Finite Group Databasebeta

← #11235196

Subgroups — #11235196

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×#2357C211yesmaximal, characteristic, complemented
2#82378C211yesmaximal, characteristic, complemented
3#82329C211yesmaximal, characteristic, complemented
4#76353C212yesmaximal, complemented
5#76348C212yesmaximal, complemented
6#75924C214yesmaximal, complemented
7#75597C214yesmaximal, complemented
8#74160C211yesmaximal, characteristic, complemented
9#74153C211yesmaximal, characteristic, complemented
10#74097C214yesmaximal, complemented
11#73799C211yesmaximal, characteristic, complemented
12#73712C212yesmaximal, complemented
13#73669C211yesmaximal, characteristic, complemented
14#73646C212yesmaximal, complemented
15#73600C214yesmaximal, complemented
16#45437—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#940C2212yes—
20C2×#930C2214yes—
21C2×#895C2214yes—
22C2×#893C2212yes—
23C2×#885C2214yes—
24C2×#884C2214yes—
25C2×#884C2214yes—
26C2×#875C2214yes—
27C2×#874C2211yescharacteristic
28C2×#874C2212yes—
29C2×C4×D12C2211yescharacteristic
30C2×C4×Dic6C2212yes—
31C3×#390C2212yes—
32C3×#388C2214yes—
33C3×#382C2211yescharacteristic
34C3×#381C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C4⋊C4C2212yes—
37C3×C4×C22⋊C4C2212yes—
38#5221C2211yescharacteristic
39#5213C2214yes—
40#4988C2212yes—
41#4988C2212yes—
42#4988C2211yescharacteristic
43#4988C2212yes—
44#4982C2212yes—
45#4979C2212yes—
46#4977C2211yescharacteristic
47Dic3×C4⋊C4C2212yes—
48#4961C2214yes—
49#4960C2214yes—
50#4960C2214yes—
51#4960C2212yes—
52#4960C2212yes—
53#4959C2214yes—
54#4957C2214yes—
55#4956C2214yes—
56#4940C2214yes—
57#4939C2212yes—
58#4939C2214yes—
59C4×(C2×D3)⋊C4C2214yes—
60C4×C4⋊Dic3C2211yescharacteristic
61#4932C2211yescharacteristic
62C4×Dic3⋊C4C2212yes—
63#4672C2214yes—
64#4672C2214yes—
65#4671C2212yes—
66#4671C2212yes—
67#4670C2214yes—
68#4669C2214yes—
69#4666C2212yes—
70#4665C2214yes—
71#4665C2214yes—
72#4664C2214yes—
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#2541—31no—
85#2399—31no—
86#2357—31no—
87#2357—32no—
88#2357D311yescharacteristic, complemented
89#2328—31no—
90#2328—31no—
91C22×C3×D4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C22×#67C2314yes—
94C22×#67C2314yes—
95C22×#67C2314yes—
96C22×D12C2312yes—
97C22×D12C2311yescharacteristic
98C22×D12C2312yes—
99C22×D12C2312yes—
100C22×D12C2314yes—
101C22×D12C2314yes—
102C22×C4×D3C2314yes—
103C22×C4×D3C2312yes—
104C22×Dic6C2312yes—
105C3×C4×D4—24no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C42C2312yes—
113C2×C3×C42C2314yes—
114C2×C3×C42C2311yescharacteristic
115C2×#217C2314yes—
116#940—24no—
117#940—44no—
118#930—48no—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×C4⋊Dic3C2312yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2312yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2312yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2314yes—
140C2×C4×Dic3C2311yescharacteristic
141#895—48no—
142#893—28no—
143#885—48no—
144#884—48no—
145#884—48no—
146#875—48no—
147#874—24no—
148#874—44no—
149C4×D12—24no—
150C4×Dic6—44no—
151C3×#95C2311yescharacteristic
152C3×#95C2311yescharacteristic
153C3×#95C2312yes—
154C3×#95C2314yes—
155C3×#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2314yes—
160C3⋊#95C2311yescharacteristic
161C3⋊#95C2314yes—
162C3⋊#95C2312yes—
