Finite Group Databasebeta

← #11234103

Subgroups — #11234103

Order 768.

2194 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2408C211yesmaximal, characteristic
2D4×#217C212yesmaximal, complemented
3#82644C212yesmaximal, complemented
4#82636C212yesmaximal, complemented
5#82590C214yesmaximal, complemented
6#82374C211yesmaximal, characteristic, complemented
7#82348C211yesmaximal, characteristic, complemented
8#77054C214yesmaximal, complemented
9#76718C212yesmaximal, complemented
10#76681C212yesmaximal, complemented
11#76346C212yesmaximal, complemented
12#76295C212yesmaximal, complemented
13#73925C212yesmaximal, complemented
14#73842C212yesmaximal, complemented
15#73828C212yesmaximal, complemented
16#45377—31nomaximal, Sylow
17C23×C3×D4C2211yescharacteristic
18C2×C3×#121C2212yes—
19C2×C3×#120C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C22×#217C2214yes—
22C2×#935C2214yescomplemented
23C2×D4×Dic3C2212yes—
24C2×D4×Dic3C2212yes—
25C2×D4×Dic3C2214yescomplemented
26C22×C4⋊Dic3C2211yescharacteristic, complemented
27C2×#878C2214yescomplemented
28C2×C4×Dic6C2211yescharacteristic, complemented
29C3×#394C2212yes—
30C3×#392C2212yes—
31C3×#390C2212yes—
32C3×#380C2211yescharacteristic
33C3×#379C2212yes—
34#5247C2214yes—
35#5222C2212yes—
36#5222C2214yescomplemented
37#5221C2212yes—
38#5221C2212yes—
39#5220C2214yes—
40#5220C2214yes—
41#5220C2212yes—
42#5220C2214yescomplemented
43#5212C2212yes—
44#5212C2211yescharacteristic, complemented
45C4×#217C2212yes—
46C4×#217C2212yescomplemented
47#4986C2212yescomplemented
48#4985C2212yescomplemented
49#4984C2212yescomplemented
50#4975C2212yescomplemented
51Dic3×C4⋊C4C2212yescomplemented
52#4972C2212yescomplemented
53#4953C2211yescharacteristic
54#4953C2211yescharacteristic, complemented
55#4952C2214yescomplemented
56#4951C2214yescomplemented
57#4950C2214yescomplemented
58#4949C2212yes—
59#4948C2214yes—
60#4948C2214yescomplemented
61#4948C2214yescomplemented
62#4947C2211yescharacteristic
63#4947C2211yescharacteristic, complemented
64#4946C2214yescomplemented
65#4946C2212yescomplemented
66#4945C2214yes—
67#4945C2214yescomplemented
68#4944C2214yescomplemented
69#4943C2214yescomplemented
70#4942C2214yescomplemented
71#4942C2214yescomplemented
72#4932C2211yescharacteristic, complemented
73#4928C2212yescomplemented
74#4658C2214yescomplemented
75#4653C2214yescomplemented
76#2750—34no—
77#2735—32no—
78#2693—32no—
79#2538—32no—
80#2534—34no—
81#2497—32no—
82#2487—31no—
83#2463—32no—
84#2437—32no—
85#2422—32no—
86#2408D311yescharacteristic
87#2398—31no—
88#2361—32no—
89D4×C22⋊C4—32no—
90#2319—32no—
91C25×C3C2312yes—
92C22×C3×D4—28no—
93C22×C3×D4—42no—
94C22×C3×D4C2312yes—
95C22×C3×D4C2311yescharacteristic
96C22×C3×D4C2314yes—
97C22×C3×D4C2312yes—
98C23×C3×C4C2311yescharacteristic
99C23×C3×C4C2311yescharacteristic
100C23×Dic3C2314yes—
101C23×Dic3C2314yes—
102C23×Dic3C2312yes—
103C22×Dic6C2311yescharacteristic
104C3×#121—28no—
105C3×#120—44no—
106C3×C4×D4—42no—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×#217—48no—
117C2×#217C2314yes—
118C2×#217C2312yes—
119C2×#217C2314yes—
120C2×#217C2312yes—
121C2×#217C2312yes—
122C2×#217C2312yes—
123C2×#217C2312yes—
124C2×#217C2314yes—
125C2×#217C2314yes—
126C2×#217C2314yes—
127C2×#217C2314yes—
