Finite Group Databasebeta

← #11235195

Subgroups — #11235195

Order 768.

2170 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2541C211yesmaximal, characteristic, complemented
2#83188C211yesmaximal, characteristic, complemented
3#83179C212yesmaximal, complemented
4#83177C212yesmaximal, complemented
5#83172C211yesmaximal, characteristic, complemented
6#82691C214yesmaximal, complemented
7#82672C214yesmaximal, complemented
8#82338C211yesmaximal, characteristic, complemented
9#82329C211yesmaximal, characteristic, complemented
10#76720C214yesmaximal, complemented
11#76690C214yesmaximal, complemented
12#76353C212yesmaximal, complemented
13#76303C212yesmaximal, complemented
14#73669C211yesmaximal, characteristic, complemented
15#73645C211yesmaximal, characteristic, complemented
16#45437—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#127C2212yes—
19C2×C3×#121C2214yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#940C2211yescharacteristic
22C2×#940C2212yes—
23C2×#939C2214yes—
24C2×#939C2214yes—
25C2×#938C2214yes—
26C2×#936C2214yes—
27C2×D4×Dic3C2212yes—
28C2×#885C2214yes—
29C2×C4×D12C2211yescharacteristic
30C2×C4×Dic6C2212yes—
31C3×#392C2212yes—
32C3×#392C2212yes—
33C3×#390C2212yes—
34C3×C4×C4⋊C4C2211yescharacteristic
35#5223C2212yes—
36#5223C2214yes—
37#5223C2214yes—
38#5223C2212yes—
39#5222C2214yes—
40#5222C2214yes—
41#5221C2214yes—
42#5221C2212yes—
43#5221C2212yes—
44#5221C2211yescharacteristic
45#5221C2212yes—
46#5220C2214yes—
47#5220C2214yes—
48#5213C2214yes—
49#5211C2212yes—
50C4×#217C2212yes—
51#4990C2214yes—
52#4988C2211yescharacteristic
53#4982C2212yes—
54#4979C2212yes—
55#4977C2211yescharacteristic
56Dic3×C4⋊C4C2212yes—
57#4961C2214yes—
58#4959C2214yes—
59#4957C2214yes—
60#4956C2214yes—
61#4951C2214yes—
62#4945C2214yes—
63#4944C2214yes—
64#4943C2214yes—
65Dic3×C22⋊C4C2214yes—
66#4939C2212yes—
67#4936C2211yescharacteristic
68C4×C4⋊Dic3C2212yes—
69C4×C4⋊Dic3C2211yescharacteristic
70#4932C2211yescharacteristic
71C4×Dic3⋊C4C2212yes—
72#4666C2212yes—
73#4664C2212yes—
74#4657C2212yes—
75#4656C2212yes—
76#2699—34no—
77#2699—34no—
78#2555—34no—
79#2555—34no—
80#2548—32no—
81#2548—32no—
82#2541—32no—
83#2541—31no—
84#2541D311yescharacteristic, complemented
85#2399—31no—
86#2357—32no—
87#2357—31no—
88#2357—31no—
89#2328—31no—
90#2328—31no—
91C22×C3×D4C2311yescharacteristic
92C22×C3×D4C2312yes—
93C22×C3×D4C2312yes—
94C22×C3×D4C2312yes—
95C22×C3×D4C2314yes—
96C22×C3×D4C2314yes—
97C23×C3×C4C2312yes—
98C22×#67C2314yes—
99C22×#67C2314yes—
100C22×#67C2314yes—
101C23×Dic3C2314yes—
102C22×D12C2311yescharacteristic
103C22×C4×D3C2312yes—
104C22×Dic6C2312yes—
105C3×#127—24no—
106C3×#127—44no—
107C3×#121—48no—
108C3×C4×D4—24no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2314yes—
114C2×C3×C42C2312yes—
115C2×C3×C42C2311yescharacteristic
116C2×#217C2314yes—
117C2×#217C2314yes—
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×#217C2314yes—
121C2×#217C2314yes—
122C2×#217C2314yes—
123C2×#217C2314yes—
124#940—24no—
