Finite Group Databasebeta

← #10732142

Subgroups — #10732142

Order 768.

1746 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
1C2×C3×#400C211yesmaximal, characteristic, complemented
2C2×#4991C211yesmaximal, characteristic
3C2×#4985C211yesmaximal, characteristic, complemented
4C2×#4988C211yesmaximal, characteristic
5C2×#4976C211yesmaximal, characteristic, complemented
6C2×#4672C211yesmaximal, characteristic
7#69428C212yesmaximal, complemented
8#69428C212yesmaximal, complemented
9#69408C212yesmaximal, complemented
10#69408C212yesmaximal, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#15170—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C3×#400—24no—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4991—24no—
38#4988—24no—
39#4985—24no—
40#4976—24no—
41#4672—24no—
42C2×#400—31no—
43C2×#400D311yescharacteristic, complemented
44C2×#394—31no—
45C2×#394—31no—
46C2×#390—31no—
47C2×#389—31no—
48#1439—32no—
49#1439—32no—
50#1437—32no—
51#1437—32no—
52C4×#95—31no—
53C24×D3C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C22×D12—24no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—42no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
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—42no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4—42no—
92C2×(C2×D3)⋊C4—42no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—42no—
101C2×C4×Dic3—24no—
102C2×C4×Dic3—42no—
103C2×C4×Dic3—42no—
104C2×C4×Dic3—42no—
105C2×C4×Dic3D414yes—
106C3×#95—24no—
107C3×#95—24no—
108C3×#95D414yes—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95D414yes—
115C23×D4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—31no—
119C22×C4⋊C4C2×D311yescharacteristic
120C22×C4⋊C4C2×D311yescharacteristic
121C22×C22⋊C4—32no—
122C22×C22⋊C4—32no—
123C22×C22⋊C4—32no—
124C22×C22⋊C4—32no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C42—31no—
128C22×C42—32no—
129#400—24no—
130#400—64no—
131#394—64no—
132#394—64no—
133#390—64no—
134#389—64no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—31no—
140C2×#95—31no—
141C2×#95C2×D312yes—
142C2×#95C2×D312yes—
143C2×#95C2×D311yescharacteristic
144C24×C3C2411yescharacteristic
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×D3—22no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3#4114yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
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×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4#4112yes—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3C2×D414yes—
215C22×Dic3C2×D412yes—
216C22×Dic3C2×D412yes—
217C22×Dic3#4112yes—
218C22×Dic3#4114yes—
219C22×Dic3#4114yes—
220C2×D12—28no—
221C2×D12—48no—
222C2×D12—48no—
223C2×D12—44no—
224C2×D12—44no—
225C3×C4⋊C4—44no—
226C3×C4⋊C4—44no—
227(C2×D3)⋊C4—48no—
228(C2×D3)⋊C4—48no—
229(C2×D3)⋊C4—48no—
230(C2×D3)⋊C4—48no—
231(C2×D3)⋊C4—48no—
232(C2×D3)⋊C4—48no—
233Dic3⋊C4—28no—
234Dic3⋊C4—44no—
235Dic3⋊C4—44no—
236C4×Dic3—44no—
237C4×Dic3—48no—
238C25—32no—
239C22×D4—64no—
240C22×D4—62no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4—122no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—32no—
248C23×C4—31no—
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D312yes—
253C23×C4C22×D311yescharacteristic
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—122no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
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—122no—
283C2×C22⋊C4—122no—
284C2×C22⋊C4—122no—
285C2×C22⋊C4—122no—
286C2×C42—34no—
287C2×C42—64no—
288C2×C42—122no—
289C2×C42—122no—
290C2×C42—122no—
291#95—24no—
292#95—24no—
293#95—34no—
294#95—64no—
295#95—62no—
296#95—62no—
297#95—64no—
298#95—64no—
299#95#6714yes—
300C23×C3C22×D411yescharacteristic
301C23×C3C22×D411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C2×#4111yescharacteristic
304C23×C3C2×#4112yes—
305C23×C3C2×#4111yescharacteristic
306C23×C3C2×#4112yes—
307C23×C3C2×#4112yes—
308C23×C3H(5,𝔽2)12yes—
309C23×C3H(5,𝔽2)11yescharacteristic
310C23×C32−1+411yescharacteristic
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
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—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4#12714yes—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—22no—
391C2×Dic3—42no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—42no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3#12414yes—
405C2×Dic3#12414yes—
406D12—416no—
407C24—34no—
408C24—62no—
409C24—62no—
410C24—62no—
411C24—62no—
412C24—62no—
413C24—62no—
414C24—62no—
415C24—62no—
416C24—62no—
417C24—62no—
418C24—62no—
419C24—62no—
420C24—62no—
421C24—62no—
422C24C23×D311yescharacteristic
423C2×D4—68no—
424C2×D4—128no—
425C2×D4—128no—
426C2×D4—124no—
