Finite Group Databasebeta

← #10944073

Subgroups — #10944073

Order 768.

1646 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5545C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic, complemented
3C2×#4843C211yesmaximal, characteristic, complemented
4C2×#4773C212yesmaximal, complemented
5#74244C211yesmaximal, characteristic, complemented
6C3×#1804C211yesmaximal, characteristic, complemented
7#70476C211yesmaximal, characteristic, complemented
8#70172C212yesmaximal, complemented
9#69556C211yesmaximal, characteristic, complemented
10#68897C212yesmaximal, complemented
11#68881C212yesmaximal, complemented
12#32911—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C22×#213C2212yes—
17C2×C4×#67—21no—
18C22×(C2×D3)⋊C4—21no—
19C22×D24C2212yes—
20#5545—24no—
21#5545—24no—
22C2×#895C2212yes—
23C2×#893C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×#67⋊C4—21no—
26C2×D3×C22⋊C4—21no—
27C2×#874C2211yescharacteristic, complemented
28C2×C4×D12C2211yescharacteristic
29C2×#872C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic, complemented
31C2×C3×#102C2212yes—
32C3×#382C2211yescharacteristic, complemented
33C2×#821C2212yescomplemented
34C2×(C2×D3)⋊C8C2212yes—
35#5015—24no—
36#4988C2212yescomplemented
37C2×#809C2212yes—
38C2×#809C2212yescomplemented
39C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
40#4843—24no—
41#4773—28no—
42#4672—21no—
43#4672—21no—
44#4671—21no—
45#4671—21no—
46C3×#340C2212yescomplemented
47#4550C2212yescomplemented
48#4529C2212yescomplemented
49C2×#518—31no—
50C2×#459—31no—
51C2×#459—31no—
52C2×#447—32no—
53#2619—31no—
54#1879—32no—
55#1879—32no—
56#1875—32no—
57#1804—31no—
58#1804D311yescharacteristic, complemented
59#1798—31no—
60C24×D3—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C2×D3×D4—22no—
66C2×D3×D4—22no—
67C2×D3×D4—22no—
68C2×D3×D4—22no—
69C2×D3×D4—22no—
70C2×D3×D4D414yescomplemented
71C22×D12—22no—
72C22×D12—21no—
73C22×D12—21no—
74C22×D12C2312yes—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2312yes—
80C3×C4×D4—22no—
81C3×C4×D4D414yescomplemented
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87D4×Dic3—22no—
88D4×Dic3—22no—
89C2×#213—24no—
90C2×#213—24no—
91C2×#213—24no—
92C4×#67—28no—
93C2×(C2×D3)⋊C4—28no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4D411yescharacteristic
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×D24—24no—
109C2×D24—24no—
110C2×D24—24no—
111#895—24no—
112#895—24no—
113#893—22no—
114#893—22no—
115#892—22no—
116#892—22no—
117#67⋊C4—28no—
118D3×C22⋊C4—28no—
119#874—22no—
120#874—22no—
121C4×D12—22no—
122C4×D12D414yescomplemented
123#872—22no—
124#872—22no—
125C3×C4⋊C8—22no—
126C3×#102—24no—
127C3×#95—21no—
128C3×#95—21no—
129#821—24no—
130(C2×D3)⋊C8—24no—
131#809—24no—
132#809—24no—
133C4⋊(C3⋊C8)—22no—
134C23×D4—31no—
135C22×D8—32no—
136C22×D8—32no—
137C2×#127—31no—
138C2×#121—32no—
139#518—64no—
140#518—64no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—31no—
149C22×C22⋊C4—61no—
150C22×C22⋊C4—61no—
151#459—64no—
152#459—64no—
153#447—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic, complemented
156C2×#102—32no—
157C2×#102—32no—
158C2×#102—32no—
159C2×#102C2×D312yes—
160C2×C22⋊C8—32no—
161#394—61no—
162#394—61no—
163#392—61no—
164#392—61no—
165#390—32no—
166#382C2×D311yescharacteristic, complemented
167#340—32no—
168#340C2×D312yescomplemented
169#336—32no—
170C24×C3—21no—
171C23×D3—21no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—22no—
176C23×D3—82no—
177C23×D3—82no—
178C23×D3C2×D411yescharacteristic
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4C2×D412yes—
183C2×C3×D4C2×D412yes—
184C2×C3×D4#4112yes—
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—82no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C2×#67—24no—
193C2×#67—24no—
194C2×#67—24no—
195C22×Dic3—21no—
196C22×Dic3—24no—
197C22×Dic3—82no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201D3×D4—28no—
202D3×D4—28no—
203D3×D4—28no—
204D3×D4—28no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—44no—
215C2×D12—42no—
216C2×D12—42no—
217C2×D12—82no—
218C2×D12—82no—
219C2×D12—82no—
220C2×D12—82no—
