Finite Group Databasebeta

← #10830818

Subgroups — #10830818

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic, complemented
2C2×#5244C212yesmaximal, complemented
3#82723C212yesmaximal, complemented
4C4×#937C211yesmaximal, characteristic, complemented
5#72142C212yesmaximal, complemented
6#71851C212yesmaximal, complemented
7#71850C212yesmaximal, complemented
8#71686C212yesmaximal, complemented
9#71619C211yesmaximal, characteristic, complemented
10#24260—31nomaximal, Sylow
11C2×C3×H(5,𝔽2)C2212yes—
12C23×C4×D3C2211yescharacteristic, complemented
13C3×#547—22no—
14C3×#546—24no—
15C3×D42—22no—
16C3×#537—21no—
17C3×#534—21no—
18C3×#534C2212yes—
19C3×#532—21no—
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×#938—22no—
24C2×#937C2211yescharacteristic
25C2×#937C2214yescomplemented
26C2×C4×#67—21no—
27#5244—28no—
28#5220C2211yescharacteristic
29#5220C2214yescomplemented
30C2×#834C2212yes—
31C2×#834C2214yescomplemented
32C2×#834C2214yescomplemented
33#5212C2212yescomplemented
34C4×#217C2211yescharacteristic, complemented
35#4989—21no—
36#4987—22no—
37#4956—24no—
38#4956—21no—
39#4955C2211yescharacteristic, complemented
40#4946C2211yescharacteristic, complemented
41C4×(C2×D3)⋊C4—21no—
42#4526C2212yescomplemented
43#3496D311yescharacteristic, complemented
44C2×B(4,𝔽2)—32no—
45#2847—32no—
46C4×#120—31no—
47#2044—32no—
48#2021—32no—
49#1964—32no—
50#1799—31no—
51#1790—32no—
52C24×D3C2311yescharacteristic
53C3×H(5,𝔽2)—24no—
54C3×H(5,𝔽2)—24no—
55C3×H(5,𝔽2)—44no—
56C2×C3×#41—22no—
57C2×C3×#41—24no—
58C22×C3×D4—22no—
59C22×C3×D4—24no—
60C22×C3×D4—21no—
61C22×C3×D4D412yescomplemented
62C22×C3×D4C2314yes—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4D411yescharacteristic, complemented
65C23×C3×C4C2311yescharacteristic
66C22×#67—21no—
67C22×#67—24no—
68C23×Dic3C2311yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3—22no—
71C22×C4×D3—22no—
72C22×C4×D3—21no—
73C22×C4×D3—44no—
74C3×C4⋊Q8—21no—
75C3×#127—21no—
76C3×#124—22no—
77C3×#123—24no—
78C3×#123—24no—
79C3×C22⋊Q8—24no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—24no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#120—24no—
90C3×#120—24no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D412yescomplemented
97C2×C3×C4⋊C4D411yescharacteristic, complemented
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4D411yescharacteristic, complemented
100C2×C3×C22⋊C4D414yescomplemented
101C2×C3×C22⋊C4C2311yescharacteristic
102C2×C3×C42D411yescharacteristic, complemented
103C2×#217—21no—
104C2×#217—24no—
105C2×#217C2311yescharacteristic
106C2×#217C2314yes—
107C2×#217C2311yescharacteristic
108#938—48no—
109#937—22no—
110#937—28no—
111#937—44no—
112#937D414yescomplemented
113C4×#67—44no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—24no—
118C2×C4⋊Dic3—22no—
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121#834—24no—
122#834—28no—
123#834—44no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—22no—
128C3⋊#95—24no—
129C2×H(5,𝔽2)C2×D312yes—
130C24×C4—31no—
131#547—22no—
132#546—24no—
133D42—22no—
134#537—21no—
135#534—21no—
136#534C2×D312yes—
137#532—21no—
138C2×#121—21no—
139C2×#121—62no—
140C2×#121C2×D312yes—
141C2×#120—31no—
142C2×#120—34no—
143C2×C4×D4—61no—
144C2×C4×D4C2×D311yescharacteristic
145B(4,𝔽2)—68no—
146C2×#99—32no—
147C2×#99—34no—
148C2×#99—34no—
149#386—34no—
150#386—31no—
151#386—64no—
152#386—61no—
153#385—31no—
154#385—61no—
155#380—32no—
156#380—62no—
157#379—31no—
158C4×C22⋊C4—31no—
