331#102—62no—
332#102—62no—
333#102—62no—
334#102—62no—
335#97—62no—
336C4×C8—62no—
337#95—21no—
338#95—21no—
339#95—31no—
340#95—31no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—84no—
378C3×D4—84no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
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—84no—
420C2×Dic3—84no—
421C2×Dic3#12112yes—
422C2×Dic3#12112yes—
423C3×C8—22no—
424C3×C8—22no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—21no—
428C24—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8D3×D412yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—81no—
451C2×D4#23712yes—
452C2×D4#23712yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—241no—
466C22×C4—241no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—28no—
473SD16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8#23712yes—
485C2×C8#23712yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—32no—
489C4⋊C4—64no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—241no—
498C4⋊C4—241no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C42—21no—
506C42—21no—
507C42—32no—
508C42—32no—
509C42—64no—
510C42—61no—
511C42—61no—
512C42—241no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3C2×#12111yescharacteristic
528C22×C3#52511yescharacteristic
529C22×C3#53211yescharacteristic
530C22×C3#57611yescharacteristic
531C22×C3#57711yescharacteristic
532C22×C3#57711yescharacteristic
533C22×C3#57711yescharacteristic
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—81no—
538C3×C4#52511yescharacteristic
539C3×C4#52511yescharacteristic
540C3×C4#52911yescharacteristic
541C3×C4#52911yescharacteristic
542Dic3—22no—
543Dic3—22no—
544Dic3—41no—
545Dic3—41no—
546Dic3—42no—
547Dic3—81no—
548Dic3—81no—
549Dic3—81no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×D3×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578D4—84no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—32no—
587C2×C4—32no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—244no—
616C2×C4—244no—
617C2×C4C2×D3×D411yescharacteristic
618C2×C4C2×#23711yescharacteristic
619C2×C4C2×#23711yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4C2×#23711yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—22no—
624C8—22no—
625C8—121no—
626C8—121no—
627C2×C3—82no—
628C2×C3—82no—
629C2×C3#239211yescharacteristic
630C2×C3#304311yescharacteristic
631C2×C3#304511yescharacteristic
632C2×C3#311811yescharacteristic
633C2×C3#312311yescharacteristic
634C2×C3#317411yescharacteristic
635C2×C3#317911yescharacteristic
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22#558611yescharacteristic
651C22D3×#12111yescharacteristic
652C22#571611yescharacteristic
653C22#576211yescharacteristic
654C22#576211yescharacteristic
655C22#577411yescharacteristic
656C22#577511yescharacteristic
657C4—41no—
658C4—41no—
659C4—42no—
660C4—62no—
661C4—62no—
662C4—81no—
663C4—122no—
664C4—121no—
665C4—121no—
666C4—241no—
667C4—241no—
668C4—241no—
669C4#558611yescharacteristic
670C4#558611yescharacteristic
671C4#560011yescharacteristic
672C4#560011yescharacteristic
673C3#2369011yescharacteristic, complemented, Sylow
674C2—82no—
675C2—82no—
676C2#7666611yescharacteristic
677C2#8013611yescharacteristic
678C2#8013711yescharacteristic
679C2#8038511yescharacteristic
680C2#8038611yescharacteristic
681C2#8175311yescharacteristic
682C2#8183711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2014C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2014D3
hypocenterC3#23690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2014

Complements

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

NH ≅ G/Ntype
1. C2×#5164C2semidirect
2. C2×Dic3×SD16C2semidirect
4. #76483C2semidirect
6. #72051C2semidirect
7. #72051C2semidirect
9. #70180C2semidirect
10. #70179C2semidirect
11. #70152C2semidirect
12. Dic3×#102C2semidirect
13. #69671C2semidirect
14. #68641C2semidirect
15. #68604C2semidirect
103. Q8×Dic3D4semidirect
673. C3#23690semidirect

Maximal subgroups

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

subgroupindex
C2×#51642
C2×Dic3×SD162
#831722
#764832
C3×#20142
#720512
#717622
#701802
#701792
#701522
Dic3×#1022
#696712
#686412
#686042
#236903