Finite Group Databasebeta

← #10827341

Subgroups — #10827341

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
2C3×#2408C211yesmaximal, characteristic, complemented
3C2×#5217C211yesmaximal, characteristic, complemented
4C22×#832C211yesmaximal, characteristic, complemented
5#71684C212yesmaximal, complemented
6#71677C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69657C212yesmaximal, complemented
9#69638C212yesmaximal, complemented
10#69515C211yesmaximal, characteristic, complemented
11#69501C211yesmaximal, characteristic, complemented
12#23960—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#237—21no—
15C2×C3×#121C2212yes—
16C2×C3×#120—21no—
17C2×C3×C4×D4C2211yescharacteristic
18C22×#213C2212yes—
19C2×#930C2211yescharacteristic
20C2×#930C2211yescharacteristic, complemented
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic, complemented
23C23×C3⋊C8C2211yescharacteristic
24#5555—24no—
25#5555—24no—
26C2×#897—21no—
27C2×#886—21no—
28C2×#886—21no—
29C2×C4×D12C2211yescharacteristic, complemented
30C2×#869C2211yescharacteristic, complemented
31C3×#394—21no—
32C3×#392—21no—
33C3×#390C2212yes—
34C3×#380C2211yescharacteristic
35C3×#379—21no—
36#5217—24no—
37C2×#832—24no—
38C2×#832C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2212yescomplemented
43C2×#809C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C2×#547—31no—
49C2×#447—31no—
50C22×#102—31no—
51#2408D311yescharacteristic, complemented
52#2023—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1874—32no—
57#1869—31no—
58#1823—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—42no—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C22×#67—21no—
72C23×Dic3—21no—
73C23×Dic3—21no—
74C2×#237—28no—
75C2×#237—44no—
76C2×#237—44no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C3×#121—24no—
81C3×#121—42no—
82C3×#120—44no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#213—24no—
95C2×#213—24no—
96C2×#213—42no—
97#930—22no—
98#930—22no—
99#930—22no—
100#930D414yescomplemented
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
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⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C22×C3⋊C8—22no—
115C22×C3⋊C8—42no—
116C22×C3⋊C8C2312yes—
117C22×C3⋊C8C2312yes—
118#897—28no—
119#886—28no—
120#886—28no—
121C4×D12—22no—
122C4×D12—22no—
123#869—22no—
124#869—22no—
125C3×#95—21no—
126#832—24no—
127#832—24no—
128#832—24no—
129#830—22no—
130#830—42no—
131#809—42no—
132C22×#41—61no—
133C23×D4C2×D311yescharacteristic
134C22×D8—32no—
135C23×C8—31no—
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×#121C2×D312yes—
145C2×#120—21no—
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⋊C8—31no—
157C2×C22⋊C8—32no—
158#394—21no—
159#392—21no—
160#390C2×D312yes—
161#380C2×D311yescharacteristic
162#379—21no—
163#340—32no—
164#340—32no—
165#340—32no—
166#336—32no—
167C24×C3—22no—
168C24×C3—44no—
169C24×C3—42no—
170C24×C3—42no—
171C24×C3—42no—
172C24×C3—82no—
173C24×C3C2×D411yescharacteristic
174C24×C3C2×D411yescharacteristic
175C23×D3—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—28no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—48no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—42no—
190C2×C3×D4—82no—
191C2×C3×D4D814yescomplemented
192C2×C3×D4#4114yescomplemented
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—24no—
208C2×#67—24no—
209C2×#67—42no—
210C2×#67—42no—
211C2×#67—42no—
212C2×#67—42no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—42no—
222#237—432no—
223C2×D12—22no—
224C2×D12—22no—
225C2×D12—21no—
226C2×D12C2×D412yes—
227C2×C4×D3—24no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×Dic6—24no—
231C2×Dic6—42no—
232C2×Dic6—42no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—44no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—44no—
241C3×C22⋊C4—82no—
242C3×C42—21no—
243C3×C42—21no—
244#213—48no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248C4⋊Dic3—24no—
249C4⋊Dic3—24no—
250C4⋊Dic3—24no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3C2×D412yes—
255C4⋊Dic3C2×D412yes—
256C4⋊Dic3C2×D412yes—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—44no—
264C25—21no—
265C2×#41—68no—
266C2×#41—124no—
267C2×#41—124no—
268C22×Q8—61no—
269C22×D4—24no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—31no—
275C22×D4—42no—
276C22×D4—61no—
277C22×D4—61no—
278C22×D4#6711yescharacteristic
279C22×D4C22×D312yes—
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—32no—
289C22×C8—32no—
290C22×C8—62no—
291C22×C8—122no—
292C4⋊Q8—62no—
293C4⋊Q8—62no—
294#123—68no—
295#123—68no—
296C22⋊Q8—68no—
297#121—24no—
298#121—34no—
299#121—42no—
300#121—62no—
301#121—62no—
302#121—62no—
303#120—44no—
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—62no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4#6711yescharacteristic
319C2×C4⋊C4C22×D312yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4#6711yescharacteristic
328C2×C22⋊C4#6711yescharacteristic
329C2×C42C22×D311yescharacteristic
330#102—64no—
331#102—64no—
332#102—64no—
333#102—122no—
334C22⋊C8—62no—
335C22⋊C8—122no—
336#95—21no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3—24no—
344C23×C3—48no—
345C23×C3—42no—
346C23×C3—42no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—88no—
