Finite Group Databasebeta

← #10729270

Subgroups — #10729270

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#875C211yesmaximal, characteristic
2C2×C3×#387C211yesmaximal, characteristic, complemented
3C2×#4990C212yesmaximal, complemented
4#71570C212yesmaximal, complemented
5C4×C3⋊#95C211yesmaximal, characteristic, complemented
6#69429C212yesmaximal, complemented
7#69426C211yesmaximal, characteristic
8#69426C211yesmaximal, characteristic
9#69424C211yesmaximal, characteristic, complemented
10#69424C211yesmaximal, characteristic, complemented
11#69408C212yesmaximal, complemented
12#14932—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×C4×Dic3C2212yes—
25C2×#875—24no—
26C2×#875—28no—
27C3×#387—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4990—28no—
40C22×#124—31no—
41C2×#388—32no—
42C2×#387D311yescharacteristic, complemented
43#1446—31no—
44#1446—31no—
45#1440—32no—
46#1439—32no—
47#1438—31no—
48#1438—31no—
49#1437—32no—
50C4×#95—31no—
51C24×D3C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C22×D12—24no—
63C22×D12—22no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—42no—
67C22×Dic6—22no—
68C22×Dic6—24no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—42no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C42—24no—
78C2×C3×C42D414yes—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—42no—
93C2×(C2×D3)⋊C4—42no—
94C2×(C2×D3)⋊C4—42no—
95C2×(C2×D3)⋊C4—42no—
96C2×(C2×D3)⋊C4D418yes—
97C2×C4×Dic3—24no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3—42no—
100#875—232no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C23×Q8—31no—
113C23×D4—31no—
114C2×#124—68no—
115C2×#124—64no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4—32no—
122C22×C22⋊C4—32no—
123C22×C22⋊C4—32no—
124C22×C42—32no—
125C22×C42C2×D311yescharacteristic
126#388—68no—
127#387—24no—
128C2×#95—31no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D311yescharacteristic
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D311yescharacteristic
139C24×C3C2411yescharacteristic
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—22no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—22no—
154C23×D3C2×D414yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4112yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4112yes—
189C22×C3×C4#4112yes—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3C2×D414yes—
209C22×Dic3#4114yes—
210C22×Dic3#4114yes—
211C2×D12—28no—
212C2×D12—48no—
213C2×D12—48no—
214C2×D12—44no—
215C2×D12—44no—
216C2×Dic6—28no—
217C2×Dic6—48no—
218C2×Dic6—48no—
219C2×Dic6—44no—
220C2×Dic6—44no—
221C3×C4⋊C4—28no—
222C3×C4⋊C4—44no—
223C3×C42—28no—
224(C2×D3)⋊C4—216no—
225(C2×D3)⋊C4—216no—
226(C2×D3)⋊C4—216no—
227(C2×D3)⋊C4—48no—
228(C2×D3)⋊C4—48no—
229(C2×D3)⋊C4—48no—
230C4×Dic3—48no—
231C25—32no—
232C22×Q8—64no—
233C22×Q8—62no—
234C22×Q8—62no—
235C22×Q8—62no—
236C22×Q8—122no—
237C22×D4—64no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4—122no—
242C23×C4—32no—
243C23×C4—32no—
244C23×C4—32no—
245C23×C4C22×D311yescharacteristic
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D311yescharacteristic
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252#124—632no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C22⋊C4—38no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—122no—
273C2×C22⋊C4—122no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4—122no—
276C2×C42—24no—
277C2×C42—64no—
278C2×C42—64no—
279C2×C42—122no—
280C2×C42#6714yes—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—64no—
286#95—64no—
287#95—64no—
288#95—62no—
289#95—62no—
290#95—122no—
291#95—122no—
292C23×C3C22×D412yes—
293C23×C3C22×D411yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4111yescharacteristic
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4112yes—
299C23×C3C2×#4112yes—
300C23×C3H(5,𝔽2)12yes—
301C23×C3H(5,𝔽2)11yescharacteristic
302C23×C32−1+411yescharacteristic
303C22×D3—24no—
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4#12014yes—
372C2×C3×C4#12414yes—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394D12—416no—
395Dic6—416no—
396C24—34no—
397C24—62no—
398C24—62no—
399C24—62no—
400C24—62no—
401C24—62no—
402C24—62no—
403C24—62no—
404C24—62no—
405C24—62no—
406C24—62no—
407C24—62no—
408C24—62no—
409C24—62no—
410C24—62no—
411C24C23×D311yescharacteristic
412C2×Q8—68no—
413C2×Q8—128no—
414C2×Q8—128no—
415C2×Q8—124no—
416C2×Q8—124no—
417C2×D4—68no—
418C2×D4—128no—
419C2×D4—128no—
420C2×D4—124no—
421C2×D4—124no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
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—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—34no—
