Finite Group Databasebeta

← #10740365

Subgroups — #10740365

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×#5222C212yesmaximal
2C2×#4951C212yesmaximal, complemented
3C3×#1438C211yesmaximal, characteristic
4#71549C212yesmaximal, complemented
5#71546C212yesmaximal, complemented
6#71545C212yesmaximal, complemented
7#69423C211yesmaximal, characteristic, complemented
8#69420C211yesmaximal, characteristic, complemented
9#69406C212yesmaximal, complemented
10#15782—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×C4×Dic3C2212yes—
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24#5222—28no—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33#4951—28no—
34C2×#394—32no—
35C2×#388—32no—
36#1445—31no—
37#1444—31no—
38#1440—32no—
39#1439—32no—
40#1438D311yescharacteristic
41#1437—32no—
42#1436—32no—
43C25×C3C2312yes—
44C22×C3×D4—24no—
45C22×C3×D4—22no—
46C22×C3×D4—41no—
47C22×C3×D4—41no—
48C22×C3×D4—42no—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—42no—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—42no—
70C2×#217—42no—
71C2×#217—42no—
72C2×#217—42no—
73C2×#217—42no—
74C2×#217D418yes—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—41no—
78C2×C4⋊Dic3—41no—
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—41no—
87C3×#95—41no—
88C3⋊#95—24no—
89C3⋊#95—22no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—42no—
95C3⋊#95—41no—
96C23×D4C2×D311yescharacteristic
97C22×C4⋊C4—32no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4—31no—
100C22×C22⋊C4—32no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4—32no—
103C22×C22⋊C4—32no—
104C22×C22⋊C4C2×D312yes—
105C22×C22⋊C4C2×D312yes—
106C22×C42—32no—
107#394—68no—
108#388—68no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—31no—
114C2×#95—32no—
115C2×#95—31no—
116C2×#95—32no—
117C2×#95C2×D311yescharacteristic
118C2×#95C2×D311yescharacteristic
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
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×C3C2×D414yes—
134C24×C3C2411yescharacteristic
135C2×C3×D4—48no—
136C2×C3×D4—44no—
137C2×C3×D4—44no—
138C2×C3×D4—48no—
139C2×C3×D4—44no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4#4112yes—
159C22×C3×C4#4112yes—
160C22×C3×C4#4114yes—
161C22×C3×C4#4112yes—
162C22×Dic3—22no—
163C22×Dic3—22no—
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—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3C2×D414yes—
188C22×Dic3C2×D414yes—
189C22×Dic3#4114yes—
190C3×C22⋊C4—216no—
191C3×C22⋊C4—48no—
192#217—48no—
193#217—48no—
194#217—48no—
195#217—48no—
196C4⋊Dic3—44no—
197C4⋊Dic3—44no—
198Dic3⋊C4—48no—
199C4×Dic3—48no—
200C25C22×D312yes—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—42no—
206C23×C4—32no—
207C23×C4—32no—
208C23×C4—32no—
209C23×C4—32no—
210C23×C4C22×D311yescharacteristic
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D311yescharacteristic
213C23×C4C22×D312yes—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4—122no—
219C2×C4⋊C4—121no—
220C2×C4⋊C4—121no—
221C2×C4⋊C4—121no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—38no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—64no—
233C2×C22⋊C4—64no—
234C2×C22⋊C4—64no—
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—41no—
244#95—41no—
245#95—64no—
246#95—64no—
247#95—64no—
248#95—64no—
249#95—64no—
250#95—62no—
251#95—122no—
252#95—121no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
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×C3C22×D412yes—
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4111yescharacteristic
291C23×C3C2×#4112yes—
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4112yes—
294C23×C3H(5,𝔽2)11yescharacteristic
295C23×C3H(5,𝔽2)12yes—
296C23×C3H(5,𝔽2)11yescharacteristic
297C23×C3H(5,𝔽2)11yescharacteristic
298C23×C32−1+411yescharacteristic
