Finite Group Databasebeta

← #10740368

Subgroups — #10740368

Order 768.

1630 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4672C212yesmaximal
2C2×#4670C212yesmaximal, complemented
3C3×#1445C211yesmaximal, characteristic, complemented
4#71570C212yesmaximal, complemented
5#69429C212yesmaximal, complemented
6#69426C211yesmaximal, characteristic
7#69420C211yesmaximal, characteristic, complemented
8#69408C212yesmaximal, complemented
9#69406C212yesmaximal, complemented
10#15782—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×(C2×D3)⋊C4C2212yes—
14C22×(C2×D3)⋊C4C2212yes—
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32#4672—28no—
33#4670—28no—
34C2×#394—32no—
35C2×#388—32no—
36#1445D311yescharacteristic, complemented
37#1444—31no—
38#1440—32no—
39#1439—32no—
40#1438—31no—
41#1437—32no—
42#1436—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—22no—
54C22×D12—42no—
55C22×D12—41no—
56C22×D12—41no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—41no—
59C2×C3×C4⋊C4—41no—
60C2×(C2×D3)⋊C4—24no—
61C2×(C2×D3)⋊C4—24no—
62C2×(C2×D3)⋊C4—24no—
63C2×(C2×D3)⋊C4—24no—
64C2×(C2×D3)⋊C4—24no—
65C2×(C2×D3)⋊C4—24no—
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—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)⋊C4D418yes—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—41no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—42no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—42no—
85C2×C4×Dic3—42no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—41no—
89C3×#95—41no—
90C3⋊#95—22no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—41no—
95C3⋊#95—42no—
96C23×D4—31no—
97C22×C4⋊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—
105C22×C22⋊C4—32no—
106C22×C42—32no—
107#394—68no—
108#388—68no—
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×D3—22no—
133C23×D3—22no—
134C23×D3C2×D414yes—
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—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
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×D414yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
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×Dic3C2×D414yes—
183C22×Dic3#4114yes—
184C22×Dic3#4114yes—
185C2×D12—48no—
186C2×D12—44no—
187C2×D12—44no—
188C2×D12—48no—
189C2×D12—44no—
190C3×C4⋊C4—44no—
191(C2×D3)⋊C4—216no—
192(C2×D3)⋊C4—48no—
193(C2×D3)⋊C4—48no—
194(C2×D3)⋊C4—48no—
195(C2×D3)⋊C4—48no—
196(C2×D3)⋊C4—48no—
197C4⋊Dic3—44no—
198Dic3⋊C4—48no—
199C4×Dic3—48no—
200C25—32no—
201C22×D4—64no—
202C22×D4—62no—
203C22×D4—121no—
204C22×D4—121no—
205C22×D4—122no—
206C23×C4—32no—
207C23×C4—32no—
208C23×C4—32no—
209C23×C4C22×D312yes—
210C23×C4C22×D311yescharacteristic
211C23×C4C22×D312yes—
212C23×C4C22×D311yescharacteristic
213C23×C4C22×D311yescharacteristic
214C2×C4⋊C4—22no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—122no—
221C2×C4⋊C4—121no—
222C2×C22⋊C4—38no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4—64no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—64no—
228C2×C22⋊C4—64no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—64no—
233C2×C22⋊C4—64no—
234C2×C22⋊C4—122no—
235C2×C22⋊C4—122no—
236C2×C22⋊C4—122no—
237C2×C22⋊C4—122no—
238C2×C22⋊C4—122no—
239C2×C22⋊C4—122no—
240C2×C42—64no—
241C2×C42—122no—
242C2×C42—122no—
243#95—24no—
244#95—24no—
245#95—41no—
246#95—41no—
247#95—64no—
248#95—64no—
249#95—64no—
250#95—62no—
251#95—122no—
252#95—121no—
253C23×C3C22×D412yes—
254C23×C3C2×#4111yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4111yescharacteristic
257C23×C3C2×#4112yes—
258C23×C3C2×#4112yes—
259C23×C3H(5,𝔽2)11yescharacteristic
260C23×C3H(5,𝔽2)11yescharacteristic
261C23×C3H(5,𝔽2)11yescharacteristic
262C23×C3H(5,𝔽2)12yes—
263C23×C32−1+411yescharacteristic
264C22×D3—24no—
265C22×D3—24no—
266C22×D3—24no—
267C22×D3—24no—
268C22×D3—24no—
269C22×D3—24no—
270C22×D3—24no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
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—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
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×C4—42no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
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—
349D12—416no—
350C24—34no—
351C24—62no—
352C24—62no—
353C24—62no—
354C24—62no—
355C24—62no—
356C24—62no—
357C24—62no—
358C24—62no—
359C24—62no—