159C4×C22⋊C4—61no—
160#342—32no—
161C24×C3—22no—
162C24×C3C2×D414yes—
163C24×C3C2×D411yescharacteristic
164C24×C3C2411yescharacteristic
165C23×D3—21no—
166C23×D3—22no—
167C23×D3—21no—
168C23×D3—22no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3C2×D412yescomplemented
172C3×#41—48no—
173C3×#41—48no—
174C3×#41—44no—
175C3×#41—44no—
176C2×C3×Q8—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—44no—
192C2×C3×D4—48no—
193C2×C3×D4—48no—
194C2×C3×D4—44no—
195C2×C3×D4—44no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D412yes—
198C2×C3×D4#4112yes—
199C22×C3×C4—22no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D411yescharacteristic
213C2×#67—24no—
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—48no—
217C2×#67—48no—
218C2×#67—48no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—22no—
222C22×Dic3—21no—
223C22×Dic3—24no—
224C22×Dic3—42no—
225C2×C4×D3—24no—
226C2×C4×D3—28no—
227C2×C4×D3—42no—
228C2×C4×D3—44no—
229C2×C4×D3—42no—
230C2×C4×D3—44no—
231C2×C4×D3—42no—
232C2×C4×D3—44no—
233C2×C4×D3—41no—
234C2×C4×D3—42no—
235C2×C4×D3—41no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—24no—
247C3×C42—21no—
248C3×C42—21no—
249#217—21no—
250#217—24no—
251#217—22no—
252#217—24no—
253#217—42no—
254#217—48no—
255#217C2×D412yescomplemented
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—48no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260C4⋊Dic3—44no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
263C25—31no—
264H(5,𝔽2)—24no—
265H(5,𝔽2)—24no—
266H(5,𝔽2)—44no—
267C2×#41—22no—
268C2×#41—24no—
269C22×D4—22no—
270C22×D4—24no—
271C22×D4—21no—
272C22×D4—61no—
273C22×D4—64no—
274C22×D4#6712yescomplemented
275C22×D4C22×D314yes—
276C22×D4C22×D311yescharacteristic
277C23×C4—31no—
278C23×C4—61no—
279C23×C4—62no—
280C23×C4—61no—
281C23×C4—62no—
282C23×C4—124no—
283C23×C4#6711yescharacteristic, complemented
284C23×C4C22×D311yescharacteristic
285C4⋊Q8—21no—
286#127—21no—
287#124—22no—
288#123—24no—
289#123—24no—
290C22⋊Q8—24no—
291#121—22no—
292#121—22no—
293#121—24no—
294#121—22no—
295#121—22no—
296#121—22no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—128no—
301#120—24no—
302#120—24no—
303#120—34no—
304#120—62no—
305#120—68no—
306#120—124no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—124no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4#6711yescharacteristic, complemented
316C2×C4⋊C4#6712yescomplemented
317C2×C22⋊C4—22no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—31no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—64no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—64no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4#6711yescharacteristic, complemented
328C2×C22⋊C4#6714yescomplemented
329C2×C22⋊C4C22×D311yescharacteristic
330C2×C42—61no—
331C2×C42#6711yescharacteristic, complemented
332#99—68no—
333#99—64no—
334#99—124no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—62no—
339#95—64no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—24no—
343C23×C3—24no—
344C23×C3—24no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—44no—
350C23×C3—44no—
351C23×C3—44no—
352C23×C3—82no—
353C23×C3—82no—
354C23×C3#12112yescomplemented
355C23×C3C22×D411yescharacteristic
356C23×C3C22×D411yescharacteristic
357C23×C3C22×D412yes—
358C23×C3C2×#4111yescharacteristic
359C23×C3H(5,𝔽2)12yes—
360C22×D3—22no—
361C22×D3—22no—
362C22×D3—24no—
363C22×D3—24no—
364C22×D3—22no—
365C22×D3—24no—
366C22×D3—44no—
367C22×D3—44no—
