Finite Group Databasebeta

← #10726320

Subgroups — #10726320

Order 768.

1486 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4987C211yesmaximal, characteristic
2C2×#4664C211yesmaximal, characteristic
3C2×#4667C211yesmaximal, characteristic
4C3×#1442C211yesmaximal, characteristic, complemented
5#71567C211yesmaximal, characteristic, complemented
6#69428C212yesmaximal
7#69427C212yesmaximal
8#69420C212yesmaximal, complemented
9#69420C211yesmaximal, characteristic, complemented
10#69419C211yesmaximal, characteristic, complemented
11#69414C212yesmaximal, complemented
12#14674—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4987—24no—
37#4667—24no—
38#4664C418yescomplemented
39C2×#381—31no—
40C2×#380—31no—
41C2×#380—31no—
42#1444—31no—
43#1444—32no—
44#1444—31no—
45#1442—32no—
46#1442—31no—
47#1442D311yescharacteristic, complemented
48#1439—32no—
49#1439—32no—
50C24×D3C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C22×C4×D3—42no—
62C22×C4×D3—42no—
63C22×C4×D3C2×C414yes—
64C22×C4×D3D414yes—
65C22×C4×D3Q814yes—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—42no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—42no—
72C2×(C2×D3)⋊C4—42no—
73C2×(C2×D3)⋊C4—42no—
74C2×(C2×D3)⋊C4—42no—
75C2×(C2×D3)⋊C4—42no—
76C2×(C2×D3)⋊C4—42no—
77C2×(C2×D3)⋊C4C2×C418yes—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—42no—
80C2×C4⋊Dic3—42no—
81C2×C4⋊Dic3—42no—
82C2×C4⋊Dic3—42no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—42no—
86C2×Dic3⋊C4—42no—
87C2×Dic3⋊C4—42no—
88C2×Dic3⋊C4—42no—
89C2×Dic3⋊C4—42no—
90C3×#95—22no—
91C3×#95—42no—
92C3×#95—42no—
93C3×#95C2×C414yes—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—42no—
97C3⋊#95—42no—
98C3⋊#95—42no—
99C3⋊#95—42no—
100C3⋊#95C2×C414yes—
101C3⋊#95C2×C418yes—
102C24×C4—31no—
103C22×C4⋊C4—31no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—32no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C22⋊C4—32no—
110C22×C22⋊C4—31no—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—31no—
113#381—38no—
114#380—64no—
115#380—64no—
116C2×#95—32no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95C2×D311yescharacteristic
125C2×#95C2×D312yes—
126C2×#95C2×D312yes—
127C2×#95C2×D311yescharacteristic
128C24×C3C2411yescharacteristic
129C23×D3—21no—
130C23×D3—22no—
131C23×D3—21no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—21no—
135C23×D3—22no—
136C23×D3—21no—
137C23×D3C22×C412yes—
138C23×D3C2×D412yes—
139C23×D3C2×Q812yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4C22×C414yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×Q812yes—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C22×C414yes—
195C22×Dic3C22×C412yes—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×Q812yes—
198C2×C4×D3—48no—
199C2×C4×D3—44no—
200C2×C4×D3—44no—
201C2×C4×D3—48no—
202C2×C4×D3C4⋊C4116yescomplemented
203C3×C4⋊C4—44no—
204(C2×D3)⋊C4—48no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—48no—
208C4⋊Dic3—44no—
209C4⋊Dic3—48no—
210Dic3⋊C4—44no—
211Dic3⋊C4—44no—
212Dic3⋊C4—48no—
213C25—31no—
214C23×C4—31no—
215C23×C4—34no—
216C23×C4—34no—
217C23×C4—34no—
218C23×C4—32no—
219C23×C4—32no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—122no—
223C23×C4—122no—
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D312yes—
227C23×C4C22×D312yes—
228C23×C4C22×D311yescharacteristic
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—122no—
235C2×C4⋊C4—122no—
236C2×C4⋊C4—122no—
237C2×C4⋊C4—122no—
238C2×C4⋊C4—122no—
239C2×C4⋊C4—122no—
240C2×C4⋊C4—122no—
241C2×C4⋊C4—122no—
242C2×C4⋊C4—122no—
243C2×C22⋊C4—38no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4—122no—
248C2×C22⋊C4—122no—
249C2×C22⋊C4—122no—
250C2×C22⋊C4—122no—
251C2×C22⋊C4—122no—
252C2×C22⋊C4—122no—
253#95—22no—
254#95—38no—
255#95—34no—
256#95—42no—
257#95—42no—
258#95—64no—
259#95—62no—
260#95—122no—
261#95—122no—
262#95—122no—
263#95—122no—
264#95C4×D314yes—
265C23×C3C23×C411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3C22×Q811yescharacteristic
268C23×C3H(5,𝔽2)12yes—
269C23×C3H(5,𝔽2)11yescharacteristic
270C23×C3H(5,𝔽2)12yes—
271C23×C3H(5,𝔽2)12yes—
272C23×C3H(5,𝔽2)11yescharacteristic
273C23×C32−1+411yescharacteristic
274C23×C32−1+412yes—
275C23×C32−1+411yescharacteristic
276C22×D3—24no—
277C22×D3—24no—
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—24no—
287C22×D3—22no—
288C22×D3—24no—
289C22×D3—22no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3C2×C4⋊C414yes—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4C2×C4⋊C414yes—
