Finite Group Databasebeta

← #10700608

Subgroups — #10700608

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#67C211yesmaximal, characteristic
2C22×#806C212yesmaximal
3C3×C22⋊#95C211yesmaximal, characteristic, complemented
4#72127C211yesmaximal, characteristic, complemented
5C2×#4527C212yesmaximal, complemented
6#10761—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×#67C2211yescharacteristic
9C23×C4×D3C411yescharacteristic, complemented
10C22×C3×C22⋊C4C2212yes—
11C22×#217C2212yes—
12C22×#217C2211yescharacteristic
13C2×C4×#67—22no—
14C2×C4×#67—22no—
15C2×C4×#67—24no—
16C2×C4×#67—24no—
17C22×(C2×D3)⋊C4C411yescharacteristic, complemented
18C22×Dic3⋊C4C411yescharacteristic, complemented
19C22×C4×Dic3C411yescharacteristic, complemented
20C2×C3×#95—22no—
21C2×C3⋊#95—22no—
22C2×#806—24no—
23C2×#806—24no—
24C2×#806C418yescomplemented
25#4527—24no—
26C22×C4×D4—31no—
27C22×#99—32no—
28C2×#342—32no—
29C22⋊#95—31no—
30C22⋊#95D311yescharacteristic, complemented
31C25×C3C2311yescharacteristic
32C24×D3C2×C411yescharacteristic
33C23×C3×C4—22no—
34C23×C3×C4—22no—
35C23×C3×C4—22no—
36C23×C3×C4—44no—
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4C2×C411yescharacteristic
39C23×C3×C4D412yes—
40C23×C3×C4D412yes—
41C22×#67—21no—
42C22×#67—22no—
43C22×#67—21no—
44C22×#67—22no—
45C22×#67—21no—
46C22×#67—21no—
47C22×#67C2×C414yes—
48C22×#67D414yes—
49C22×#67D412yes—
50C22×#67Q812yes—
51C23×Dic3—22no—
52C23×Dic3C2×C411yescharacteristic
53C23×Dic3C2×C411yescharacteristic
54C23×Dic3C2×C411yescharacteristic
55C22×C4×D3—22no—
56C22×C4×D3—22no—
57C22×C4×D3—22no—
58C22×C4×D3—44no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4C2×C414yescomplemented
65C2×#217—21no—
66C2×#217—22no—
67C2×#217—21no—
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217C2×C414yescomplemented
74C2×#217D414yes—
75C4×#67—216no—
76C4×#67—216no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—22no—
86C3×#95—44no—
87C3⋊#95—44no—
88#806—216no—
89C23×D4—31no—
90C24×C4—31no—
91C24×C4C2×D311yescharacteristic
92C2×C4×D4—64no—
93C2×C4×D4—64no—
94C2×C4×D4—62no—
95C2×C4×D4—62no—
96C22×C4⋊C4—31no—
97C22×C22⋊C4—31no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4—31no—
100C22×C22⋊C4C2×D312yes—
101C22×C42—31no—
102C2×#99—38no—
103C2×#99—64no—
104C2×#99—64no—
105C2×#95—22no—
106C2×#95—62no—
107#342—64no—
108C24×C3—42no—
109C24×C3—42no—
110C24×C3C22×C411yescharacteristic
111C24×C3C22×C412yes—
112C24×C3C2×D411yescharacteristic
113C24×C3C2×D412yes—
114C24×C3C2×D411yescharacteristic
115C24×C3C2×D412yes—
116C24×C3C2×D411yescharacteristic
117C24×C3C2×Q811yescharacteristic
118C24×C3#4112yes—
119C24×C3#4111yescharacteristic
120C24×C3#4111yescharacteristic
121C23×D3—21no—
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—21no—
125C23×D3—42no—
126C23×D3—42no—
127C23×D3C4212yes—
128C23×D3C22⋊C412yes—
129C23×D3C4⋊C412yes—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4—48no—
144C22×C3×C4—48no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4C22⋊C412yes—
150C22×C3×C4C22⋊C412yes—
151C2×#67—24no—
152C2×#67—24no—
153C2×#67—24no—
154C2×#67—24no—
155C2×#67—24no—
156C2×#67—24no—
157C2×#67—24no—
158C2×#67—22no—
159C2×#67—22no—
160C2×#67—24no—
161C2×#67—22no—
162C2×#67—22no—
163C2×#67—24no—
164C2×#67—24no—
165C2×#67C22⋊C418yes—
166C2×#67C4⋊C418yes—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—24no—
176C22×Dic3—42no—
177C22×Dic3—42no—
178C22×Dic3—44no—
179C22×Dic3—44no—
180C22×Dic3—42no—
181C22×Dic3—42no—
182C22×Dic3C4212yes—
183C22×Dic3C22⋊C412yes—
184C22×Dic3C22⋊C412yes—
185C22×Dic3C22⋊C412yes—
186C22×Dic3C4⋊C412yes—
187C2×C4×D3—28no—
188C2×C4×D3—48no—
189C2×C4×D3—44no—
190C2×C4×D3—48no—
