Finite Group Databasebeta

← #10724463

Subgroups — #10724463

Order 768.

1550 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#385C211yesmaximal, characteristic
2C2×#4951C212yesmaximal
3#72127C212yesmaximal
4#71565C212yesmaximal, complemented
5#71563C212yesmaximal, complemented
6#71546C211yesmaximal, characteristic
7#71546C211yesmaximal, characteristic
8#69412C212yesmaximal, complemented
9#69407C212yesmaximal, complemented
10#14514—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4×Dic3C2212yes—
21C3×#385C418yescomplemented
22C2×C3×#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33#4951—28no—
34C2×#388—32no—
35C2×#385D311yescharacteristic
36#1442—32no—
37#1441—32no—
38#1439—31no—
39#1439—31no—
40#1436—32no—
41C4⋊#95—32no—
42C22⋊#95—32no—
43C25×C3C2311yescharacteristic
44C23×C3×C4—41no—
45C23×C3×C4—41no—
46C23×C3×C4—42no—
47C23×C3×C4—42no—
48C23×C3×C4C2×C414yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C2×C3×C4⋊C4—42no—
58C2×C3×C4⋊C4—42no—
59C2×C3×C4⋊C4C2×C418yes—
60C2×C3×C22⋊C4—41no—
61C2×C3×C22⋊C4—41no—
62C2×C3×C22⋊C4C2×C414yes—
63C2×C3×C22⋊C4C2×C414yes—
64C2×C3×C22⋊C4D414yes—
65C2×C3×C22⋊C4Q814yes—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—42no—
71C2×#217—42no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—41no—
76C2×C4⋊Dic3—42no—
77C2×C4⋊Dic3—41no—
78C2×C4⋊Dic3—42no—
79C2×C4×Dic3—24no—
80C2×C4×Dic3—24no—
81C2×C4×Dic3—42no—
82C3×#95C2×C418yes—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—24no—
86C3⋊#95—41no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89C3⋊#95—42no—
90C3⋊#95—42no—
91C3⋊#95—41no—
92C3⋊#95—42no—
93C24×C4C2×D311yescharacteristic
94C22×C4⋊C4—32no—
95C22×C4⋊C4—31no—
96C22×C4⋊C4—31no—
97C22×C4⋊C4C2×D312yes—
98C22×C22⋊C4—32no—
99C22×C22⋊C4—32no—
100C22×C22⋊C4C2×D311yescharacteristic
101C22×C22⋊C4C2×D311yescharacteristic
102C22×C42—32no—
103#388—68no—
104#385Dic318yescomplemented
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—31no—
113C2×#95—32no—
114C2×#95—31no—
115C2×#95C2×D312yes—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3C22×C412yes—
125C24×C3C2×D412yes—
126C24×C3C2×Q812yes—
127C24×C3C2411yescharacteristic
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—48no—
147C22×C3×C4—44no—
148C22×C3×C4—48no—
149C22×C3×C4—44no—
150C22×C3×C4—44no—
151C22×C3×C4C22×C412yes—
152C22×C3×C4C22×C414yes—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C22×C414yes—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×Q814yes—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
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#4114yes—
184C22×Dic3#4114yes—
185C3×C4⋊C4—48no—
186C3×C22⋊C4—44no—
187C3×C22⋊C4C4⋊C4116yescomplemented
188#217—48no—
189#217—48no—
190C4⋊Dic3—44no—
191C4⋊Dic3—48no—
192C4⋊Dic3—44no—
193C4×Dic3—48no—
194C25C22×D311yescharacteristic
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—32no—
198C23×C4—32no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2×Dic314yes—
204C23×C4C22×D311yescharacteristic
205C23×C4C22×D311yescharacteristic
206C23×C4C22×D312yes—
207C23×C4C22×D312yes—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—64no—
211C2×C4⋊C4—64no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—122no—
214C2×C4⋊C4—121no—
215C2×C4⋊C4—121no—
216C2×C4⋊C4—122no—
217C2×C4⋊C4C2×Dic318yes—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—64no—
221C2×C22⋊C4—64no—
222C2×C22⋊C4—64no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—122no—
225C2×C22⋊C4—122no—
226C2×C22⋊C4Dic614yes—
227C2×C22⋊C4D1214yes—
228C2×C22⋊C4C2×Dic314yes—
229C2×C22⋊C4C2×Dic314yes—
230C2×C42—64no—
231C2×C42—64no—
232C2×C42—122no—
233#95—64no—
234#95—64no—
235#95—64no—
236#95—121no—
237#95—121no—
238#95—122no—
239#95—122no—
240#95—122no—
241#95—122no—
242#95—122no—
243#95C2×Dic318yes—
244C23×C3—24no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3C2×C4⋊C414yes—
267C23×C3C23×C411yescharacteristic
268C23×C3C22×D411yescharacteristic
269C23×C3C22×Q811yescharacteristic
270C23×C3C2×#4112yes—
271C23×C3C2×#4112yes—
272C23×C3H(5,𝔽2)11yescharacteristic
273C23×C3H(5,𝔽2)11yescharacteristic
274C23×C3H(5,𝔽2)12yes—
