Finite Group Databasebeta

← #10751715

Subgroups — #10751715

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×#4656C212yesmaximal
3C2×#4655C211yesmaximal, characteristic
4C3×#1442C211yesmaximal, characteristic
5#71568C211yesmaximal, characteristic
6#69424C212yesmaximal
7#69420C212yesmaximal
8#69419C211yesmaximal, characteristic
9#69415C211yesmaximal, characteristic
10#69409C212yesmaximal
11#69401C211yesmaximal, characteristic
12#16603—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×#888—28no—
23C2×#888—24no—
24C2×C3×#95C2212yes—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4656—28no—
39#4655—24no—
40C22×C4⋊Q8—31no—
41C2×#395—32no—
42C2×#387—31no—
43#1446—32no—
44#1444—31no—
45#1444—32no—
46#1444—31no—
47#1442—31no—
48#1442D311yescharacteristic
49#1441—32no—
50#1434—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—42no—
65C22×Dic6—42no—
66C22×Dic6—42no—
67C22×Dic6—42no—
68C22×Dic6D418yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—42no—
77C2×C4⋊Dic3—42no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—42no—
85C2×Dic3⋊C4—42no—
86C2×Dic3⋊C4Q814yescomplemented
87C2×C4×Dic3—24no—
88C2×C4×Dic3D414yes—
89#888—232no—
90C3×#95—22no—
91C3×#95—24no—
92C3×#95—42no—
93C3×#95Q814yescomplemented
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—42no—
101C3⋊#95—42no—
102C3⋊#95—42no—
103C3⋊#95Q814yescomplemented
104C23×Q8—32no—
105C2×C4⋊Q8—64no—
106C2×C4⋊Q8—68no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C42—31no—
115#395—68no—
116#387—64no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—31no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95C2×D312yes—
128C2×#95C2×D311yescharacteristic
129C2×#95C2×D312yes—
130C2×#95C2×D311yescharacteristic
131C24×C3C2411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
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—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×Q814yes—
158C22×C3×C4C2×Q812yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4C2×Q812yes—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—21no—
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—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3C2×D414yes—
188C22×Dic3C2×D414yes—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×Q814yes—
192C22×Dic3C2×Q812yes—
193C22×Dic3C2×Q812yes—
194C22×Dic3#4112yes—
195C2×Dic6—216no—
196C2×Dic6—48no—
197C2×Dic6—48no—
198C2×Dic6—48no—
199C2×Dic6—48no—
200C2×Dic6—48no—
201C2×Dic6—48no—
202C3×C4⋊C4—28no—
203C4⋊Dic3—28no—
204C4⋊Dic3—48no—
205C4⋊Dic3—44no—
206Dic3⋊C4—216no—
207Dic3⋊C4—28no—
208Dic3⋊C4—48no—
209C4×Dic3—28no—
210C22×Q8—38no—
211C22×Q8—64no—
212C22×Q8—64no—
213C22×Q8—122no—
214C22×Q8—122no—
215C22×Q8—122no—
216C22×Q8—122no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—32no—
220C23×C4—32no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D312yes—
225C23×C4C22×D312yes—
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C4⋊Q8—632no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—122no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4—122no—
246C2×C4⋊C4—122no—
247C2×C42—34no—
248C2×C42—64no—
249#95—22no—
250#95—24no—
251#95—34no—
252#95—42no—
253#95—64no—
254#95—62no—
255#95—62no—
256#95—64no—
257#95—64no—
258#95—62no—
259#95—122no—
260#95—122no—
261#95—122no—
262#95Dic614yescomplemented
263C23×C3C22×D411yescharacteristic
264C23×C3C22×D412yes—
265C23×C3C22×Q811yescharacteristic
266C23×C3C22×Q811yescharacteristic
267C23×C3C22×Q811yescharacteristic
268C23×C3C2×#4111yescharacteristic
269C23×C3H(5,𝔽2)11yescharacteristic
270C23×C3H(5,𝔽2)12yes—
271C23×C3H(5,𝔽2)11yescharacteristic
272C23×C32−1+412yes—
273C23×C32−1+412yes—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4C4⋊Q814yes—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3#12014yes—
336C2×Dic3C22⋊Q814yes—
337Dic6—432no—
