Finite Group Databasebeta

← #10750404

Subgroups — #10750404

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×#947C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic, complemented
3C2×#4994C212yesmaximal, complemented
4C2×#4991C211yesmaximal, characteristic
5C2×#4980C211yesmaximal, characteristic, complemented
6#72134C211yesmaximal, characteristic, complemented
7#71570C212yesmaximal, complemented
8#69429C212yesmaximal, complemented
9#69428C212yesmaximal, complemented
10#69426C211yesmaximal, characteristic
11Dic3⋊#95C211yesmaximal, characteristic, complemented
12#16505—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#947—28no—
18C2×#947—24no—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C22×Dic3⋊C4C2212yes—
27C22×C4×Dic3C2211yescharacteristic
28C3×#395—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4994—28no—
40#4991—24no—
41#4980—24no—
42C22×#124—31no—
43C2×#396—32no—
44C2×#395D311yescharacteristic, complemented
45C2×#394—31no—
46C2×#387—31no—
47#1446—31no—
48#1440—32no—
49#1439—32no—
50#1439—32no—
51#1438—31no—
52C4⋊#95—31no—
53C24×D3C2312yes—
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2312yes—
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×D12—24no—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—42no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4D414yes—
80#947—232no—
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)⋊C4—42no—
97C2×(C2×D3)⋊C4—42no—
98C2×(C2×D3)⋊C4D418yes—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—42no—
104C2×C4×Dic3—24no—
105C2×C4×Dic3D414yes—
106C3×#95—22no—
107C3×#95—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—24no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114C3⋊#95—42no—
115C3⋊#95—42no—
116C23×Q8C2×D311yescharacteristic
117C23×D4—31no—
118C2×#124—64no—
119C2×#124—68no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4C2×D312yes—
123C22×C4⋊C4C2×D311yescharacteristic
124C22×C22⋊C4—32no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C42—31no—
131#396—68no—
132#395—24no—
133#394—64no—
134#387—64no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—31no—
138C2×#95—31no—
139C2×#95—31no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95C2×D311yescharacteristic
144C2×#95C2×D312yes—
145C24×C3C2411yescharacteristic
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×D3—22no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3—22no—
160C23×D3C2×D414yes—
161C2×C3×Q8—28no—
162C2×C3×Q8—48no—
163C2×C3×Q8—48no—
164C2×C3×Q8—44no—
165C2×C3×Q8—44no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4#4112yes—
196C22×C3×C4#4114yes—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—22no—
221C22×Dic3C2×D412yes—
222C22×Dic3C2×D412yes—
223C22×Dic3C2×D414yes—
224C2×D12—28no—
225C2×D12—48no—
226C2×D12—48no—
227C2×D12—44no—
228C2×D12—44no—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—48no—
231(C2×D3)⋊C4—216no—
232(C2×D3)⋊C4—216no—
233(C2×D3)⋊C4—48no—
234(C2×D3)⋊C4—48no—
235(C2×D3)⋊C4—48no—
236(C2×D3)⋊C4—48no—
237C4⋊Dic3—44no—
238Dic3⋊C4—48no—
239C4×Dic3—28no—
240C25—32no—
241C22×Q8—24no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×D4—64no—
247C22×D4—62no—
248C22×D4—62no—
249C22×D4—62no—
250C22×D4—122no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—31no—
256C23×C4C22×D312yes—
257C23×C4C22×D312yes—
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261#124—632no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—122no—
272C2×C4⋊C4#6714yes—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—64no—
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—64no—
285C2×C22⋊C4—122no—
286C2×C22⋊C4—122no—
287C2×C22⋊C4—122no—
288C2×C22⋊C4—122no—
289C2×C22⋊C4—122no—
290C2×C22⋊C4—122no—
291C2×C42—34no—
292C2×C42—64no—
293#95—22no—
294#95—24no—
295#95—62no—
296#95—62no—
297#95—62no—
298#95—62no—
299#95—64no—
300#95—62no—
301#95—122no—
302#95—122no—
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D412yes—
305C23×C3C22×D411yescharacteristic
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3H(5,𝔽2)12yes—
309C23×C3H(5,𝔽2)11yescharacteristic
310C23×C3H(5,𝔽2)11yescharacteristic
311C23×C32−1+411yescharacteristic
312C23×C32−1+412yes—
313C23×C32−1+411yescharacteristic
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—24no—
320C22×D3—24no—
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—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C3×Q8—416no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4#12114yes—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—42no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—42no—
397C2×Dic3—41no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3#12014yes—
408D12—416no—
409C24—34no—
410C24—62no—
411C24—62no—
412C24—62no—
413C24—62no—
414C24—62no—
415C24—62no—
416C24—62no—
417C24—62no—
418C24—62no—
419C24—62no—
420C24—62no—
421C24—62no—
422C24—62no—