163C3⋊#95C2311yescharacteristic
164C2×#127—31no—
165C2×#127—32no—
166C2×#127—31no—
167C2×#127—32no—
168C2×#124—34no—
169C2×#124—34no—
170C2×#121—34no—
171C2×#121—34no—
172C2×#121—34no—
173C2×#121—34no—
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#392—32no—
191#392—32no—
192#390—31no—
193#390—32no—
194#390—31no—
195#390—32no—
196#390—32no—
197#390—34no—
198#390—32no—
199#390—34no—
200#390C2×D312yes—
201#388—34no—
202#388—34no—
203#388—34no—
204#388C2×D314yes—
205#386—34no—
206#386—34no—
207#386—34no—
208#386—34no—
209#382—31no—
210#382C2×D311yescharacteristic
211#381—34no—
212#381—32no—
213#381C2×D312yes—
214C4×C4⋊C4—32no—
215C4×C4⋊C4—32no—
216C4×C4⋊C4—31no—
217C4×C4⋊C4C2×D312yes—
218C4×C4⋊C4C2×D311yescharacteristic
219C4×C22⋊C4—34no—
220C4×C22⋊C4—32no—
221C4×C22⋊C4C2×D312yes—
222#376—31no—
223C24×C3C2412yes—
224C23×D3C2414yes—
225C23×D3C2412yes—
226C2×C3×D4—22no—
227C2×C3×D4—22no—
228C2×C3×D4—22no—
229C22×C3×C4—24no—
230C22×C3×C4—44no—
231C22×C3×C4C2412yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2412yes—
237C22×C3×C4C2411yescharacteristic
238C2×#67—28no—
239C2×#67—28no—
240C2×#67—44no—
241C2×#67—44no—
242C2×#67—44no—
243C2×#67—44no—
244C2×#67—44no—
245C2×#67—44no—
246C2×#67—44no—
247C22×Dic3C2412yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C22×Dic3C2412yes—
252C2×D12—22no—
253C2×D12—24no—
254C2×D12—24no—
255C2×D12—22no—
256C2×D12—22no—
257C2×D12—24no—
258C2×D12—24no—
259C2×D12—24no—
260C2×D12—28no—
261C2×D12—44no—
262C2×D12—44no—
263C2×D12—44no—
264C2×D12—44no—
265C2×D12—44no—
266C2×D12—44no—
267C2×D12—44no—
268C2×C4×D3—24no—
269C2×C4×D3—28no—
270C2×C4×D3—44no—
271C2×C4×D3—44no—
272C2×C4×D3—44no—
273C2×Dic6—24no—
274C2×Dic6—24no—
275C2×Dic6—42no—
276C3×C4⋊C4—24no—
277C3×C4⋊C4—22no—
278C3×C4⋊C4—24no—
279C3×C22⋊C4—24no—
280C3×C22⋊C4—28no—
281C3×C22⋊C4—44no—
282C3×C42—24no—
283C3×C42—44no—
284C3×C42#4114yes—
285#217—44no—
286(C2×D3)⋊C4—24no—
287(C2×D3)⋊C4—28no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—28no—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—44no—
292(C2×D3)⋊C4—44no—
293(C2×D3)⋊C4—44no—
294(C2×D3)⋊C4—44no—
295(C2×D3)⋊C4—44no—
296(C2×D3)⋊C4—44no—
297C4⋊Dic3—22no—
298C4⋊Dic3—24no—
299C4⋊Dic3—42no—
300Dic3⋊C4—24no—
301Dic3⋊C4—24no—
302Dic3⋊C4—42no—
303Dic3⋊C4—42no—
304C4×Dic3—22no—
305C4×Dic3—24no—
306C4×Dic3—44no—
307C4×Dic3#4114yes—
308C22×Q8—32no—
309C22×D4—34no—
310C22×D4—34no—
311C22×D4—34no—
312C22×D4—31no—
313C22×D4—32no—
314C22×D4—34no—
315C22×D4—32no—
316C22×D4—32no—
317C22×D4—34no—
318C22×D4C22×D311yescharacteristic
319C23×C4—34no—
320C23×C4—32no—
321C23×C4C22×D312yes—
322#127—64no—
323#127—64no—
324#127—124no—
325#127—124no—
326#124—128no—
327#124—128no—
328#121—128no—
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⋊C4C22×D312yes—
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D312yes—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
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×C42C22×D312yes—
366C2×C42C22×D314yes—
367C2×C42C22×D311yescharacteristic
368#95—32no—
369#95—32no—
370#95—32no—
371#95—31no—
372#95—34no—
373#95—34no—
374#95—32no—
375#95—31no—
376#95C22×D312yes—
377#95C22×D311yescharacteristic
378#95C22×D312yes—
379#95C22×D314yes—
380#95C22×D311yescharacteristic
381C23×C3—22no—
382C23×C3—22no—
383C23×C3—24no—
384C23×C3—24no—
385C23×C3—22no—
386C23×C3C2511yescharacteristic
