Finite Group Databasebeta

← #10742734

Subgroups — #10742734

Order 768.

1862 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4665C212yesmaximal
2C2×#4664C212yesmaximal, complemented
3C3×#1442C211yesmaximal, characteristic, complemented
4#71570C212yesmaximal, complemented
5#69427C212yesmaximal, complemented
6#69426C211yesmaximal, characteristic
7(C2×D3)⋊#95C212yesmaximal, complemented
8#69422C211yesmaximal, characteristic, complemented
9#69418C212yesmaximal, complemented
10#15952—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C23×C4×D3C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×(C2×D3)⋊C4C2212yes—
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C2×C3×#95C2212yes—
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32#4665—28no—
33#4664—28no—
34C2×#386—32no—
35C2×#381—32no—
36#1445—31no—
37#1443—32no—
38#1442D311yescharacteristic, complemented
39#1439—32no—
40#1439—32no—
41#1438—31no—
42C22⋊#95—32no—
43C24×D3C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C22×D12—24no—
53C22×D12—24no—
54C22×D12—41no—
55C22×D12—41no—
56C22×D12D414yes—
57C22×C4×D3—24no—
58C22×C4×D3—24no—
59C22×C4×D3—24no—
60C22×C4×D3—24no—
61C22×C4×D3—24no—
62C22×C4×D3—42no—
63C22×C4×D3—42no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—42no—
66C2×(C2×D3)⋊C4—24no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—24no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—42no—
76C2×(C2×D3)⋊C4—42no—
77C2×(C2×D3)⋊C4—42no—
78C2×(C2×D3)⋊C4—42no—
79C2×(C2×D3)⋊C4—42no—
80C2×(C2×D3)⋊C4—42no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—41no—
83C2×C4⋊Dic3—41no—
84C3×#95—24no—
85C3×#95—41no—
86C3×#95—42no—
87C3×#95D414yescomplemented
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—41no—
94C3⋊#95—41no—
95C3⋊#95—42no—
96C23×D4—31no—
97C24×C4—32no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4C2×D311yescharacteristic
100C22×C22⋊C4—32no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4—32no—
103C22×C22⋊C4—32no—
104C22×C22⋊C4—32no—
105#386—68no—
106#381—68no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—31no—
113C2×#95—31no—
114C2×#95—32no—
115C2×#95C2×D312yes—
116C2×#95C2×D312yes—
117C2×#95C2×D311yescharacteristic
118C2×#95C2×D311yescharacteristic
119C24×C3C2411yescharacteristic
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—22no—
132C23×D3C2×D414yes—
133C23×D3#4114yes—
134C23×D3#4114yes—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4#4112yes—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
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—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3#4114yes—
185C2×D12—216no—
186C2×D12—44no—
187C2×D12—44no—
188C2×D12—44no—
189C2×D12—48no—
190C2×C4×D3—216no—
191C2×C4×D3—48no—
192C2×C4×D3—48no—
193C2×C4×D3—48no—
194C2×C4×D3—48no—
195C2×C4×D3—48no—
196C2×C4×D3—48no—
197C3×C4⋊C4—44no—
198(C2×D3)⋊C4—48no—
199(C2×D3)⋊C4—48no—
200(C2×D3)⋊C4—48no—
201(C2×D3)⋊C4—48no—
202(C2×D3)⋊C4—48no—
203C4⋊Dic3—44no—
204C25—32no—
205C22×D4—34no—
206C22×D4—64no—
207C22×D4—64no—
208C22×D4—121no—
209C22×D4—121no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—32no—
213C23×C4—64no—
214C23×C4—64no—
215C23×C4—64no—
216C23×C4—64no—
217C23×C4—64no—
218C23×C4—122no—
219C23×C4—122no—
220C23×C4C22×D312yes—
221C23×C4C22×D311yescharacteristic
222C23×C4C22×D312yes—
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—121no—
229C2×C4⋊C4—121no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—64no—
233C2×C22⋊C4—64no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—122no—
240C2×C22⋊C4—122no—
241C2×C22⋊C4—122no—
242C2×C22⋊C4—122no—
243C2×C22⋊C4—122no—
244C2×C22⋊C4—122no—
245#95—24no—
246#95—42no—
247#95—41no—
248#95—64no—
249#95—64no—
250#95—64no—
251#95—64no—
252#95—64no—
253#95—121no—
254#95—121no—
255#95—122no—
256#95D1214yescomplemented
257C23×C3C22×D411yescharacteristic
258C23×C3C22×D411yescharacteristic
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4112yes—
261C23×C3C2×#4112yes—
262C23×C3C2×#4112yes—
263C23×C3H(5,𝔽2)11yescharacteristic
264C23×C3H(5,𝔽2)12yes—
265C23×C3H(5,𝔽2)11yescharacteristic
266C23×C3H(5,𝔽2)11yescharacteristic
267C23×C32−1+411yescharacteristic
268C22×D3—24no—
269C22×D3—24no—
270C22×D3—24no—
271C22×D3—24no—
272C22×D3—24no—
273C22×D3—24no—
274C22×D3—24no—
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—24no—
278C22×D3—24no—
279C22×D3—24no—
280C22×D3—24no—
281C22×D3—24no—
282C22×D3—24no—
283C22×D3—24no—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
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×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3#12318yes—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353D12—416no—
354C4×D3—432no—
355C24—34no—
