Finite Group Databasebeta

← #10750428

Subgroups — #10750428

Order 768.

1638 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×#4657C212yesmaximal
3C2×#4656C212yesmaximal
4C2×#4654C211yesmaximal, characteristic
5C3×#1442C211yesmaximal, characteristic
6#71568C211yesmaximal, characteristic
7#69424C212yesmaximal
8#69421C212yesmaximal
9#69419C211yesmaximal, characteristic
10#69415C211yesmaximal, characteristic
11#69401C211yesmaximal, characteristic
12#16507—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2212yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×#888—28no—
23C2×#888—24no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4657—28no—
39#4656—28no—
40#4654—24no—
41C22×C4⋊Q8—31no—
42C2×#398—32no—
43C2×#395—32no—
44C2×#387—31no—
45#1446—32no—
46#1445—32no—
47#1444—31no—
48#1444—31no—
49#1442—31no—
50#1442D311yescharacteristic
51#1434—31no—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×Dic6—24no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6—42no—
69C22×Dic6—42no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—42no—
79C2×C4⋊Dic3D414yes—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—42no—
87C2×Dic3⋊C4D418yes—
88C2×C4×Dic3—24no—
89C2×C4×Dic3D414yes—
90#888—232no—
91C3×#95—22no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95Q814yescomplemented
95C3⋊#95—22no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95Q814yescomplemented
105C23×Q8—32no—
106C2×C4⋊Q8—64no—
107C2×C4⋊Q8—68no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C42—31no—
116#398—68no—
117#395—68no—
118#387—64no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95C2×D312yes—
130C2×#95C2×D312yes—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D311yescharacteristic
133C24×C3C2411yescharacteristic
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—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
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—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×Q814yes—
161C22×C3×C4C2×Q812yes—
162C22×C3×C4C2×Q812yes—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×Q814yes—
196C22×Dic3#4112yes—
197C2×Dic6—216no—
198C2×Dic6—48no—
199C2×Dic6—48no—
200C2×Dic6—48no—
201C2×Dic6—48no—
202C2×Dic6—48no—
203C2×Dic6—48no—
204C3×C4⋊C4—28no—
205C4⋊Dic3—28no—
206C4⋊Dic3—28no—
207C4⋊Dic3—48no—
208Dic3⋊C4—216no—
209Dic3⋊C4—48no—
210Dic3⋊C4—44no—
211C4×Dic3—28no—
212C22×Q8—64no—
213C22×Q8—64no—
214C22×Q8—64no—
215C22×Q8—64no—
216C22×Q8—64no—
217C22×Q8—122no—
218C22×Q8—122no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—32no—
224C23×C4—31no—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D312yes—
227C23×C4C22×D312yes—
228C23×C4C22×D311yescharacteristic
229C23×C4C22×D311yescharacteristic
230C4⋊Q8—632no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—122no—
248C2×C4⋊C4—122no—
249C2×C42—34no—
250C2×C42—64no—
251#95—22no—
252#95—24no—
253#95—24no—
254#95—34no—
255#95—62no—
256#95—62no—
257#95—64no—
258#95—64no—
259#95—64no—
260#95—62no—
261#95—64no—
262#95—122no—
263#95—122no—
264#95Dic614yescomplemented
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D412yes—
267C23×C3C22×D411yescharacteristic
268C23×C3C22×Q811yescharacteristic
269C23×C3C22×Q811yescharacteristic
270C23×C3C2×#4111yescharacteristic
271C23×C3H(5,𝔽2)12yes—
272C23×C32−1+411yescharacteristic
273C23×C32−1+412yes—
274C23×C32−1+411yescharacteristic
275C23×C32−1+412yes—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
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—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4C4⋊Q814yes—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—22no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3#12014yes—
338C2×Dic3#12114yes—
339Dic6—432no—
340C24C23×D311yescharacteristic