427C2×D4—124no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
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—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—32no—
455C22×C4—32no—
456C22×C4—34no—
457C22×C4—34no—
458C22×C4—34no—
459C22×C4—32no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—62no—
466C22×C4—61no—
467C22×C4—61no—
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—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4D3×D412yes—
484C22×C4D3×D412yes—
485C22×C4#23912yes—
486C22×C4C2×#6714yes—
487C22×C4C2×#6712yes—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—68no—
491C4⋊C4—124no—
492C4⋊C4—124no—
493C22⋊C4—128no—
494C22⋊C4—128no—
495C22⋊C4—128no—
496C22⋊C4—128no—
497C22⋊C4—128no—
498C22⋊C4—128no—
499C42—124no—
500C42—128no—
501C22×C3#52511yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3C2×#12411yescharacteristic
504C22×C3C2×#12411yescharacteristic
505C22×C3#52912yes—
506C22×C3#52912yes—
507C22×C3C2×#12711yescharacteristic
508C22×C3#53212yes—
509C22×C3#53212yes—
510C22×C3#53411yescharacteristic
511C22×C3#53711yescharacteristic
512C22×C3#53812yes—
513C22×C3#53912yes—
514C22×C3#54612yes—
515C22×C3#54612yes—
516C22×C3#54611yescharacteristic
517C22×C3#54611yescharacteristic
518C22×C3#54811yescharacteristic
519C22×C3#54811yescharacteristic
520C22×C3#55212yes—
521C22×C3#55312yes—
522C22×C3#55312yes—
523C22×C3#55312yes—
524C2×D3—44no—
525C2×D3—44no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—82no—
532C2×D3—82no—
533C2×D3—82no—
534C2×D3—82no—
535C2×D3—82no—
536C2×D3—82no—
537C2×D3—82no—
538C2×D3—82no—
539C3×C4—24no—
540C3×C4—42no—
541C3×C4—44no—
542C3×C4—82no—
543C3×C4—81no—
544Dic3—24no—
545Dic3—24no—
546Dic3—42no—
547Dic3—44no—
548Dic3—81no—
549C23—64no—
550C23—64no—
551C23—64no—
552C23—64no—
553C23—64no—
554C23—64no—
555C23—64no—
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—
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—
584C23C2×D3×D411yescharacteristic
585C23C2×D3×D411yescharacteristic
586C23C2×#23911yescharacteristic
587C23#100711yescharacteristic
588C23D3×#4111yescharacteristic
589C23D3×#4112yes—
590C23D3×#4112yes—
591C23D3×#4112yes—
592C23#100912yes—
593C23#100911yescharacteristic
594C23C22×#6711yescharacteristic
595D4—1216no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—34no—
607C2×C4—34no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—64no—
631C2×C4—64no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—64no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—122no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—122no—
655C2×C4#94014yes—
656C2×C3#239811yescharacteristic
657C2×C3#253911yescharacteristic
658C2×C3#254011yescharacteristic
659C2×C3#255812yes—
660C2×C3#255812yes—
661C2×C3#255912yes—
662C2×C3#255912yes—
663C2×C3#256211yescharacteristic
664C2×C3#259111yescharacteristic
665C2×C3#263911yescharacteristic
666C2×C3#285811yescharacteristic
667D3—84no—
668C22—124no—
669C22—124no—
670C22—124no—
671C22—124no—
672C22—124no—
673C22—124no—
674C22—124no—
675C22—242no—
676C22—242no—
677C22—242no—
678C22—242no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22#562711yescharacteristic
684C22#562911yescharacteristic
685C22#563011yescharacteristic
686C22#563311yescharacteristic
687C22#565311yescharacteristic
688C22#565412yes—
689C22#565611yescharacteristic
690C22#566812yes—
691C22D3×#12411yescharacteristic
692C22D3×#12411yescharacteristic
693C22#569112yes—
694C22#569212yes—
695C22#569312yes—
696C22#569512yes—
697C22#570412yes—
698C22#570612yes—
699C22#570912yes—
700C22#570912yes—
701C22C2×#94011yescharacteristic
702C22#581711yescharacteristic
703C22#582911yescharacteristic
704C22#582912yes—
705C22#583212yes—
706C4—24no—
707C4—42no—
708C4—44no—
709C4—64no—
710C4—64no—
711C4—82no—
712C4—81no—
713C4—122no—
714C4—124no—
715C4—241no—
716C3#1517011yescharacteristic, complemented, Sylow
717C2—244no—
718C2#7813111yescharacteristic
719C2#7817812yes—
720C2#7817912yes—
721C2#7818011yescharacteristic
722C2#7835811yescharacteristic
723C2#7836111yescharacteristic
724C2#8277911yescharacteristic
725C2#8290111yescharacteristic
726C2#8299311yescharacteristic
727C2#8306212yes—
728C2#8310912yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#400C2
prime radicalC23×D12C22
hypercenterC2×#400D3
hypocenterC3#15170
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
1. C2×C3×#400C2semidirect
3. C2×#4985C2semidirect
5. C2×#4976C2semidirect
7. #69428C2semidirect
8. #69428C2semidirect
9. #69408C2semidirect
10. #69408C2semidirect
11. Dic3×#95C2semidirect
43. C2×#400D3semidirect
716. C3#15170semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4002
C2×#49912
C2×#49852
C2×#49882
C2×#49762
C2×#46722
#694282
#694082
Dic3×#952
#151703