Finite Group Databasebeta

← #10827343

Subgroups — #10827343

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5555C211yesmaximal, characteristic, complemented
2C22×#821C211yesmaximal, characteristic, complemented
3#82403C211yesmaximal, characteristic, complemented
4C2×#4732C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69961C212yesmaximal, complemented
8#69932C212yesmaximal, complemented
9#69924C211yesmaximal, characteristic, complemented
10#68678C212yesmaximal, complemented
11#68673C212yesmaximal, complemented
12#23960—31nomaximal, Sylow
13C22×#237—21no—
14C23×D12C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#937—21no—
18C2×#930C2212yes—
19C2×#930C2211yescharacteristic
20C2×#930C2211yescharacteristic, complemented
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic, complemented
23C22×D24C2212yes—
24#5555—24no—
25#5555—24no—
26C2×#897—21no—
27C2×#886—21no—
28C2×#886—21no—
29C2×C4×D12C2211yescharacteristic, complemented
30C2×#869C2211yescharacteristic, complemented
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2212yescomplemented
33#5212C2211yescharacteristic
34C2×#821—24no—
35C2×#821C2212yes—
36C2×#821C2212yes—
37C2×#821C2212yes—
38C2×#821C2212yescomplemented
39#4991—21no—
40#4960—21no—
41#4955—21no—
42#4939C2212yes—
43#4732—24no—
44C3×#336C2212yescomplemented
45#4551C2212yescomplemented
46#4551C2212yescomplemented
47#4551C2212yescomplemented
48C2×#547—31no—
49C2×#447—31no—
50C22×#102—31no—
51#2408—31no—
52#2023—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1874—32no—
57#1869D311yescharacteristic, complemented
58#1823—31no—
59C24×D3—21no—
60C22×C3×D4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C22×#67—22no—
66C23×Dic3—21no—
67C23×Dic3—21no—
68C2×#237—28no—
69C2×#237—44no—
70C2×#237—44no—
71C22×D12—22no—
72C22×D12—24no—
73C22×D12—22no—
74C22×D12—21no—
75C22×D12—42no—
76C22×D12C2311yescharacteristic
77C22×D12C2312yes—
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×C3×C8—22no—
81C22×C3×C8—42no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91#937—44no—
92#930—22no—
93#930—22no—
94#930—24no—
95#930—22no—
96#930—42no—
97#930D414yescomplemented
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4—22no—
103C2×(C2×D3)⋊C4—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—22no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—22no—
110C2×C4⋊Dic3D414yescomplemented
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×D24—24no—
116C2×D24—24no—
117C2×D24—42no—
118#897—28no—
119#886—28no—
120#886—28no—
121C4×D12—22no—
122C4×D12—22no—
123#869—22no—
124#869—22no—
125C3×C22⋊C8—22no—
126C3×C22⋊C8—42no—
127C3⋊#95—21no—
128#821—24no—
129#821—24no—
130#821—24no—
131#821—42no—
132C22×#41—61no—
133C23×D4—31no—
134C22×D8—32no—
135C23×C8C2×D311yescharacteristic
136#547—64no—
137#547—64no—
138C2×C4⋊Q8—31no—
139C2×#123—61no—
140C2×#123—61no—
141C2×C22⋊Q8—61no—
142C2×#121—31no—
143C2×#121—31no—
144C2×#121—32no—
145C2×#120—61no—
146C2×C4×D4—31no—
147C2×C4×D4C2×D311yescharacteristic
148C22×C4⋊C4—31no—
149C22×C4⋊C4—31no—
150#447—64no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—64no—
156C2×C22⋊C8C2×D311yescharacteristic
157C2×C22⋊C8C2×D312yescomplemented
158#394—61no—
159#392—61no—
160#390—32no—
161#380—31no—
162#379—61no—
163#340—32no—
164#340—32no—
165#340—32no—
166#336C2×D312yescomplemented
167C24×C3C2×D411yescharacteristic
168C24×C3C2×D411yescharacteristic
169C23×D3—21no—
170C23×D3—22no—
171C23×D3—44no—
172C23×D3—42no—
173C23×D3—42no—
174C23×D3—42no—
175C23×D3—82no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4D814yescomplemented
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2411yescharacteristic
192C2×#67—24no—
193C2×#67—24no—
194C2×#67—42no—
195C2×#67—42no—
196C2×#67—42no—
197C2×#67—42no—
198C2×#67—44no—
199C2×#67—44no—
200C2×#67—82no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—42no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211#237—432no—
