Finite Group Databasebeta

← #10742737

Subgroups — #10742737

Order 768.

1862 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5220C212yesmaximal
2C2×#4943C212yesmaximal, complemented
3C3×#1438C211yesmaximal, characteristic
4#71546C212yesmaximal, complemented
5#71546C212yesmaximal, complemented
6#71544C212yesmaximal, complemented
7#69422C211yesmaximal, characteristic, complemented
8#69418C212yesmaximal, complemented
9#69415C211yesmaximal, characteristic, complemented
10#15952—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×Dic3C2212yes—
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×C3×#95C2211yescharacteristic
21C2×C3×#95C2211yescharacteristic
22#5220—28no—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33#4943—28no—
34C2×#386—32no—
35C2×#381—32no—
36#1445—31no—
37#1443—32no—
38#1442—31no—
39#1439—32no—
40#1439—32no—
41#1438D311yescharacteristic
42C22⋊#95—32no—
43C25×C3C2312yes—
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—41no—
47C22×C3×D4—41no—
48C22×C3×D4D414yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×Dic3—24no—
54C23×Dic3—24no—
55C23×Dic3—24no—
56C23×Dic3—24no—
57C23×Dic3—24no—
58C23×Dic3—42no—
59C23×Dic3—42no—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—42no—
69C2×C3×C22⋊C4—42no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217—24no—
73C2×#217—24no—
74C2×#217—24no—
75C2×#217—42no—
76C2×#217—42no—
77C2×#217—42no—
78C2×#217—42no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—41no—
81C2×C4⋊Dic3—41no—
82C2×C4⋊Dic3—42no—
83C2×C4⋊Dic3—41no—
84C3×#95—41no—
85C3×#95—41no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—41no—
93C3⋊#95—42no—
94C3⋊#95—42no—
95C3⋊#95D414yescomplemented
96C23×D4C2×D311yescharacteristic
97C24×C4—32no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4—31no—
100C22×C22⋊C4—32no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4—32no—
103C22×C22⋊C4C2×D312yes—
104C22×C22⋊C4C2×D312yes—
105#386—68no—
106#381—68no—
107C2×#95—31no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—31no—
115C2×#95—32no—
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×C3C2×D414yes—
132C24×C3#4114yes—
133C24×C3#4114yes—
134C24×C3C2411yescharacteristic
135C2×C3×D4—216no—
136C2×C3×D4—44no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4—48no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
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×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4#4112yes—
162C22×Dic3—216no—
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×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—48no—
190C22×Dic3—48no—
191C22×Dic3—48no—
192C22×Dic3—48no—
193C22×Dic3—48no—
194C22×Dic3—48no—
195C22×Dic3C2×D414yes—
196C22×Dic3#4114yes—
197C3×C22⋊C4—48no—
198C3×C22⋊C4—48no—
199#217—48no—
200#217—48no—
201#217—48no—
202C4⋊Dic3—44no—
203C4⋊Dic3—44no—
204C25C22×D312yes—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4D1214yes—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—32no—
213C23×C4—32no—
214C23×C4—64no—
215C23×C4—64no—
216C23×C4—64no—
217C23×C4—64no—
218C23×C4—64no—
219C23×C4—122no—
220C23×C4—122no—
221C23×C4C22×D312yes—
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C2×C4⋊C4—62no—
226C2×C4⋊C4—121no—
227C2×C4⋊C4—121no—
228C2×C4⋊C4—121no—
229C2×C4⋊C4—122no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—122no—
242C2×C22⋊C4—122no—
243C2×C22⋊C4—122no—
244C2×C22⋊C4—122no—
245#95—34no—
246#95—41no—
247#95—41no—
248#95—64no—
249#95—64no—
250#95—64no—
251#95—64no—
252#95—64no—
253#95—64no—
254#95—122no—
255#95—121no—
256#95—122no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—24no—
264C23×C3—24no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3—24no—
270C23×C3—24no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—24no—
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#12318yes—
292C23×C3C22×D411yescharacteristic
293C23×C3C22×D411yescharacteristic
294C23×C3C2×#4112yes—
295C23×C3C2×#4112yes—
296C23×C3C2×#4111yescharacteristic
297C23×C3C2×#4112yes—
298C23×C3H(5,𝔽2)11yescharacteristic
299C23×C3H(5,𝔽2)11yescharacteristic
300C23×C3H(5,𝔽2)12yes—
301C23×C3H(5,𝔽2)11yescharacteristic
302C23×C32−1+411yescharacteristic
303C3×D4—416no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
