Finite Group Databasebeta

← #10732912

Subgroups — #10732912

Order 768.

2942 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#940C211yesmaximal, characteristic
2C22×#935C211yesmaximal, characteristic, complemented
3C2×#4958C212yesmaximal, complemented
4C2×#4672C212yesmaximal, complemented
5C3×#1438C211yesmaximal, characteristic, complemented
6#71550C214yesmaximal, complemented
7#71550C212yesmaximal, complemented
8#69405C212yesmaximal, complemented
9#15261—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C23×#67C2212yes—
12C23×#67C2212yes—
13C24×Dic3C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C2×#940—24no—
20C2×#940—28no—
21C2×#935—24no—
22C2×#935—28no—
23C22×(C2×D3)⋊C4C2214yes—
24C22×(C2×D3)⋊C4C2214yes—
25C22×Dic3⋊C4C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30#4958—28no—
31#4672—28no—
32C22×#127—31no—
33C22×#123—31no—
34C2×#394—32no—
35C2×#386—32no—
36#1438—34no—
37#1438—32no—
38#1438D311yescharacteristic, complemented
39#1436—32no—
40C25×C3C2311yescharacteristic
41C25×C3C2311yescharacteristic
42C24×D3C2312yes—
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—42no—
46C22×C3×D4D414yes—
47C23×C3×C4C2314yes—
48C23×C3×C4C2311yescharacteristic
49C22×#67—24no—
50C22×#67—24no—
51C22×#67—24no—
52C22×#67—24no—
53C22×#67—24no—
54C22×#67—28no—
55C22×#67—28no—
56C22×#67—44no—
57C22×#67—44no—
58C22×#67D418yes—
59C23×Dic3—22no—
60C23×Dic3—24no—
61C23×Dic3—24no—
62C23×Dic3—42no—
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C22×D12—24no—
67C22×D12—22no—
68C22×D12—24no—
69C22×D12—42no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—44no—
72C2×C3×C22⋊C4—42no—
73C2×C3×C22⋊C4—44no—
74C2×#217—22no—
75C2×#217—24no—
76C2×#217—24no—
77C2×#217—24no—
78C2×#217—24no—
79#940—232no—
80#935—232no—
81C2×(C2×D3)⋊C4—28no—
82C2×(C2×D3)⋊C4—28no—
83C2×(C2×D3)⋊C4—28no—
84C2×(C2×D3)⋊C4—44no—
85C2×(C2×D3)⋊C4—44no—
86C2×(C2×D3)⋊C4—44no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4D418yescomplemented
90C2×C4×Dic3—22no—
91C2×C4×Dic3—24no—
92C3×#95—24no—
93C3⋊#95—28no—
94C3⋊#95—28no—
95C23×D4—32no—
96C23×D4—32no—
97C23×D4—31no—
98C23×D4C2×D311yescharacteristic
99C24×C4—31no—
100C2×#127—68no—
101C2×#127—64no—
102C2×#123—68no—
103C2×#123—64no—
104C22×C4⋊C4—32no—
105C22×C22⋊C4—31no—
106C22×C22⋊C4—32no—
107C22×C22⋊C4—34no—
108C22×C22⋊C4—34no—
109C22×C22⋊C4C2×D312yes—
110C22×C22⋊C4C2×D312yes—
111C22×C42—31no—
112#394—68no—
113#386—68no—
114C2×#95—34no—
115C2×#95—34no—
116C2×#95C2×D312yes—
117C24×C3—22no—
118C24×C3—24no—
119C24×C3—24no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—24no—
124C24×C3—24no—
125C24×C3—22no—
126C24×C3C2×D412yes—
127C24×C3C2×D412yes—
128C24×C3C2×D414yes—
129C24×C3C2411yescharacteristic
130C23×D3—24no—
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—24no—
134C23×D3—24no—
135C23×D3—24no—
136C23×D3—24no—
137C23×D3—24no—
138C23×D3C2×D414yes—
139C2×C3×D4—28no—
140C2×C3×D4—28no—
141C2×C3×D4—44no—
142C2×C3×D4—416no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D414yes—
154C2×#67—216no—
155C2×#67—216no—
156C2×#67—216no—
157C2×#67—48no—
158C2×#67—416no—
159C2×#67—416no—
160C2×#67—416no—
161C2×#67—416no—
162C2×#67—48no—
163C2×#67—48no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—24no—
167C22×Dic3—28no—
168C22×Dic3—24no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—416no—
177C22×Dic3—44no—
178C22×Dic3—44no—
179C22×Dic3C2×D414yes—
180C22×Dic3C2×D414yes—
181C22×Dic3C2×D414yes—
182C22×Dic3#4118yes—
