Finite Group Databasebeta

← #10824657

Subgroups — #10824657

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#121C211yesmaximal, characteristic
2C2×#4758C211yesmaximal, characteristic, complemented
3C2×#4718C211yesmaximal, characteristic, complemented
4#73866C211yesmaximal, characteristic, complemented
5C3×#2033C211yesmaximal, characteristic
6#71685C211yesmaximal, characteristic
7#70643C211yesmaximal, characteristic, complemented
8Dic3×#107C211yesmaximal, characteristic, complemented
9#70217C211yesmaximal, characteristic, complemented
10#70180C211yesmaximal, characteristic, complemented
11Dic3×#102C211yesmaximal, characteristic, complemented
12#69937C211yesmaximal, characteristic, complemented
13#69520C211yesmaximal, characteristic, complemented
14#68809C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#23701—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5221C2211yescharacteristic
30#5220—21no—
31C2×#832C2211yescharacteristic
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#4984C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4951—21no—
49#4946—21no—
50Dic3×C22⋊C4—21no—
51Dic3×C22⋊C4—21no—
52#4758—24no—
53#4718—24no—
54#4652—21no—
55#4649—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×#121—31no—
66#2088—31no—
67#2077—31no—
68#2068—31no—
69#2067—31no—
70#2055—31no—
71#2036—31no—
72#2033D311yescharacteristic
73#2000—31no—
74C4×#107—31no—
75C4×#103—31no—
76C4×#102—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×Dic3—21no—
81C23×Dic3D411yescharacteristic
82C23×Dic3D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—22no—
87C3×#121—22no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—44no—
101C2×#214—22no—
102C2×#214—22no—
103C2×#214—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108#924—22no—
109#924—22no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—22no—
114C2×C4×Dic3—22no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#887—22no—
125#887—22no—
126C3×#107—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×C22⋊C8—22no—
130C3×#95—21no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#830—22no—
140#818—22no—
141#816—22no—
142#816—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C8×Dic3D414yescomplemented
147#810—22no—
148#808—22no—
149C22×SD16—31no—
150C2×C22⋊Q8—31no—
151C2×#121C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—61no—
155C22×C42—31no—
156C4×SD16—64no—
157#431—64no—
158C2×#107—31no—
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#102C2×D311yescharacteristic
169C2×C22⋊C8—31no—
170C2×C22⋊C8C2×D311yescharacteristic
171C2×C4×C8—31no—
172C2×C4×C8—31no—
173#391—31no—
174#390—31no—
175#389—31no—
176#389—31no—
177#388—61no—
178#387—61no—
179#386—61no—
180#385—61no—
181#380—31no—
182#380C2×D311yescharacteristic
183C4×C4⋊C4—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186C4×C22⋊C4—61no—
187C4×C22⋊C4—61no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—42no—
203C2×C3×D4—42no—
204C2×C3×D4#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—22no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—42no—
227C22×Dic3—42no—
228C22×Dic3—42no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6#4112yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4#4112yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—42no—
253#217—24no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—42no—
258#214—28no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—22no—
272C4×Dic3—22no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C2×D412yes—
284C22×Q8—31no—
285C22×D4—21no—
286C22×D4C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4C22×D311yescharacteristic
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300#121—22no—
301#121—22no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—124no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—62no—
329C2×C42—62no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42—61no—
333#107—22no—
334#107—62no—
335#106—62no—
336C4⋊C8—62no—
337C4⋊C8—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—22no—
342#102—22no—
343#102—62no—
344C22⋊C8—22no—
345C22⋊C8—62no—
346C4×C8—34no—
347C4×C8—62no—
348#95—21no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—22no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3#13512yes—
370C23×C3C22×D411yescharacteristic
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—48no—
374C3×D4—41no—
375C3×D4—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×Dic3—28no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—44no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—84no—
420C2×Dic3#12112yes—
421C2×Dic3#12112yes—
422C2×Dic3#13512yes—
423C2×Dic3AGL(1,ℤ/8)12yes—
424C2×Dic3#13712yes—
425Dic6—22no—
426Dic6—22no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—41no—
431C3⋊C8—22no—
432C3⋊C8—41no—
433C24—21no—
434C24D3×D411yescharacteristic
435C2×Q8—32no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4#23712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—68no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8D3×D412yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—64no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4#23712yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—64no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C42—32no—
515C42—32no—
516C42—62no—
517C42—62no—
518C42—61no—
519C42—61no—
520C42—122no—
521C42—122no—
522C42—122no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#52511yescharacteristic
539C22×C3#52511yescharacteristic
540C22×C3C2×#13511yescharacteristic
541C22×C3C2×#13511yescharacteristic
542C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
543C22×C3C2×#13711yescharacteristic
544C3×C4—42no—
545C3×C4—41no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4#52511yescharacteristic
549C3×C4#52911yescharacteristic
550C3×C4#52911yescharacteristic
551C3×C4#53211yescharacteristic
552Dic3—22no—
553Dic3—22no—
554Dic3—42no—
555Dic3—42no—
556Dic3—41no—
557Dic3—81no—
558Dic3—81no—
559Dic3#46814yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23#91212yes—
574C23C2×D3×D411yescharacteristic
575Q8—62no—
576Q8—62no—
577Q8—121no—
578Q8—121no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—41no—
583D4—41no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—32no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
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—41no—
605C2×C4—68no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—124no—
616C2×C4—121no—
617C2×C4—121no—
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—244no—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4C2×#23711yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—22no—
634C8—41no—
635C8—62no—
636C8—121no—
637C2×C3—42no—
638C2×C3—82no—
639C2×C3#239211yescharacteristic
640C2×C3#305011yescharacteristic
641C2×C3C2×#46811yescharacteristic
642C2×C3#310511yescharacteristic
643C2×C3#310611yescharacteristic
644C2×C3#310711yescharacteristic
645C2×C3#310811yescharacteristic
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#558611yescharacteristic
661C22D3×#12111yescharacteristic
662C22#565611yescharacteristic
663C22C2×#91211yescharacteristic
664C22D3×#13511yescharacteristic
665C22D3×AGL(1,ℤ/8)11yescharacteristic
666C22D3×#13711yescharacteristic
667C4—34no—
668C4—41no—
669C4—42no—
670C4—62no—
671C4—62no—
672C4—81no—
673C4—81no—
674C4—122no—
675C4—121no—
676C4—122no—
677C4—241no—
678C4—241no—
679C4#559611yescharacteristic
680C4#560611yescharacteristic
681C4#564911yescharacteristic
682C4#568011yescharacteristic
683C3#2370111yescharacteristic, complemented, Sylow
684C2—42no—
685C2—82no—
686C2#7689011yescharacteristic
687C2#7998911yescharacteristic
688C2#8021311yescharacteristic
689C2#8071411yescharacteristic
690C2D3×#46811yescharacteristic
691C2#8126911yescharacteristic
692C2#8134911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2033C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2033D3
hypocenterC3#23701
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. C2×#4758C2semidirect
3. C2×#4718C2semidirect
4. #73866C2semidirect
7. #70643C2semidirect
8. Dic3×#107C2semidirect
9. #70217C2semidirect
10. #70180C2semidirect
11. Dic3×#102C2semidirect
12. #69937C2semidirect
13. #69520C2semidirect
14. #68809C2semidirect
15. #68781C2semidirect
146. C8×Dic3D4semidirect
683. C3#23701semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1212
C2×#47582
C2×#47182
#738662
C3×#20332
#716852
#706432
Dic3×#1072
#702172
#701802
Dic3×#1022
#699372
#695202
#688092
#687812
#237013