368C22×D3—48no—
369C22×D3—41no—
370C22×D3—42no—
371C22×D3—41no—
372C22×D3—42no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—42no—
376C3×Q8—44no—
377C3×D4—24no—
378C3×D4—24no—
379C3×D4—44no—
380C3×D4—48no—
381C3×D4—44no—
382C3×D4—44no—
383C3×D4—44no—
384C3×D4—44no—
385C3×D4—44no—
386C3×D4—82no—
387C3×D4—82no—
388C3×D4—82no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—24no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—24no—
396C2×C3×C4—24no—
397C2×C3×C4—24no—
398C2×C3×C4—42no—
399C2×C3×C4—41no—
400C2×C3×C4—48no—
401C2×C3×C4—44no—
402C2×C3×C4—42no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—81no—
406C2×C3×C4#12011yescharacteristic
407C2×C3×C4#12012yes—
408C2×C3×C4#12011yescharacteristic
409C2×C3×C4#12112yescomplemented
410#67—48no—
411#67—816no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—22no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—42no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—44no—
422C2×Dic3—41no—
423C2×Dic3—44no—
424C2×Dic3—41no—
425C2×Dic3—44no—
426C4×D3—44no—
427C4×D3—48no—
428C4×D3—44no—
429C24—22no—
430C24—32no—
431C24—61no—
432C24—62no—
433C24—61no—
434C24—62no—
435C24—122no—
436C24—122no—
437C24C2×#6714yes—
438C24C2×#6711yescharacteristic
439C24C23×D311yescharacteristic
440#41—48no—
441#41—48no—
442#41—44no—
443#41—44no—
444C2×Q8—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—48no—
461C2×D4—48no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—64no—
465C2×D4—82no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—128no—
469C2×D4—128no—
470C2×D4—128no—
471C2×D4D3×D412yes—
472C2×D4#23712yes—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—64no—
480C22×C4—68no—
481C22×C4—61no—
482C22×C4—62no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—64no—
486C22×C4—122no—
487C22×C4—124no—
488C22×C4—124no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—124no—
492C22×C4—122no—
493C22×C4—121no—
494C22×C4—122no—
495C22×C4—121no—
496C22×C4C2×#6714yes—
497C22×C4C2×#6711yescharacteristic
498C22×C4C2×#6711yescharacteristic
499C22×C4C2×#6712yes—
500C22×C4C2×#6711yescharacteristic
501C22×C4C2×#6711yescharacteristic
502C22×C4C2×#6712yes—
503C22×C4C2×#6711yescharacteristic
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—124no—
510C4⋊C4—122no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—24no—
517C22⋊C4—32no—
518C22⋊C4—61no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—62no—
522C22⋊C4—64no—
523C22⋊C4—128no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—128no—
528C42—21no—
529C42—21no—
530C42—122no—
531C22×C3—21no—
532C22×C3—22no—
533C22×C3—21no—
534C22×C3—22no—
535C22×C3—22no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—44no—
543C22×C3—44no—
544C22×C3—44no—
545C22×C3—81no—
546C22×C3—84no—
547C22×C3—84no—
548C22×C3—164no—
549C22×C3C2×#12011yescharacteristic
550C22×C3C2×#12111yescharacteristic
551C22×C3#53511yescharacteristic
552C22×C3D4212yes—
553C22×C3#54612yes—
554C2×D3—24no—
555C2×D3—28no—
556C2×D3—42no—
557C2×D3—44no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—44no—
561C2×D3—44no—
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—42no—
565C3×C4—22no—
566C3×C4—21no—
567C3×C4—22no—
568C3×C4—44no—
569C3×C4—41no—
570C3×C4—41no—
571C3×C4B(4,𝔽2)12yescomplemented
572Dic3—42no—
573Dic3—42no—
574Dic3—81no—
575Dic3—84no—
576C23—21no—
577C23—21no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—62no—
589C23—62no—
590C23—64no—
591C23—64no—
592C23—62no—
593C23—64no—
594C23—82no—