212C2×D12—28no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12—21no—
220C2×D12—22no—
221C2×D12—48no—
222C2×D12—44no—
223C2×D12—44no—
224C2×D12—42no—
225C2×D12C2×D412yes—
226C2×D12#4114yescomplemented
227C2×C4×D3—24no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×Dic6—24no—
231C2×Dic6—42no—
232C2×Dic6—42no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C2×C3×C8—44no—
240C3×C4⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C42—21no—
244C3×C42—21no—
245#217—42no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—44no—
250(C2×D3)⋊C4—44no—
251(C2×D3)⋊C4—82no—
252C4⋊Dic3—24no—
253C4⋊Dic3—24no—
254C4⋊Dic3—24no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—22no—
259C4⋊Dic3—22no—
260C4⋊Dic3C2×D412yes—
261C4⋊Dic3C2×D412yes—
262C4⋊Dic3C2×D412yes—
263D24—48no—
264C25—61no—
265C2×#41—68no—
266C2×#41—124no—
267C2×#41—124no—
268C22×Q8—61no—
269C22×D4—31no—
270C22×D4—32no—
271C22×D4—64no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—62no—
275C22×D4—61no—
276C22×D4—62no—
277C22×D4—62no—
278C22×D4—122no—
279C22×D4D1211yescharacteristic
280C23×C4—61no—
281C23×C4—61no—
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C2×D8—64no—
286C2×D8—64no—
287C2×D8—122no—
288C22×C8—22no—
289C22×C8—42no—
290C22×C8C22×D312yes—
291C22×C8C22×D312yes—
292C4⋊Q8—62no—
293C4⋊Q8—62no—
294#123—68no—
295#123—68no—
296C22⋊Q8—68no—
297#121—34no—
298#121—62no—
299#121—62no—
300#121—62no—
301#121—64no—
302#121—122no—
303#120—124no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—62no—
307C4×D4—62no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—62no—
319C2×C4⋊C4D1211yescharacteristic
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—62no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4—62no—
327C2×C22⋊C4D1211yescharacteristic
328C2×C22⋊C4D1211yescharacteristic
329C2×C42C22×D311yescharacteristic
330#102—64no—
331#102—64no—
332#102—64no—
333#102—122no—
334C22⋊C8—22no—
335C22⋊C8—42no—
336#95—61no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3#12012yes—
344C23×C3C2×D812yes—
345C23×C3C2×D812yes—
346C23×C3AGL(1,ℤ/8)12yes—
347C23×C3C22×D411yescharacteristic
348C22×D3—22no—
349C22×D3—24no—
350C22×D3—48no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3—42no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—42no—
358C22×D3—42no—
359C22×D3—42no—
360C22×D3—42no—
361C22×D3—88no—
362C22×D3—88no—
363C22×D3—84no—
364C22×D3—84no—
365C3×D4—28no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—42no—
379C2×C3×C4#12012yes—
380C2×C3×C4C2×D812yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4112yes—
384C2×C3×C4H(5,𝔽2)12yes—
385#67—48no—
386#67—48no—
387#67—88no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—48no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—48no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402D12—24no—
403D12—24no—
404D12—24no—
405D12—48no—
406D12—41no—
407D12—41no—
408D12—42no—
409D12—42no—
410C4×D3—48no—
411Dic6—48no—
412C3×C8—42no—
413C3×C8—42no—
414C24—61no—
415C24—62no—
416C24—124no—
417C24—122no—
418C24—122no—
419C24—122no—
420C24—242no—
421C24C2×D1211yescharacteristic
422C24C2×D1211yescharacteristic
423#41—1232no—
424C2×Q8—64no—
425C2×Q8—122no—
426C2×Q8—122no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—32no—
430C2×D4—34no—
431C2×D4—68no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—61no—
437C2×D4—62no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—128no—
443C2×D4—124no—
444C2×D4—124no—
445C2×D4—122no—
446C2×D4—122no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—122no—
452C2×D4—242no—
453C2×D4D2414yescomplemented
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4C2×D1211yescharacteristic
474C22×C4C2×D1211yescharacteristic
475C22×C4C2×D1211yescharacteristic
476C22×C4C2×D1211yescharacteristic
477C22×C4D3×D412yes—
478C22×C4D3×D412yes—
479C22×C4C23×D311yescharacteristic
480D8—128no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C4⋊C4—22no—
489C4⋊C4—32no—