360C24—62no—
361C24—62no—
362C24—62no—
363C24—62no—
364C24—62no—
365C24C23×D311yescharacteristic
366C2×D4—128no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4—128no—
370C2×D4—124no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—62no—
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—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#23512yes—
418C22×C4D3×D414yes—
419C22×C4#23912yes—
420C22×C4#23912yes—
421C4⋊C4—44no—
422C4⋊C4—128no—
423C4⋊C4—124no—
424C22⋊C4—616no—
425C22⋊C4—128no—
426C22⋊C4—128no—
427C22⋊C4—128no—
428C22⋊C4—128no—
429C22⋊C4—128no—
430C42—128no—
431C22×C3#52512yes—
432C22×C3#52912yes—
433C22×C3#53212yes—
434C22×C3#53412yes—
435C22×C3#53811yescharacteristic
436C22×C3#53811yescharacteristic
437C22×C3#53911yescharacteristic
438C22×C3#54012yes—
439C22×C3#54212yes—
440C22×C3D4211yescharacteristic
441C22×C3#54612yes—
442C22×C3#54612yes—
443C22×C3#54712yes—
444C22×C3#55112yes—
445C22×C3#55312yes—
446C22×C3#55312yes—
447C22×C3#55511yescharacteristic
448C22×C3#55511yescharacteristic
449C22×C3#55611yescharacteristic
450C22×C3#55911yescharacteristic
451C22×C3#55911yescharacteristic
452C22×C3#56011yescharacteristic
453C22×C3#56211yescharacteristic
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—44no—
457C2×D3—44no—
458C2×D3—44no—
459C2×D3—44no—
460C2×D3—44no—
461C2×D3—82no—
462C2×D3—82no—
463C2×D3—82no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—82no—
467C2×D3—82no—
468C2×D3—82no—
469C3×C4—42no—
470C3×C4—44no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—82no—
474Dic3—44no—
475Dic3—44no—
476Dic3—44no—
477C23—64no—
478C23—64no—
479C23—64no—
480C23—64no—
481C23—64no—
482C23—64no—
483C23—64no—
484C23—122no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23—122no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23C2×#23511yescharacteristic
513C23C2×D3×D412yes—
514C23#100411yescharacteristic
515C23#100411yescharacteristic
516C23C2×#23911yescharacteristic
517C23C2×#23911yescharacteristic
518C23D3×#4112yes—
519C23D3×#4112yes—
520C23#100911yescharacteristic
521C23#100912yes—
522C23#101011yescharacteristic
523D4—1216no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
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×C3#245511yescharacteristic
575C2×C3#247612yes—
576C2×C3#252112yes—
577C2×C3#255511yescharacteristic
578C2×C3#259311yescharacteristic
579C2×C3#260212yes—
580C2×C3#266811yescharacteristic
581C2×C3#270112yes—
582C2×C3#275211yescharacteristic
583C2×C3#279511yescharacteristic
584C2×C3#280811yescharacteristic
585D3—84no—
586C22—124no—
587C22—124no—
588C22—124no—
589C22—124no—
590C22—124no—
591C22—124no—
592C22—124no—
593C22—242no—
594C22—242no—
595C22—242no—
596C22—242no—
597C22—242no—
598C22—242no—
599C22—242no—
600C22—242no—
601C22#549411yescharacteristic
602C22#552712yes—
603C22#554812yes—
604C22#558411yescharacteristic
605C22#558511yescharacteristic
606C22#561012yes—
607C22#562712yes—
608C22#563012yes—
609C22#564411yescharacteristic
610C22#564511yescharacteristic
611C22#565312yes—
612C22#565512yes—
613C22#566312yes—
614C22#566411yescharacteristic
615C22#566611yescharacteristic
616C22#567011yescharacteristic
617C22#568411yescharacteristic
618C22#569112yes—
619C22#569512yes—
620C22#570912yes—
621C22#571012yes—
622C22#571711yescharacteristic
623C22#573411yescharacteristic
624C4—42no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—82no—
629C4—124no—
630C4—124no—
631C4—124no—
632C3#1578211yescharacteristic, complemented, Sylow
633C2—244no—
634C2#7410911yescharacteristic
635C2#7504412yes—
636C2#7537412yes—
637C2#7635611yescharacteristic
638C2#7660111yescharacteristic
639C2#7769812yes—
640C2#7776012yes—
641C2#7802411yescharacteristic
642C2#7811211yescharacteristic
643C2#7819811yescharacteristic
644C2#7830311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1445C2
prime radicalC23×D12C22
hypercenter#1445D3
hypocenterC3#15782
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#4670C2semidirect
3. C3×#1445C2semidirect
4. #71570C2semidirect
5. #69429C2semidirect
7. #69420C2semidirect
8. #69408C2semidirect
9. #69406C2semidirect
36. #1445D3semidirect
632. C3#15782semidirect

Maximal subgroups

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

subgroupindex
C2×#46722
C2×#46702
C3×#14452
#715702
#694292
#694262
#694202
#694082
#694062
#157823