Finite Group Databasebeta

← #10727933

Subgroups — #10727933

Order 768.

3350 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×C3×#380C211yesmaximal, characteristic, complemented
3C2×#4992C212yesmaximal, complemented
4C2×#4955C212yesmaximal
5#72127C211yesmaximal, characteristic, complemented
6#71570C212yesmaximal, complemented
7#71544C212yesmaximal, complemented
8#69429C214yesmaximal, complemented
9#14820—31nomaximal, Sylow
10C25×D3C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×Dic3C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic, complemented
14C22×C3×C4⋊C4C2211yescharacteristic, complemented
15C22×C3×C22⋊C4C2212yes—
16C22×#217C2212yes—
17C22×(C2×D3)⋊C4—24no—
18C22×(C2×D3)⋊C4—28no—
19C22×(C2×D3)⋊C4C2214yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2214yes—
23C3×#380—24no—
24C2×C3×#95C2212yescomplemented
25C2×C3⋊#95C2214yescomplemented
26C2×C3⋊#95C2212yescomplemented
27C2×C3⋊#95C2212yescomplemented
28C2×C3⋊#95C2214yescomplemented
29#4992—28no—
30#4955—28no—
31C23×C22⋊C4—31no—
32C2×#397—32no—
33C2×#380D311yescharacteristic, complemented
34C2×#379—32no—
35#1440—34no—
36#1439—32no—
37C22⋊#95—31no—
38C22⋊#95—32no—
39C25×C3C2311yescharacteristic
40C24×D3—22no—
41C24×D3—22no—
42C24×D3—24no—
43C24×D3—24no—
44C24×D3—24no—
45C24×D3—24no—
46C24×D3—42no—
47C24×D3—42no—
48C24×D3D414yes—
49C24×D3C2312yes—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—42no—
53C23×C3×C4D414yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—22no—
59C23×Dic3—24no—
60C23×Dic3—24no—
61C23×Dic3—42no—
62C23×Dic3C2314yes—
63C23×Dic3C2312yes—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×#217—24no—
72C2×#217—24no—
73C2×#217—24no—
74C2×(C2×D3)⋊C4—232no—
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—28no—
81C2×(C2×D3)⋊C4—28no—
82C2×(C2×D3)⋊C4—28no—
83C2×(C2×D3)⋊C4—28no—
84C2×(C2×D3)⋊C4—44no—
85C2×(C2×D3)⋊C4—44no—
86C2×(C2×D3)⋊C4D418yes—
87C3×#95—24no—
88C3⋊#95—24no—
89C3⋊#95—28no—
90C3⋊#95—44no—
91C3⋊#95—42no—
92C26—31no—
93C24×C4—31no—
94C24×C4C2×D311yescharacteristic
95C22×C4⋊C4C2×D311yescharacteristic, complemented
96C22×C4⋊C4C2×D311yescharacteristic, complemented
97C22×C22⋊C4—32no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4—34no—
100C22×C22⋊C4—32no—
101C22×C22⋊C4—34no—
102C22×C22⋊C4—68no—
103C22×C22⋊C4—64no—
104C22×C22⋊C4C2×D312yes—
105#397—68no—
106#380—24no—
107#379—68no—
108C2×#95—32no—
109C2×#95—34no—
110C2×#95—34no—
111C2×#95—32no—
112C2×#95C2×D312yescomplemented
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3C2×D412yes—
118C24×C3C2×D412yes—
119C24×C3#4114yes—
120C24×C3#4112yes—
121C24×C3#4114yes—
122C24×C3C2411yescharacteristic
123C23×D3—28no—
124C23×D3—216no—
125C23×D3—28no—
126C23×D3—28no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—24no—
130C23×D3—22no—
131C23×D3—24no—
132C23×D3—22no—
133C23×D3—24no—
134C23×D3—24no—
135C23×D3—216no—
136C23×D3—216no—
137C23×D3—48no—
138C23×D3—416no—
139C23×D3—416no—
140C23×D3—416no—
141C23×D3—416no—
142C23×D3—44no—
143C23×D3—44no—
144C23×D3—48no—
145C23×D3—44no—
146C23×D3—44no—
147C23×D3—48no—
148C23×D3C2×D414yes—
149C23×D3C2×D414yes—
150C23×D3#4114yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—28no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—28no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—44no—
169C22×C3×C4—416no—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×Dic3—28no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—416no—
186C22×Dic3—44no—
187C22×Dic3—44no—
188C22×Dic3C2×D414yes—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—44no—
191C3×C22⋊C4—48no—
192#217—48no—
193(C2×D3)⋊C4—216no—
194(C2×D3)⋊C4—216no—
195(C2×D3)⋊C4—416no—
196(C2×D3)⋊C4—416no—
197C25—32no—
198C25—34no—
199C25—64no—
200C25—64no—
