Finite Group Databasebeta

← #10690576

Subgroups — #10690576

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5213C211yesmaximal, characteristic
2C2×#5207C211yesmaximal, characteristic, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4C2×#4468C212yesmaximal, complemented
5#67674C212yesmaximal, complemented
6#9083—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×#67C411yescharacteristic
9C22×#217C411yescharacteristic
10C22×(C2×D3)⋊C4C411yescharacteristic
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C22×#208—21no—
13C2×C3×C22⋊C8—21no—
14C2×C3×C22⋊C8—21no—
15C2×C3×C22⋊C8C2212yes—
16#5213—21no—
17#5213—21no—
18#5213—21no—
19#5213—21no—
20C2×C3⋊#95C411yescharacteristic
21C2×C3⋊#95C2211yescharacteristic
22#5207—24no—
23C2×#830—21no—
24C2×#830C2212yes—
25#4468—24no—
26C2×#407—31no—
27C2×#386—31no—
28#1779D311yescharacteristic, complemented
29C2×#323—32no—
30#1280—32no—
31C25×C3C2×C411yescharacteristic
32C24×D3C2×C411yescharacteristic
33C23×C3×C4—22no—
34C23×C3×C4—22no—
35C23×C3×C4—42no—
36C23×C3×C4—42no—
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—21no—
45C22×#67—22no—
46C22×#67—22no—
47C22×#67—42no—
48C22×#67—42no—
49C23×Dic3C2×C411yescharacteristic
50C23×Dic3C2×C411yescharacteristic
51C23×Dic3C2×C411yescharacteristic
52C22×C3×C8—22no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—42no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—42no—
63C2×#208—24no—
64C2×#208—42no—
65C2×#208—42no—
66C22×C3⋊C8—22no—
67C3×C22⋊C8—22no—
68C3×C22⋊C8—22no—
69C3×C22⋊C8—42no—
70C3×C22⋊C8—42no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75#830—24no—
76#830—22no—
77#830D414yescomplemented
78C23×D4—31no—
79C24×C4C2×D311yescharacteristic
80C22×M16—61no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—31no—
83C22×C22⋊C4—31no—
84#407—64no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—32no—
88C2×C22⋊C8—61no—
89C2×C22⋊C8C2×D312yes—
90#386—61no—
91#386—61no—
92#386—61no—
93#386—61no—
94C2×#95—31no—
95C2×#95—31no—
96#323—64no—
97C24×C3—22no—
98C24×C3—22no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—22no—
102C24×C3—22no—
103C24×C3—41no—
104C24×C3—41no—
105C24×C3C22⋊C411yescharacteristic
106C24×C3C22⋊C411yescharacteristic
107C24×C3C22×C411yescharacteristic
108C23×D3—21no—
109C23×D3—21no—
110C23×D3—22no—
111C23×D3—22no—
112C23×D3—41no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3—41no—
116C23×D3M1612yescomplemented
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—48no—
137C22×C3×C4—48no—
138C22×C3×C4C22⋊C411yescharacteristic
139C22×C3×C4C22⋊C411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4112yes—
144C2×#67—24no—
145C2×#67—24no—
146C2×#67—48no—
147C2×#67—42no—
148C2×#67—42no—
149C2×#67—48no—
150C2×#67—42no—
151C2×#67—42no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3M1612yescomplemented
165C2×C3×C8—42no—
166C2×C3×C8—42no—
167C2×C3×C8—42no—
168#217—42no—
169#217—42no—
170(C2×D3)⋊C4—42no—
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—41no—
175#208—48no—
176C2×C3⋊C8—24no—
177C2×C3⋊C8—42no—
178C2×C3⋊C8—42no—
179C25—31no—
180C25C4×D311yescharacteristic
181C22×D4—62no—
182C22×D4—62no—
183C22×D4—61no—
184C22×D4—61no—
185C22×D4—122no—
186C22×D4—122no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—31no—
190C23×C4—31no—
191C23×C4—31no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4C4×D311yescharacteristic
195C23×C4D1211yescharacteristic
196C23×C4D1211yescharacteristic
197C23×C4#6711yescharacteristic
198C23×C4#6711yescharacteristic
199C23×C4C22×D311yescharacteristic
200C2×M16—64no—
201C2×M16—122no—
202C2×M16—122no—
203C22×C8—22no—
204C22×C8—62no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—122no—
214C2×C22⋊C4—122no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—34no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C22⋊C8—64no—
221C22⋊C8—62no—
222#95—61no—
223#95—61no—
224#95—61no—
225#95—61no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—24no—
231C23×C3—24no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—44no—
245C23×C3—44no—
246C23×C3—44no—
247C23×C3—44no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3#9912yes—
253C23×C3#10012yes—
254C23×C3C2×C22⋊C411yescharacteristic
255C23×C3C4×D412yes—
256C23×C3C2×M1611yescharacteristic
257C23×C3C2×M1611yescharacteristic
258C23×C3#13112yes—
259C22×D3—22no—
260C22×D3—24no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—82no—
273C22×D3—82no—
274C22×D3—82no—
275C22×D3—82no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—44no—
290C2×C3×C4—44no—
