Finite Group Databasebeta

← #10750427

Subgroups — #10750427

Order 768.

1638 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic
2C2×#5230C211yesmaximal, characteristic
3C2×#4657C212yesmaximal
4C2×#4656C211yesmaximal, characteristic
5C2×#4652C211yesmaximal, characteristic
6C3×#1446C211yesmaximal, characteristic
7#69424C211yesmaximal, characteristic
8#69421C212yesmaximal
9#69420C212yesmaximal
10#69414C212yesmaximal
11#69403C211yesmaximal, characteristic
12#16507—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×#944—28no—
16C2×#944—24no—
17C22×C4⋊Dic3C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28#5230—24no—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4657—28no—
40#4656—24no—
41#4652—24no—
42C22×C4⋊Q8—31no—
43C2×#398—32no—
44C2×#395—31no—
45C2×#395—31no—
46C2×#387—31no—
47#1446—31no—
48#1446D311yescharacteristic
49#1445—32no—
50#1444—32no—
51#1442—32no—
52#1434—31no—
53C22×C3×Q8—24no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—42no—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—24no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—42no—
74#944—232no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—42no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—42no—
90C2×Dic3⋊C4D414yes—
91C2×Dic3⋊C4D418yes—
92C2×C4×Dic3—24no—
93C2×C4×Dic3D414yes—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—24no—
97C3×#95—42no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—42no—
107C3⋊#95Q818yescomplemented
108C23×Q8—31no—
109C23×Q8C2×D311yescharacteristic
110C2×C4⋊Q8—64no—
111C2×C4⋊Q8—68no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—32no—
118C22×C42—31no—
119#398—68no—
120#395—64no—
121#395—64no—
122#387—64no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—31no—
130C2×#95—31no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D312yes—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D312yes—
137C24×C3C2411yescharacteristic
138C2×C3×Q8—28no—
139C2×C3×Q8—48no—
140C2×C3×Q8—48no—
141C2×C3×Q8—44no—
142C2×C3×Q8—44no—
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—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—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×Q814yes—
171C22×C3×C4#4112yes—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D414yes—
202C22×Dic3C2×D412yes—
203C22×Dic3C2×D414yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×Q814yes—
207C22×Dic3C2×Q814yes—
208C2×Dic6—28no—
209C2×Dic6—48no—
210C2×Dic6—48no—
211C2×Dic6—44no—
212C2×Dic6—44no—
213C4⋊Dic3—48no—
214Dic3⋊C4—216no—
215Dic3⋊C4—28no—
216Dic3⋊C4—216no—
217Dic3⋊C4—48no—
218Dic3⋊C4—44no—
219C4×Dic3—28no—
220C22×Q8—22no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—64no—
226C22×Q8—62no—
227C22×Q8—62no—
228C22×Q8—62no—
229C22×Q8—122no—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—32no—
234C23×C4—31no—
235C23×C4—32no—
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D312yes—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C4⋊Q8—632no—
242C2×C4⋊C4—38no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—122no—
258C2×C4⋊C4—122no—
259C2×C42—34no—
260C2×C42—64no—
261#95—22no—
262#95—22no—
263#95—24no—
264#95—38no—
265#95—42no—
266#95—62no—
267#95—62no—
268#95—62no—
269#95—64no—
270#95—64no—
271#95—64no—
272#95—64no—
273#95—62no—
274#95—122no—
275C23×C3C22×D411yescharacteristic
276C23×C3C22×D412yes—
277C23×C3C22×D411yescharacteristic
278C23×C3C22×Q812yes—
279C23×C3C2×#4111yescharacteristic
280C23×C3H(5,𝔽2)12yes—
281C23×C32−1+411yescharacteristic
282C23×C32−1+411yescharacteristic
283C23×C32−1+411yescharacteristic
284C23×C32−1+412yes—
285C23×C32−1+411yescharacteristic
286C3×Q8—416no—
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—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12114yes—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3#12014yes—