183C2×D12—28no—
184C2×D12—416no—
185C2×D12—44no—
186C2×D12—44no—
187C3×C22⋊C4—48no—
188C3×C22⋊C4—48no—
189#217—28no—
190#217—216no—
191(C2×D3)⋊C4—416no—
192(C2×D3)⋊C4—416no—
193Dic3⋊C4—216no—
194C4×Dic3—28no—
195C25—32no—
196C25C22×D311yescharacteristic
197C25C22×D311yescharacteristic
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—38no—
201C22×D4—42no—
202C22×D4—64no—
203C22×D4—64no—
204C22×D4—68no—
205C22×D4—68no—
206C22×D4—64no—
207C22×D4—64no—
208C22×D4—64no—
209C22×D4—64no—
210C22×D4—62no—
211C22×D4—64no—
212C22×D4—122no—
213C22×D4—124no—
214C22×D4—124no—
215C22×D4D1214yes—
216C23×C4—32no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4—64no—
220C23×C4—62no—
221C23×C4—64no—
222C23×C4—122no—
223C23×C4C22×D314yes—
224C23×C4C22×D311yescharacteristic
225#127—632no—
226#123—632no—
227C2×C4⋊C4—38no—
228C2×C4⋊C4—64no—
229C2×C4⋊C4—64no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—68no—
240C2×C22⋊C4—68no—
241C2×C22⋊C4—68no—
242C2×C22⋊C4—124no—
243C2×C22⋊C4—124no—
244C2×C22⋊C4—124no—
245C2×C42—64no—
246C2×C42—62no—
247#95—24no—
248#95—68no—
249#95—68no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—22no—
254C23×C3—28no—
255C23×C3—28no—
256C23×C3—28no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—28no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3—44no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—44no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3#12014yes—
278C23×C3C22×D411yescharacteristic
279C23×C3C22×D412yes—
280C23×C3C22×D412yes—
281C23×C3C2×#4114yes—
282C23×C3H(5,𝔽2)14yes—
283C23×C3H(5,𝔽2)12yes—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—28no—
288C22×D3—28no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—44no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—44no—
295C22×D3—44no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3—44no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3—44no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3—44no—
305C3×D4—416no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—28no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—44no—
314C2×C3×C4—44no—
315C2×C3×C4—44no—
316C2×C3×C4#12014yes—
317#67—432no—
318#67—432no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—28no—
322C2×Dic3—28no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—28no—
328C2×Dic3—416no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—44no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C2×Dic3#12318yescomplemented
335D12—416no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—34no—
346C24—62no—
347C24—64no—
348C24—62no—
349C24—64no—
350C24—64no—
351C24—64no—
352C24—64no—
353C24—64no—
354C24C2×D1212yes—
355C24C2×D1212yes—
356C24D3×D414yes—
357C24C23×D311yescharacteristic
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—44no—
361C2×D4—416no—
362C2×D4—68no—
363C2×D4—616no—
364C2×D4—616no—
365C2×D4—616no—
366C2×D4—1216no—
367C2×D4—124no—
368C2×D4—128no—
369C2×D4—128no—
370C2×D4—1216no—
371C2×D4—1216no—
372C2×D4—1216no—
373C2×D4—1216no—
374C2×D4—128no—
375C2×D4—124no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—34no—
386C22×C4—38no—
387C22×C4—34no—
388C22×C4—34no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—64no—
392C22×C4—68no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—64no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—1216no—
402C22×C4—124no—
403C22×C4—124no—
404C22×C4C2×D1212yes—
405C22×C4D3×D414yes—