341C2×Q8—616no—
342C2×Q8—128no—
343C2×Q8—128no—
344C2×Q8—128no—
345C2×Q8—128no—
346C2×Q8—128no—
347C2×Q8—128no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
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—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—34no—
375C22×C4—32no—
376C22×C4—34no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4C2×Dic614yes—
406C22×C4C2×Dic612yes—
407C22×C4D3×D412yes—
408C22×C4D3×D414yes—
409C22×C4D3×D412yes—
410C22×C4D3×Q812yes—
411C4⋊C4—28no—
412C4⋊C4—616no—
413C4⋊C4—68no—
414C4⋊C4—68no—
415C4⋊C4—128no—
416C4⋊C4—124no—
417C4⋊C4—128no—
418C42—68no—
419C22×C3C2×#12011yescharacteristic
420C22×C3C2×#12111yescharacteristic
421C22×C3C2×C4⋊Q811yescharacteristic
422C22×C3#53511yescharacteristic
423C22×C3#53611yescharacteristic
424C22×C3#53612yes—
425C22×C3#53611yescharacteristic
426C22×C3#53612yes—
427C22×C3#53611yescharacteristic
428C22×C3#53612yes—
429C22×C3#54112yes—
430C22×C3#54412yes—
431C22×C3#54412yes—
432C22×C3D4212yes—
433C22×C3#54612yes—
434C22×C3D4×Q812yes—
435C22×C3D4×Q811yescharacteristic
436C22×C3D4×Q811yescharacteristic
437C22×C3D4×Q812yes—
438C22×C3#55711yescharacteristic
439C22×C3#55711yescharacteristic
440C22×C3#56312yes—
441C22×C3#56312yes—
442C3×C4—24no—
443C3×C4—42no—
444C3×C4—44no—
445C3×C4—44no—
446C3×C4—81no—
447Dic3—24no—
448Dic3—24no—
449Dic3—44no—
450Dic3—42no—
451Dic3—44no—
452Dic3—81no—
453C23C22×Dic611yescharacteristic
454C23C2×D3×D412yes—
455C23C2×D3×D411yescharacteristic
456C23C2×D3×D411yescharacteristic
457C23C2×D3×Q811yescharacteristic
458C23#100711yescharacteristic
459C23D3×#4111yescharacteristic
460C23#100912yes—
461C23#101012yes—
462C23#101012yes—
463C23#101011yescharacteristic
464Q8—1232no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—34no—
478C2×C4—34no—
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—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—64no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—121no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4#88814yes—
528C2×C3#240811yescharacteristic
529C2×C3#243411yescharacteristic
530C2×C3#254611yescharacteristic
531C2×C3#261411yescharacteristic
532C2×C3#262111yescharacteristic
533C2×C3#266011yescharacteristic
534C2×C3#267712yes—
535C2×C3#267712yes—
536C2×C3#271312yes—
537C2×C3#271312yes—
538C2×C3#282111yescharacteristic
539C22C2×#88811yescharacteristic
540C22#551912yes—
541C22D4×Dic611yescharacteristic
542C22D4×Dic612yes—
543C22#556911yescharacteristic
544C22D3×#12011yescharacteristic
545C22D3×#12111yescharacteristic
546C22#560912yes—
547C22#562811yescharacteristic
548C22#562812yes—
549C22#563212yes—
550C22#564812yes—
551C22#564811yescharacteristic
552C22#565412yes—
553C22#567411yescharacteristic
554C22#567512yes—
555C22#567511yescharacteristic
556C22#568511yescharacteristic
557C22#568912yes—
558C22#569912yes—
559C22#570112yes—
560C22#571212yes—
561C22#572711yescharacteristic
562C4—24no—
563C4—44no—
564C4—44no—
565C4—42no—
566C4—64no—
567C4—64no—
568C4—81no—
569C4—124no—
570C4—122no—
571C4—124no—
572C4—241no—
573C3#1650711yescharacteristic, complemented, Sylow
574C2#7419611yescharacteristic
575C2#7441211yescharacteristic
576C2#7515612yes—
577C2#7566712yes—
578C2#7606011yescharacteristic
579C2#7687011yescharacteristic
580C2#7689511yescharacteristic
581C2#7753811yescharacteristic
582C2#7796812yes—
583C2#7824312yes—
584C2#7831711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1442C2
hypercenter#1442D3
hypocenterC3#16507
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
94. C3×#95Q8semidirect
104. C3⋊#95Q8semidirect
264. #95Dic6semidirect
573. C3#16507semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#46572
C2×#46562
C2×#46542
C3×#14422
#715682
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