128C2×#217C2314yes—
129#935—48no—
130D4×Dic3—28no—
131D4×Dic3—44no—
132D4×Dic3—48no—
133C2×C4⋊Dic3—42no—
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147#878—48no—
148C4×Dic6—42no—
149C3×#95C2312yes—
150C3⋊#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2311yescharacteristic
158C3⋊#95C2312yes—
159C3⋊#95C2311yescharacteristic
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C23×D4C2×D311yescharacteristic
164C2×#123—34no—
165C2×C22⋊Q8—34no—
166C2×#121C2×D312yes—
167C2×#120C2×D312yes—
168C2×C4×Q8—31no—
169C2×C4×D4—32no—
170C2×C4×D4—32no—
171C2×C4×D4—34no—
172C2×C4×D4C2×D311yescharacteristic
173C22×C4⋊C4—31no—
174C22×C22⋊C4—34no—
175#400—32no—
176#400—32no—
177#400—34no—
178#399—32no—
179#397—34no—
180#397—31no—
181#397—31no—
182#396—34no—
183#396—34no—
184#396—34no—
185#394—32no—
186#394—34no—
187#394C2×D312yes—
188#393—31no—
189#393—31no—
190#392C2×D312yes—
191#390—32no—
192#390—32no—
193#390C2×D312yes—
194#389—32no—
195#389—32no—
196#389—32no—
197#388—34no—
198#388—34no—
199#388—34no—
200#388—34no—
201#387—34no—
202#386—34no—
203#386—34no—
204#386—34no—
205#386—32no—
206#385—34no—
207#385—34no—
208#385—32no—
209#384—32no—
210#381—34no—
211#380—32no—
212#380—31no—
213#380—34no—
214#380—34no—
215#380C2×D311yescharacteristic
216#379—34no—
217#379C2×D312yes—
218C4×C4⋊C4—32no—
219C4×C22⋊C4—32no—
220C4×C22⋊C4—32no—
221#376—31no—
222C24×C3—28no—
223C24×C3—24no—
224C24×C3—44no—
225C24×C3—42no—
226C24×C3—42no—
227C24×C3C2414yes—
228C24×C3C2411yescharacteristic
229C24×C3C2411yescharacteristic
230C2×C3×D4—28no—
231C2×C3×D4—28no—
232C2×C3×D4—28no—
233C2×C3×D4—22no—
234C2×C3×D4—22no—
235C2×C3×D4—24no—
236C2×C3×D4—24no—
237C2×C3×D4—44no—
238C2×C3×D4—41no—
239C2×C3×D4—42no—
240C2×C3×D4—416no—
241C2×C3×D4#4118yes—
242C22×C3×C4—24no—
243C22×C3×C4—24no—
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2414yes—
250C22×C3×C4C2411yescharacteristic
251C22×C3×C4C2411yescharacteristic
252C22×Dic3—28no—
253C22×Dic3—28no—
254C22×Dic3—44no—
255C22×Dic3—42no—
256C22×Dic3—44no—
257C22×Dic3—44no—
258C22×Dic3—44no—
259C22×Dic3—44no—
260C22×Dic3C2412yes—
261C22×Dic3C2412yes—
262C22×Dic3C2412yes—
263C22×Dic3C2414yes—
264C22×Dic3C2414yes—
265C22×Dic3C2412yes—
266C2×Dic6—24no—
267C2×Dic6—41no—
268C3×C4⋊C4—24no—
269C3×C4⋊C4—41no—
270C3×C22⋊C4—28no—
271C3×C22⋊C4—44no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—41no—
274C3×C22⋊C4—42no—
275C3×C22⋊C4—44no—
276C3×C42—22no—
277#217—28no—
278#217—24no—
279#217—24no—
280#217—24no—
281#217—28no—
282#217—24no—
283#217—24no—
284#217—44no—
285#217—44no—
286#217—44no—
287#217—44no—
288#217—44no—
289C4⋊Dic3—24no—
290C4⋊Dic3—24no—
291C4⋊Dic3—28no—
292C4⋊Dic3—41no—
293C4⋊Dic3—42no—
294Dic3⋊C4—42no—
295Dic3⋊C4—44no—
296Dic3⋊C4—44no—
297Dic3⋊C4—44no—
298Dic3⋊C4—44no—
299C4×Dic3—24no—
300C4×Dic3—24no—
301C4×Dic3—44no—
302C25C22×D312yes—
303C22×Q8—31no—
304C22×D4—28no—
305C22×D4—42no—
306C22×D4C22×D314yes—
307C22×D4C22×D311yescharacteristic
308C22×D4C22×D312yes—
309C22×D4C22×D312yes—
310C23×C4—34no—
311C23×C4—34no—
312C23×C4—32no—
313C23×C4C22×D311yescharacteristic
314C23×C4C22×D311yescharacteristic
315#123—128no—
316C22⋊Q8—128no—
317#121—28no—
318#120—44no—
319C4×Q8—122no—
320C4×D4—42no—
321C4×D4—68no—
322C4×D4—124no—
