Finite Group Databasebeta

← #10801421

Subgroups — #10801421

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4852C211yesmaximal, characteristic
2C2×#4798C211yesmaximal, characteristic
3C2×#4790C211yesmaximal, characteristic, complemented
4#76480C211yesmaximal, characteristic
5C4×#887C211yesmaximal, characteristic, complemented
6C3×#1927C211yesmaximal, characteristic, complemented
7#70564C211yesmaximal, characteristic, complemented
8#70230C211yesmaximal, characteristic
9Dic3×#103C211yesmaximal, characteristic, complemented
10#69690C211yesmaximal, characteristic, complemented
11#69462C211yesmaximal, characteristic
12#68905C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic
14#68686C211yesmaximal, characteristic
15#68602C211yesmaximal, characteristic, complemented
16C4×#464—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4977C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4939C2211yescharacteristic
45C4×Dic3⋊C4—21no—
46C42×Dic3—21no—
47C42×Dic3C2211yescharacteristic
48#4852—24no—
49#4798C418yescomplemented
50#4790—24no—
51#4670—21no—
52#4670—21no—
53#4666—21no—
54#4666—21no—
55#4649—21no—
56#4649—21no—
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63C2×#464—31no—
64C2×C4×Q16—31no—
65C2×C4×SD16—31no—
66C4×#124—31no—
67C42×Q8—31no—
68#1983—31no—
69#1949—31no—
70#1927D311yescharacteristic, complemented
71#1925—31no—
72#1880—31no—
73#1878—31no—
74#1805—31no—
75C4×C4⋊C8—31no—
76C4×#103—31no—
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947C2×C414yes—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93C2×#216—22no—
94C2×#216—22no—
95C2×#216C2×C414yescomplemented
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C2×#204—22no—
122C2×#204—22no—
123C2×#204C2×C414yes—
124#893—22no—
125#893—22no—
126#887—22no—
127#887—22no—
128#887C2×C414yescomplemented
129#887D414yes—
130C4×Dic6—22no—
131C4×Dic6—22no—
132C3×#106—22no—
133C3×#103—22no—
134C3×#103C2×C414yescomplemented
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138#835—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#821—22no—
144#816—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8C2×C414yescomplemented
147C8×Dic3—22no—
148C8×Dic3—22no—
149#810—22no—
150#809—22no—
151#809C2×C414yes—
152#807—22no—
153C22×Q16—31no—
154C22×SD16—31no—
155C2×#124—31no—
156C2×C4×Q8—31no—
157C2×C4×Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C2×C4×D4—31no—
161#464—38no—
162C4×Q16—64no—
163C4×SD16—64no—
164C2×#107—31no—
165C2×#106C2×D311yescharacteristic
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103C2×D311yescharacteristic
172C2×#103C2×D311yescharacteristic
173C2×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×C4×C8—31no—
177C2×C4×C8—31no—
178#391C2×D311yescharacteristic
179#390—31no—
180#388—61no—
181#388—61no—
182#382—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186C4×C4⋊C4—61no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189C4×C22⋊C4—61no—
190C4×C22⋊C4—61no—
191C43—31no—
192C43—61no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340C2×D311yescharacteristic
197#336—31no—
198#336C2×D311yescharacteristic
199C23×D3—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C22×C412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C22×C412yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6C22×C412yes—
233C2×Dic6C2×D412yes—
234C2×Dic6#4112yes—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8C22×C412yes—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C22×C412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250#216—28no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
260C4⋊Dic3#4112yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264C4×Dic3—24no—
265C4×Dic3—22no—
266C4×Dic3—24no—
267C4×Dic3—24no—
268C4×Dic3—22no—
269C4×Dic3—22no—
270C4×Dic3—41no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C4×Dic3C22×C412yes—
275C4×Dic3C2×D412yes—
276C4×Dic3#4112yes—
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⋊C8C22×C412yes—
284#204—28no—
285C22×Q8—31no—
286C22×Q8C22×D311yescharacteristic
287C22×D4—31no—
288C23×C4—61no—
289C2×Q16—34no—
290C2×Q16—62no—
291C2×Q16—62no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299#124—34no—
300#124—62no—
301C4×Q8—34no—
302C4×Q8—34no—
303C4×Q8—62no—
304C4×Q8—62no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×D4—62no—
310C4×D4—62no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C42—21no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—22no—
341C4⋊C8—34no—
342C4⋊C8—62no—
343#103—22no—