423C24—62no—
424C24C23×D311yescharacteristic
425C2×Q8—28no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×D4—68no—
431C2×D4—128no—
432C2×D4—128no—
433C2×D4—124no—
434C2×D4—124no—
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—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—32no—
460C22×C4—34no—
461C22×C4—32no—
462C22×C4—62no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4D3×D414yes—
487C22×C4D3×D412yes—
488C22×C4#23712yes—
489C22×C4#23914yes—
490C22×C4C2×#6712yes—
491C22×C4C2×#6712yes—
492C22×C4C2×#6712yes—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—128no—
496C4⋊C4—124no—
497C22⋊C4—616no—
498C22⋊C4—616no—
499C22⋊C4—128no—
500C22⋊C4—128no—
501C22⋊C4—128no—
502C22⋊C4—128no—
503C42—68no—
504C22×C3C2×#12011yescharacteristic
505C22×C3C2×#12111yescharacteristic
506C22×C3#53211yescharacteristic
507C22×C3#53511yescharacteristic
508C22×C3#53511yescharacteristic
509C22×C3#53611yescharacteristic
510C22×C3#53612yes—
511C22×C3#53611yescharacteristic
512C22×C3#53711yescharacteristic
513C22×C3#53711yescharacteristic
514C22×C3#53712yes—
515C22×C3#53812yes—
516C22×C3#54211yescharacteristic
517C22×C3#54211yescharacteristic
518C22×C3#54212yes—
519C22×C3D4212yes—
520C22×C3#54612yes—
521C22×C3#54612yes—
522C22×C3#54712yes—
523C22×C3#54812yes—
524C22×C3#55612yes—
525C22×C3#56212yes—
526C22×C3#56312yes—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—82no—
535C2×D3—82no—
536C2×D3—82no—
537C2×D3—82no—
538C2×D3—82no—
539C2×D3—82no—
540C2×D3—82no—
541C2×D3—82no—
542C3×C4—24no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—44no—
546C3×C4—44no—
547Dic3—24no—
548Dic3—42no—
549Dic3—81no—
550Dic3—82no—
551Dic3—81no—
552C23—64no—
553C23—64no—
554C23—64no—
555C23—64no—
556C23—64no—
557C23—64no—
558C23—64no—
559C23—122no—
560C23—122no—
561C23—122no—
562C23—122no—
563C23—122no—
564C23—122no—
565C23—122no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—122no—
570C23—122no—
571C23—122no—
572C23—122no—
573C23—122no—
574C23—122no—
575C23—122no—
576C23—122no—
577C23—122no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23C2×D3×D411yescharacteristic
588C23C2×D3×D412yes—
589C23C2×#23711yescharacteristic
590C23#100412yes—
591C23C2×#23912yes—
592C23#100711yescharacteristic
593C23#100711yescharacteristic
594C23#100712yes—
595C23#100911yescharacteristic
596C23#100911yescharacteristic
597C23C22×#6711yescharacteristic
598Q8—416no—
599D4—1216no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—34no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—62no—
632C2×C4—64no—
633C2×C4—62no—
634C2×C4—64no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—122no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4#93914yes—
658C2×C3#240811yescharacteristic
659C2×C3#243011yescharacteristic
660C2×C3#254511yescharacteristic
661C2×C3#261611yescharacteristic
662C2×C3#262011yescharacteristic
663C2×C3#266211yescharacteristic
664C2×C3#267412yes—
665C2×C3#267612yes—
666C2×C3#270012yes—
667C2×C3#271512yes—
668C2×C3#282011yescharacteristic
669D3—84no—
670C22—124no—
671C22—124no—
672C22—124no—
673C22—124no—
674C22—124no—
675C22—124no—
676C22—124no—
677C22—242no—
678C22—242no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22—242no—
685C22D3×#12011yescharacteristic
686C22#559112yes—
687C22#559211yescharacteristic
688C22#562911yescharacteristic
689C22#562911yescharacteristic
690C22#562912yes—
691C22#563012yes—
692C22#564311yescharacteristic
693C22#564311yescharacteristic
694C22#564412yes—
695C22#564712yes—
696C22#572411yescharacteristic
697C22#572412yes—
698C22#572511yescharacteristic
699C22#572612yes—
700C22#573212yes—
701C22#573312yes—
702C22#573512yes—
703C22D4×#6712yes—
704C22C2×#93911yescharacteristic
705C22#581611yescharacteristic
706C22#581712yes—
707C22#583011yescharacteristic
708C4—24no—
709C4—42no—
710C4—44no—
711C4—44no—
712C4—42no—
713C4—64no—
714C4—122no—
715C4—242no—
716C4—241no—
717C4—241no—
718C3#1650511yescharacteristic, complemented, Sylow
719C2—244no—
720C2#7720111yescharacteristic
721C2#7720911yescharacteristic
722C2#7721912yes—
723C2#7725011yescharacteristic
724C2#7726712yes—
725C2#7727511yescharacteristic
726C2#8258211yescharacteristic
727C2#8282111yescharacteristic
728C2#8282712yes—
729C2#8323711yescharacteristic
730C2#8326512yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#395C2
prime radicalC23×D12C22
hypercenterC2×#395D3
hypocenterC3#16505
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×C3×#395C2semidirect
3. C2×#4994C2semidirect
5. C2×#4980C2semidirect
6. #72134C2semidirect
7. #71570C2semidirect
8. #69429C2semidirect
9. #69428C2semidirect
11. Dic3⋊#95C2semidirect
44. C2×#395D3semidirect
718. C3#16505semidirect

Maximal subgroups

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

subgroupindex
C22×#9472
C2×C3×#3952
C2×#49942
C2×#49912
C2×#49802
#721342
#715702
#694292
#694282
#694262
Dic3⋊#952
#165053