299C3×D4—416no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
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—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24D3×D414yes—
365C24C23×D311yescharacteristic
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—44no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—34no—
390C22×C4—34no—
391C22×C4—34no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
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#23512yes—
418C22×C4#23712yes—
419C22×C4#23714yes—
420C22×C4#23712yes—
421C4⋊C4—128no—
422C4⋊C4—124no—
423C4⋊C4—124no—
424C22⋊C4—216no—
425C22⋊C4—48no—
426C22⋊C4—128no—
427C22⋊C4—128no—
428C22⋊C4—128no—
429C22⋊C4—128no—
430C42—128no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3#52512yes—
447C22×C3#52912yes—
448C22×C3#53212yes—
449C22×C3#53412yes—
450C22×C3#53811yescharacteristic
451C22×C3#53811yescharacteristic
452C22×C3#53911yescharacteristic
453C22×C3#54012yes—
454C22×C3#54212yes—
455C22×C3D4211yescharacteristic
456C22×C3#54612yes—
457C22×C3#54612yes—
458C22×C3#54712yes—
459C22×C3#55112yes—
460C22×C3#55312yes—
461C22×C3#55312yes—
462C22×C3#55511yescharacteristic
463C22×C3#55511yescharacteristic
464C22×C3#55611yescharacteristic
465C22×C3#55911yescharacteristic
466C22×C3#55911yescharacteristic
467C22×C3#56011yescharacteristic
468C22×C3#56211yescharacteristic
469C3×C4—44no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473Dic3—44no—
474Dic3—44no—
475Dic3—44no—
476Dic3—82no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
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—
512C23C2×#23511yescharacteristic
513C23C2×D3×D412yes—
514C23C2×#23711yescharacteristic
515C23C2×#23712yes—
516C23C2×#23711yescharacteristic
517C23#100412yes—
518C23#100411yescharacteristic
519C23#100411yescharacteristic
520C23D3×#4112yes—
521C23#100911yescharacteristic
522C23#101011yescharacteristic
523D4—416no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
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—84no—
575C2×C3#245511yescharacteristic
576C2×C3#247612yes—
577C2×C3#252112yes—
578C2×C3#255511yescharacteristic
579C2×C3#259311yescharacteristic
580C2×C3#260212yes—
581C2×C3#266811yescharacteristic
582C2×C3#270112yes—
583C2×C3#275211yescharacteristic
584C2×C3#279511yescharacteristic
585C2×C3#280811yescharacteristic
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22#549012yes—
602C22#549411yescharacteristic
603C22#554211yescharacteristic
604C22#555712yes—
605C22#556311yescharacteristic
606C22#558612yes—
607C22#558712yes—
608C22#558912yes—
609C22#559011yescharacteristic
610C22#559511yescharacteristic
611C22#559612yes—
612C22#560112yes—
613C22#560211yescharacteristic
614C22#561112yes—
615C22#561412yes—
616C22#561711yescharacteristic
617C22#561912yes—
618C22#565412yes—
619C22#566212yes—
620C22#567011yescharacteristic
621C22#568411yescharacteristic
622C22#571311yescharacteristic
623C22#571311yescharacteristic
624C4—42no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—124no—
629C4—124no—
630C4—124no—
631C4—242no—
632C3#1578211yescharacteristic, complemented, Sylow
633C2—84no—
634C2#7385711yescharacteristic
635C2#7473112yes—
636C2#7474812yes—
637C2#7636611yescharacteristic
638C2#7642211yescharacteristic
639C2#7667411yescharacteristic
640C2#7669412yes—
641C2#7673411yescharacteristic
642C2#7694911yescharacteristic
643C2#7702012yes—
644C2#7707211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15782
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#4951C2semidirect
4. #71549C2semidirect
5. #71546C2semidirect
6. #71545C2semidirect
7. #69423C2semidirect
8. #69420C2semidirect
9. #69406C2semidirect
632. C3#15782semidirect

Maximal subgroups

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

subgroupindex
C2×#52222
C2×#49512
C3×#14382
#715492
#715462
#715452
#694232
#694202
#694062
#157823