344#103—62no—
345#103—62no—
346#103—62no—
347#103C4×D314yescomplemented
348#102—34no—
349#102—62no—
350#102—62no—
351C4×C8—62no—
352C4×C8—62no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—22no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
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#13512yes—
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3C4×D412yes—
412C2×Dic3C4×D412yes—
413C2×Dic3#12112yes—
414C2×Dic3#12112yes—
415C2×Dic3#13512yes—
416C2×Dic3#13712yes—
417D12—22no—
418D12—22no—
419D12—41no—
420D12—41no—
421C4×D3—84no—
422Dic6—22no—
423Dic6—22no—
424Dic6—22no—
425Dic6C4×D414yes—
426C3×C8—22no—
427C3×C8—41no—
428C3⋊C8—22no—
429C3⋊C8—41no—
430C24—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—32no—
437C2×Q8—32no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8C2×C4×D312yes—
443C2×D4—32no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466Q16—68no—
467SD16—68no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—32no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8C2×C4×D312yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—32no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—122no—
492C4⋊C4C2×C4×D312yes—
493C4⋊C4D3×D412yes—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C42—21no—
500C42—21no—
501C42—32no—
502C42—32no—
503C42—32no—
504C42—32no—
505C42—42no—
506C42—62no—
507C42—62no—
508C42—64no—
509C42—64no—
510C42—64no—
511C42—62no—
512C42—121no—
513C42—122no—
514C42—122no—
515C42—122no—
516C22×C3C2×C4×D411yescharacteristic
517C22×C3C2×#12111yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3C2×#13511yescharacteristic
520C22×C3C2×#13511yescharacteristic
521C22×C3C2×#13711yescharacteristic
522C22×C3#57511yescharacteristic
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C3×C4—22no—
531C3×C4—41no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4C2×C4×D411yescharacteristic
537C3×C4C4×#4111yescharacteristic
538C3×C4#52911yescharacteristic
539C3×C4#53211yescharacteristic
540Dic3—22no—
541Dic3—22no—
542Dic3—22no—
543Dic3—24no—
544Dic3—42no—
545Dic3—81no—
546Dic3#46414yescomplemented
547C23—62no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23C2×D3×D411yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—34no—
558Q8—41no—
559Q8—41no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563D4—62no—
564D4—62no—
565D4—121no—
566D4—121no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—244no—
606C2×C4#91612yes—
607C2×C4C22×C4×D311yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C8—22no—
614C8—41no—
615C8—62no—
616C8—121no—
617C2×C3C4×#12111yescharacteristic
618C2×C3C4×#13511yescharacteristic
619C2×C3C4×#13711yescharacteristic
620C2×C3C2×#46411yescharacteristic
621C2×C3#303511yescharacteristic
622C2×C3#310211yescharacteristic
623C2×C3#310811yescharacteristic
624D3—82no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22—241no—
632C22C4×D3×D411yescharacteristic
633C22D3×#12111yescharacteristic
634C22#565611yescharacteristic
635C22C2×#91611yescharacteristic
636C22D3×#13511yescharacteristic
637C22D3×#13711yescharacteristic
638C22#577711yescharacteristic
639C4—22no—
640C4—34no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—62no—
646C4—62no—
647C4—64no—
648C4—62no—
649C4—81no—
650C4—122no—
651C4—241no—
652C4#552211yescharacteristic
653C4C4×D3×D411yescharacteristic
654C4#566811yescharacteristic
655C4#570611yescharacteristic
656C3C4×#46411yescharacteristic, complemented, Sylow
657C2—242no—
658C2#7545211yescharacteristic
659C2C4×#91611yescharacteristic
660C2#7982911yescharacteristic
661C2#8042711yescharacteristic
662C2D3×#46411yescharacteristic
663C2#8141311yescharacteristic
664C2#8149211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1927C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1927D3
hypocenterC3C4×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
3. C2×#4790C2semidirect
5. C4×#887C2semidirect
6. C3×#1927C2semidirect
7. #70564C2semidirect
9. Dic3×#103C2semidirect
10. #69690C2semidirect
12. #68905C2semidirect
15. #68602C2semidirect
49. #4798C4semidirect
70. #1927D3semidirect
95. C2×#216C2×C4semidirect
128. #887C2×C4semidirect
134. C3×#103C2×C4semidirect
146. Dic3⋊C8C2×C4semidirect
347. #103C4×D3semidirect
546. Dic3#464semidirect
656. C3C4×#464semidirect

Maximal subgroups

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

subgroupindex
C2×#48522
C2×#47982
C2×#47902
#764802
C4×#8872
C3×#19272
#705642
#702302
Dic3×#1032
#696902
#694622
#689052
#687802
#686862
#686022
C4×#4643