595C23—82no—
596C23—121no—
597C23—122no—
598C23—121no—
599C23—122no—
600C23—124no—
601C23—124no—
602C23—128no—
603C23—121no—
604C23—122no—
605C23—121no—
606C23#93812yescomplemented
607C23C2×D3×D411yescharacteristic
608C23C2×#23711yescharacteristic
609C23#100412yes—
610C23C22×#6711yescharacteristic
611C23C22×#6712yes—
612Q8—44no—
613D4—24no—
614D4—24no—
615D4—44no—
616D4—48no—
617D4—44no—
618D4—44no—
619D4—44no—
620D4—44no—
621D4—44no—
622D4—82no—
623D4—82no—
624D4—82no—
625D4—128no—
626D4—2416no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—24no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—64no—
644C2×C4—62no—
645C2×C4—62no—
646C2×C4—81no—
647C2×C4—124no—
648C2×C4—128no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—122no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—124no—
655C2×C4—121no—
656C2×C4—124no—
657C2×C4—121no—
658C2×C4—124no—
659C2×C4—124no—
660C2×C4—121no—
661C2×C4#93812yescomplemented
662C2×C4#95311yescharacteristic
663C2×C4#95312yes—
664C2×C4#95311yescharacteristic
665C2×C3—21no—
666C2×C3—22no—
667C2×C3—21no—
668C2×C3—22no—
669C2×C3—42no—
670C2×C3—84no—
671C2×C3#240811yescharacteristic
672C2×C3C2×B(4,𝔽2)11yescharacteristic
673C2×C3#302511yescharacteristic
674D3—44no—
675C22—21no—
676C22—22no—
677C22—21no—
678C22—22no—
679C22—22no—
680C22—41no—
681C22—41no—
682C22—42no—
683C22—42no—
684C22—41no—
685C22—41no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—64no—
690C22—68no—
691C22—81no—
692C22—84no—
693C22—84no—
694C22—122no—
695C22—124no—
696C22—122no—
697C22—122no—
698C22—124no—
699C22—124no—
700C22—122no—
701C22—122no—
702C22—122no—
703C22—164no—
704C22C2×#93811yescharacteristic
705C22D4×#6712yes—
706C22#580511yescharacteristic
707C22#580612yes—
708C22C2×#95311yescharacteristic
709C4—22no—
710C4—21no—
711C4—22no—
712C4—44no—
713C4—41no—
714C4—41no—
715C4—122no—
716C4—122no—
717C4—244no—
718C4—241no—
719C4#524412yescomplemented
720C3#2426011yescharacteristic, complemented, Sylow
721C2—21no—
722C2—22no—
723C2—21no—
724C2—22no—
725C2—42no—
726C2—84no—
727C2—124no—
728C2#8405311yescharacteristic
729C2C2×#524411yescharacteristic
730C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#24260
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C3×#3496C2semidirect
2. C2×#5244C2semidirect
3. #82723C2semidirect
4. C4×#937C2semidirect
5. #72142C2semidirect
6. #71851C2semidirect
7. #71850C2semidirect
8. #71686C2semidirect
9. #71619C2semidirect
12. C23×C4×D3C22semidirect
25. C2×#937C22semidirect
29. #5220C22semidirect
31. C2×#834C22semidirect
32. C2×#834C22semidirect
33. #5212C22semidirect
34. C4×#217C22semidirect
39. #4955C22semidirect
40. #4946C22semidirect
42. #4526C22semidirect
43. #3496D3semidirect
61. C22×C3×D4D4semidirect
64. C23×C3×C4D4semidirect
96. C2×C3×C4⋊C4D4semidirect
97. C2×C3×C4⋊C4D4semidirect
99. C2×C3×C22⋊C4D4semidirect
100. C2×C3×C22⋊C4D4semidirect
102. C2×C3×C42D4semidirect
112. #937D4semidirect
171. C23×D3C2×D4semidirect
255. #217C2×D4semidirect
274. C22×D4#67semidirect
283. C23×C4#67semidirect
315. C2×C4⋊C4#67semidirect
316. C2×C4⋊C4#67semidirect
327. C2×C22⋊C4#67semidirect
328. C2×C22⋊C4#67semidirect
331. C2×C42#67semidirect
354. C23×C3#121semidirect
409. C2×C3×C4#121semidirect
571. C3×C4B(4,𝔽2)semidirect
606. C23#938semidirect
661. C2×C4#938semidirect
719. C4#5244semidirect
720. C3#24260semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
C2×#52442
#827232
C4×#9372
#721422
#718512
#718502
#716862
#716192
#242603