201C25—62no—
202C25—62no—
203C25—64no—
204C25—64no—
205C25—122no—
206C25—122no—
207C25C22×D311yescharacteristic
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—32no—
211C23×C4—34no—
212C23×C4—42no—
213C23×C4—64no—
214C23×C4—62no—
215C23×C4—64no—
216C23×C4—122no—
217C23×C4#6714yes—
218C23×C4C22×D312yes—
219C23×C4C22×D312yes—
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—38no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—632no—
233C2×C22⋊C4—64no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—68no—
237C2×C22⋊C4—68no—
238C2×C22⋊C4—68no—
239C2×C22⋊C4—68no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—64no—
243C2×C22⋊C4—124no—
244C2×C22⋊C4—124no—
245#95—24no—
246#95—68no—
247#95—64no—
248#95—122no—
249#95—124no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—22no—
257C23×C3—24no—
258C23×C3—28no—
259C23×C3—28no—
260C23×C3—28no—
261C23×C3—28no—
262C23×C3#12114yes—
263C23×C3#12318yes—
264C23×C3#12314yes—
265C23×C3#12414yes—
266C23×C3#12618yes—
267C23×C3C22×D412yes—
268C23×C3C22×D411yescharacteristic
269C23×C3C22×D411yescharacteristic
270C23×C3C2×#4112yes—
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4112yes—
273C23×C3C2×#4112yes—
274C23×C3H(5,𝔽2)12yes—
275C23×C3H(5,𝔽2)12yes—
276C22×D3—232no—
277C22×D3—24no—
278C22×D3—24no—
279C22×D3—24no—
280C22×D3—28no—
281C22×D3—24no—
282C22×D3—24no—
283C22×D3—24no—
284C22×D3—28no—
285C22×D3—28no—
286C22×D3—28no—
287C22×D3—28no—
288C22×D3—28no—
289C22×D3—464no—
290C22×D3—416no—
291C22×D3—416no—
292C22×D3—416no—
293C22×D3—416no—
294C22×D3—464no—
295C22×D3—416no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—44no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—44no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3—44no—
305C22×D3—44no—
306C22×D3—416no—
307C22×D3#12118yes—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—28no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—416no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—44no—
326C2×C3×C4#12014yes—
327C2×Dic3—24no—
328C2×Dic3—28no—
329C2×Dic3—416no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C2×Dic3—44no—
335C2×Dic3—44no—
336C2×Dic3—44no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3—44no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—34no—
345C24—34no—
346C24—34no—
347C24—616no—
348C24—68no—
349C24—68no—
350C24—616no—
351C24—68no—
352C24—64no—
353C24—62no—
354C24—62no—
355C24—64no—
356C24—64no—
357C24—62no—
358C24—64no—
359C24—62no—
360C24—616no—
361C24—128no—
362C24—1216no—
363C24—1216no—
364C24—1216no—
365C24—1216no—
366C24—124no—
367C24—124no—
368C24—128no—
369C24—128no—
370C24—124no—
371C24—124no—
372C24#23514yes—
373C24D3×D412yes—
374C24#23712yes—
375C24#23914yes—
376C24C2×#6712yes—
377C24C23×D311yescharacteristic
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—34no—
396C22×C4—44no—
397C22×C4—416no—
398C22×C4—68no—
399C22×C4—62no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—1216no—
411C22×C4—124no—
412C22×C4—124no—
413C22×C4D3×D414yes—
414C22×C4C2×#6712yes—
415C22×C4C2×#6712yes—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—616no—
420C22⋊C4—616no—
421C22⋊C4—1216no—
422C22⋊C4—1216no—
423C22⋊C4—128no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—216no—
429C22×C3—24no—
430C22×C3—216no—
431C22×C3—24no—
432C22×C3#39718yescomplemented
433C22×C3C2×#12011yescharacteristic
434C22×C3C2×#12111yescharacteristic
435C22×C3C2×#12112yes—
436C22×C3C2×#12312yes—
437C22×C3C2×#12311yescharacteristic
438C22×C3#52511yescharacteristic
439C22×C3#52512yes—
440C22×C3C2×#12411yescharacteristic
441C22×C3C2×#12612yes—
442C22×C3#53411yescharacteristic
443C22×C3#53412yes—
444C22×C3#53511yescharacteristic
445C22×C3#53812yes—
446C22×C3#54614yes—
447C22×C3#54614yes—