450C22×C4—34no—
451C22×C4—34no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4#23512yes—
471C22×C4#23512yes—
472C22×C4D3×D414yes—
473C22×C4#23912yes—
474C22×C4#23912yes—
475C22×C4C2×#6712yes—
476C22×C4C2×#6712yes—
477C22×C4C2×#6712yes—
478C4⋊C4—28no—
479C4⋊C4—44no—
480C22⋊C4—616no—
481C22⋊C4—616no—
482C22⋊C4—616no—
483C22⋊C4—128no—
484C22⋊C4—128no—
485C22⋊C4—128no—
486C42—28no—
487C42—128no—
488C22×C3C2×#12011yescharacteristic
489C22×C3#52511yescharacteristic
490C22×C3#52512yes—
491C22×C3C2×#12411yescharacteristic
492C22×C3#52912yes—
493C22×C3#52912yes—
494C22×C3#53212yes—
495C22×C3#53211yescharacteristic
496C22×C3#53211yescharacteristic
497C22×C3#53511yescharacteristic
498C22×C3#53711yescharacteristic
499C22×C3#53812yes—
500C22×C3#54212yes—
501C22×C3#54612yes—
502C22×C3#54612yes—
503C22×C3#54612yes—
504C22×C3#54712yes—
505C22×C3#55112yes—
506C22×C3#55512yes—
507C22×C3#55611yescharacteristic
508C22×C3#55611yescharacteristic
509C22×C3#55911yescharacteristic
510C22×C3#55911yescharacteristic
511C2×D3—44no—
512C2×D3—44no—
513C2×D3—44no—
514C2×D3—44no—
515C2×D3—44no—
516C2×D3—44no—
517C2×D3—44no—
518C2×D3—82no—
519C2×D3—82no—
520C2×D3—82no—
521C2×D3—82no—
522C2×D3—82no—
523C2×D3—82no—
524C2×D3—82no—
525C2×D3—82no—
526C3×C4—24no—
527C3×C4—24no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—81no—
532C3×C4—81no—
533Dic3—44no—
534Dic3—44no—
535Dic3—44no—
536C23—64no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23—64no—
541C23—64no—
542C23—64no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23—122no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—122no—
556C23—122no—
557C23—122no—
558C23—122no—
559C23—122no—
560C23—122no—
561C23—122no—
562C23—122no—
563C23—122no—
564C23—122no—
565C23—122no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—122no—
570C23—122no—
571C23C2×#23511yescharacteristic
572C23C2×#23511yescharacteristic
573C23C2×D3×D412yes—
574C23C2×#23911yescharacteristic
575C23C2×#23911yescharacteristic
576C23D3×#4112yes—
577C23D3×#4112yes—
578C23#100911yescharacteristic
579C23#100912yes—
580C23#101011yescharacteristic
581C23C22×#6711yescharacteristic
582Q8—1216no—
583D4—1216no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—64no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—64no—
623C2×C4—64no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4#93714yes—
639C2×C4#94714yes—
640C2×C3#238811yescharacteristic
641C2×C3#243011yescharacteristic
642C2×C3#243511yescharacteristic
643C2×C3#245912yes—
644C2×C3#247412yes—
645C2×C3#252112yes—
646C2×C3#255712yes—
647C2×C3#259211yescharacteristic
648C2×C3#261011yescharacteristic
649C2×C3#266211yescharacteristic
650C2×C3#286111yescharacteristic
651D3—84no—
652C22—124no—
653C22—124no—
654C22—124no—
655C22—124no—
656C22—124no—
657C22—124no—
658C22—124no—
659C22—242no—
660C22—242no—
661C22—242no—
662C22—242no—
663C22—242no—
664C22—242no—
665C22—242no—
666C22—242no—
667C22#552712yes—
668C22#553212yes—
669C22#554712yes—
670C22#554812yes—
671C22#558411yescharacteristic
672C22#558411yescharacteristic
673C22#558511yescharacteristic
674C22#558511yescharacteristic
675C22#563012yes—
676C22#563012yes—
677C22#565412yes—
678C22#565512yes—
679C22#569112yes—
680C22#569112yes—
681C22#570412yes—
682C22#571012yes—
683C22#579111yescharacteristic
684C22C2×#93711yescharacteristic
685C22C2×#94711yescharacteristic
686C22#583011yescharacteristic
687C22#583211yescharacteristic
688C22#583211yescharacteristic
689C22#583311yescharacteristic
690C4—24no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—81no—
696C4—81no—
697C4—124no—
698C4—124no—
699C4—124no—
700C3#1493211yescharacteristic, complemented, Sylow
701C2—244no—
702C2#7501012yes—
703C2#7504412yes—
704C2#7532612yes—
705C2#7535812yes—
706C2#8226011yescharacteristic
707C2#8240011yescharacteristic
708C2#8248411yescharacteristic
709C2#8276111yescharacteristic
710C2#8284811yescharacteristic
711C2#8305711yescharacteristic
712C2#8311311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#387C2
prime radicalC23×D12C22
hypercenterC2×#387D3
hypocenterC3#14932
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
2. C2×C3×#387C2semidirect
3. C2×#4990C2semidirect
4. #71570C2semidirect
5. C4×C3⋊#95C2semidirect
6. #69429C2semidirect
9. #69424C2semidirect
10. #69424C2semidirect
11. #69408C2semidirect
42. C2×#387D3semidirect
700. C3#14932semidirect

Maximal subgroups

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

subgroupindex
C22×#8752
C2×C3×#3872
C2×#49902
#715702
C4×C3⋊#952
#694292
#694262
#694242
#694082
#149323