Finite Group Databasebeta

← #10720457

Subgroups — #10720457

Order 768.

2058 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#894C211yesmaximal, characteristic
2C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
3C2×#4666C211yesmaximal, characteristic, complemented
4C3×#1436C211yesmaximal, characteristic, complemented
5#71564C211yesmaximal, characteristic, complemented
6#71537C211yesmaximal, characteristic, complemented
7#69429C212yesmaximal, complemented
8#69426C212yesmaximal
9#69408C211yesmaximal, characteristic, complemented
10#69408C212yesmaximal, complemented
11#69406C212yesmaximal, complemented
12#14167—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×#894—24no—
28C2×#894—24no—
29C2×#894C418yescomplemented
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C4×(C2×D3)⋊C4—24no—
40#4666—24no—
41C22×#123—31no—
42C2×C4×C22⋊C4—31no—
43C2×C4×C22⋊C4—31no—
44#1440—32no—
45#1438—32no—
46#1437—31no—
47#1437—31no—
48#1437—32no—
49#1436—32no—
50#1436—31no—
51#1436D311yescharacteristic, complemented
52C24×D3C2311yescharacteristic
53C24×D3C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C22×D12—22no—
64C22×D12—22no—
65C22×D12—42no—
66C22×D12—42no—
67C22×D12C2×C414yes—
68C22×C4×D3—42no—
69C22×C4×D3—42no—
70C22×C4×D3C2×C414yes—
71C22×C4×D3D414yes—
72C22×C4×D3D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2×C414yes—
76C2×C3×C42—22no—
77C2×C3×C42—42no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—24no—
94C2×(C2×D3)⋊C4—42no—
95C2×(C2×D3)⋊C4—42no—
96C2×(C2×D3)⋊C4C2×C414yes—
97C2×(C2×D3)⋊C4C2×C414yes—
98C2×(C2×D3)⋊C4C2×C414yes—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4C2×C414yes—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—42no—
105C2×C4×Dic3—42no—
106C2×C4×Dic3—42no—
107#894—232no—
108C3×#95—22no—
109C3×#95—24no—
110C3×#95—42no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—42no—
117C23×D4—31no—
118C24×C4—31no—
119C2×#123—38no—
120C2×#123—64no—
121C2×#123—64no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4C2×D311yescharacteristic
124C22×C22⋊C4—31no—
125C22×C22⋊C4—31no—
126C22×C22⋊C4—31no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—32no—
131C22×C42—32no—
132C22×C42—31no—
133C22×C42C2×D311yescharacteristic
134C4×C22⋊C4—64no—
135C4×C22⋊C4—64no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95C2×D312yes—
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—21no—
153C23×D3—21no—
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3—22no—
160C23×D3C22×C412yes—
161C23×D3C22×C412yes—
162C23×D3C2×D412yes—
163C23×D3C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C22×C412yes—
187C22×C3×C4C22×C412yes—
188C22×C3×C4C22×C412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4#4114yes—
192C22×C3×C4#4114yes—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
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×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3C22×C412yes—
212C22×Dic3C22×C412yes—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D412yes—
215C22×Dic3#4114yes—
216C22×Dic3#4114yes—
217C2×D12—28no—
218C2×D12—48no—
219C2×D12—44no—
220C2×D12—44no—
221C2×D12—48no—
222C2×C4×D3—48no—
223C2×C4×D3—48no—
224C2×C4×D3—44no—
225C2×C4×D3—44no—
226C2×C4×D3C22⋊C4116yescomplemented
227C3×C4⋊C4—28no—
228C3×C42—44no—
229(C2×D3)⋊C4—216no—
230(C2×D3)⋊C4—216no—
231(C2×D3)⋊C4—28no—
232(C2×D3)⋊C4—28no—
233(C2×D3)⋊C4—28no—
234(C2×D3)⋊C4—48no—
235(C2×D3)⋊C4—48no—
