Finite Group Databasebeta

← #10732576

Subgroups — #10732576

Order 768.

1838 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2C2×#5221C211yesmaximal, characteristic
3C2×#5221C211yesmaximal, characteristic
4C2×#4972C211yesmaximal, characteristic, complemented
5C2×#4936C211yesmaximal, characteristic, complemented
6C2×C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
7#71563C211yesmaximal, characteristic, complemented
8#71547C214yesmaximal, complemented
9#71547C214yesmaximal, complemented
10#15222—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2214yes—
14C22×C3×C42C2211yescharacteristic
15C22×#217C2212yes—
16C22×#217C2214yes—
17C22×#217C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C3×#390—24no—
25#5221—24no—
26#5221—24no—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2214yes—
31#4972—24no—
32#4936—24no—
33C4×Dic3⋊C4—24no—
34C2×#390—31no—
35C2×#390—31no—
36C2×#390D311yescharacteristic
37C2×#389—31no—
38C2×#382—31no—
39C2×C4×C4⋊C4—31no—
40#1439—34no—
41#1439—34no—
42C4⋊#95—31no—
43C25×C3C2312yes—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4Q814yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C22⋊C4—28no—
63C2×C3×C22⋊C4—44no—
64C2×C3×C22⋊C4—44no—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×C3×C42D414yes—
68C2×#217—28no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—28no—
72C2×#217—28no—
73C2×#217—44no—
74C2×#217—44no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D414yes—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—44no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3D414yes—
92C3⋊#95—24no—
93C3⋊#95—28no—
94C3⋊#95—24no—
95C3⋊#95—44no—
96C23×D4C2×D311yescharacteristic
97C22×C4⋊C4—31no—
98C22×C4⋊C4—32no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4C2×D311yescharacteristic
102C22×C22⋊C4—32no—
103C22×C22⋊C4—34no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4C2×D314yes—
106C22×C42—31no—
107C22×C42—31no—
108C22×C42C2×D311yescharacteristic
109#390—24no—
110#390—64no—
111#390—64no—
112#389—64no—
113#382—64no—
114C4×C4⋊C4—64no—
115C2×#95—32no—
116C2×#95—34no—
117C2×#95—32no—
118C2×#95—34no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—24no—
123C24×C3—24no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—24no—
128C24×C3—24no—
129C24×C3C2×Q814yes—
130C24×C3C2411yescharacteristic
131C2×C3×D4—28no—
132C2×C3×D4—28no—
133C2×C3×D4—28no—
134C2×C3×D4—44no—
135C2×C3×D4—44no—
136C2×C3×D4—44no—
137C2×C3×D4—44no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
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—24no—
148C22×C3×C4—24no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×Q812yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4112yes—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—24no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3C2×D414yes—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×D412yes—
182C22×Dic3#4114yes—
183C22×Dic3#4114yes—
184C3×C4⋊C4—44no—
185C3×C22⋊C4—416no—
186C3×C42—28no—
187#217—416no—
188#217—48no—
189#217—48no—
190C4⋊Dic3—28no—
191C4⋊Dic3—44no—
192Dic3⋊C4—216no—
193Dic3⋊C4—48no—
194C4×Dic3—28no—
195C4×Dic3—44no—
196C25C22×D312yes—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4Dic614yes—
204C23×C4—31no—
205C23×C4—32no—
206C23×C4—31no—
207C23×C4—32no—
208C23×C4—32no—
209C23×C4C22×D312yes—
210C23×C4C22×D312yes—
211C23×C4C22×D311yescharacteristic
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—34no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—124no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4—68no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—68no—
233C2×C22⋊C4—68no—
234C2×C22⋊C4—124no—
235C2×C22⋊C4—124no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—34no—
239C2×C42—62no—
240C2×C42—62no—
241C2×C42—62no—
242C2×C42—62no—
243C2×C42—62no—
244C2×C42#6714yes—
245#95—68no—
246#95—64no—
247#95—64no—
248#95—124no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—28no—
253C23×C3—28no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—44no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—44no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3—42no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—44no—
269C23×C3—44no—
270C23×C3—44no—
271C23×C3—44no—
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D411yescharacteristic
274C23×C3C22×D411yescharacteristic
275C23×C3C22×Q811yescharacteristic
276C23×C3C2×#4111yescharacteristic
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4112yes—
280C23×C3C2×#4112yes—
281C23×C3H(5,𝔽2)12yes—
282C23×C3H(5,𝔽2)12yes—
283C3×D4—416no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—42no—
295C2×C3×C4—44no—
296C2×C3×C4#12114yes—
297C2×C3×C4#12114yes—