191C2×C4×D3—44no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196#217—24no—
197#217—24no—
198#217—24no—
199#217—24no—
200#217—42no—
201#217—42no—
202(C2×D3)⋊C4—28no—
203Dic3⋊C4—28no—
204C4×Dic3—28no—
205C25—31no—
206C25C22×D311yescharacteristic
207C22×D4—34no—
208C22×D4—34no—
209C22×D4—32no—
210C22×D4—32no—
211C22×D4—61no—
212C22×D4—62no—
213C22×D4—62no—
214C22×D4—61no—
215C22×D4—61no—
216C22×D4—61no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—44no—
224C23×C4—62no—
225C23×C4—62no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4—124no—
229C23×C4C4×D311yescharacteristic
230C23×C4C4×D311yescharacteristic
231C23×C4D1212yes—
232C23×C4#6712yes—
233C4×D4—616no—
234C4×D4—616no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—34no—
244C2×C22⋊C4—34no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4C4×D314yescomplemented
257C2×C42—62no—
258C2×C42—62no—
259C2×C42—62no—
260#99—616no—
261#95—44no—
262#95—124no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—44no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C2×C4211yescharacteristic
280C23×C3C2×C22⋊C411yescharacteristic
281C23×C3C2×C22⋊C411yescharacteristic
282C23×C3C2×C22⋊C412yes—
283C23×C3C2×C22⋊C411yescharacteristic
284C23×C3C2×C4⋊C411yescharacteristic
285C23×C3C2×C4⋊C412yes—
286C23×C3#11711yescharacteristic
287C23×C3#11711yescharacteristic
288C23×C3C4×D412yes—
289C23×C3C4×D412yes—
290C23×C3C4×D412yes—
291C23×C3C4×D412yes—
292C23×C3#12012yes—
293C23×C3#12012yes—
294C23×C3#12112yes—
295C23×C3#12111yescharacteristic
296C23×C3#12111yescharacteristic
297C23×C3C22⋊Q812yes—
298C23×C3C22⋊Q811yescharacteristic
299C23×C3C22⋊Q811yescharacteristic
300C23×C3#12312yes—
301C23×C3#12312yes—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—42no—
309C22×D3—44no—
310C22×D3—44no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—44no—
316C22×D3—42no—
317C22×D3—44no—
318C22×D3#9514yescomplemented
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—48no—
327C2×C3×C4—48no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—48no—
331C2×C3×C4—44no—
332C2×C3×C4—84no—
333C2×C3×C4—82no—
334C2×C3×C4—82no—
335C2×C3×C4#9914yescomplemented
336#67—28no—
337#67—28no—
338#67—28no—
339#67—28no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—48no—
355C2×Dic3—44no—
356C2×Dic3—84no—
357C2×Dic3—82no—
358C2×Dic3—82no—
359C2×Dic3#9514yescomplemented
360C4×D3—416no—
361C24—32no—
362C24—32no—
363C24—32no—
364C24—42no—
365C24—42no—
366C24—61no—
367C24—61no—
368C24—61no—
369C24—61no—
370C24—122no—
371C24—122no—
372C24C2×C4×D311yescharacteristic
373C24C2×C4×D312yes—
374C24C2×D1211yescharacteristic
375C24#23512yes—
376C24D3×D412yes—
377C24D3×D411yescharacteristic
378C24D3×D412yes—
379C24#23711yescharacteristic
380C24D3×Q811yescharacteristic
381C24#23911yescharacteristic
382C24C2×#6711yescharacteristic
383C2×D4—38no—
384C2×D4—38no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—64no—
388C2×D4—64no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—64no—
392C2×D4—62no—
393C2×D4—64no—
394C2×D4—62no—
395C2×D4—64no—
396C2×D4—64no—
397C2×D4—62no—
398C2×D4—62no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—68no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—64no—
433C22×C4—128no—
434C22×C4—128no—
435C22×C4—124no—
436C22×C4—124no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4—124no—
440C22×C4—124no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4(C2×D3)⋊C412yes—
444C22×C4(C2×D3)⋊C412yes—
445C4⋊C4—68no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—68no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—122no—
456C22⋊C4—122no—
457C42—68no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3C2×#9511yescharacteristic
