Finite Group Databasebeta

← #10750405

Subgroups — #10750405

Order 768.

1942 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
1C22×C3×#124C211yesmaximal, characteristic
2C2×#5230C211yesmaximal, characteristic, complemented
3C2×#5222C211yesmaximal, characteristic
4C2×#4948C212yesmaximal, complemented
5C2×#4933C211yesmaximal, characteristic, complemented
6#72134C211yesmaximal, characteristic, complemented
7#72131C211yesmaximal, characteristic
8#71549C212yesmaximal, complemented
9#71547C212yesmaximal, complemented
10#71546C212yesmaximal, complemented
11#71535C211yesmaximal, characteristic, complemented
12#16505—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C2×C3×#124—28no—
16C2×C3×#124—24no—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×#217C2212yes—
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2211yescharacteristic
28#5230—24no—
29#5222—24no—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4948—28no—
41#4933—24no—
42C22×#124D311yescharacteristic
43C2×#396—32no—
44C2×#395—31no—
45C2×#394—31no—
46C2×#387—31no—
47#1446—31no—
48#1440—32no—
49#1439—32no—
50#1439—32no—
51#1438—31no—
52C4⋊#95—31no—
53C25×C3C2312yes—
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C22×C3×D4—24no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—42no—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2312yes—
74C3×#124—232no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4D418yes—
81C2×C3×C42—24no—
82C2×C3×C42D414yes—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—24no—
88C2×#217—24no—
89C2×#217—42no—
90C2×#217—42no—
91C2×#217—42no—
92C2×#217—42no—
93C2×#217—42no—
94C2×#217—42no—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3—42no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—42no—
105C2×Dic3⋊C4D414yes—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—24no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113C3⋊#95—24no—
114C3⋊#95—42no—
115C3⋊#95—42no—
116C23×Q8C2×D311yescharacteristic
117C23×D4C2×D311yescharacteristic
118C2×#124—28no—
119C2×#124—24no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—32no—
124C22×C22⋊C4—32no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4C2×D312yes—
129C22×C22⋊C4C2×D312yes—
130C22×C42C2×D311yescharacteristic
131#396—68no—
132#395—64no—
133#394—64no—
134#387—64no—
135C2×#95—31no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—31no—
140C2×#95—31no—
141C2×#95—31no—
142C2×#95—31no—
143C2×#95—32no—
144C2×#95—32no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3C2×D414yes—
160C24×C3C2411yescharacteristic
161C2×C3×Q8—28no—
162C2×C3×Q8—48no—
163C2×C3×Q8—48no—
164C2×C3×Q8—44no—
165C2×C3×Q8—44no—
166C2×C3×D4—28no—
167C2×C3×D4—48no—
168C2×C3×D4—48no—
169C2×C3×D4—44no—
170C2×C3×D4—44no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4112yes—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—22no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—22no—
226C22×Dic3C2×D412yes—
227C22×Dic3C2×D412yes—
228C22×Dic3#4114yes—
229C3×C22⋊C4—216no—
230C3×C22⋊C4—216no—
231C3×C42—28no—
232#217—48no—
233#217—48no—
234#217—48no—
235#217—48no—
236C4⋊Dic3—48no—
237Dic3⋊C4—28no—
238Dic3⋊C4—48no—
239Dic3⋊C4—44no—
240C25C22×D312yes—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×D4—24no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—42no—
251C23×C4—31no—
252C23×C4—32no—
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4C22×D312yes—
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261#124—232no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—122no—
272C2×C4⋊C4—122no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—122no—
285C2×C22⋊C4—122no—
286C2×C22⋊C4—122no—
287C2×C22⋊C4—122no—
288C2×C22⋊C4—122no—
289C2×C22⋊C4—122no—
290C2×C22⋊C4#6718yes—
291C2×C42—24no—
292C2×C42#6714yes—
293#95—64no—
294#95—62no—
295#95—62no—
296#95—62no—
297#95—62no—
298#95—62no—
299#95—62no—
300#95—64no—
301#95—122no—
302#95—122no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3C22×D411yescharacteristic
339C23×C3C22×D411yescharacteristic
340C23×C3C22×D412yes—
341C23×C3C2×#4111yescharacteristic
342C23×C3C2×#4112yes—
343C23×C3H(5,𝔽2)12yes—
344C23×C3H(5,𝔽2)11yescharacteristic
345C23×C3H(5,𝔽2)11yescharacteristic
346C23×C32−1+411yescharacteristic
347C23×C32−1+412yes—
348C23×C32−1+411yescharacteristic
349C3×Q8—416no—
350C3×D4—416no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4#12014yes—
372C2×C3×C4#12114yes—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—41no—
382C2×Dic3—42no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—42no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—42no—