352C23×C3—88no—
353C23×C3—84no—
354C23×C3—84no—
355C23×C3#12012yes—
356C23×C3C2×D812yes—
357C23×C3C2×D812yes—
358C23×C3AGL(1,ℤ/8)12yes—
359C23×C3C22×D411yescharacteristic
360C22×D3—22no—
361C22×D3—42no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—42no—
365C3×D4—28no—
366C3×D4—24no—
367C3×D4—24no—
368C3×D4—48no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—88no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4#12012yes—
389C2×C3×C4C2×D812yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4112yes—
393C2×C3×C4H(5,𝔽2)12yes—
394#67—48no—
395#67—48no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—48no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—48no—
407D12—24no—
408D12—41no—
409D12—41no—
410C4×D3—48no—
411Dic6—48no—
412C3⋊C8—42no—
413C3⋊C8—42no—
414C24—22no—
415C24—44no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—61no—
420C24—82no—
421C24C2×#6711yescharacteristic
422C24C2×#6711yescharacteristic
423#41—1232no—
424C2×Q8—64no—
425C2×Q8—122no—
426C2×Q8—122no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—28no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—32no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—61no—
445C2×D4—62no—
446C2×D4—62no—
447C2×D4—82no—
448C2×D4—122no—
449C2×D4—122no—
450C2×D4—122no—
451C2×D4—122no—
452C2×D4#21314yescomplemented
453C2×D4#23714yescomplemented
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4D3×D412yes—
474C22×C4D3×D412yes—
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C23×D311yescharacteristic
480D8—128no—
481C2×C8—62no—
482C2×C8—62no—
483C2×C8—62no—
484C2×C8—62no—
485C2×C8—62no—
486C2×C8—62no—
487C2×C8—124no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—32no—
492C4⋊C4—32no—
493C4⋊C4—32no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—64no—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—82no—
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—44no—
521C22×C3—44no—
522C22×C3—44no—
523C22×C3—816no—
524C22×C3—82no—
525C22×C3—82no—
526C22×C3—82no—
527C22×C3—82no—
528C22×C3#44714yes—
529C22×C3C2×#12011yescharacteristic
530C22×C3#54612yes—
531C22×C3C22×D811yescharacteristic
532C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
533C22×C3#57712yes—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—82no—
538C2×D3—82no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—82no—
544C3×C4#52511yescharacteristic
545C3×C4#53511yescharacteristic
546C3×C4#54612yes—
547Dic3—42no—
548Dic3—42no—
549Dic3—42no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—22no—
554C23—21no—
555C23—22no—
556C23—24no—
557C23—48no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—62no—
565C23—88no—
566C23—88no—
567C23—84no—
568C23—84no—
569C23—121no—
570C23—122no—
571C23—121no—
572C23—122no—
573C23C2×#21312yes—
574C23C2×#21312yes—
575C23#93712yes—
576C23#94912yes—
577C23C22×#6711yescharacteristic
578Q8—128no—
579D4—28no—
580D4—24no—
581D4—24no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—64no—
586D4—88no—
587D4—128no—
588D4—128no—
589D4—121no—
590D4—121no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—128no—
611C2×C4—128no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—128no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4C2×#21312yes—
620C2×C4#93712yes—
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4C2×#23712yes—
624C2×C4#100412yes—
625C8—122no—
626C8—122no—
627C2×C3—24no—
628C2×C3—42no—
629C2×C3—84no—
630C2×C3#240411yescharacteristic
631C2×C3C2×#44711yescharacteristic
632C2×C3#301011yescharacteristic
633C2×C3#329412yes—
634C2×C3#330612yes—
635D3—82no—
636C22—24no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—22no—
642C22—42no—
643C22—41no—
644C22—41no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—242no—
657C22—242no—
658C22#503614yes—
659C22C22×#21311yescharacteristic
660C22C2×#93711yescharacteristic
661C22#580612yes—
662C22C2×#94911yescharacteristic
663C22#583612yes—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—82no—
669C4—122no—
670C4—122no—
671C4—122no—
672C4#558611yescharacteristic
673C4#559112yes—
674C4#559211yescharacteristic
675C3#2396011yescharacteristic, complemented, Sylow
676C2—24no—
677C2—42no—
678C2—84no—
679C2—242no—
680C2#8258111yescharacteristic
681C2C2×#503611yescharacteristic
682C2#8339011yescharacteristic
683C2#8348212yes—
684C2#8349612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#23960
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#5555C2semidirect
2. C3×#2408C2semidirect
3. C2×#5217C2semidirect
4. C22×#832C2semidirect
5. #71684C2semidirect
6. #71677C2semidirect
7. #71610C2semidirect
8. #69657C2semidirect
9. #69638C2semidirect
10. #69515C2semidirect
11. #69501C2semidirect
20. C2×#930C22semidirect
22. C22×C4⋊Dic3C22semidirect
29. C2×C4×D12C22semidirect
30. C2×#869C22semidirect
42. C2×#830C22semidirect
43. C2×#809C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
51. #2408D3semidirect
100. #930D4semidirect
110. C2×C4⋊Dic3D4semidirect
191. C2×C3×D4D8semidirect
192. C2×C3×D4#41semidirect
452. C2×D4#213semidirect
453. C2×D4#237semidirect
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
C3×#24082
C2×#52172
C22×#8322
#716842
#716772
#716102
#696572
#696382
#695152
#695012
#239603