125#940—44no—
126#939—48no—
127#939—48no—
128#938—48no—
129#936—48no—
130D4×Dic3—28no—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2311yescharacteristic
148#885—48no—
149C4×D12—24no—
150C4×Dic6—44no—
151C3×#95C2311yescharacteristic
152C3×#95C2311yescharacteristic
153C3⋊#95C2312yes—
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2311yescharacteristic
160C3⋊#95C2311yescharacteristic
161C3⋊#95C2312yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C2×#127—31no—
165C2×#127—32no—
166C2×#127C2×D311yescharacteristic
167C2×#127C2×D312yes—
168C2×#124—34no—
169C2×#124—34no—
170C2×#121—34no—
171C2×#121—34no—
172C2×#121—34no—
173C2×#121C2×D314yes—
174C2×C4×Q8—32no—
175C2×C4×D4—32no—
176C2×C4×D4—31no—
177C2×C4×D4C2×D311yescharacteristic
178#398—32no—
179#398—32no—
180#395—32no—
181#395—32no—
182#394—34no—
183#394—34no—
184#394—34no—
185#394—34no—
186#392—34no—
187#392—32no—
188#392—34no—
189#392—32no—
190#392C2×D312yes—
191#392C2×D312yes—
192#390—32no—
193#390—31no—
194#390—32no—
195#390—32no—
196#390—34no—
197#390—31no—
198#390—34no—
199#390—32no—
200#390C2×D312yes—
201#388—34no—
202#388—34no—
203#388—34no—
204#388—34no—
205#386—34no—
206#386—34no—
207#386—34no—
208#386—34no—
209#382—31no—
210#382—31no—
211#381—32no—
212#381—34no—
213#381—32no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4—32no—
216C4×C4⋊C4—32no—
217C4×C4⋊C4—31no—
218C4×C4⋊C4C2×D311yescharacteristic
219C4×C22⋊C4—32no—
220C4×C22⋊C4—34no—
221C4×C22⋊C4—32no—
222#376—31no—
223C24×C3C2412yes—
224C24×C3C2414yes—
225C23×D3C2412yes—
226C2×C3×D4—24no—
227C2×C3×D4—22no—
228C2×C3×D4—22no—
229C2×C3×D4—24no—
230C2×C3×D4—24no—
231C2×C3×D4—24no—
232C2×C3×D4—24no—
233C2×C3×D4—22no—
234C2×C3×D4—28no—
235C2×C3×D4—44no—
236C2×C3×D4—44no—
237C2×C3×D4—44no—
238C2×C3×D4—44no—
239C2×C3×D4—44no—
240C2×C3×D4—44no—
241C2×C3×D4—44no—
242C22×C3×C4—24no—
243C22×C3×C4—44no—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2412yes—
249C22×C3×C4C2411yescharacteristic
250C2×#67—28no—
251C2×#67—28no—
252C2×#67—44no—
253C2×#67—44no—
254C2×#67—44no—
255C2×#67—44no—
256C2×#67—44no—
257C2×#67—44no—
258C2×#67—44no—
259C22×Dic3—28no—
260C22×Dic3—44no—
261C22×Dic3—44no—
262C22×Dic3C2414yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C22×Dic3C2412yes—
267C22×Dic3C2412yes—
268C2×D12—22no—
269C2×D12—22no—
270C2×D12—22no—
271C2×C4×D3—24no—
272C2×C4×D3—44no—
273C2×Dic6—24no—
274C2×Dic6—24no—
275C2×Dic6—42no—
276C3×C4⋊C4—22no—
277C3×C4⋊C4—24no—
278C3×C22⋊C4—24no—
279C3×C22⋊C4—28no—
280C3×C22⋊C4—44no—
281C3×C42—24no—
282C3×C42#4114yes—
283#217—28no—
284#217—28no—
285#217—28no—
286#217—44no—
287#217—44no—
288#217—44no—
289#217—44no—
290#217—44no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—28no—
293(C2×D3)⋊C4—44no—
294(C2×D3)⋊C4—44no—
295C4⋊Dic3—22no—
296C4⋊Dic3—24no—
297C4⋊Dic3—24no—
298C4⋊Dic3—42no—
299Dic3⋊C4—24no—
300Dic3⋊C4—24no—
301Dic3⋊C4—42no—
302Dic3⋊C4—42no—
303C4×Dic3—22no—
304C4×Dic3—24no—
305C4×Dic3—44no—
306C4×Dic3—44no—
307C4×Dic3#4114yes—
308C22×Q8—32no—