351C2×Dic3C4⋊Q814yes—
352Dic6—416no—
353C24C23×D311yescharacteristic
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—68no—
360C2×Q8—128no—
361C2×Q8—128no—
362C2×Q8—124no—
363C2×Q8—124no—
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—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—34no—
391C22×C4—34no—
392C22×C4—32no—
393C22×C4—34no—
394C22×C4—32no—
395C22×C4—34no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4#23512yes—
425C22×C4D3×D414yes—
426C22×C4D3×D412yes—
427C22×C4D3×D412yes—
428C22×C4D3×Q814yes—
429C4⋊C4—616no—
430C4⋊C4—616no—
431C4⋊C4—68no—
432C4⋊C4—128no—
433C4⋊C4—124no—
434C4⋊C4—128no—
435C42—68no—
436C22×C3C2×#12011yescharacteristic
437C22×C3C2×#12111yescharacteristic
438C22×C3C2×C4⋊Q811yescharacteristic
439C22×C3#53511yescharacteristic
440C22×C3#53611yescharacteristic
441C22×C3#53612yes—
442C22×C3#53611yescharacteristic
443C22×C3#53611yescharacteristic
444C22×C3#53611yescharacteristic
445C22×C3#53611yescharacteristic
446C22×C3#53612yes—
447C22×C3#54111yescharacteristic
448C22×C3#54111yescharacteristic
449C22×C3#54412yes—
450C22×C3#54412yes—
451C22×C3D4212yes—
452C22×C3#54612yes—
453C22×C3D4×Q812yes—
454C22×C3D4×Q812yes—
455C22×C3D4×Q812yes—
456C22×C3#55712yes—
457C22×C3#56312yes—
458C22×C3#56312yes—
459C3×C4—24no—
460C3×C4—44no—
461C3×C4—44no—
462C3×C4—81no—
463C3×C4—81no—
464Dic3—24no—
465Dic3—24no—
466Dic3—42no—
467Dic3—44no—
468Dic3—42no—
469Dic3—44no—
470C23C2×#23511yescharacteristic
471C23C2×D3×D411yescharacteristic
472C23C2×D3×D412yes—
473C23C2×D3×D411yescharacteristic
474C23#100412yes—
475C23C2×D3×Q812yes—
476C23#100711yescharacteristic
477C23#100711yescharacteristic
478C23#101011yescharacteristic
479C23#101012yes—
480C23#101011yescharacteristic
481Q8—416no—
482Q8—1216no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—34no—
494C2×C4—34no—
495C2×C4—42no—
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—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—62no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4#88414yes—
548C2×C3#240811yescharacteristic
549C2×C3#243411yescharacteristic
550C2×C3#254611yescharacteristic
551C2×C3#261411yescharacteristic
552C2×C3#262111yescharacteristic
553C2×C3#266011yescharacteristic
554C2×C3#267712yes—
555C2×C3#267712yes—
556C2×C3#271312yes—
557C2×C3#271312yes—
558C2×C3#282111yescharacteristic
559C22C2×#88411yescharacteristic
560C22#549012yes—
561C22#556712yes—
562C22#558311yescharacteristic
563C22#558311yescharacteristic
564C22D3×#12011yescharacteristic
565C22#559012yes—
566C22#559211yescharacteristic
567C22#562312yes—
568C22#562812yes—
569C22#562811yescharacteristic
570C22#562811yescharacteristic
571C22#563212yes—
572C22#563212yes—
573C22#564612yes—
574C22#567212yes—
575C22#567511yescharacteristic
576C22#567512yes—
577C22#567511yescharacteristic
578C22D3×C4⋊Q811yescharacteristic
579C22#572411yescharacteristic
580C22#572712yes—
581C22#572912yes—
582C4—24no—
583C4—44no—
584C4—44no—
585C4—64no—
586C4—64no—
587C4—81no—
588C4—81no—
589C4—124no—
590C4—122no—
591C4—122no—
592C4—124no—
593C3#1650711yescharacteristic, complemented, Sylow
594C2#7382411yescharacteristic
595C2#7653511yescharacteristic
596C2#7656712yes—
597C2#7656812yes—
598C2#7664111yescharacteristic
599C2#7682312yes—
600C2#7685411yescharacteristic
601C2#7687011yescharacteristic
602C2#7717311yescharacteristic
603C2#7720811yescharacteristic
604C2#7726112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1446C2
hypercenter#1446D3
hypocenterC3#16507
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
107. C3⋊#95Q8semidirect
593. C3#16507semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×#52302
C2×#46572
C2×#46562
C2×#46522
C3×#14462
#694242
#694212
#694202
#694142
#694032
#165073