485C22×C3C4×C22⋊C411yescharacteristic
486C22×C3C4×C22⋊C411yescharacteristic
487C22×C3#37911yescharacteristic
488C22×C3#38011yescharacteristic
489C22×C3#38011yescharacteristic
490C22×C3#38111yescharacteristic
491C22×C3#38512yes—
492C22×C3#38512yes—
493C22×C3#38612yes—
494C22×C3#38612yes—
495C22×C3C2×#9912yes—
496C22×C3#41012yes—
497C2×D3—24no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—44no—
506C3×C4—24no—
507C3×C4—42no—
508C3×C4—84no—
509Dic3—24no—
510Dic3—42no—
511Dic3—42no—
512Dic3—84no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—34no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—122no—
537C23—124no—
538C23—124no—
539C23—122no—
540C23—122no—
541C23—124no—
542C23—124no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23C42×D311yescharacteristic
547C23#87211yescharacteristic
548C23C4×D1212yes—
549C23D3×C22⋊C411yescharacteristic
550C23D3×C22⋊C411yescharacteristic
551C23D3×C22⋊C412yes—
552C23#67⋊C412yes—
553C23#88212yes—
554C23#88312yes—
555C23#88412yes—
556C23D3×C4⋊C411yescharacteristic
557C23D3×C4⋊C412yes—
558C23#89211yescharacteristic
559C23#89312yes—
560C23#89412yes—
561C23#89511yescharacteristic
562C23#89612yes—
563C23#89711yescharacteristic
564C23C2×(C2×D3)⋊C411yescharacteristic
565C23C4×#6712yes—
566C23#93712yes—
567C23#93811yescharacteristic
568C23#94611yescharacteristic
569D4—68no—
570D4—68no—
571D4—68no—
572D4—68no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—34no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—82no—
598C2×C4—1216no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—128no—
606C2×C4—124no—
607C2×C4—242no—
608C2×C4—244no—
609C2×C4—242no—
610C2×C4#80614yescomplemented
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—22no—
618C2×C3—22no—
619C2×C3C22⋊#9511yescharacteristic
620C2×C3C4×#9912yes—
621C2×C3#178111yescharacteristic
622C2×C3#178211yescharacteristic
623C2×C3#179912yes—
624D3—44no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—21no—
641C22—21no—
642C22—22no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—64no—
652C22—124no—
653C22—124no—
654C22—124no—
655C22—124no—
656C22—124no—
657C22—122no—
658C22—124no—
659C22—122no—
660C22D3×#9511yescharacteristic
661C22#466411yescharacteristic
662C22#466512yes—
663C22#466611yescharacteristic
664C22#466711yescharacteristic
665C22#466812yes—
666C22C4×(C2×D3)⋊C411yescharacteristic
667C22C2×#80612yes—
668C22#495511yescharacteristic
669C22#495612yes—
670C22#498711yescharacteristic
671C22#498912yes—
672C22#500612yes—
673C4—24no—
674C4—42no—
675C4—64no—
676C4—84no—
677C4—122no—
678C4—122no—
679C4—244no—
680C3#1076111yescharacteristic, complemented, Sylow
681C2—22no—
682C2—22no—
683C2—21no—
684C2—21no—
685C2—21no—
686C2—21no—
687C2—22no—
688C2—22no—
689C2—124no—
690C2(C2×D3)⋊#9511yescharacteristic
691C2C4×#80612yes—
692C2#6949811yescharacteristic
693C2#6953012yes—
694C2#6953711yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC3×C22⋊#95C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#67C22
cosocleC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#10761
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C3×C22⋊#95C2semidirect
4. #72127C2semidirect
5. C2×#4527C2semidirect
9. C23×C4×D3C4semidirect
17. C22×(C2×D3)⋊C4C4semidirect
18. C22×Dic3⋊C4C4semidirect
19. C22×C4×Dic3C4semidirect
24. C2×#806C4semidirect
30. C22⋊#95D3semidirect
64. C2×C3×C22⋊C4C2×C4semidirect
73. C2×#217C2×C4semidirect
256. C2×C22⋊C4C4×D3semidirect
318. C22×D3#95semidirect
335. C2×C3×C4#99semidirect
359. C2×Dic3#95semidirect
610. C2×C4#806semidirect
680. C3#10761semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#672
C22×#8062
C3×C22⋊#952
#721272
C2×#45272
#107613