338C24C23×D311yescharacteristic
339C2×Q8—616no—
340C2×Q8—128no—
341C2×Q8—128no—
342C2×Q8—128no—
343C2×Q8—128no—
344C2×Q8—128no—
345C2×Q8—128no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—34no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—34no—
375C22×C4—34no—
376C22×C4—32no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—62no—
403C22×C4C2×Dic614yes—
404C22×C4C2×Dic612yes—
405C22×C4D3×D414yes—
406C22×C4D3×D412yes—
407C22×C4D3×Q812yes—
408C22×C4D3×Q812yes—
409C4⋊C4—28no—
410C4⋊C4—68no—
411C4⋊C4—616no—
412C4⋊C4—68no—
413C4⋊C4—128no—
414C4⋊C4—124no—
415C4⋊C4—128no—
416C42—68no—
417C22×C3C2×#12011yescharacteristic
418C22×C3C2×C22⋊Q811yescharacteristic
419C22×C3C2×C4⋊Q811yescharacteristic
420C22×C3#53412yes—
421C22×C3#53412yes—
422C22×C3#53511yescharacteristic
423C22×C3#53511yescharacteristic
424C22×C3#53611yescharacteristic
425C22×C3#53611yescharacteristic
426C22×C3#54212yes—
427C22×C3#54412yes—
428C22×C3#54412yes—
429C22×C3#54412yes—
430C22×C3#54812yes—
431C22×C3D4×Q812yes—
432C22×C3D4×Q812yes—
433C22×C3D4×Q812yes—
434C22×C3#55411yescharacteristic
435C22×C3#55411yescharacteristic
436C22×C3Q8211yescharacteristic
437C22×C3Q8211yescharacteristic
438C22×C3#56312yes—
439C22×C3#56312yes—
440C3×C4—24no—
441C3×C4—44no—
442C3×C4—42no—
443C3×C4—44no—
444C3×C4—81no—
445Dic3—24no—
446Dic3—24no—
447Dic3—44no—
448Dic3—42no—
449Dic3—44no—
450Dic3—81no—
451C23C22×Dic611yescharacteristic
452C23C2×D3×D412yes—
453C23C2×D3×D411yescharacteristic
454C23#100411yescharacteristic
455C23C2×D3×Q811yescharacteristic
456C23C2×D3×Q811yescharacteristic
457C23D3×#4111yescharacteristic
458C23#100911yescharacteristic
459C23#100912yes—
460C23#101012yes—
461C23#101012yes—
462Q8—1232no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—34no—
476C2×C4—34no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—64no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—62no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—121no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4#88814yes—
526C2×C3#240911yescharacteristic
527C2×C3#243411yescharacteristic
528C2×C3#256711yescharacteristic
529C2×C3#261311yescharacteristic
530C2×C3#262811yescharacteristic
531C2×C3#266011yescharacteristic
532C2×C3#272912yes—
533C2×C3#272912yes—
534C2×C3#273412yes—
535C2×C3#273412yes—
536C2×C3#282111yescharacteristic
537C22C2×#88811yescharacteristic
538C22#551912yes—
539C22D4×Dic612yes—
540C22#553911yescharacteristic
541C22Q8×Dic611yescharacteristic
542C22D3×#12011yescharacteristic
543C22#560512yes—
544C22#560711yescharacteristic
545C22D3×C22⋊Q811yescharacteristic
546C22#563212yes—
547C22#563212yes—
548C22#564811yescharacteristic
549C22#565112yes—
550C22#565312yes—
551C22#567411yescharacteristic
552C22#567511yescharacteristic
553C22#567712yes—
554C22#568912yes—
555C22#568912yes—
556C22#569011yescharacteristic
557C22#571012yes—
558C22#571212yes—
559C22#572211yescharacteristic
560C4—24no—
561C4—44no—
562C4—44no—
563C4—42no—
564C4—64no—
565C4—64no—
566C4—81no—
567C4—124no—
568C4—124no—
569C4—122no—
570C4—241no—
571C3#1660311yescharacteristic, complemented, Sylow
572C2#7419611yescharacteristic
573C2#7441211yescharacteristic
574C2#7517912yes—
575C2#7570712yes—
576C2#7607111yescharacteristic
577C2#7678011yescharacteristic
578C2#7712211yescharacteristic
579C2#7764811yescharacteristic
580C2#7787212yes—
581C2#7820911yescharacteristic
582C2#7837412yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1442C2
hypercenter#1442D3
hypocenterC3#16603
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
86. C2×Dic3⋊C4Q8semidirect
93. C3×#95Q8semidirect
103. C3⋊#95Q8semidirect
262. #95Dic6semidirect
571. C3#16603semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#46562
C2×#46552
C3×#14422
#715682
#694242
#694202
#694192
#694152
#694092
#694012
#166033