Finite Group Databasebeta

← #10945108

Subgroups — #10945108

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4870C211yesmaximal, characteristic
2C2×#4818C211yesmaximal, characteristic
3#76607C211yesmaximal, characteristic, complemented
4C4×#893C211yesmaximal, characteristic
5C3×#1927C211yesmaximal, characteristic, complemented
6#70570C211yesmaximal, characteristic, complemented
7#70276C211yesmaximal, characteristic
8#70241C211yesmaximal, characteristic, complemented
9#70229C211yesmaximal, characteristic, complemented
10#69687C211yesmaximal, characteristic
11#69464C211yesmaximal, characteristic, complemented
12#68896C211yesmaximal, characteristic
13#68799C211yesmaximal, characteristic, complemented
14#68672C211yesmaximal, characteristic
15#68605C211yesmaximal, characteristic, complemented
16#33026—31nomaximal, Sylow
17C2×#944C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C42×D3—21no—
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4973C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C4×(C2×D3)⋊C4—21no—
44C42×Dic3C2211yescharacteristic
45#4929C2211yescharacteristic
46#4870—24no—
47#4818—24no—
48#4666—21no—
49#4666—21no—
50#4650—21no—
51#4650—21no—
52#4646—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#476—31no—
63C2×#461—31no—
64#2862—31no—
65C42×D4—31no—
66#1987—31no—
67#1981—31no—
68#1927—31no—
69#1927—31no—
70#1927—31no—
71#1927D311yescharacteristic, complemented
72#1877—31no—
73#1877—31no—
74#1877—31no—
75#1877—31no—
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#944—22no—
92#944—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—22no—
117#893—22no—
118#893—22no—
119#893—22no—
120#893D414yes—
121#888—22no—
122#888—22no—
123C4×D12—22no—
124C4×D12—22no—
125C42×D3—44no—
126C3×#106—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#821—22no—
142#816—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#810—22no—
146#810—22no—
147#809—22no—
148#808—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×C4⋊Q8—31no—
152C2×C4⋊Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C22×C42—61no—
157#476—64no—
158#461—64no—
159C2×#106—31no—
160C2×#106C2×D311yescharacteristic
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103C2×D311yescharacteristic
168C2×#103C2×D311yescharacteristic
169C2×#102—31no—
170C2×#102—31no—
171#391—31no—
172#391—31no—
173#391—31no—
174#391—61no—
175#391—61no—
176#391C2×D311yescharacteristic
177#382—61no—
178#382—61no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4C2×D311yescharacteristic
182C4×C22⋊C4—61no—
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185C43—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×D412yes—
219C2×D12#4112yes—
220C2×D12#4112yes—
221C2×C4×D3—24no—
222C2×C4×D3—24no—
223C2×C4×D3—24no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—42no—
237C2×Dic6—42no—
238C2×Dic6—42no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D412yes—
250C3×C4⋊C4#4112yes—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—42no—
254#215—44no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3#4112yes—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—22no—
268C4×Dic3—41no—
269C4×Dic3—41no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3SD1614yescomplemented
273C4×Dic3C2×D412yes—
274C4×Dic3#4112yes—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281#204—44no—
282C22×Q8—31no—
283C22×Q8—61no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C23×C4—61no—
287C23×C4—61no—
288C23×C4—61no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C4⋊Q8—62no—
300C4⋊Q8—62no—
301C4⋊Q8—62no—
302C4⋊Q8—62no—
303C4×D4—34no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
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—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C42—21no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—124no—
330C2×C42C22×D311yescharacteristic
331#106—22no—
332#106—62no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#103—22no—
336#103—22no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#102—62no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—22no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
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×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4C2×SD1612yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—41no—