356C24—34no—
357C24—34no—
358C24—62no—
359C24—62no—
360C24—62no—
361C24—62no—
362C24—62no—
363C24—62no—
364C24—62no—
365C24—62no—
366C24—62no—
367C24—62no—
368C24—62no—
369C24—62no—
370C24C23×D311yescharacteristic
371C2×D4—616no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—128no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—34no—
402C22×C4—616no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—128no—
424C22×C4—128no—
425C22×C4—128no—
426C22×C4—128no—
427C22×C4—128no—
428C22×C4—128no—
429C22×C4C2×D1212yes—
430C22×C4C2×D1214yes—
431C22×C4D3×D412yes—
432C22×C4#23912yes—
433C4⋊C4—44no—
434C4⋊C4—124no—
435C22⋊C4—128no—
436C22⋊C4—128no—
437C22⋊C4—128no—
438C22⋊C4—128no—
439C22⋊C4—128no—
440C22×C3C2×#12312yes—
441C22×C3#52512yes—
442C22×C3#52912yes—
443C22×C3#53411yescharacteristic
444C22×C3#53711yescharacteristic
445C22×C3#53811yescharacteristic
446C22×C3#53812yes—
447C22×C3#53912yes—
448C22×C3#54012yes—
449C22×C3D4211yescharacteristic
450C22×C3#54612yes—
451C22×C3#54612yes—
452C22×C3#54711yescharacteristic
453C22×C3#55011yescharacteristic
454C22×C3#55112yes—
455C22×C3#55111yescharacteristic
456C22×C3#55111yescharacteristic
457C22×C3#55112yes—
458C22×C3#55112yes—
459C22×C3#55212yes—
460C22×C3#55511yescharacteristic
461C22×C3#56011yescharacteristic
462C22×C3#56211yescharacteristic
463C2×D3—28no—
464C2×D3—28no—
465C2×D3—28no—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—44no—
470C2×D3—44no—
471C2×D3—44no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—44no—
475C2×D3—44no—
476C2×D3—44no—
477C2×D3—44no—
478C3×C4—44no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—82no—
483Dic3—44no—
484Dic3—82no—
485Dic3—82no—
486C23—38no—
487C23—64no—
488C23—64no—
489C23—64no—
490C23—64no—
491C23—64no—
492C23—64no—
493C23—64no—
494C23—64no—
495C23—64no—
496C23—64no—
497C23—64no—
498C23—64no—
499C23—64no—
500C23—64no—
501C23—64no—
502C23—64no—
503C23—64no—
504C23—64no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—122no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—122no—
518C23—122no—
519C23—122no—
520C23—122no—
521C23C22×D1211yescharacteristic
522C23C2×D3×D411yescharacteristic
523C23#100412yes—
524C23#100411yescharacteristic
525C23C2×#23911yescharacteristic
526C23#100711yescharacteristic
527C23D3×#4112yes—
528C23D3×#4112yes—
529C23D3×#4112yes—
530C23#100911yescharacteristic
531C23#100911yescharacteristic
532D4—1216no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—1232no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C3#241812yes—
585C2×C3#241912yes—
586C2×C3#247112yes—
587C2×C3#248311yescharacteristic
588C2×C3#257211yescharacteristic
589C2×C3#260911yescharacteristic
590C2×C3#268211yescharacteristic
591C2×C3#269311yescharacteristic
592C2×C3#269911yescharacteristic
593C2×C3#274312yes—
594C2×C3#276011yescharacteristic
595D3—48no—
596C22—68no—
597C22—68no—
598C22—68no—
599C22—124no—
600C22—124no—
601C22—124no—
602C22—124no—
603C22—124no—
604C22—124no—
605C22—124no—
606C22—124no—
607C22—124no—
608C22—124no—
609C22—124no—
610C22—124no—
611C22#551011yescharacteristic
612C22#552412yes—
613C22D4×D1211yescharacteristic
614C22#555412yes—
615C22#557711yescharacteristic
616C22#559511yescharacteristic
617C22#560511yescharacteristic
618C22#561212yes—
619C22#563712yes—
620C22#563812yes—
621C22#564411yescharacteristic
622C22D3×#12312yes—
623C22#565911yescharacteristic
624C22#566312yes—
625C22#567011yescharacteristic
626C22#567812yes—
627C22#567911yescharacteristic
628C22#569112yes—
629C22#569312yes—
630C22#570011yescharacteristic
631C22#570512yes—
632C22#570512yes—
633C22#572611yescharacteristic
634C4—42no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—82no—
639C4—124no—
640C4—242no—
641C4—242no—
642C3#1595211yescharacteristic, complemented, Sylow
643C2—128no—
644C2#7424311yescharacteristic
645C2#7451711yescharacteristic
646C2#7533212yes—
647C2#7584412yes—
648C2#7626011yescharacteristic
649C2#7684211yescharacteristic
650C2#7714711yescharacteristic
651C2#7767711yescharacteristic
652C2#7794712yes—
653C2#7818411yescharacteristic
654C2#7838212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1442C2
prime radicalC23×D12C22
hypercenter#1442D3
hypocenterC3#15952
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#4664C2semidirect
3. C3×#1442C2semidirect
4. #71570C2semidirect
5. #69427C2semidirect
7. (C2×D3)⋊#95C2semidirect
8. #69422C2semidirect
9. #69418C2semidirect
38. #1442D3semidirect
87. C3×#95D4semidirect
256. #95D12semidirect
642. C3#15952semidirect

Maximal subgroups

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

subgroupindex
C2×#46652
C2×#46642
C3×#14422
#715702
#694272
#694262
(C2×D3)⋊#952
#694222
#694182
#159523