Finite Group Databasebeta

← #10945129

Subgroups — #10945129

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×C3×#463C211yesmaximal, characteristic
2#82994C211yesmaximal, characteristic, complemented
3C4×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic, complemented
5#72047C211yesmaximal, characteristic, complemented
6#72047C211yesmaximal, characteristic, complemented
7#72037C211yesmaximal, characteristic
8#72037C211yesmaximal, characteristic
9#71721C211yesmaximal, characteristic, complemented
10#70461C211yesmaximal, characteristic, complemented
11#70443C211yesmaximal, characteristic, complemented
12#70242C211yesmaximal, characteristic, complemented
13#70241C211yesmaximal, characteristic, complemented
14#70174C211yesmaximal, characteristic, complemented
15#70173C211yesmaximal, characteristic, complemented
16#33028—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C22×C4×Dic3—21no—
24C2×#870C2211yescharacteristic
25C3×#463—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C4×#217—21no—
40#5009—24no—
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4945—21no—
46#4945—21no—
47#4945—21no—
48#4945—21no—
49Dic3×C22⋊C4—21no—
50Dic3×C22⋊C4—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C42×Dic3C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#478—31no—
63C2×#463D311yescharacteristic
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#1879—31no—
74#1877—31no—
75#1877—31no—
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—22no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#124—22no—
92C3×#124—22no—
93C3×C4×D4—22no—
94C3×C4×D4D414yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106D4×Dic3—22no—
107D4×Dic3—22no—
108D4×Dic3—22no—
109D4×Dic3—22no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
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#870—22no—
129#870—22no—
130C3×C4⋊C8—22no—
131C3×#103—22no—
132C3×#102—22no—
133#835—22no—
134#835—22no—
135#832—22no—
136#832—22no—
137#832—22no—
138#832—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146#808—22no—
147#807—22no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16C2×D311yescharacteristic
150C22×D8C2×D311yescharacteristic
151C2×#125—31no—
152C2×#124C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4C2×D311yescharacteristic
156C22×C42—61no—
157#478—64no—
158#463—24no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8C2×D311yescharacteristic
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171#391—31no—
172#391—31no—
173#390—31no—
174#390—31no—
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#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C2×D412yes—
212C2×C3×D4#4112yes—
213C2×C3×D4#4112yes—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—24no—
223C22×Dic3—24no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—42no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3—42no—
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C3×SD16—44no—
237C3×D8—44no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D412yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—42no—
249C3×C42C2×D412yes—
250C3×C42#4112yes—
251#217—24no—
252#217—24no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3#4112yes—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—22no—
272C4×Dic3—22no—
273C4×Dic3—41no—
274C4×Dic3—41no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C4×Dic3—42no—
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×Q8C22×D311yescharacteristic
285C22×D4C22×D311yescharacteristic
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—61no—
289C23×C4—61no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×SD16—22no—
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—22no—
303#124—22no—
304C4×D4—22no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4#6714yes—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
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—22no—
340C4⋊C8—62no—
341#103—22no—
342#103—62no—
343#103—62no—
344#102—22no—
345#102—62no—
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×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4#12414yes—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—48no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4#12112yes—
395C2×C3×C4#12112yes—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—48no—
415C2×Dic3—48no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3#13512yes—
427C2×Dic3AGL(1,ℤ/8)12yes—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—21no—
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
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#23712yes—
450C2×D4#23712yes—
451C2×D4C2×#6712yes—
452C22×C4—21no—
453C22×C4—24no—
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—44no—
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—32no—
490C4⋊C4—32no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4C2×#6712yes—
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—62no—
510C42—62no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—121no—
516C42—121no—
517C42—122no—
518C42—122no—
519C42—122no—
520C42#23712yes—
521C42C2×#6712yes—
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—22no—
545C3×C4—42no—
546C3×C4—81no—
547C3×C4C2×#12411yescharacteristic
548C3×C4#52911yescharacteristic
549C3×C4#52911yescharacteristic
550C3×C4#53211yescharacteristic
551Dic3—22no—
552Dic3—22no—
553Dic3—41no—
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—41no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4#93614yes—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—61no—
600C2×C4—61no—
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—62no—
610C2×C4—62no—
611C2×C4—128no—
612C2×C4—128no—
613C2×C4—121no—
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#93912yes—
624C2×C4#93912yes—
625C2×C4C2×#23711yescharacteristic
626C2×C4C2×#23711yescharacteristic
627C2×C4C2×#23711yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4C22×#6711yescharacteristic
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#560611yescharacteristic
659C22#560611yescharacteristic
660C22D3×#13511yescharacteristic
661C22#576811yescharacteristic
662C22D3×AGL(1,ℤ/8)11yescharacteristic
663C22#577311yescharacteristic
664C22C2×#93911yescharacteristic
665C4—22no—
666C4—22no—
667C4—42no—
668C4—62no—
669C4—62no—
670C4—81no—
671C4—121no—
672C4—122no—
673C4—121no—
674C4—122no—
675C4—241no—
676C4—241no—
677C4#560011yescharacteristic
678C4#560011yescharacteristic
679C4C2×#93611yescharacteristic
680C4#583211yescharacteristic
681C3#3302811yescharacteristic, complemented, Sylow
682C2—44no—
683C2—161no—
684C2#8050411yescharacteristic
685C2#8050511yescharacteristic
686C2#8050611yescharacteristic
687C2#8050711yescharacteristic
688C2#8269811yescharacteristic
689C2#8390011yescharacteristic
690C2#8393111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#463C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#463D3
hypocenterC3#33028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
2. #82994C2semidirect
3. C4×D4×Dic3C2semidirect
4. C2×#5009C2semidirect
5. #72047C2semidirect
6. #72047C2semidirect
9. #71721C2semidirect
10. #70461C2semidirect
11. #70443C2semidirect
12. #70242C2semidirect
13. #70241C2semidirect
14. #70174C2semidirect
15. #70173C2semidirect
681. C3#33028semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4632
#829942
C4×D4×Dic32
C2×#50092
#720472
#720372
#717212
#704612
#704432
#702422
#702412
#701742
#701732
#330283