399C2×Dic3—41no—
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×Dic3C2×SD1612yes—
409C2×Dic3C2×SD1612yes—
410C2×Dic3#13712yes—
411D12—22no—
412D12—22no—
413D12—22no—
414D12#12414yes—
415C4×D3—48no—
416C4×D3—48no—
417C4×D3—48no—
418Dic6—41no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—84no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—61no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—122no—
441C2×Q8—122no—
442C2×Q8—122no—
443C2×D4—32no—
444C2×D4—32no—
445C2×D4—32no—
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—64no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—124no—
473SD16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—32no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—122no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4#23512yes—
498C4⋊C4D3×D412yes—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—64no—
502C42—21no—
503C42—21no—
504C42—34no—
505C42—32no—
506C42—32no—
507C42—42no—
508C42—62no—
509C42—61no—
510C42—61no—
511C42—121no—
512C42—121no—
513C42—122no—
514C42—122no—
515C22×C3C2×#12111yescharacteristic
516C22×C3#52511yescharacteristic
517C22×C3#52511yescharacteristic
518C22×C3C22×SD1611yescharacteristic
519C22×C3C2×#13711yescharacteristic
520C22×C3#57511yescharacteristic
521C22×C3#57511yescharacteristic
522C2×D3—24no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C3×C4—22no—
530C3×C4—42no—
531C3×C4—41no—
532C3×C4—42no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4C2×#12411yescharacteristic
536C3×C4#52911yescharacteristic
537C3×C4#52911yescharacteristic
538C3×C4#53211yescharacteristic
539Dic3—22no—
540Dic3—22no—
541Dic3—42no—
542Dic3—41no—
543Dic3—81no—
544Dic3—81no—
545Dic3#46114yes—
546C23—62no—
547C23—62no—
548C23—62no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23C2×D3×D411yescharacteristic
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—121no—
559Q8—121no—
560Q8—121no—
561Q8—121no—
562Q8—244no—
563D4—34no—
564D4—62no—
565D4—62no—
566D4—62no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—41no—
580C2×C4—41no—
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—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—128no—
598C2×C4—128no—
599C2×C4—128no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4D3×SD1612yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3#285011yescharacteristic
620C2×C3C2×#46111yescharacteristic
621C2×C3#304011yescharacteristic
622C2×C3#310111yescharacteristic
623C2×C3#310111yescharacteristic
624C2×C3#311611yescharacteristic
625C2×C3#311611yescharacteristic
626D3—44no—
627C22—64no—
628C22—122no—
629C22—122no—
630C22—122no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22#554711yescharacteristic
635C22D3×#12111yescharacteristic
636C22#565611yescharacteristic
637C22C2×D3×SD1611yescharacteristic
638C22#576811yescharacteristic
639C22D3×#13711yescharacteristic
640C22#577711yescharacteristic
641C4—22no—
642C4—34no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—62no—
647C4—62no—
648C4—81no—
649C4—81no—
650C4—122no—
651C4—121no—
652C4—241no—
653C4—241no—
654C4#552811yescharacteristic
655C4#554811yescharacteristic
656C4D3×#12411yescharacteristic
657C4#570611yescharacteristic
658C3#3302611yescharacteristic, complemented, Sylow
659C2—124no—
660C2#7595711yescharacteristic
661C2#7965111yescharacteristic
662C2#7984811yescharacteristic
663C2D3×#46111yescharacteristic
664C2#8059911yescharacteristic
665C2#8141211yescharacteristic
666C2#8149511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1927C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1927D3
hypocenterC3#33026
LCSG ▷ C22×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. #76607C2semidirect
5. C3×#1927C2semidirect
6. #70570C2semidirect
8. #70241C2semidirect
9. #70229C2semidirect
11. #69464C2semidirect
13. #68799C2semidirect
15. #68605C2semidirect
71. #1927D3semidirect
272. C4×Dic3SD16semidirect
658. C3#33026semidirect

Maximal subgroups

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

subgroupindex
C2×#48702
C2×#48182
#766072
C4×#8932
C3×#19272
#705702
#702762
#702412
#702292
#696872
#694642
#688962
#687992
#686722
#686052
#330263