448C22×C3#55114yes—
449C22×C3#55114yes—
450C2×D3—28no—
451C2×D3—216no—
452C2×D3—464no—
453C2×D3—44no—
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—44no—
457C2×D3—48no—
458C2×D3—44no—
459C2×D3—48no—
460C2×D3—48no—
461C2×D3—44no—
462C3×C4—24no—
463C3×C4—42no—
464C3×C4—82no—
465C3×C4—82no—
466Dic3—44no—
467Dic3—84no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—38no—
481C23—632no—
482C23—68no—
483C23—68no—
484C23—68no—
485C23—68no—
486C23—64no—
487C23—64no—
488C23—64no—
489C23—64no—
490C23—68no—
491C23—64no—
492C23—64no—
493C23—68no—
494C23—1264no—
495C23—1216no—
496C23—1216no—
497C23—1216no—
498C23—1216no—
499C23—1264no—
500C23—1216no—
501C23—124no—
502C23—124no—
503C23—1216no—
504C23—124no—
505C23—124no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—124no—
511C23—124no—
512C23#89418yes—
513C23#89818yes—
514C23#92914yes—
515C23#93814yes—
516C23#94714yes—
517C23C2×#23512yes—
518C23C2×D3×D412yes—
519C23C2×D3×D411yescharacteristic
520C23C2×#23711yescharacteristic
521C23#100412yes—
522C23C2×#23912yes—
523C23D3×#4112yes—
524C23#100912yes—
525C23C22×#6711yescharacteristic
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—416no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—64no—
545C2×C4—68no—
546C2×C4—1216no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—124no—
550C2×C4—124no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—124no—
555C2×C4—124no—
556C2×C4—124no—
557C2×C4#93714yes—
558C2×C3—28no—
559C2×C3C2×#39711yescharacteristic
560C2×C3#240511yescharacteristic
561C2×C3#241214yes—
562C2×C3#243511yescharacteristic
563C2×C3#248014yes—
564C2×C3#248211yescharacteristic
565C2×C3#256111yescharacteristic
566C2×C3#261711yescharacteristic
567C2×C3#284711yescharacteristic
568D3—48no—
569C22—24no—
570C22—216no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—216no—
576C22—24no—
577C22—616no—
578C22—68no—
579C22—1264no—
580C22—124no—
581C22—124no—
582C22—128no—
583C22—124no—
584C22—124no—
585C22—124no—
586C22—128no—
587C22—128no—
588C22—124no—
589C22#499218yescomplemented
590C22#549014yes—
591C22C2×#89412yes—
592C22C2×#89812yes—
593C22#552714yes—
594C22D3×#12112yes—
595C22#560512yes—
596C22#560612yes—
597C22#561712yes—
598C22#563014yes—
599C22#563714yes—
600C22C2×#92911yescharacteristic
601C22C2×#93711yescharacteristic
602C22C2×#93811yescharacteristic
603C22#580411yescharacteristic
604C22C2×#94711yescharacteristic
605C22#582811yescharacteristic
606C22#583011yescharacteristic
607C4—24no—
608C4—42no—
609C4—82no—
610C4—82no—
611C4—124no—
612C4—244no—
613C3#1482011yescharacteristic, complemented, Sylow
614C2—28no—
615C2—128no—
616C2#7390914yes—
617C2#7401814yes—
618C2#8212611yescharacteristic
619C2C2×#499211yescharacteristic
620C2#8228211yescharacteristic
621C2#8271011yescharacteristic
622C2#8272311yescharacteristic
623C2#8276111yescharacteristic
624C2#8281911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#380C2
prime radicalC25×D3C22
hypercenterC2×#380D3
hypocenterC3#14820
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×C3×#380C2semidirect
3. C2×#4992C2semidirect
5. #72127C2semidirect
6. #71570C2semidirect
7. #71544C2semidirect
8. #69429C2semidirect
13. C22×C3×C4⋊C4C22semidirect
14. C22×C3×C4⋊C4C22semidirect
24. C2×C3×#95C22semidirect
25. C2×C3⋊#95C22semidirect
26. C2×C3⋊#95C22semidirect
27. C2×C3⋊#95C22semidirect
28. C2×C3⋊#95C22semidirect
33. C2×#380D3semidirect
95. C22×C4⋊C4C2×D3semidirect
96. C22×C4⋊C4C2×D3semidirect
112. C2×#95C2×D3semidirect
432. C22×C3#397semidirect
589. C22#4992semidirect
613. C3#14820semidirect

Maximal subgroups

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

subgroupindex
C23×(C2×D3)⋊C42
C2×C3×#3802
C2×#49922
C2×#49552
#721272
#715702
#715442
#694292
#148203