406C4⋊C4—616no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—68no—
410C22⋊C4—616no—
411C22⋊C4—1216no—
412C22⋊C4—1216no—
413C42—68no—
414C22×C3—24no—
415C22×C3—28no—
416C22×C3—48no—
417C22×C3—44no—
418C22×C3—48no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—84no—
425C22×C3—84no—
426C22×C3C2×#12011yescharacteristic
427C22×C3C2×#12011yescharacteristic
428C22×C3C2×#12312yes—
429C22×C3#53212yes—
430C22×C3#53411yescharacteristic
431C22×C3#53412yes—
432C22×C3#53512yes—
433C22×C3#54014yes—
434C22×C3D4214yes—
435C22×C3#54614yes—
436C22×C3#54614yes—
437C22×C3#55014yes—
438C22×C3#55912yes—
439C22×C3#56012yes—
440C2×D3—44no—
441C2×D3—44no—
442C2×D3—44no—
443C2×D3—48no—
444C2×D3—48no—
445C2×D3—84no—
446C2×D3—84no—
447C2×D3—84no—
448C2×D3—84no—
449C3×C4—24no—
450C3×C4—84no—
451Dic3—28no—
452Dic3—44no—
453Dic3—44no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—28no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—28no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—64no—
482C23—64no—
483C23—64no—
484C23—68no—
485C23—68no—
486C23—122no—
487C23—124no—
488C23—124no—
489C23—124no—
490C23—122no—
491C23—122no—
492C23—124no—
493C23—124no—
494C23—122no—
495C23—124no—
496C23—124no—
497C23—124no—
498C23—124no—
499C23—124no—
500C23—124no—
501C23—124no—
502C23#88214yes—
503C23C22×D1211yescharacteristic
504C23C2×D3×D412yes—
505C23C2×D3×D412yes—
506C23#100412yes—
507C23#100414yes—
508C23D3×#4114yes—
509D4—416no—
510D4—1216no—
511D4—1232no—
512D4—1232no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—38no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—64no—
525C2×C4—68no—
526C2×C4—64no—
527C2×C4—68no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—68no—
533C2×C4—1216no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—124no—
537C2×C4—124no—
538C2×C4—124no—
539C2×C4#88214yes—
540C2×C3—44no—
541C2×C3—82no—
542C2×C3#240311yescharacteristic
543C2×C3#240512yes—
544C2×C3#243112yes—
545C2×C3#244514yes—
546C2×C3#267314yes—
547C2×C3#270912yes—
548D3—84no—
549C22—24no—
550C22—28no—
551C22—44no—
552C22—44no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—44no—
559C22—84no—
560C22—84no—
561C22—128no—
562C22—124no—
563C22—128no—
564C22—124no—
565C22—124no—
566C22—244no—
567C22—244no—
568C22—244no—
569C22—244no—
570C22C2×#88211yescharacteristic
571C22C2×#88211yescharacteristic
572C22#549311yescharacteristic
573C22#555414yes—
574C22#558912yes—
575C22#559014yes—
576C22#559512yes—
577C22#559914yes—
578C22#561014yes—
579C22#561514yes—
580C22D3×#12312yes—
581C22#566412yes—
582C22#566812yes—
583C22#571512yes—
584C4—24no—
585C4—68no—
586C4—84no—
587C4—124no—
588C4—124no—
589C3#1526111yescharacteristic, complemented, Sylow
590C2—44no—
591C2—82no—
592C2—244no—
593C2#7385211yescharacteristic
594C2#7468612yes—
595C2#7482012yes—
596C2#7631312yes—
597C2#7671114yes—
598C2#7690714yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC22×#940C2
hypercenter#1438D3
hypocenterC3#15261
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C22×#935C2semidirect
3. C2×#4958C2semidirect
4. C2×#4672C2semidirect
5. C3×#1438C2semidirect
6. #71550C2semidirect
7. #71550C2semidirect
8. #69405C2semidirect
38. #1438D3semidirect
89. C2×Dic3⋊C4D4semidirect
334. C2×Dic3#123semidirect
589. C3#15261semidirect

Maximal subgroups

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

subgroupindex
C22×#9402
C22×#9352
C2×#49582
C2×#46722
C3×#14382
#715502
#694052
#152613