Finite Group Databasebeta

← #10824631

Subgroups — #10824631

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C22⋊Q8C211yesmaximal, characteristic
2C2×#4790C211yesmaximal, characteristic, complemented
3C2×#4718C211yesmaximal, characteristic
4#73866C211yesmaximal, characteristic
5C3×#2035C211yesmaximal, characteristic
6#71700C211yesmaximal, characteristic
7#70589C211yesmaximal, characteristic, complemented
8Dic3×#106C211yesmaximal, characteristic, complemented
9#70285C211yesmaximal, characteristic, complemented
10#70247C211yesmaximal, characteristic, complemented
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69936C211yesmaximal, characteristic
13#69518C211yesmaximal, characteristic
14#68807C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#23699—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#4984C2211yescharacteristic
40#4980—21no—
41Dic3×C4⋊C4—21no—
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4950—21no—
49#4946—21no—
50Dic3×C22⋊C4—21no—
51Dic3×C22⋊C4—21no—
52#4790—24no—
53#4718—24no—
54#4652—21no—
55#4649—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×C22⋊Q8—31no—
66#2086—31no—
67#2075—31no—
68#2066—31no—
69#2066—31no—
70#2056—31no—
71#2035—31no—
72#2035D311yescharacteristic
73#2001—31no—
74C4×#106—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×Dic3D411yescharacteristic
80C23×Dic3D411yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×#217—21no—
93C2×#217—21no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96C2×#216—22no—
97C2×#216—22no—
98C2×#216—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104#924—22no—
105#924—22no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#887—22no—
122#887—22no—
123C3×#106—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×C22⋊C8—22no—
127C3×#95—21no—
128#835—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#830—22no—
139#817—22no—
140#816—22no—
141#816—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144C8×Dic3—22no—
145C8×Dic3D414yes—
146#810—22no—
147#807—22no—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C22×C42—31no—
154C4×Q16—64no—
155#431—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171#391—31no—
172#391—31no—
173#389—31no—
174#389—31no—
175#387—61no—
176#387—61no—
177#385—61no—
178#385—61no—
179#380—31no—
180#380C2×D311yescharacteristic
181C4×C4⋊C4—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185C4×C22⋊C4—61no—
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4#4112yes—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—42no—
247#217—24no—
248#217—42no—
249#216—28no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—42no—
256C4⋊Dic3—42no—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—24no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—22no—
264C4×Dic3—22no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3C2×D412yes—
270C4×Dic3#4112yes—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D412yes—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C23×C4—31no—
280C23×C4—31no—
281C23×C4C22×D311yescharacteristic
282C2×Q16—62no—
283C2×Q16—62no—
284C2×Q16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—62no—
318C2×C42—62no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42—61no—
323#107—62no—
324#106—22no—
325#106—62no—
326C4⋊C8—62no—
327C4⋊C8—62no—
328#103—22no—
329#103—22no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334C22⋊C8—22no—
335C22⋊C8—62no—
336C4×C8—34no—
337C4×C8—62no—
338#95—21no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3#13512yes—
355C23×C3C22×D411yescharacteristic
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
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—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×Dic3—28no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—44no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3#12112yes—
407C2×Dic3#12112yes—
408C2×Dic3#13512yes—
409C2×Dic3#13712yes—
410C2×Dic3#13712yes—
411Dic6—22no—
412Dic6—22no—
413Dic6—41no—
414Dic6—41no—
415C3×C8—22no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—41no—
419C24D3×D411yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8#23712yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—32no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8D3×D412yes—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—64no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4#23712yes—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—64no—
492C22⋊C4—122no—
493C42—32no—
494C42—32no—
495C42—62no—
496C42—62no—
497C42—61no—
498C42—61no—
499C42—122no—
500C42—122no—
501C42—122no—
502C42—122no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C2×#12111yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#52511yescharacteristic
513C22×C3C2×#13511yescharacteristic
514C22×C3C2×#13511yescharacteristic
515C22×C3C2×#13711yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—42no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4#52511yescharacteristic
523C3×C4#52911yescharacteristic
524C3×C4#52911yescharacteristic
525C3×C4#53211yescharacteristic
526Dic3—22no—
527Dic3—22no—
528Dic3—42no—
529Dic3—42no—
530Dic3—41no—
531Dic3—81no—
532Dic3—81no—
533Dic3#47014yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#91612yes—
541C23C2×D3×D411yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
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—41no—
574C2×C4—68no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—124no—
585C2×C4—121no—
586C2×C4—121no—
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×C4C2×D3×D411yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4C2×#23711yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—22no—
602C8—41no—
603C8—62no—
604C8—121no—
605C2×C3—42no—
606C2×C3#239211yescharacteristic
607C2×C3#305011yescharacteristic
608C2×C3C2×#47011yescharacteristic
609C2×C3#310711yescharacteristic
610C2×C3#310711yescharacteristic
611C2×C3#310811yescharacteristic
612C2×C3#310811yescharacteristic
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22#558611yescharacteristic
621C22D3×#12111yescharacteristic
622C22#565611yescharacteristic
623C22C2×#91611yescharacteristic
624C22D3×#13511yescharacteristic
625C22D3×#13711yescharacteristic
626C22D3×#13711yescharacteristic
627C4—34no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—62no—
632C4—62no—
633C4—81no—
634C4—81no—
635C4—121no—
636C4—122no—
637C4—122no—
638C4—241no—
639C4—241no—
640C4#559611yescharacteristic
641C4#560611yescharacteristic
642C4#564911yescharacteristic
643C4#568011yescharacteristic
644C3#2369911yescharacteristic, complemented, Sylow
645C2—42no—
646C2#7689011yescharacteristic
647C2#8002911yescharacteristic
648C2#8021311yescharacteristic
649C2#8065611yescharacteristic
650C2D3×#47011yescharacteristic
651C2#8141311yescharacteristic
652C2#8149311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2035C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2035D3
hypocenterC3#23699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
2. C2×#4790C2semidirect
7. #70589C2semidirect
8. Dic3×#106C2semidirect
9. #70285C2semidirect
10. #70247C2semidirect
11. Dic3×#103C2semidirect
14. #68807C2semidirect
15. #68781C2semidirect
644. C3#23699semidirect

Maximal subgroups

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

subgroupindex
Dic3×C22⋊Q82
C2×#47902
C2×#47182
#738662
C3×#20352
#717002
#705892
Dic3×#1062
#702852
#702472
Dic3×#1032
#699362
#695182
#688072
#687812
#236993