291C2×C3×C4—44no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4#9912yes—
295C2×C3×C4#12011yescharacteristic
296C2×C3×C4#12112yes—
297C2×C3×C4#12311yescharacteristic
298#67—48no—
299#67—48no—
300C2×Dic3—22no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—82no—
312C2×Dic3—82no—
313C3×C8—82no—
314C3⋊C8—44no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—32no—
322C24—41no—
323C24—41no—
324C24—61no—
325C24—61no—
326C24—62no—
327C24—62no—
328C24—121no—
329C24—121no—
330C24—121no—
331C24—121no—
332C24(C2×D3)⋊C411yescharacteristic
333C24(C2×D3)⋊C411yescharacteristic
334C24C2×C4×D311yescharacteristic
335C2×D4—64no—
336C2×D4—64no—
337C2×D4—128no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—128no—
341C2×D4—122no—
342C2×D4—122no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—32no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—121no—
372C22×C4—121no—
373C22×C4—121no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4—121no—
377C22×C4(C2×D3)⋊C411yescharacteristic
378C22×C4(C2×D3)⋊C411yescharacteristic
379C22×C4C2×D1211yescharacteristic
380C22×C4D3×D412yes—
381C22×C4#23712yes—
382C22×C4C2×#6711yescharacteristic
383M16—128no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—64no—
388C2×C8—122no—
389C2×C8—122no—
390C22⋊C4—122no—
391C22⋊C4—122no—
392C22⋊C4—122no—
393C22⋊C4—122no—
394C22⋊C4—121no—
395C22⋊C4—121no—
396C22⋊C4—122no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—48no—
412C22×C3—48no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3#38611yescharacteristic
422C22×C3#40711yescharacteristic
423C22×C3#40811yescharacteristic
424C22×C3C2×#9911yescharacteristic
425C22×C3C2×#9911yescharacteristic
426C22×C3C2×#10011yescharacteristic
427C22×C3#41311yescharacteristic
428C22×C3#43512yes—
429C22×C3#43612yes—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—84no—
434C2×D3—81no—
435C2×D3—81no—
436C2×D3—81no—
437C2×D3—81no—
438C2×D3—84no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—41no—
444Dic3—42no—
445Dic3—81no—
446Dic3—81no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—24no—
452C23—24no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—62no—
474C23—64no—
475C23—122no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—122no—
485C23—121no—
486C23—242no—
487C23—242no—
488C23—242no—
489C23—242no—
490C23#80612yes—
491C23#82312yes—
492C23#67⋊C412yes—
493C23#90112yes—
494C23D3×M1611yescharacteristic
495C23D3×M1611yescharacteristic
496C23C2×(C2×D3)⋊C411yescharacteristic
497D4—128no—
498D4—128no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—62no—
518C2×C4—122no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—242no—
529C2×C4—242no—
530C2×C4#80612yes—
531C2×C4#88211yescharacteristic
532C2×C4#88611yescharacteristic
533C2×C4#93912yes—
534C8—82no—
535C8—124no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—42no—
539C2×C3—42no—
540C2×C3#146211yescharacteristic
541C2×C3#146511yescharacteristic
542C2×C3#146811yescharacteristic
543C2×C3#147111yescharacteristic
544C2×C3#187011yescharacteristic
545C2×C3#189811yescharacteristic
546C2×C3#190011yescharacteristic
547D3—82no—
548C22—22no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—48no—
563C22—48no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—122no—
573C22—122no—
574C22—122no—
575C22—244no—
576C22—244no—
577C22—241no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22#472812yes—
582C22#472912yes—
583C22C2×#80611yescharacteristic
584C22C2×#80611yescharacteristic
585C22#495811yescharacteristic
586C22#511011yescharacteristic
587C22#512611yescharacteristic
588C22C2×#82311yescharacteristic
589C22#513211yescharacteristic
590C4—22no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—41no—
595C4—122no—
596C4—241no—
597C4—241no—
598C3#908311yescharacteristic, complemented, Sylow
599C2—22no—
600C2—22no—
601C2—42no—
602C2—42no—
603C2—242no—
604C2#6949611yescharacteristic
605C2#6992011yescharacteristic
606C2#6992111yescharacteristic
607C2#6995311yescharacteristic
608C2#6996711yescharacteristic
609C2#6996811yescharacteristic
610C2#6998511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#67C4
cosocleC23×C3×C4C23
hypercenter#1779D3
hypocenterC3#9083
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#5207C2semidirect
3. C3×#1779C2semidirect
4. C2×#4468C2semidirect
5. #67674C2semidirect
28. #1779D3semidirect
77. #830D4semidirect
116. C23×D3M16semidirect
164. C22×Dic3M16semidirect
598. C3#9083semidirect

Maximal subgroups

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

subgroupindex
C2×#52132
C2×#52072
C3×#17792
C2×#44682
#676742
#90833