Finite Group Databasebeta

← #10731960

Subgroups — #10731960

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4983C211yesmaximal, characteristic
2C2×#4975C211yesmaximal, characteristic
3C2×#4661C211yesmaximal, characteristic
4C2×#4657C212yesmaximal
5C2×#4647C212yesmaximal
6C2×#4649C211yesmaximal, characteristic
7C3×#1445C211yesmaximal, characteristic
8#69422C211yesmaximal, characteristic
9#69421C212yesmaximal
10Dic3⋊#95C211yesmaximal, characteristic
11Dic3⋊#95C212yesmaximal
12#15151—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36#4983—24no—
37#4975—24no—
38#4661—24no—
39#4657—28no—
40#4649—24no—
41#4647—28no—
42C2×#398—32no—
43C2×#398—31no—
44C2×#398—31no—
45C2×#389—32no—
46C2×#384—31no—
47C2×C4×C4⋊C4—31no—
48#1445—32no—
49#1445—31no—
50#1445D311yescharacteristic
51C4⋊#95—31no—
52C4⋊#95—32no—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—24no—
72C2×C4⋊Dic3—24no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4D414yes—
89C2×Dic3⋊C4D414yes—
90C2×Dic3⋊C4Q818yes—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3D414yes—
97C2×C4×Dic3D414yes—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—24no—
101C3×#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C42—31no—
119C22×C42—31no—
120C22×C42—31no—
121#398—68no—
122#398—64no—
123#398—64no—
124#389—68no—
125#384—64no—
126C4×C4⋊C4—64no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95C2×D312yes—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D311yescharacteristic
137C2×#95C2×D312yes—
138C24×C3C2411yescharacteristic
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
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×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×Q814yes—
167C22×C3×C4#4112yes—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—21no—
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—22no—
190C22×Dic3—22no—
191C22×Dic3C2×D412yes—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D412yes—
198C22×Dic3C2×D412yes—
199C22×Dic3C2×Q814yes—
200C22×Dic3C2×Q814yes—
201C22×Dic3#4112yes—
202C22×Dic3#4112yes—
203C22×Dic3#4114yes—
204C3×C4⋊C4—44no—
205C4⋊Dic3—44no—
206C4⋊Dic3—48no—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—216no—
210Dic3⋊C4—48no—
211Dic3⋊C4—48no—
212Dic3⋊C4—48no—
213C4×Dic3—28no—
214C4×Dic3—28no—
215C4×Dic3—44no—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—32no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4C22×D312yes—
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D312yes—
226C23×C4C22×D311yescharacteristic
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—38no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C42—34no—
255C2×C42—34no—
256C2×C42—62no—
257C2×C42—64no—
258C2×C42—62no—
259C2×C42—62no—
260C2×C42—64no—
261#95—22no—
262#95—24no—
263#95—22no—
264#95—24no—
265#95—62no—
266#95—64no—
267#95—64no—
268#95—64no—
269#95—64no—
270#95—62no—
271#95—64no—
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D411yescharacteristic
274C23×C3C22×D411yescharacteristic
275C23×C3C22×D411yescharacteristic
276C23×C3C22×Q812yes—
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4111yescharacteristic
280C23×C3C2×#4112yes—
281C23×C32−1+412yes—
282C23×C32−1+412yes—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4#12114yes—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
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#12114yes—
342C2×Dic3#12114yes—
343C2×Dic3C22⋊Q818yes—
344C2×Dic3#12314yes—
345C2×Dic3#12518yes—
346C2×Dic3#12714yes—
347C2×Dic3C4⋊Q814yes—
348C24C23×D311yescharacteristic
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—34no—