236Dic3⋊C4—28no—
237C4×Dic3—48no—
238C4×Dic3—44no—
239C25—31no—
240C25—31no—
241C22×D4—34no—
242C22×D4—62no—
243C22×D4—62no—
244C22×D4—122no—
245C22×D4—122no—
246C23×C4—31no—
247C23×C4—34no—
248C23×C4—34no—
249C23×C4—34no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4—122no—
254C23×C4—122no—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D312yes—
258C23×C4C22×D312yes—
259C23×C4C22×D311yescharacteristic
260#123—632no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C4×D314yes—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—34no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—122no—
287C2×C22⋊C4—122no—
288C2×C42—22no—
289C2×C42—42no—
290C2×C42—64no—
291C2×C42—62no—
292C2×C42—122no—
293C2×C42—122no—
294C2×C42—122no—
295#95—24no—
296#95—22no—
297#95—42no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—62no—
302#95—64no—
303#95—122no—
304C23×C3C23×C411yescharacteristic
305C23×C3C22×D411yescharacteristic
306C23×C3C22×D411yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3H(5,𝔽2)12yes—
312C23×C3H(5,𝔽2)12yes—
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—24no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—24no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—24no—
329C22×D3—24no—
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—41no—
341C22×D3—41no—
342C22×D3—42no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3C2×C22⋊C414yes—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4C2×C22⋊C414yes—
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—42no—
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×Dic3C2×C22⋊C414yes—
408D12—416no—
409C4×D3—416no—
410C24—32no—
411C24—32no—
412C24—32no—
413C24—32no—
414C24—62no—
415C24—62no—
416C24—62no—
417C24—62no—
418C24—62no—
419C24—62no—
420C24—62no—
421C24—61no—
422C24—61no—
423C24—62no—
424C24—62no—
425C24—62no—
426C24—62no—
427C24—62no—
428C24C23×D311yescharacteristic
429C2×D4—68no—
430C2×D4—128no—
431C2×D4—124no—
432C2×D4—124no—
433C2×D4—128no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—32no—
459C22×C4—316no—
460C22×C4—32no—
461C22×C4—34no—
462C22×C4—34no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
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—62no—
480C22×C4—62no—
481C22×C4—124no—
482C22×C4—128no—
483C22×C4—124no—
484C22×C4—128no—
485C22×C4C2×C4×D312yes—
486C22×C4C2×C4×D312yes—
487C22×C4C2×C4×D312yes—
488C22×C4#23514yes—
489C22×C4D3×D412yes—
490C22×C4D3×D412yes—
491C22×C4#23914yes—
492C4⋊C4—28no—
493C4⋊C4—68no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—616no—
497C22⋊C4—616no—
498C22⋊C4—68no—
499C22⋊C4—128no—
500C22⋊C4—128no—
501C42—44no—
502C42—124no—
503C42—128no—
504C22×C3C22×C22⋊C411yescharacteristic
505C22×C3C4×#4111yescharacteristic
506C22×C3C4×#4111yescharacteristic
507C22×C3C4×#4112yes—
508C22×C3#51812yes—
509C22×C3#52512yes—
510C22×C3#52512yes—
511C22×C3#53211yescharacteristic
512C22×C3#53211yescharacteristic
513C22×C3#53212yes—
514C22×C3#53411yescharacteristic
515C22×C3#53411yescharacteristic
516C22×C3#53412yes—
517C22×C3#55112yes—
518C22×C3#55112yes—
519C22×C3#55312yes—
520C22×C3#55312yes—
521C22×C3#55312yes—
522C22×C3#55312yes—
523C22×C3#55912yes—
524C22×C3#55912yes—
525C2×D3—24no—
526C2×D3—24no—
527C2×D3—24no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—42no—
535C2×D3—44no—
536C2×D3—44no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—82no—