309C22×D4—34no—
310C22×D4—34no—
311C22×D4—34no—
312C22×D4—31no—
313C22×D4C22×D312yes—
314C22×D4C22×D311yescharacteristic
315C22×D4C22×D312yes—
316C22×D4C22×D314yes—
317C22×D4C22×D312yes—
318C22×D4C22×D314yes—
319C23×C4—34no—
320C23×C4—32no—
321C23×C4C22×D312yes—
322#127—24no—
323#127—44no—
324#127—64no—
325#127—124no—
326#124—128no—
327#124—128no—
328#121—48no—
329#121—128no—
330#121—128no—
331#121—128no—
332C4×Q8—124no—
333C4×D4—24no—
334C4×D4—68no—
335C4×D4—64no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D312yes—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—34no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4C22×D312yes—
359C2×C22⋊C4C22×D314yes—
360C2×C22⋊C4C22×D314yes—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—32no—
364C2×C42—34no—
365C2×C42—34no—
366C2×C42C22×D312yes—
367C2×C42C22×D311yescharacteristic
368#95—32no—
369#95—32no—
370#95—32no—
371#95—31no—
372#95—34no—
373#95—32no—
374#95—32no—
375#95—34no—
376#95—34no—
377#95—32no—
378#95—31no—
379#95C22×D311yescharacteristic
380#95C22×D311yescharacteristic
381C23×C3—24no—
382C23×C3—22no—
383C23×C3—22no—
384C23×C3—24no—
385C23×C3—24no—
386C23×C3—24no—
387C23×C3—24no—
388C23×C3—24no—
389C23×C3—24no—
390C23×C3—24no—
391C23×C3—24no—
392C23×C3—22no—
393C23×C3C2511yescharacteristic
394C22×D3—22no—
395C22×D3—22no—
396C22×D3—24no—
397C22×D3—24no—
398C22×D3—22no—
399C3×D4—48no—
400C3×D4—44no—
401C3×D4—84no—
402C3×D4—84no—
403C3×D4—88no—
404C3×D4—88no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—84no—
415C2×C3×C4C2×#4112yes—
416C2×C3×C4C2×#4112yes—
417C2×C3×C4C2×#4112yes—
418C2×C3×C4C2×#4112yes—
419C2×C3×C4H(5,𝔽2)14yes—
420C2×C3×C4H(5,𝔽2)12yes—
421#67—88no—
422#67—88no—
423#67—88no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—24no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—24no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—88no—
440C2×Dic3C2×#4114yes—
441C2×Dic3H(5,𝔽2)14yes—
442D12—44no—
443C4×D3—84no—
444Dic6—48no—
445C24—32no—
446C24C23×D312yes—
447C24C23×D314yes—
448C2×Q8—64no—
449C2×Q8—64no—
450C2×Q8—122no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—28no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—68no—
468C2×D4—68no—
469C2×D4—62no—
470C2×D4—62no—
471C2×D4—62no—
472C2×D4—124no—
473C2×D4—124no—
474C2×D4—124no—
475C2×D4—124no—
476C2×D4—124no—
477C2×D4—124no—
478C2×D4—124no—
479C22×C4—24no—
480C22×C4—32no—
481C22×C4—32no—
482C22×C4—32no—
483C22×C4—34no—
484C22×C4—32no—
485C22×C4—32no—
486C22×C4—44no—
487C22×C4—68no—
488C22×C4—64no—
489C22×C4—124no—
490C22×C4—124no—
491C22×C4—124no—
492C22×C4C23×D312yes—
493C22×C4C23×D312yes—
494C22×C4C23×D311yescharacteristic
495C22×C4C23×D311yescharacteristic
496C22×C4C23×D312yes—
497C22×C4C23×D311yescharacteristic
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—62no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C22⋊C4—24no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—68no—
514C22⋊C4—64no—