221C2×D12C2×D412yes—
222C2×D12C2×D412yes—
223C2×D12#4112yes—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—24no—
229C2×C4×D3—24no—
230C2×C4×D3—82no—
231C2×C4×D3—82no—
232C2×C4×D3C2×D412yes—
233C2×C4×D3#4112yes—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4C2×D412yes—
241C3×C22⋊C4—24no—
242C3×C42C2×D412yes—
243C3×C42C2×D412yes—
244#217—24no—
245#213—48no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—44no—
252(C2×D3)⋊C4—82no—
253(C2×D3)⋊C4—82no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258C4×Dic3—24no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264D24—48no—
265C25—61no—
266C22×D4—34no—
267C22×D4—31no—
268C22×D4—32no—
269C22×D4—62no—
270C22×D4—62no—
271C22×D4—61no—
272C22×D4—62no—
273C22×D4—62no—
274C22×D4—62no—
275C22×D4—62no—
276C22×D4—61no—
277C22×D4—61no—
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—31no—
281C23×C4—61no—
282C23×C4—61no—
283C23×C4—61no—
284C2×D8—64no—
285C2×D8—64no—
286C2×D8—64no—
287C2×D8—64no—
288C2×D8—64no—
289C2×D8—64no—
290C22×C8—32no—
291C22×C8C22×D312yes—
292#127—62no—
293#127—62no—
294#121—64no—
295#121—64no—
296C4×D4—22no—
297C4×D4—34no—
298C4×D4—68no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—68no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4D1214yescomplemented
306#117—62no—
307#117—62no—
308#117—62no—
309#117—62no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—68no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—68no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—62no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4—62no—
327C2×C42—31no—
328C2×C42—61no—
329C2×C42C22×D311yescharacteristic
330C4⋊C8—22no—
331C4⋊C8—62no—
332#102—24no—
333#102—64no—
334#102—64no—
335#102—64no—
336C22⋊C8—64no—
337#95—21no—
338#95—21no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3C22×D411yescharacteristic
345C22×D3—21no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—21no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3—22no—
352C22×D3—216no—
353C22×D3—42no—
354C22×D3—42no—
355C22×D3—42no—
356C22×D3—42no—
357C22×D3—42no—
358C22×D3—42no—
359C22×D3—42no—
360C22×D3—42no—
361C22×D3—42no—
362C22×D3—88no—
363C22×D3—88no—
364C22×D3—84no—
365C22×D3—84no—
366C22×D3#12112yes—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4#12114yes—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—84no—
379C2×C3×C4#12012yes—
380C2×C3×C4#12012yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387#67—216no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—42no—
396C2×Dic3—84no—
397C2×Dic3#12112yes—
398D12—22no—
399D12—22no—
400D12—22no—
401D12—42no—
402D12—42no—
403D12—42no—
404D12—42no—
405D12—84no—
406D12—84no—
407D12—88no—
408D12#12114yes—
409C4×D3—22no—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—84no—
414C4×D3—84no—
415C3×C8—42no—
416C3⋊C8—42no—
417C24—21no—
418C24—31no—
419C24—64no—
420C24—64no—
421C24—64no—
422C24—61no—
423C24—62no—
424C24—242no—
425C24—242no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—32no—
430C2×D4—32no—
431C2×D4—32no—
432C2×D4—64no—
433C2×D4—68no—
434C2×D4—68no—
435C2×D4—68no—
436C2×D4—64no—
437C2×D4—64no—
438C2×D4—68no—
439C2×D4—61no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—62no—
443C2×D4—61no—
444C2×D4—62no—
445C2×D4—61no—
446C2×D4—62no—
447C2×D4—62no—
448C2×D4—124no—
449C2×D4—122no—
450C2×D4—122no—
451C2×D4—242no—
452C2×D4—242no—
453C2×D4—242no—
454C2×D4—242no—
455C2×D4C2×D1212yes—
456C2×D4D3×D412yes—
457C2×D4#23912yes—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—31no—
462C22×C4—32no—
463C22×C4—32no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—64no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—64no—
473C22×C4—64no—
474C22×C4—82no—
475C22×C4—242no—
476C22×C4—242no—
477C22×C4—242no—
478C22×C4D3×D411yescharacteristic
479C22×C4D3×D411yescharacteristic
480C22×C4C23×D311yescharacteristic
481D8—128no—
482D8—128no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—62no—
487C2×C8—62no—
488C2×C8—62no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—32no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—61no—
496C4⋊C4C2×D1212yes—