328C2×Dic3—24no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3C2×C4⋊C414yes—
357C4×D3—416no—
358C24—32no—
359C24—32no—
360C24—32no—
361C24—61no—
362C24—62no—
363C24—61no—
364C24—62no—
365C24—61no—
366C24—62no—
367C24—62no—
368C24—61no—
369C24C23×D311yescharacteristic
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—316no—
399C22×C4—34no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—128no—
426C22×C4—124no—
427C22×C4—128no—
428C22×C4—124no—
429C22×C4C2×C4×D314yes—
430C22×C4C2×C4×D312yes—
431C22×C4D3×D412yes—
432C22×C4D3×Q812yes—
433C4⋊C4—44no—
434C4⋊C4—124no—
435C4⋊C4—124no—
436C4⋊C4—124no—
437C4⋊C4—128no—
438C4⋊C4—128no—
439C22⋊C4—128no—
440C22⋊C4—124no—
441C22⋊C4—124no—
442C22⋊C4—128no—
443C22×C3C22×C4⋊C411yescharacteristic
444C22×C3#51812yes—
445C22×C3#52012yes—
446C22×C3#52011yescharacteristic
447C22×C3#52011yescharacteristic
448C22×C3#53411yescharacteristic
449C22×C3#53411yescharacteristic
450C22×C3#53511yescharacteristic
451C22×C3#53511yescharacteristic
452C22×C3#53611yescharacteristic
453C22×C3#53611yescharacteristic
454C22×C3#54312yes—
455C22×C3#54411yescharacteristic
456C22×C3#54412yes—
457C22×C3#54411yescharacteristic
458C22×C3#56012yes—
459C22×C3#56012yes—
460C22×C3#56212yes—
461C22×C3#56212yes—
462C22×C3#56212yes—
463C22×C3#56212yes—
464C22×C3#56312yes—
465C22×C3#56312yes—
466C2×D3—24no—
467C2×D3—24no—
468C2×D3—24no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—44no—
475C2×D3—44no—
476C2×D3—42no—
477C3×C4—24no—
478C3×C4—44no—
479C3×C4—82no—
480C3×C4—81no—
481C3×C4—81no—
482Dic3—24no—
483Dic3—44no—
484Dic3—81no—
485Dic3—81no—
486Dic3—82no—
487C23—34no—
488C23—64no—
489C23—62no—
490C23—62no—
491C23—62no—
492C23—62no—
493C23—64no—
494C23—62no—
495C23—64no—
496C23—64no—
497C23—62no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23—62no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23C22×C4×D311yescharacteristic
511C23C2×D3×D411yescharacteristic
512C23#100412yes—
513C23#100411yescharacteristic
514C23C2×D3×Q811yescharacteristic
515C23#100711yescharacteristic
516C23#100912yes—
517C23#100912yes—
518C23#100911yescharacteristic
519C23#101012yes—
520C23#101011yescharacteristic
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—34no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—1216no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4D3×C4⋊C414yes—
580C2×C3#231511yescharacteristic
581C2×C3#231711yescharacteristic
582C2×C3#231711yescharacteristic
583C2×C3#237612yes—
584C2×C3#237612yes—
585C2×C3#281511yescharacteristic
586C2×C3#281611yescharacteristic
587C2×C3#281611yescharacteristic
588C2×C3#281711yescharacteristic
589C2×C3#284112yes—
590C2×C3#284112yes—
591D3—44no—
592C22—64no—
593C22—64no—
594C22—64no—
595C22—122no—
596C22—124no—
597C22—122no—
598C22—124no—
599C22—124no—
600C22—122no—
601C22—124no—
602C22—122no—
603C22C2×D3×C4⋊C411yescharacteristic
604C22#552312yes—
605C22#554512yes—
606C22#554611yescharacteristic
607C22#557411yescharacteristic
608C22#558911yescharacteristic
609C22#560711yescharacteristic
610C22#562112yes—
611C22#562811yescharacteristic
612C22#563412yes—
613C22#564712yes—
614C22#565311yescharacteristic
615C22#566612yes—
616C22#566712yes—
617C22#567411yescharacteristic
618C22#567511yescharacteristic
619C22#568412yes—
620C22#568911yescharacteristic
621C22#568912yes—
622C22#569912yes—
623C22#571112yes—
624C22#571212yes—
625C22#572911yescharacteristic
626C4—24no—
627C4—44no—
628C4—64no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—124no—
633C4—241no—
634C4—241no—
635C4—242no—
636C3#1467411yescharacteristic, complemented, Sylow
637C2—124no—
638C2#7422511yescharacteristic
639C2#7459111yescharacteristic
640C2#7531212yes—
641C2#7584112yes—
642C2#7596811yescharacteristic
643C2#7685311yescharacteristic
644C2#7708511yescharacteristic
645C2#7772911yescharacteristic
646C2#7801512yes—
647C2#7831611yescharacteristic
648C2#7840812yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1442C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenter#1442D3
hypocenterC3#14674
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. C3×#1442C2semidirect
5. #71567C2semidirect
8. #69420C2semidirect
9. #69420C2semidirect
10. #69419C2semidirect
11. #69414C2semidirect
38. #4664C4semidirect
47. #1442D3semidirect
202. C2×C4×D3C4⋊C4semidirect
636. C3#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#49872
C2×#46642
C2×#46672
C3×#14422
#715672
#694282
#694272
#694202
#694192
#694142
#146743