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×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—432no—
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—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C2×D1214yes—
368C24#23714yes—
369C24#23714yes—
370C24C23×D311yescharacteristic
371C2×D4—216no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
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—34no—
396C22×C4—34no—
397C22×C4—616no—
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—62no—
423C22×C4—62no—
424C22×C4—128no—
425C22×C4—128no—
426C22×C4—128no—
427C22×C4—128no—
428C22×C4—128no—
429C22×C4—128no—
430C22×C4C2×D1212yes—
431C22×C4D3×D412yes—
432C22×C4#23712yes—
433C4⋊C4—124no—
434C4⋊C4—124no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—128no—
438C22⋊C4—128no—
439C22⋊C4—128no—
440C22×C3—28no—
441C22×C3—28no—
442C22×C3—28no—
443C22×C3—44no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3C2×#12312yes—
456C22×C3#52512yes—
457C22×C3#52912yes—
458C22×C3#53411yescharacteristic
459C22×C3#53711yescharacteristic
460C22×C3#53811yescharacteristic
461C22×C3#53812yes—
462C22×C3#53912yes—
463C22×C3#54012yes—
464C22×C3D4211yescharacteristic
465C22×C3#54612yes—
466C22×C3#54612yes—
467C22×C3#54711yescharacteristic
468C22×C3#55011yescharacteristic
469C22×C3#55112yes—
470C22×C3#55111yescharacteristic
471C22×C3#55111yescharacteristic
472C22×C3#55112yes—
473C22×C3#55112yes—
474C22×C3#55212yes—
475C22×C3#55511yescharacteristic
476C22×C3#56011yescharacteristic
477C22×C3#56211yescharacteristic
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—82no—
482Dic3—44no—
483Dic3—44no—
484Dic3—82no—
485Dic3—82no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
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—
518C23—42no—
519C23—42no—
520C23#88618yes—
521C23C22×D1211yescharacteristic
522C23C2×D3×D411yescharacteristic
523C23C2×#23712yes—
524C23C2×#23712yes—
525C23C2×#23711yescharacteristic
526C23#100411yescharacteristic
527C23#100411yescharacteristic
528C23#100412yes—
529C23D3×#4112yes—
530C23#100911yescharacteristic
531C23#101011yescharacteristic
532D4—416no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
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—42no—
554C2×C4—42no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—1232no—
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×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C3—48no—
585C2×C3#241812yes—
586C2×C3#241912yes—
587C2×C3#247112yes—
588C2×C3#248311yescharacteristic
589C2×C3#257211yescharacteristic
590C2×C3#260911yescharacteristic
591C2×C3#268211yescharacteristic
592C2×C3#269311yescharacteristic
593C2×C3#269911yescharacteristic
594C2×C3#274312yes—
595C2×C3#276011yescharacteristic
596C22—28no—
597C22—28no—
598C22—28no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22C2×#88612yes—
612C22#549311yescharacteristic
613C22D4×D1211yescharacteristic
614C22#555412yes—
615C22#555511yescharacteristic
616C22#558712yes—
617C22#558711yescharacteristic
618C22#558711yescharacteristic
619C22#558712yes—
620C22#559112yes—
621C22#559711yescharacteristic
622C22#560012yes—
623C22#560212yes—
624C22#560612yes—
625C22#561512yes—
626C22#561711yescharacteristic
627C22#561712yes—
628C22#565012yes—
629C22#565812yes—
630C22#567011yescharacteristic
631C22#568411yescharacteristic
632C22#571811yescharacteristic
633C22#572011yescharacteristic
634C4—42no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—124no—
639C4—124no—
640C4—242no—
641C4—242no—
642C3#1595211yescharacteristic, complemented, Sylow
643C2—48no—
644C2#7383311yescharacteristic
645C2#7469412yes—
646C2#7489112yes—
647C2#7637911yescharacteristic
648C2#7638511yescharacteristic
649C2#7668512yes—
650C2#7671811yescharacteristic
651C2#7673711yescharacteristic
652C2#7695611yescharacteristic
653C2#7696811yescharacteristic
654C2#7705212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15952
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#4943C2semidirect
4. #71546C2semidirect
5. #71546C2semidirect
6. #71544C2semidirect
7. #69422C2semidirect
8. #69418C2semidirect
9. #69415C2semidirect
95. C3⋊#95D4semidirect
642. C3#15952semidirect

Maximal subgroups

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

subgroupindex
C2×#52202
C2×#49432
C3×#14382
#715462
#715442
#694222
#694182
#694152
#159523