Finite Group Databasebeta

← #10944019

Subgroups — #10944019

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#237C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×#5173C212yesmaximal
4#83269C211yesmaximal, characteristic, complemented
5C2×#5090C211yesmaximal, characteristic, complemented
6#72047C212yesmaximal, complemented
7#71604C211yesmaximal, characteristic, complemented
8#70485C211yesmaximal, characteristic, complemented
9#70475C211yesmaximal, characteristic, complemented
10#70281C212yesmaximal, complemented
11#70269C212yesmaximal, complemented
12#32907—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#946C2212yes—
18C22×#216C2212yes—
19C2×D4×Dic3C2211yescharacteristic
20C2×C4×#67—21no—
21C22×C4×Dic3—21no—
22C4×#237—24no—
23C4×#237—24no—
24C2×#892C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×#67⋊C4—21no—
27C2×#877—21no—
28C2×C42×D3C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#461—24no—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33#5229C2212yes—
34C2×#835C2212yes—
35C2×#832C2212yes—
36#5173—28no—
37C2×(C2×D3)⋊C8C2212yes—
38C2×#816C2212yes—
39#5090—24no—
40#4987—21no—
41#4987—21no—
42#4972—21no—
43#4972—21no—
44#4935C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×C4×#41—31no—
50C2×#462—31no—
51C2×#461D311yescharacteristic, complemented
52C2×#452—32no—
53#2857—31no—
54#1878—32no—
55#1877—32no—
56#1873—32no—
57#1803—31no—
58#1802—31no—
59#1797—31no—
60C22×C3×Q8C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C23×Dic3—21no—
66C2×#237—22no—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—22no—
70C2×#237—22no—
71C2×#237D414yes—
72C22×C4×D3D411yescharacteristic
73C22×C4×D3D411yescharacteristic
74C22×C4×D3C2311yescharacteristic
75C22×Dic6C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×C4×D4—22no—
83C3×C4×D4D414yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90#946—24no—
91#946—24no—
92C2×#216—24no—
93C2×#216—24no—
94C2×#216—24no—
95D4×Dic3—22no—
96D4×Dic3—22no—
97C4×#67—28no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3—28no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2312yes—
115#892—22no—
116#892—22no—
117#887—22no—
118#887—22no—
119#67⋊C4—28no—
120#877—28no—
121C42×D3—22no—
122C42×D3—22no—
123C4×Dic6—22no—
124C4×Dic6D414yes—
125C3×C4⋊C8—22no—
126C3×#103—24no—
127#835—24no—
128#832—24no—
129C3⋊#95—22no—
130C3⋊#95—22no—
131(C2×D3)⋊C8—24no—
132#816—24no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16C2×D312yes—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×C22⋊Q8—32no—
139C4×#41—64no—
140C4×#41—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—61no—
149C22×C42—31no—
150C22×C42—61no—
151#462—64no—
152#461—24no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103C2×D312yes—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#391—32no—
162#389—61no—
163#389—61no—
164#384—31no—
165#380—61no—
166#380—61no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—21no—
171C23×D3C2×D411yescharacteristic
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D412yes—
183C2×C3×D4#4112yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C2×#67—24no—
193C2×#67—24no—
194C2×#67—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—21no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207#237—28no—
208#237—28no—
209#237—28no—
210#237—28no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—41no—
221C2×C4×D3C2×D412yes—
222C2×C4×D3#4112yes—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C2×D412yes—
227C2×Dic6C2×D412yes—
228C2×Dic6#4112yes—
229C3×SD16—48no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—24no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240#217—24no—
241#216—48no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—44no—
244C4⋊Dic3—21no—
245C4⋊Dic3—22no—
246C4⋊Dic3—22no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249C4⋊Dic3C2×D412yes—
250Dic3⋊C4—24no—
251Dic3⋊C4—24no—
252Dic3⋊C4—44no—
253C4×Dic3—24no—
254C4×Dic3—24no—
255C4×Dic3—24no—
256C4×Dic3—22no—
257C4×Dic3—22no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C22×Q8—31no—
270C22×Q8C22×D312yes—
271C22×D4—61no—
272C22×D4C22×D311yescharacteristic
273C23×C4—21no—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4—31no—
277C23×C4—61no—
278C23×C4—61no—
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—24no—
285C22×C8—32no—
286C22×C8C22×D312yes—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C22⋊Q8—64no—
290C22⋊Q8—64no—
291C4×Q8—34no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×D4—22no—
296C4×D4—68no—
297C4×D4—62no—
298C4×D4—68no—
299C4×D4—62no—
300C4×D4#6714yes—
301#117—68no—
302#117—62no—
303#117—62no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—68no—
324C2×C42—61no—
325C2×C42—62no—
326C2×C42—62no—
327C2×C42C22×D311yescharacteristic
328C4⋊C8—22no—
329C4⋊C8—62no—
330#103—24no—
331#103—64no—
332#103—64no—
333#102—64no—
334C22⋊C8—64no—
335#95—62no—
