Finite Group Databasebeta

← #10740341

Subgroups — #10740341

Order 768.

1730 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5222C212yesmaximal
2C2×#4945C212yesmaximal, complemented
3C2×#4951C212yesmaximal, complemented
4C3×#1438C211yesmaximal, characteristic
5#71547C212yesmaximal, complemented
6#71546C212yesmaximal, complemented
7#69420C211yesmaximal, characteristic, complemented
8#69415C211yesmaximal, characteristic, complemented
9Dic3⋊#95C212yesmaximal, complemented
10#15780—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×C3×C22⋊C4C2212yes—
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25#5222—28no—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33#4951—28no—
34#4945—28no—
35C2×#394—32no—
36C2×#388—32no—
37C2×#388—32no—
38#1444—31no—
39#1442—31no—
40#1439—32no—
41#1439—32no—
42#1438D311yescharacteristic
43C4⋊#95—32no—
44C25×C3C2312yes—
45C22×C3×D4—24no—
46C22×C3×D4—22no—
47C22×C3×D4—41no—
48C22×C3×D4—41no—
49C22×C3×D4—42no—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—42no—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217—42no—
73C2×#217—42no—
74C2×#217—42no—
75C2×#217D418yes—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—41no—
78C2×C4⋊Dic3—42no—
79C2×C4⋊Dic3—41no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—42no—
85C2×Dic3⋊C4—42no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C3×#95—41no—
90C3×#95—41no—
91C3⋊#95—24no—
92C3⋊#95—22no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—42no—
97C3⋊#95D414yescomplemented
98C23×D4C2×D311yescharacteristic
99C22×C4⋊C4—32no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—31no—
103C22×C22⋊C4—32no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4—32no—
106C22×C22⋊C4—32no—
107C22×C22⋊C4C2×D312yes—
108C22×C22⋊C4C2×D312yes—
109C22×C42—32no—
110#394—68no—
111#388—68no—
112#388—68no—
113C2×#95—31no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—31no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95C2×D311yescharacteristic
121C2×#95C2×D311yescharacteristic
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3C2×D414yes—
137C24×C3C2411yescharacteristic
138C2×C3×D4—48no—
139C2×C3×D4—44no—
140C2×C3×D4—44no—
141C2×C3×D4—48no—
142C2×C3×D4—44no—
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—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4114yes—
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—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D414yes—
191C22×Dic3C2×D414yes—
192C22×Dic3#4114yes—
193C3×C22⋊C4—216no—
194C3×C22⋊C4—48no—
195#217—216no—
196#217—48no—
197#217—48no—
198#217—48no—
199C4⋊Dic3—44no—
200C4⋊Dic3—44no—
201Dic3⋊C4—48no—
202Dic3⋊C4—48no—
203C4×Dic3—48no—
204C25C22×D312yes—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4—42no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—32no—
213C23×C4—32no—
214C23×C4C22×D311yescharacteristic
215C23×C4C22×D311yescharacteristic
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D312yes—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—64no—
223C2×C4⋊C4—122no—
224C2×C4⋊C4—122no—
225C2×C4⋊C4—121no—
226C2×C4⋊C4—121no—
227C2×C4⋊C4—122no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—38no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—64no—
243C2×C22⋊C4—122no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42—64no—
247C2×C42—64no—
248C2×C42—64no—
249#95—34no—
250#95—41no—
251#95—41no—
252#95—64no—
253#95—64no—
254#95—64no—
255#95—64no—
256#95—62no—
257#95—122no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—24no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3C22×D411yescharacteristic
294C23×C3C22×D412yes—
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4111yescharacteristic
297C23×C3C2×#4112yes—
298C23×C3C2×#4112yes—
299C23×C3H(5,𝔽2)11yescharacteristic
300C23×C3H(5,𝔽2)12yes—
301C23×C3H(5,𝔽2)11yescharacteristic
302C23×C32−1+411yescharacteristic
303C23×C32−1+411yescharacteristic
304C3×D4—416no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
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—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×Dic3—24no—