275C23×C3H(5,𝔽2)12yes—
276C23×C32−1+411yescharacteristic
277C23×C32−1+411yescharacteristic
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—416no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4C2×C4⋊C418yes—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24C2×Dic612yes—
340C24C2×D1212yes—
341C24C22×Dic312yes—
342C24C23×D311yescharacteristic
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—34no—
362C22×C4—34no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4C2×Dic614yes—
395C22×C4C2×D1214yes—
396C22×C4C22×Dic312yes—
397C22×C4C22×Dic314yes—
398C22×C4C22×Dic312yes—
399C22×C4C22×Dic314yes—
400C4⋊C4—48no—
401C4⋊C4—128no—
402C4⋊C4—124no—
403C4⋊C4—124no—
404C22⋊C4—44no—
405C22⋊C4—128no—
406C22⋊C4—128no—
407C22⋊C4C4⋊Dic3116yescomplemented
408C42—128no—
409C22×C3—24no—
410C22×C3—28no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—44no—
419C22×C3—42no—
420C22×C3C22×C4⋊C411yescharacteristic
421C22×C3C4×#4112yes—
422C22×C3#51811yescharacteristic
423C22×C3#51812yes—
424C22×C3#51911yescharacteristic
425C22×C3#53711yescharacteristic
426C22×C3#53711yescharacteristic
427C22×C3#53912yes—
428C22×C3#53912yes—
429C22×C3#54212yes—
430C22×C3#54212yes—
431C22×C3#54411yescharacteristic
432C22×C3#54411yescharacteristic
433C22×C3#54612yes—
434C22×C3#54612yes—
435C22×C3#55111yescharacteristic
436C22×C3#55312yes—
437C22×C3#55412yes—
438C22×C3#55412yes—
439C22×C3#55911yescharacteristic
440C22×C3#56012yes—
441C22×C3#56011yescharacteristic
442C22×C3H(3,𝔽4)11yescharacteristic
443C3×C4—28no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—44no—
447Dic3—44no—
448Dic3—44no—
449Dic3—82no—
450Dic3—82no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23C2×C4⋊Dic314yes—
474C23C22×Dic611yescharacteristic
475C23C22×D1211yescharacteristic
476C23#100412yes—
477C23#100412yes—
478C23#100411yescharacteristic
479C23#100411yescharacteristic
480C23#100711yescharacteristic
481C23#100711yescharacteristic
482C23D3×#4112yes—
483C23D3×#4112yes—
484C23C23×Dic311yescharacteristic
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—416no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4C2×C4⋊Dic318yes—
538C2×C3—44no—
539C2×C3#231611yescharacteristic
540C2×C3#233212yes—
541C2×C3#234912yes—
542C2×C3#237411yescharacteristic
543C2×C3#237511yescharacteristic
544C2×C3#269112yes—
545C2×C3#269311yescharacteristic
546C2×C3#271211yescharacteristic
547C2×C3#276311yescharacteristic
548C2×C3#279012yes—
549C2×C3#280711yescharacteristic
550C22—28no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22#550011yescharacteristic
562C22#550011yescharacteristic
563C22#551011yescharacteristic
564C22#551011yescharacteristic
565C22#553912yes—
566C22#553912yes—
567C22#555412yes—
568C22#555412yes—
569C22#559411yescharacteristic
570C22#559511yescharacteristic
571C22#559512yes—
572C22#563812yes—
573C22#563812yes—
574C22#564012yes—
575C22#564012yes—
576C22#565911yescharacteristic
577C22#566112yes—
578C22#566411yescharacteristic
579C22C22×C4⋊Dic311yescharacteristic
580C22#579812yes—
581C22#579811yescharacteristic
582C22#581211yescharacteristic
583C22Dic3×#4112yes—
584C4—28no—
585C4—44no—
586C4—42no—
587C4—42no—
588C4—124no—
589C4—124no—
590C4—242no—
591C4—242no—
592C3#1451411yescharacteristic, complemented, Sylow
593C2—44no—
594C2#7451711yescharacteristic
595C2#7452911yescharacteristic
596C2#7455411yescharacteristic
597C2#7457712yes—
598C2#7458311yescharacteristic
599C2#7463512yes—
600C2#8215611yescharacteristic
601C2#8233312yes—
602C2#8258011yescharacteristic
603C2#8274612yes—
604C2#8283911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#385C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#385D3
hypocenterC3#14514
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
4. #71565C2semidirect
5. #71563C2semidirect
8. #69412C2semidirect
9. #69407C2semidirect
21. C3×#385C4semidirect
104. #385Dic3semidirect
187. C3×C22⋊C4C4⋊C4semidirect
407. C22⋊C4C4⋊Dic3semidirect
592. C3#14514semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3852
C2×#49512
#721272
#715652
#715632
#715462
#694122
#694072
#145143