497C22⋊C4—24no—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C22⋊C4—64no—
504C22⋊C4—124no—
505C22⋊C4—242no—
506C22⋊C4—242no—
507C42—64no—
508C42—61no—
509C42—61no—
510C42C2×D1212yes—
511C42D3×D412yes—
512C22×C3—24no—
513C22×C3—44no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—44no—
517C22×C3C2×#12011yescharacteristic
518C22×C3C2×#12111yescharacteristic
519C22×C3#52511yescharacteristic
520C22×C3#57612yes—
521C22×C3#57612yes—
522C2×D3—22no—
523C2×D3—24no—
524C2×D3—24no—
525C2×D3—42no—
526C2×D3—44no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—44no—
532C2×D3—816no—
533C2×D3—82no—
534C2×D3—82no—
535C2×D3—82no—
536C2×D3—82no—
537C2×D3—82no—
538C2×D3—82no—
539C2×D3—82no—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—42no—
543C3×C4—82no—
544C3×C4C2×#12111yescharacteristic
545C3×C4C2×#12111yescharacteristic
546C3×C4#52511yescharacteristic
547C3×C4#52511yescharacteristic
548Dic3—22no—
549Dic3—41no—
550Dic3—42no—
551Dic3—41no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—32no—
556C23—42no—
557C23—42no—
558C23—616no—
559C23—61no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—61no—
565C23—62no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—122no—
570C23—122no—
571C23—122no—
572C23—122no—
573C23—122no—
574C23—122no—
575C23—248no—
576C23—244no—
577C23—244no—
578C23—248no—
579C23C2×D3×D411yescharacteristic
580D4—22no—
581D4—22no—
582D4—22no—
583D4—34no—
584D4—616no—
585D4—62no—
586D4—62no—
587D4—62no—
588D4—122no—
589D4—122no—
590D4—122no—
591D4—122no—
592D4—244no—
593D4—244no—
594D4—248no—
595D4#89514yes—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—84no—
616C2×C4—122no—
617C2×C4—244no—
618C2×C4—244no—
619C2×C4—244no—
620C2×C4#88212yes—
621C2×C4#88212yes—
622C2×C4C22×D1211yescharacteristic
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4C2×#23911yescharacteristic
626C2×C4D3×#4111yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C8—42no—
629C8—122no—
630C2×C3—44no—
631C2×C3#284711yescharacteristic
632C2×C3#300811yescharacteristic
633C2×C3#300811yescharacteristic
634C2×C3#304112yes—
635C2×C3#312212yes—
636D3—42no—
637D3—44no—
638D3—162no—
639C22—24no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—44no—
644C22—62no—
645C22—64no—
646C22—64no—
647C22—122no—
648C22—121no—
649C22—124no—
650C22—122no—
651C22—121no—
652C22—122no—
653C22—124no—
654C22—2416no—
655C22—242no—
656C22—242no—
657C22—242no—
658C22—242no—
659C22—242no—
660C22—242no—
661C22—242no—
662C22C2×#88211yescharacteristic
663C22D3×#12111yescharacteristic
664C22#562711yescharacteristic
665C22#576912yes—
666C22#577412yes—
667C4—22no—
668C4—22no—
669C4—42no—
670C4—62no—
671C4—82no—
672C4—121no—
673C4—121no—
674C4—122no—
675C4C2×#89511yescharacteristic
676C4#552411yescharacteristic
677C4D3×#12111yescharacteristic
678C4#562711yescharacteristic
679C3#3291111yescharacteristic, complemented, Sylow
680C2—44no—
681C2—122no—
682C2—124no—
683C2—482no—
684C2#7404811yescharacteristic
685C2#7883411yescharacteristic
686C2#7892811yescharacteristic
687C2#8053612yes—
688C2#8114612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
hypercenter#1804D3
hypocenterC3#32911
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. C2×#5545C2semidirect
2. C2×#5015C2semidirect
3. C2×#4843C2semidirect
4. C2×#4773C2semidirect
5. #74244C2semidirect
6. C3×#1804C2semidirect
7. #70476C2semidirect
8. #70172C2semidirect
9. #69556C2semidirect
10. #68897C2semidirect
11. #68881C2semidirect
27. C2×#874C22semidirect
30. C2×C3×C4⋊C8C22semidirect
32. C3×#382C22semidirect
33. C2×#821C22semidirect
36. #4988C22semidirect
38. C2×#809C22semidirect
39. C2×C4⋊(C3⋊C8)C22semidirect
46. C3×#340C22semidirect
47. #4550C22semidirect
48. #4529C22semidirect
58. #1804D3semidirect
70. C2×D3×D4D4semidirect
81. C3×C4×D4D4semidirect
122. C4×D12D4semidirect
155. C2×C4⋊C8C2×D3semidirect
166. #382C2×D3semidirect
168. #340C2×D3semidirect
305. C4×D4D12semidirect
679. C3#32911semidirect

Maximal subgroups

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

subgroupindex
C2×#55452
C2×#50152
C2×#48432
C2×#47732
#742442
C3×#18042
#704762
#701722
#695562
#688972
#688812
#329113