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—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—
353C2×Dic3—42no—
354C2×Dic3#12118yes—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24D3×D414yes—
370C24C23×D311yescharacteristic
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—48no—
375C2×D4—44no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—34no—
395C22×C4—34no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—62no—
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—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4#23512yes—
423C22×C4D3×D412yes—
424C22×C4#23712yes—
425C22×C4#23714yes—
426C4⋊C4—128no—
427C4⋊C4—128no—
428C4⋊C4—124no—
429C4⋊C4—124no—
430C22⋊C4—216no—
431C22⋊C4—48no—
432C22⋊C4—616no—
433C22⋊C4—128no—
434C22⋊C4—128no—
435C22⋊C4—128no—
436C42—128no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3C2×#12112yes—
453C22×C3#52512yes—
454C22×C3#52912yes—
455C22×C3#53512yes—
456C22×C3#53711yescharacteristic
457C22×C3#53811yescharacteristic
458C22×C3#53912yes—
459C22×C3#53911yescharacteristic
460C22×C3#54212yes—
461C22×C3D4211yescharacteristic
462C22×C3#54612yes—
463C22×C3#54612yes—
464C22×C3#54712yes—
465C22×C3#54711yescharacteristic
466C22×C3#54712yes—
467C22×C3#55011yescharacteristic
468C22×C3#55312yes—
469C22×C3#55511yescharacteristic
470C22×C3#55512yes—
471C22×C3#55611yescharacteristic
472C22×C3#55911yescharacteristic
473C22×C3H(3,𝔽4)11yescharacteristic
474C22×C3#56311yescharacteristic
475C3×C4—44no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479Dic3—28no—
480Dic3—44no—
481Dic3—82no—
482Dic3—82no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C2×#23511yescharacteristic
519C23C2×D3×D412yes—
520C23C2×D3×D411yescharacteristic
521C23C2×#23712yes—
522C23C2×#23711yescharacteristic
523C23#100412yes—
524C23#100411yescharacteristic
525C23#100411yescharacteristic
526C23D3×#4112yes—
527C23#101011yescharacteristic
528C23#101011yescharacteristic
529D4—416no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—38no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
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×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C3—84no—
581C2×C3#241412yes—
582C2×C3#242512yes—
583C2×C3#245511yescharacteristic
584C2×C3#247412yes—
585C2×C3#253711yescharacteristic
586C2×C3#259311yescharacteristic
587C2×C3#267011yescharacteristic
588C2×C3#271211yescharacteristic
589C2×C3#271812yes—
590C2×C3#272711yescharacteristic
591C2×C3#278511yescharacteristic
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22#549012yes—
608C22#549411yescharacteristic
609C22#554211yescharacteristic
610C22#555111yescharacteristic
611C22#555712yes—
612C22#558612yes—
613C22#559011yescharacteristic
614C22#559112yes—
615C22#559212yes—
616C22#559411yescharacteristic
617C22#559711yescharacteristic
618C22#560112yes—
619C22#560211yescharacteristic
620C22#560312yes—
621C22D3×#12112yes—
622C22#561112yes—
623C22#561612yes—
624C22#565512yes—
625C22#566012yes—
626C22#567011yescharacteristic
627C22#567211yescharacteristic
628C22#571311yescharacteristic
629C22#571811yescharacteristic
630C4—42no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—68no—
635C4—124no—
636C4—242no—
637C4—242no—
638C3#1578011yescharacteristic, complemented, Sylow
639C2—84no—
640C2#7383711yescharacteristic
641C2#7472912yes—
642C2#7489512yes—
643C2#7636611yescharacteristic
644C2#7639911yescharacteristic
645C2#7667012yes—
646C2#7667411yescharacteristic
647C2#7672711yescharacteristic
648C2#7694411yescharacteristic
649C2#7703912yes—
650C2#7707011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15780
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#4945C2semidirect
3. C2×#4951C2semidirect
5. #71547C2semidirect
6. #71546C2semidirect
7. #69420C2semidirect
8. #69415C2semidirect
9. Dic3⋊#95C2semidirect
97. C3⋊#95D4semidirect
638. C3#15780semidirect

Maximal subgroups

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

subgroupindex
C2×#52222
C2×#49452
C2×#49512
C3×#14382
#715472
#715462
#694202
#694152
Dic3⋊#952
#157803