375C22×C4—34no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
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#23512yes—
409C22×C4D3×D412yes—
410C22×C4D3×D412yes—
411C22×C4D3×D412yes—
412C22×C4D3×D412yes—
413C22×C4D3×Q814yes—
414C4⋊C4—44no—
415C4⋊C4—68no—
416C4⋊C4—616no—
417C4⋊C4—68no—
418C4⋊C4—128no—
419C4⋊C4—128no—
420C4⋊C4—128no—
421C4⋊C4—128no—
422C4⋊C4—124no—
423C42—68no—
424C42—68no—
425C42—124no—
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12111yescharacteristic
428C22×C3C2×#12111yescharacteristic
429C22×C3C2×C22⋊Q812yes—
430C22×C3C2×#12311yescharacteristic
431C22×C3C2×#12512yes—
432C22×C3C2×#12711yescharacteristic
433C22×C3C2×C4⋊Q811yescharacteristic
434C22×C3#53211yescharacteristic
435C22×C3#53211yescharacteristic
436C22×C3#53211yescharacteristic
437C22×C3#53612yes—
438C22×C3#53611yescharacteristic
439C22×C3#53611yescharacteristic
440C22×C3#54112yes—
441C22×C3#54712yes—
442C22×C3#54712yes—
443C22×C3D4×Q812yes—
444C22×C3D4×Q812yes—
445C22×C3#55212yes—
446C22×C3#55212yes—
447C22×C3#55712yes—
448C22×C3#55712yes—
449C3×C4—24no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—44no—
453C3×C4—82no—
454Dic3—24no—
455Dic3—24no—
456Dic3—42no—
457Dic3—44no—
458Dic3—82no—
459Dic3#39818yescomplemented
460C23C2×#23511yescharacteristic
461C23C2×D3×D411yescharacteristic
462C23C2×D3×D411yescharacteristic
463C23C2×D3×D411yescharacteristic
464C23C2×D3×D411yescharacteristic
465C23C2×D3×Q812yes—
466C23D3×#4112yes—
467C23D3×#4111yescharacteristic
468C23D3×#4111yescharacteristic
469C23#101012yes—
470C23#101012yes—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—34no—
483C2×C4—38no—
484C2×C4—34no—
485C2×C4—34no—
486C2×C4—38no—
487C2×C4—34no—
488C2×C4—34no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—64no—
521C2×C4—121no—
522C2×C4—122no—
523C2×C4—121no—
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#88414yes—
536C2×C3C2×#39811yescharacteristic
537C2×C3#248611yescharacteristic
538C2×C3#248611yescharacteristic
539C2×C3#250612yes—
540C2×C3#250612yes—
541C2×C3#254212yes—
542C2×C3#254611yescharacteristic
543C2×C3#254611yescharacteristic
544C2×C3#256512yes—
545C2×C3#262111yescharacteristic
546C2×C3#285611yescharacteristic
547C22C2×#88411yescharacteristic
548C22#553112yes—
549C22#554811yescharacteristic
550C22#554811yescharacteristic
551C22#556712yes—
552C22D3×#12111yescharacteristic
553C22D3×#12111yescharacteristic
554C22D3×C22⋊Q812yes—
555C22#563212yes—
556C22#563212yes—
557C22#564812yes—
558C22#564811yescharacteristic
559C22D3×#12311yescharacteristic
560C22#565512yes—
561C22#565512yes—
562C22#566811yescharacteristic
563C22#567511yescharacteristic
564C22#568512yes—
565C22D3×#12512yes—
566C22#570112yes—
567C22#570812yes—
568C22D3×#12711yescharacteristic
569C22D3×C4⋊Q811yescharacteristic
570C4—24no—
571C4—38no—
572C4—42no—
573C4—44no—
574C4—42no—
575C4—64no—
576C4—64no—
577C4—82no—
578C4—122no—
579C4—124no—
580C4—242no—
581C3#1515111yescharacteristic, complemented, Sylow
582C2#7407511yescharacteristic
583C2#7502612yes—
584C2#7507612yes—
585C2#7647611yescharacteristic
586C2#7653511yescharacteristic
587C2#7748511yescharacteristic
588C2#7753811yescharacteristic
589C2#7778511yescharacteristic
590C2D3×#39811yescharacteristic
591C2#7803712yes—
592C2#7815312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1445C2
hypercenter#1445D3
hypocenterC3#15151
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
459. Dic3#398semidirect
581. C3#15151semidirect

Maximal subgroups

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

subgroupindex
C2×#49832
C2×#49752
C2×#46612
C2×#46572
C2×#46472
C2×#46492
C3×#14452
#694222
#694212
Dic3⋊#952
#151513