323C4×D4—128no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—122no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—32no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—128no—
350C2×C22⋊C4C22×D314yes—
351C2×C22⋊C4C22×D311yescharacteristic
352C2×C22⋊C4C22×D311yescharacteristic
353C2×C22⋊C4C22×D312yes—
354C2×C22⋊C4C22×D314yes—
355C2×C22⋊C4C22×D314yes—
356C2×C42—32no—
357C2×C42—32no—
358C2×C42—34no—
359C2×C42C22×D311yescharacteristic
360#95—31no—
361#95—32no—
362#95—31no—
363#95—32no—
364#95—32no—
365#95—32no—
366#95—32no—
367#95—34no—
368#95—34no—
369#95—32no—
370#95—34no—
371#95—34no—
372#95—34no—
373#95C22×D312yes—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—24no—
377C23×C3—22no—
378C23×C3—24no—
379C23×C3—22no—
380C23×C3—24no—
381C23×C3—24no—
382C23×C3—24no—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—21no—
387C23×C3—24no—
388C23×C3—48no—
389C23×C3—416no—
390C23×C3—88no—
391C23×C3—84no—
392C23×C3—84no—
393C23×C3C2×#4118yes—
394C23×C3H(5,𝔽2)12yes—
395C23×C32−1+412yes—
396C23×C3C2511yescharacteristic
397C3×D4—216no—
398C3×D4—416no—
399C3×D4—44no—
400C3×D4—48no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—24no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—44no—
414C2×C3×C4—44no—
415C2×C3×C4C2×#4114yes—
416C2×C3×C4H(5,𝔽2)12yes—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—22no—
429C2×Dic3—24no—
430C2×Dic3—24no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—88no—
435C2×Dic3—88no—
436C2×Dic3—84no—
437C2×Dic3H(5,𝔽2)14yes—
438Dic6—44no—
439C24—28no—
440C24—24no—
441C24—44no—
442C24—42no—
443C24—42no—
444C24C23×D314yes—
445C24C23×D311yescharacteristic
446C24C23×D311yescharacteristic
447C2×Q8—64no—
448C2×Q8—121no—
449C2×D4—28no—
450C2×D4—28no—
451C2×D4—22no—
452C2×D4—28no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—41no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—416no—
460C2×D4#23718yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—34no—
470C22×C4—34no—
471C22×C4—68no—
472C22×C4—68no—
473C22×C4—124no—
474C22×C4—122no—
475C22×C4—124no—
476C22×C4—124no—
477C22×C4—124no—
478C22×C4—124no—
479C22×C4C23×D311yescharacteristic
480C22×C4C23×D311yescharacteristic
481C22×C4C23×D311yescharacteristic
482C22×C4C23×D314yes—
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D311yescharacteristic
485C4⋊C4—24no—
486C4⋊C4—41no—
487C4⋊C4—68no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—121no—
493C4⋊C4—124no—
494C4⋊C4—124no—
495C4⋊C4—124no—
496C4⋊C4—124no—
497C22⋊C4—28no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—64no—
504C22⋊C4—68no—
505C22⋊C4—64no—
506C22⋊C4—68no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—64no—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—124no—
513C22⋊C4—124no—
514C22⋊C4—124no—
515C42—22no—
516C42—64no—
517C42—64no—
518C42—124no—
519C22×C3—28no—
520C22×C3—28no—
521C22×C3—24no—
522C22×C3—24no—
523C22×C3—28no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—44no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3—44no—
533C22×C3—41no—
534C22×C3—44no—
535C22×C3—41no—
536C22×C3—816no—
537C22×C3C22×#4112yes—
538C22×C3C2×H(5,𝔽2)11yescharacteristic
539C22×C3C2×H(5,𝔽2)11yescharacteristic
540C22×C3C2×H(5,𝔽2)12yes—
541C22×C3C2×2−1+411yescharacteristic