490C4⋊C4—32no—
491C4⋊C4—32no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—64no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—64no—
503C22⋊C4—64no—
504C22⋊C4—64no—
505C22⋊C4—124no—
506C22⋊C4—122no—
507C22⋊C4—124no—
508C22⋊C4—242no—
509C42—21no—
510C42—21no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—24no—
514C22×C3—24no—
515C22×C3—22no—
516C22×C3—24no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3#44714yes—
521C22×C3C2×#12011yescharacteristic
522C22×C3#54612yes—
523C22×C3C22×D811yescharacteristic
524C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
525C22×C3#57712yes—
526C2×D3—44no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—816no—
533C2×D3—82no—
534C2×D3—82no—
535C2×D3—82no—
536C2×D3—82no—
537C2×D3—82no—
538C2×D3—82no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4#52511yescharacteristic
544C3×C4#53511yescharacteristic
545C3×C4#54612yes—
546Dic3—42no—
547Dic3—42no—
548Dic3—42no—
549Dic3—82no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—62no—
557C23—64no—
558C23—128no—
559C23—122no—
560C23—121no—
561C23—122no—
562C23—121no—
563C23—122no—
564C23—122no—
565C23—122no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—248no—
570C23—248no—
571C23—244no—
572C23—244no—
573C23#88212yes—
574C23C2×D2412yes—
575C23C2×D2412yes—
576C23#90812yes—
577C23C22×D1211yescharacteristic
578Q8—128no—
579D4—28no—
580D4—64no—
581D4—64no—
582D4—64no—
583D4—128no—
584D4—128no—
585D4—128no—
586D4—121no—
587D4—122no—
588D4—121no—
589D4—122no—
590D4—248no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—128no—
608C2×C4—128no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—128no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4#88212yes—
620C2×C4C2×D2412yes—
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4#100412yes—
624C2×C4D3×#4112yes—
625C8—42no—
626C8—42no—
627C2×C3—24no—
628C2×C3—42no—
629C2×C3#240411yescharacteristic
630C2×C3C2×#44711yescharacteristic
631C2×C3#301011yescharacteristic
632C2×C3#329412yes—
633C2×C3#330612yes—
634D3—82no—
635D3—84no—
636C22—24no—
637C22—22no—
638C22—22no—
639C22—24no—
640C22—22no—
641C22—24no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—122no—
646C22—124no—
647C22—122no—
648C22—124no—
649C22—122no—
650C22—124no—
651C22—2416no—
652C22—242no—
653C22—242no—
654C22—242no—
655C22—242no—
656C22—242no—
657C22—242no—
658C22#473214yes—
659C22C2×#88211yescharacteristic
660C22#555412yes—
661C22C22×D2411yescharacteristic
662C22C2×#90811yescharacteristic
663C22#575112yes—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—122no—
669C4—122no—
670C4—122no—
671C4—242no—
672C4#559211yescharacteristic
673C4#561012yes—
674C4#565611yescharacteristic
675C3#2396011yescharacteristic, complemented, Sylow
676C2—24no—
677C2—42no—
678C2—244no—
679C2—242no—
680C2#7488511yescharacteristic
681C2C2×#473211yescharacteristic
682C2#7867811yescharacteristic
683C2#7943812yes—
684C2#7949912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
hypercenter#1869D3
hypocenterC3#23960
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
1. C2×#5555C2semidirect
2. C22×#821C2semidirect
3. #82403C2semidirect
4. C2×#4732C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69961C2semidirect
8. #69932C2semidirect
9. #69924C2semidirect
10. #68678C2semidirect
11. #68673C2semidirect
20. C2×#930C22semidirect
22. C22×C4⋊Dic3C22semidirect
29. C2×C4×D12C22semidirect
30. C2×#869C22semidirect
32. C2×C3×C22⋊C8C22semidirect
38. C2×#821C22semidirect
44. C3×#336C22semidirect
45. #4551C22semidirect
46. #4551C22semidirect
47. #4551C22semidirect
57. #1869D3semidirect
97. #930D4semidirect
110. C2×C4⋊Dic3D4semidirect
157. C2×C22⋊C8C2×D3semidirect
166. #336C2×D3semidirect
178. C2×C3×D4D8semidirect
226. C2×D12#41semidirect
453. C2×D4D24semidirect
675. C3#23960semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C22×#8212
#824032
C2×#47322
C3×#18692
#719732
#699612
#699322
#699242
#686782
#686732
#239603