336#95—62no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3C22×D411yescharacteristic
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3#12112yes—
352C3×Q8—42no—
353C3×Q8—42no—
354C3×Q8—42no—
355C3×Q8—42no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—22no—
359C3×D4#12114yes—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—48no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4#12012yes—
370C2×C3×C4#12012yes—
371C2×C3×C4#13512yes—
372C2×C3×C4#13712yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379#67—216no—
380C2×Dic3—216no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—48no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3#12112yes—
399C2×Dic3#13712yes—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—44no—
405C4×D3—44no—
406Dic6—22no—
407Dic6—22no—
408Dic6—22no—
409Dic6#12114yes—
410C3×C8—42no—
411C3⋊C8—42no—
412C24—21no—
413C24—31no—
414#41—68no—
415#41—68no—
416#41—68no—
417#41—68no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—32no—
425C2×Q8—32no—
426C2×Q8—32no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4D3×D412yes—
437C2×D4#23712yes—
438C2×D4C2×#6712yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—31no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—62no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—64no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—121no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4C23×D311yescharacteristic
471Q16—128no—
472SD16—48no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C4⋊C4—21no—
480C4⋊C4—32no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—61no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—122no—
489C4⋊C4—124no—
490C4⋊C4—122no—
491C4⋊C4C2×#6712yes—
492C22⋊C4—24no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—124no—
496C42—64no—
497C42—64no—
498C42—62no—
499C42—62no—
500C42—64no—
501C42—61no—
502C42—61no—
503C42D3×D412yes—
504C42C2×#6712yes—
505C22×C3—24no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3C2×#13511yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57511yescharacteristic
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C3×C4—22no—
527C3×C4—42no—
528C3×C4—82no—
529C3×C4#45214yescomplemented
530C3×C4C2×#12111yescharacteristic
531C3×C4C2×#12111yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#52511yescharacteristic
534Dic3—24no—
535Dic3—22no—
536Dic3—22no—
537Dic3—41no—
538Dic3—42no—
539Dic3—82no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—32no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23C2×D3×D411yescharacteristic
555Q8—34no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563D4—22no—
564D4—22no—
565D4—22no—
566D4—616no—
567D4#93814yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—616no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—124no—
596C2×C4—128no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—124no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4#91512yes—
604C2×C4#91612yes—
605C2×C4#93712yes—
606C2×C4#93712yes—
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×#23711yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4C22×#6711yescharacteristic
613C8—42no—
614C8—122no—
615C2×C3—44no—
616C2×C3#284711yescharacteristic
617C2×C3C2×#45211yescharacteristic
618C2×C3#300711yescharacteristic
619C2×C3#303511yescharacteristic
620C2×C3#303911yescharacteristic
621C2×C3#310711yescharacteristic
622C2×C3#311611yescharacteristic
623D3—42no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—62no—
632C22—62no—
633C22—62no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22D3×#12111yescharacteristic
639C22#560611yescharacteristic
640C22C2×#91511yescharacteristic
641C22C2×#91611yescharacteristic
642C22D3×#13711yescharacteristic
643C22#577711yescharacteristic
644C22C2×#93711yescharacteristic
645C4—22no—
646C4—42no—
647C4—62no—
648C4—62no—
649C4—64no—
650C4—82no—
651C4—122no—
652C4—121no—
653C4—242no—
654C4#517314yescomplemented
655C4D3×#12111yescharacteristic
656C4#560611yescharacteristic
657C4C2×#93811yescharacteristic
658C4#582811yescharacteristic
659C3#3290711yescharacteristic, complemented, Sylow
660C2—44no—
661C2—122no—
662C2#8040911yescharacteristic
663C2#8041111yescharacteristic
664C2#8042711yescharacteristic
665C2#8042911yescharacteristic
666C2#8272311yescharacteristic
667C2C2×#517311yescharacteristic
668C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenterC2×#461D3
hypocenterC3#32907
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. C2×C3×#461C2semidirect
4. #83269C2semidirect
5. C2×#5090C2semidirect
6. #72047C2semidirect
7. #71604C2semidirect
8. #70485C2semidirect
9. #70475C2semidirect
10. #70281C2semidirect
11. #70269C2semidirect
51. C2×#461D3semidirect
529. C3×C4#452semidirect
654. C4#5173semidirect
659. C3#32907semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
C2×C3×#4612
C2×#51732
#832692
C2×#50902
#720472
#716042
#704852
#704752
#702812
#702692
#329073