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—44no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4#55114yes—
548Dic3—24no—
549Dic3—44no—
550Dic3—42no—
551Dic3—42no—
552Dic3—42no—
553Dic3—44no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—21no—
558C23—24no—
559C23—21no—
560C23—21no—
561C23—24no—
562C23—21no—
563C23—24no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—48no—
569C23—416no—
570C23—88no—
571C23—84no—
572C23—84no—
573C23C2×#23718yes—
574C23#100412yes—
575C23#101012yes—
576C23C24×D311yescharacteristic
577Q8—124no—
578D4—216no—
579D4—48no—
580D4—416no—
581D4—44no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—34no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—64no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—248no—
615C2×C4—248no—
616C2×C4—244no—
617C2×C4C2×#23714yes—
618C2×C4#100412yes—
619C2×C3—42no—
620C2×C3—48no—
621C2×C3—42no—
622C2×C3C2×#55111yescharacteristic
623C2×C3#351912yes—
624C2×C3#352012yes—
625C2×C3#353611yescharacteristic
626C2×C3#353711yescharacteristic
627C22—28no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—44no—
636C22—41no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—44no—
644C22—816no—
645C22C22×#23712yes—
646C22C2×#100411yescharacteristic
647C22C2×#100411yescharacteristic
648C22C2×#101011yescharacteristic
649C22D3×H(5,𝔽2)12yes—
650C4—41no—
651C4—44no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—64no—
656C4—124no—
657C4—122no—
658C4—124no—
659C4—122no—
660C4—122no—
661C4#558714yes—
662C3#4537711yescharacteristic, complemented, Sylow
663C2—42no—
664C2—48no—
665C2—42no—
666C2C2×#558711yescharacteristic
667C2#8595511yescharacteristic
668C2#8633411yescharacteristic
669C2#8661112yes—
670C2#8664112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2408C2
hypercenter#2408D3
hypocenterC3#45377
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. D4×#217C2semidirect
3. #82644C2semidirect
4. #82636C2semidirect
5. #82590C2semidirect
6. #82374C2semidirect
7. #82348C2semidirect
8. #77054C2semidirect
9. #76718C2semidirect
10. #76681C2semidirect
11. #76346C2semidirect
12. #76295C2semidirect
13. #73925C2semidirect
14. #73842C2semidirect
15. #73828C2semidirect
22. C2×#935C22semidirect
25. C2×D4×Dic3C22semidirect
26. C22×C4⋊Dic3C22semidirect
27. C2×#878C22semidirect
28. C2×C4×Dic6C22semidirect
36. #5222C22semidirect
42. #5220C22semidirect
44. #5212C22semidirect
46. C4×#217C22semidirect
47. #4986C22semidirect
48. #4985C22semidirect
49. #4984C22semidirect
50. #4975C22semidirect
51. Dic3×C4⋊C4C22semidirect
52. #4972C22semidirect
54. #4953C22semidirect
55. #4952C22semidirect
56. #4951C22semidirect
57. #4950C22semidirect
60. #4948C22semidirect
61. #4948C22semidirect
63. #4947C22semidirect
64. #4946C22semidirect
65. #4946C22semidirect
67. #4945C22semidirect
68. #4944C22semidirect
69. #4943C22semidirect
70. #4942C22semidirect
71. #4942C22semidirect
72. #4932C22semidirect
73. #4928C22semidirect
74. #4658C22semidirect
75. #4653C22semidirect
662. C3#45377semidirect

Maximal subgroups

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

subgroupindex
C3×#24082
D4×#2172
#826442
#826362
#825902
#823742
#823482
#770542
#767182
#766812
#763462
#762952
#739252
#738422
#738282
#453773