540C2×D3—82no—
541C2×D3—82no—
542C2×D3—82no—
543C3×C4—24no—
544C3×C4—42no—
545C3×C4—44no—
546C3×C4—44no—
547C3×C4—42no—
548Dic3—24no—
549Dic3—44no—
550Dic3—44no—
551Dic3—42no—
552C23—34no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—62no—
563C23—64no—
564C23—64no—
565C23—62no—
566C23—62no—
567C23—62no—
568C23—64no—
569C23—64no—
570C23—122no—
571C23—122no—
572C23—122no—
573C23—122no—
574C23—122no—
575C23—122no—
576C23—122no—
577C23—122no—
578C23—122no—
579C23—121no—
580C23—122no—
581C23—121no—
582C23—122no—
583C23—122no—
584C23—121no—
585C23—122no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—121no—
592C23—122no—
593C23—122no—
594C23C22×C4×D311yescharacteristic
595C23C2×#23512yes—
596C23C2×D3×D411yescharacteristic
597C23C2×D3×D411yescharacteristic
598C23#100412yes—
599C23C2×#23912yes—
600C23D3×#4112yes—
601C23D3×#4112yes—
602C23#100912yes—
603D4—1216no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—34no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—62no—
634C2×C4—64no—
635C2×C4—62no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—64no—
640C2×C4—62no—
641C2×C4—64no—
642C2×C4—64no—
643C2×C4—64no—
644C2×C4—64no—
645C2×C4—1216no—
646C2×C4—122no—
647C2×C4—121no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—121no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4D3×C22⋊C414yes—
658C2×C3#229711yescharacteristic
659C2×C3#229711yescharacteristic
660C2×C3#230811yescharacteristic
661C2×C3#237412yes—
662C2×C3#237412yes—
663C2×C3#260212yes—
664C2×C3#260212yes—
665C2×C3#260212yes—
666C2×C3#260212yes—
667D3—44no—
668D3—82no—
669C22—64no—
670C22—64no—
671C22—64no—
672C22—124no—
673C22—124no—
674C22—124no—
675C22—124no—
676C22—124no—
677C22—124no—
678C22—122no—
679C22—122no—
680C22—122no—
681C22—124no—
682C22—124no—
683C22—242no—
684C22—242no—
685C22—242no—
686C22—242no—
687C22C4×#23511yescharacteristic
688C22C2×D3×C22⋊C411yescharacteristic
689C22#551412yes—
690C22#552211yescharacteristic
691C22#552712yes—
692C22#554012yes—
693C22#554512yes—
694C22#554712yes—
695C22#554812yes—
696C22C4×#23912yes—
697C22#558512yes—
698C22#558911yescharacteristic
699C22#560512yes—
700C22#562712yes—
701C22#563712yes—
702C22#565311yescharacteristic
703C22#566412yes—
704C22#566811yescharacteristic
705C22#569512yes—
706C22#570912yes—
707C22#572511yescharacteristic
708C4—24no—
709C4—42no—
710C4—44no—
711C4—42no—
712C4—44no—
713C4—64no—
714C4—124no—
715C4—124no—
716C4—122no—
717C3#1416711yescharacteristic, complemented, Sylow
718C2—124no—
719C2—242no—
720C2#7356311yescharacteristic
721C2#7366712yes—
722C2#7461812yes—
723C2#7487411yescharacteristic
724C2#7497512yes—
725C2#7537412yes—
726C2#7564512yes—
727C2#7609612yes—
728C2#7647811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1436C2
socle, cosocleC24×C3C24
prime radicalC23×D12C22
hypercenter#1436D3
hypocenterC3#14167
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
2. C2×C4×(C2×D3)⋊C4C2semidirect
3. C2×#4666C2semidirect
4. C3×#1436C2semidirect
5. #71564C2semidirect
6. #71537C2semidirect
7. #69429C2semidirect
9. #69408C2semidirect
10. #69408C2semidirect
11. #69406C2semidirect
29. C2×#894C4semidirect
51. #1436D3semidirect
226. C2×C4×D3C22⋊C4semidirect
717. C3#14167semidirect

Maximal subgroups

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

subgroupindex
C22×#8942
C2×C4×(C2×D3)⋊C42
C2×#46662
C3×#14362
#715642
#715372
#694292
#694262
#694082
#694062
#141673