399C2×Dic3—41no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24C2×#6714yes—
424C24C23×D311yescharacteristic
425C2×Q8—28no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×D4—28no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—44no—
434C2×D4—44no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—34no—
453C22×C4—62no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—62no—
479C22×C4—61no—
480C22×C4—62no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4D3×D412yes—
487C22×C4#23712yes—
488C22×C4C2×#6712yes—
489C22×C4C2×#6714yes—
490C22×C4C2×#6712yes—
491C22×C4C2×#6712yes—
492C22×C4C2×#6714yes—
493C4⋊C4—68no—
494C4⋊C4—128no—
495C4⋊C4—128no—
496C4⋊C4—124no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—128no—
500C22⋊C4—128no—
501C22⋊C4—128no—
502C22⋊C4—128no—
503C42—28no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—44no—
508C22×C3—44no—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3—82no—
512C22×C3—82no—
513C22×C3—82no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—82no—
517C22×C3—82no—
518C22×C3—82no—
519C22×C3C2×#12011yescharacteristic
520C22×C3C2×#12111yescharacteristic
521C22×C3#53211yescharacteristic
522C22×C3#53511yescharacteristic
523C22×C3#53511yescharacteristic
524C22×C3#53611yescharacteristic
525C22×C3#53611yescharacteristic
526C22×C3#53612yes—
527C22×C3#53711yescharacteristic
528C22×C3#53711yescharacteristic
529C22×C3#53712yes—
530C22×C3#53812yes—
531C22×C3#54212yes—
532C22×C3#54211yescharacteristic
533C22×C3#54211yescharacteristic
534C22×C3D4212yes—
535C22×C3#54612yes—
536C22×C3#54612yes—
537C22×C3#54712yes—
538C22×C3#54812yes—
539C22×C3#55612yes—
540C22×C3#56212yes—
541C22×C3#56312yes—
542C3×C4—24no—
543C3×C4—24no—
544C3×C4—42no—
545C3×C4—44no—
546Dic3—42no—
547Dic3—42no—
548Dic3—44no—
549Dic3—81no—
550Dic3—82no—
551Dic3—81no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C2×D3×D411yescharacteristic
588C23C2×#23711yescharacteristic
589C23#100412yes—
590C23D3×#4112yes—
591C23#100911yescharacteristic
592C23#100911yescharacteristic
593C23#101011yescharacteristic
594C23#101012yes—
595C23#101011yescharacteristic
596C23C22×#6711yescharacteristic
597C23C22×#6712yes—
598Q8—416no—
599D4—416no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—64no—
623C2×C4—64no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—64no—
627C2×C4—64no—
628C2×C4—122no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—121no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—121no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4#93814yes—
657C2×C4#95314yes—
658C2×C3—84no—
659C2×C3#240811yescharacteristic
660C2×C3#243011yescharacteristic
661C2×C3#254511yescharacteristic
662C2×C3#261611yescharacteristic
663C2×C3#262011yescharacteristic
664C2×C3#266211yescharacteristic
665C2×C3#267412yes—
666C2×C3#267612yes—
667C2×C3#270012yes—
668C2×C3#271512yes—
669C2×C3#282011yescharacteristic
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22#566912yes—
686C22#567112yes—
687C22#567212yes—
688C22#567411yescharacteristic
689C22#567511yescharacteristic
690C22#567712yes—
691C22#567812yes—
692C22#568211yescharacteristic
693C22#568211yescharacteristic
694C22#568412yes—
695C22C2×#93811yescharacteristic
696C22D4×#6712yes—
697C22#580511yescharacteristic
698C22#580612yes—
699C22#582412yes—
700C22#582511yescharacteristic
701C22#582512yes—
702C22#582912yes—
703C22#583211yescharacteristic
704C22#583311yescharacteristic
705C22#583312yes—
706C22#583311yescharacteristic
707C22C2×#95311yescharacteristic
708C4—24no—
709C4—24no—
710C4—44no—
711C4—42no—
712C4—124no—
713C4—122no—
714C4—122no—
715C4—241no—
716C4—242no—
717C4—241no—
718C3#1650511yescharacteristic, complemented, Sylow
719C2—84no—
720C2#8297111yescharacteristic
721C2#8298111yescharacteristic
722C2#8298212yes—
723C2#8298312yes—
724C2#8298512yes—
725C2#8298812yes—
726C2#8299211yescharacteristic
727C2#8300111yescharacteristic
728C2#8405311yescharacteristic
729C2#8409111yescharacteristic
730C2#8409411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#124C2
prime radicalC23×C3×D4C22
hypercenterC22×#124D3
hypocenterC3#16505
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
2. C2×#5230C2semidirect
4. C2×#4948C2semidirect
5. C2×#4933C2semidirect
6. #72134C2semidirect
8. #71549C2semidirect
9. #71547C2semidirect
10. #71546C2semidirect
11. #71535C2semidirect
718. C3#16505semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×#52302
C2×#52222
C2×#49482
C2×#49332
#721342
#721312
#715492
#715472
#715462
#715352
#165053