Finite Group Databasebeta

← #10945125

Subgroups — #10945125

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5153C211yesmaximal, characteristic
2C4×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×#4763C211yesmaximal, characteristic, complemented
4#76426C211yesmaximal, characteristic, complemented
5C3×#1879C211yesmaximal, characteristic
6#72037C211yesmaximal, characteristic
7#71612C211yesmaximal, characteristic, complemented
8#70171C211yesmaximal, characteristic, complemented
9#70162C211yesmaximal, characteristic, complemented
10#70156C211yesmaximal, characteristic, complemented
11#69636C211yesmaximal, characteristic, complemented
12#68801C211yesmaximal, characteristic, complemented
13#68794C211yesmaximal, characteristic, complemented
14#68627C211yesmaximal, characteristic, complemented
15#68605C211yesmaximal, characteristic, complemented
16#33028—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×C4×Dic3—21no—
24C2×#890C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C4×#217—21no—
33#5153—24no—
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4951—21no—
45#4951—21no—
46#4944—21no—
47#4944—21no—
48Dic3×C22⋊C4—21no—
49Dic3×C22⋊C4—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51C42×Dic3C2211yescharacteristic
52#4929C2211yescharacteristic
53#4763—24no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×#463—31no—
64#2861—31no—
65C42×D4—31no—
66#1989—31no—
67#1983—31no—
68#1928—31no—
69#1927—31no—
70#1926—31no—
71#1925—31no—
72#1879—31no—
73#1879D311yescharacteristic
74#1877—31no—
75#1877—31no—
76#1831—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×D8—22no—
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102#936—22no—
103#936—22no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3—22no—
107D4×Dic3D414yes—
108C2×#214—22no—
109C2×#214—22no—
110C2×#214—22no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—44no—
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3D411yescharacteristic
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#890—22no—
129#890—22no—
130C3×#102—22no—
131C3×#102—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#817—22no—
143#816—22no—
144#816—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147#810—22no—
148#807—22no—
149C22×SD16—31no—
150C22×D8C2×D311yescharacteristic
151C2×#125—31no—
152C2×#124—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4C2×D311yescharacteristic
156C22×C42—61no—
157#478—64no—
158#463—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×#102C2×D311yescharacteristic
171#391—31no—
172#391—31no—
173#390—31no—
174#390C2×D311yescharacteristic
175#388—61no—
176#388—61no—
177#388—61no—
178#388—61no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—31no—
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
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C2×C3×D4#4112yes—
207C2×C3×D4#4112yes—
208C22×C3×C4—24no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—24no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—24no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—42no—
223C22×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C3×D8—44no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4#4112yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C42—21no—
251C3×C42—21no—
252#217—24no—
253#217—24no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—42no—
258#214—44no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264C4⋊Dic3C2×D412yes—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—42no—
267C4×Dic3—22no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—22no—
271C4×Dic3—41no—
272C4×Dic3—41no—
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⋊C8—21no—
284C22×Q8—31no—
285C22×D4C22×D311yescharacteristic
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—61no—
289C23×C4—61no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×D8—22no—
294C2×D8—22no—
295C2×D8—22no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300#125—62no—
301#125—62no—
302#124—62no—
303#124—62no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—34no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
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—124no—
336C2×C42C22×D311yescharacteristic
337#107—62no—
338#106—62no—
339C4⋊C8—62no—
340C4⋊C8—62no—
341#103—62no—
342#103—62no—
343#103—62no—
344#102—22no—
345#102—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3C22×D411yescharacteristic
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4#12414yes—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—48no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4AGL(1,ℤ/8)12yes—
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—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—48no—
410C2×Dic3—48no—
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—41no—
420C2×Dic3—41no—
421C2×Dic3#12112yes—
422C2×Dic3#12112yes—
423C2×Dic3#13512yes—
424Dic6—41no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—21no—
433C24—21no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4#23512yes—
450C2×D4D3×D412yes—
451C2×D4#23712yes—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—124no—
475D8—44no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—32no—
492C4⋊C4—32no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4#23712yes—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—64no—
504C22⋊C4—64no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C42—21no—
510C42—21no—
511C42—32no—
512C42—32no—
513C42—62no—
514C42—62no—
515C42—61no—
516C42—61no—
517C42—121no—
518C42—121no—
519C42—122no—
520C42—122no—
521C42—122no—
522C22×C3—24no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3C2×#12111yescharacteristic
537C22×C3#52511yescharacteristic
538C22×C3#52511yescharacteristic
539C22×C3C2×#13511yescharacteristic
540C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
541C22×C3#57511yescharacteristic
542C22×C3#57511yescharacteristic
543C3×C4—22no—
544C3×C4—41no—
545C3×C4—42no—
546C3×C4—81no—
547C3×C4C2×#12411yescharacteristic
548C3×C4#52911yescharacteristic
549C3×C4#52911yescharacteristic
550C3×C4#53211yescharacteristic
551Dic3—22no—
552Dic3—22no—
553Dic3—22no—
554Dic3—42no—
555Dic3—41no—
556Dic3—42no—
557Dic3—81no—
558Dic3—81no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×D3×D411yescharacteristic
574Q8—121no—
575Q8—121no—
576Q8—121no—
577Q8—121no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4#88514yes—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—32no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—128no—
613C2×C4—128no—
614C2×C4—121no—
615C2×C4—121no—
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#91112yes—
625C2×C4C2×#23511yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4C2×#23711yescharacteristic
628C2×C4C2×#23711yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C8—41no—
632C8—41no—
633C8—121no—
634C8—121no—
635C2×C3—44no—
636C2×C3—161no—
637C2×C3#285011yescharacteristic
638C2×C3#303511yescharacteristic
639C2×C3#303611yescharacteristic
640C2×C3#310211yescharacteristic
641C2×C3#310511yescharacteristic
642C2×C3#311111yescharacteristic
643C2×C3#311411yescharacteristic
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22#554711yescharacteristic
659C22#558611yescharacteristic
660C22D3×#12111yescharacteristic
661C22C2×#91111yescharacteristic
662C22D3×#13511yescharacteristic
663C22#577311yescharacteristic
664C22#577311yescharacteristic
665C4—22no—
666C4—42no—
667C4—41no—
668C4—62no—
669C4—62no—
670C4—62no—
671C4—81no—
672C4—122no—
673C4—122no—
674C4—121no—
675C4—241no—
676C4—241no—
677C4C2×#88511yescharacteristic
678C4#553811yescharacteristic
679C4#559611yescharacteristic
680C4#564911yescharacteristic
681C3#3302811yescharacteristic, complemented, Sylow
682C2—44no—
683C2—161no—
684C2#7486911yescharacteristic
685C2#7959211yescharacteristic
686C2#7987011yescharacteristic
687C2#7999111yescharacteristic
688C2#8006811yescharacteristic
689C2#8033211yescharacteristic
690C2#8053311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1879C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenter#1879D3
hypocenterC3#33028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. C4×D4×Dic3C2semidirect
3. C2×#4763C2semidirect
4. #76426C2semidirect
7. #71612C2semidirect
8. #70171C2semidirect
9. #70162C2semidirect
10. #70156C2semidirect
11. #69636C2semidirect
12. #68801C2semidirect
13. #68794C2semidirect
14. #68627C2semidirect
15. #68605C2semidirect
681. C3#33028semidirect

Maximal subgroups

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

subgroupindex
C2×#51532
C4×D4×Dic32
C2×#47632
#764262
C3×#18792
#720372
#716122
#701712
#701622
#701562
#696362
#688012
#687942
#686272
#686052
#330283