515C22⋊C4—68no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C22⋊C4—124no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
523C42—24no—
524C42—34no—
525C42—62no—
526C42—64no—
527C42—124no—
528C42—124no—
529C42#23714yes—
530C22×C3—44no—
531C22×C3—42no—
532C22×C3—44no—
533C22×C3—44no—
534C22×C3—42no—
535C22×C3—44no—
536C22×C3—44no—
537C22×C3—42no—
538C22×C3—44no—
539C22×C3—44no—
540C22×C3—44no—
541C22×C3—44no—
542C22×C3C22×#4111yescharacteristic
543C22×C3C22×#4111yescharacteristic
544C22×C3C2×H(5,𝔽2)11yescharacteristic
545C22×C3C2×H(5,𝔽2)12yes—
546C22×C3C2×H(5,𝔽2)12yes—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—44no—
550C2×D3—42no—
551C2×D3—44no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—24no—
555C3×C4—42no—
556C3×C4—42no—
557C3×C4#55514yes—
558Dic3—24no—
559Dic3—24no—
560Dic3—42no—
561Dic3—44no—
562Dic3—42no—
563Dic3—42no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—62no—
577C23—64no—
578C23—62no—
579C23—62no—
580C23—64no—
581C23C24×D311yescharacteristic
582Q8—128no—
583D4—44no—
584D4—48no—
585D4—84no—
586D4—88no—
587D4—84no—
588D4—88no—
589D4—124no—
590D4—248no—
591D4—248no—
592D4—248no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—34no—
603C2×C4—34no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—64no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—84no—
620C2×C4—244no—
621C2×C4—248no—
622C2×C4C2×#23712yes—
623C2×C4C2×#23712yes—
624C2×C4C2×#23712yes—
625C2×C4#100414yes—
626C2×C4#100412yes—
627C2×C4D3×#4112yes—
628C2×C3—84no—
629C2×C3—82no—
630C2×C3C2×#55511yescharacteristic
631C2×C3#350811yescharacteristic
632C2×C3#350811yescharacteristic
633C2×C3#350812yes—
634C2×C3#353712yes—
635D3—82no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—122no—
649C22—124no—
650C22—122no—
651C22—122no—
652C22—124no—
653C22C22×#23711yescharacteristic
654C22C2×#100411yescharacteristic
655C22C2×#100412yes—
656C22C2×D3×#4111yescharacteristic
657C22D3×H(5,𝔽2)12yes—
658C4—22no—
659C4—24no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—64no—
664C4—64no—
665C4—122no—
666C4—122no—
667C4—122no—
668C4—124no—
669C4#561614yes—
670C3#4543711yescharacteristic, complemented, Sylow
671C2—84no—
672C2—82no—
673C2—242no—
674C2C2×#561611yescharacteristic
675C2#8619011yescharacteristic
676C2#8631911yescharacteristic
677C2#8662312yes—
678C2#8677112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2541C2
hypercenter#2541D3
hypocenterC3#45437
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C3×#2541C2semidirect
2. #83188C2semidirect
3. #83179C2semidirect
4. #83177C2semidirect
5. #83172C2semidirect
6. #82691C2semidirect
7. #82672C2semidirect
8. #82338C2semidirect
9. #82329C2semidirect
10. #76720C2semidirect
11. #76690C2semidirect
12. #76353C2semidirect
13. #76303C2semidirect
14. #73669C2semidirect
15. #73645C2semidirect
84. #2541D3semidirect
670. C3#45437semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#831882
#831792
#831772
#831722
#826912
#826722
#823382
#823292
#767202
#766902
#763532
#763032
#736692
#736452
#454373