298C2×C3×C4#12114yes—
299C2×C3×C4C22⋊Q814yes—
300C2×C3×C4C22⋊Q814yes—
301C2×C3×C4C22⋊Q814yes—
302C2×C3×C4#12614yes—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—28no—
314C2×Dic3—24no—
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—44no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—44no—
328C2×Dic3—44no—
329C2×Dic3#12414yes—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24C2×Dic614yes—
341C24C23×D311yescharacteristic
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—32no—
361C22×C4—34no—
362C22×C4—34no—
363C22×C4—32no—
364C22×C4—34no—
365C22×C4—61no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4C2×Dic612yes—
388C22×C4#23512yes—
389C22×C4D3×D412yes—
390C22×C4D3×D412yes—
391C22×C4#23712yes—
392C22×C4#23712yes—
393C22×C4C2×#6714yes—
394C22×C4C2×#6712yes—
395C4⋊C4—44no—
396C4⋊C4—616no—
397C4⋊C4—68no—
398C4⋊C4—128no—
399C4⋊C4—124no—
400C22⋊C4—416no—
401C22⋊C4—128no—
402C22⋊C4—1216no—
403C22⋊C4—128no—
404C42—28no—
405C42—68no—
406C42—124no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—48no—
410C22×C3—44no—
411C22×C3—48no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—84no—
415C22×C3—84no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3C2×#12111yescharacteristic
419C22×C3C2×#12111yescharacteristic
420C22×C3C2×#12111yescharacteristic
421C22×C3C2×C22⋊Q811yescharacteristic
422C22×C3C2×C22⋊Q811yescharacteristic
423C22×C3C2×C22⋊Q811yescharacteristic
424C22×C3#52512yes—
425C22×C3C2×#12411yescharacteristic
426C22×C3C2×#12611yescharacteristic
427C22×C3#52912yes—
428C22×C3#53211yescharacteristic
429C22×C3#53511yescharacteristic
430C22×C3#53511yescharacteristic
431C22×C3#53512yes—
432C22×C3#54012yes—
433C22×C3#54612yes—
434C22×C3#55012yes—
435C22×C3#55312yes—
436C22×C3#55412yes—
437C22×C3#55412yes—
438C22×C3#55512yes—
439C22×C3#55912yes—
440C22×C3#55912yes—
441C3×C4—44no—
442C3×C4—82no—
443C3×C4#39618yes—
444Dic3—24no—
445Dic3—42no—
446Dic3—44no—
447Dic3—44no—
448Dic3—82no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—28no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23C22×Dic611yescharacteristic
473C23C2×#23511yescharacteristic
474C23C2×D3×D411yescharacteristic
475C23C2×D3×D411yescharacteristic
476C23C2×#23711yescharacteristic
477C23C2×#23711yescharacteristic
478C23#100412yes—
479C23#100412yes—
480C23D3×#4112yes—
481C23D3×#4112yes—
482C23C22×#6711yescharacteristic
483D4—416no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—34no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—68no—
508C2×C4—64no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—124no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—124no—
522C2×C4—124no—
523C2×C4#87814yes—
524C2×C4#87814yes—
525C2×C4#87914yes—
526C2×C4#88414yes—
527C2×C4#92414yes—
528C2×C4#93814yes—
529C2×C4#93914yes—
530C2×C3—84no—
531C2×C3C2×#39611yescharacteristic
532C2×C3#241412yes—
533C2×C3#245412yes—
534C2×C3#251312yes—
535C2×C3#254411yescharacteristic
536C2×C3#254811yescharacteristic
537C2×C3#256311yescharacteristic
538C2×C3#256611yescharacteristic
539C2×C3#258012yes—
540C2×C3#261911yescharacteristic
541C2×C3#285011yescharacteristic
542C22—48no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—48no—
547C22—84no—
548C22—84no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×#87811yescharacteristic
554C22C2×#87811yescharacteristic
555C22C2×#87911yescharacteristic
556C22C2×#88411yescharacteristic
557C22#553912yes—
558C22#553912yes—
559C22#555912yes—
560C22#555912yes—
561C22#559912yes—
562C22#561212yes—
563C22#561412yes—
564C22#561612yes—
565C22#566811yescharacteristic
566C22D3×#12411yescharacteristic
567C22#568012yes—
568C22#571511yescharacteristic
569C22#571511yescharacteristic
570C22#571912yes—
571C22C2×#92411yescharacteristic
572C22C2×#93811yescharacteristic
573C22C2×#93911yescharacteristic
574C22#580512yes—
575C22#582812yes—
576C4—44no—
577C4—64no—
578C4—82no—
579C4—124no—
580C4—124no—
581C4—122no—
582C4—242no—
583C4#494818yes—
584C3#1522211yescharacteristic, complemented, Sylow
585C2—84no—
586C2#7632412yes—
587C2#7634012yes—
588C2#7635011yescharacteristic
589C2#7635211yescharacteristic
590C2#7636111yescharacteristic
591C2#7636311yescharacteristic
592C2C2×#494811yescharacteristic
593C2#8238812yes—
594C2#8270412yes—
595C2#8300811yescharacteristic
596C2#8318911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#15222
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#4972C2semidirect
5. C2×#4936C2semidirect
6. C2×C4×Dic3⋊C4C2semidirect
7. #71563C2semidirect
8. #71547C2semidirect
9. #71547C2semidirect
584. C3#15222semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C2×#52212
C2×#49722
C2×#49362
C2×C4×Dic3⋊C42
#715632
#715472
#152223