387C22×D3—22no—
388C22×D3—24no—
389C22×D3—24no—
390C22×D3—22no—
391C22×D3—24no—
392C22×D3—24no—
393C22×D3—24no—
394C22×D3—24no—
395C22×D3—24no—
396C22×D3—24no—
397C22×D3—22no—
398C22×D3—24no—
399C3×D4—44no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—24no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—84no—
413C2×C3×C4C2×#4112yes—
414C2×C3×C4C2×#4112yes—
415C2×C3×C4C2×#4112yes—
416C2×C3×C4C2×#4112yes—
417C2×C3×C4H(5,𝔽2)14yes—
418C2×C3×C4H(5,𝔽2)12yes—
419#67—88no—
420#67—88no—
421#67—88no—
422C2×Dic3—22no—
423C2×Dic3—22no—
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—22no—
433C2×Dic3—22no—
434C2×Dic3C2×#4114yes—
435C2×Dic3H(5,𝔽2)14yes—
436D12—44no—
437D12—48no—
438D12—84no—
439D12—84no—
440D12—88no—
441D12—88no—
442C4×D3—88no—
443C4×D3—84no—
444Dic6—48no—
445C24—32no—
446C24—34no—
447C24C23×D312yes—
448C2×Q8—64no—
449C2×Q8—64no—
450C2×Q8—122no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—68no—
455C2×D4—68no—
456C2×D4—62no—
457C2×D4—64no—
458C2×D4—64no—
459C2×D4—64no—
460C2×D4—64no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—64no—
464C2×D4—68no—
465C2×D4—124no—
466C2×D4—124no—
467C2×D4—124no—
468C2×D4—124no—
469C2×D4—124no—
470C2×D4—124no—
471C2×D4—124no—
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—32no—
484C22×C4—32no—
485C22×C4—44no—
486C22×C4—68no—
487C22×C4—64no—
488C22×C4—124no—
489C22×C4—124no—
490C22×C4—124no—
491C22×C4C23×D312yes—
492C22×C4C23×D314yes—
493C22×C4C23×D312yes—
494C22×C4C23×D311yescharacteristic
495C22×C4C23×D311yescharacteristic
496C22×C4C23×D312yes—
497C22×C4C23×D311yescharacteristic
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—62no—
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—44no—
526C42—62no—
527C42—64no—
528C42—124no—
529C42#23514yes—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—44no—
535C22×C3C22×#4111yescharacteristic
536C22×C3C22×#4111yescharacteristic
537C22×C3C2×H(5,𝔽2)11yescharacteristic
538C22×C3C2×H(5,𝔽2)12yes—
539C22×C3C2×H(5,𝔽2)12yes—
540C2×D3—44no—
541C2×D3—42no—
542C2×D3—44no—
543C2×D3—42no—
544C2×D3—44no—
545C2×D3—44no—
546C2×D3—44no—
547C2×D3—44no—
548C2×D3—42no—
549C2×D3—44no—
550C2×D3—44no—
551C2×D3—44no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—24no—
555C3×C4—44no—
556C3×C4—42no—
557C3×C4—42no—
558C3×C4#55514yes—
559Dic3—24no—
560Dic3—24no—
561Dic3—42no—
562Dic3—42no—
563Dic3—42no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—22no—
568C23—24no—
569C23—62no—
570C23—64no—
571C23—64no—
572C23—62no—
573C23—64no—
574C23—64no—
575C23—64no—
576C23—62no—
577C23—64no—
578C23—64no—
579C23—64no—
580C23—64no—
581C23C24×D311yescharacteristic
582Q8—128no—
583D4—44no—
584D4—124no—
585D4—128no—
586D4—244no—
587D4—244no—
588D4—248no—
589D4—248no—
590D4—248no—
591D4—248no—
592D4—248no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—34no—
606C2×C4—34no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—84no—
620C2×C4—244no—
621C2×C4—248no—
622C2×C4C2×#23512yes—
623C2×C4C2×#23512yes—
624C2×C4C2×#23512yes—
625C2×C4D3×#4112yes—
626C2×C4#100914yes—
627C2×C4#100912yes—
628C2×C3—82no—
629C2×C3C2×#55511yescharacteristic
630C2×C3#350812yes—
631C2×C3#350811yescharacteristic
632C2×C3#350811yescharacteristic
633C2×C3#353712yes—
634D3—84no—
635D3—82no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—124no—
642C22—124no—
643C22—124no—
644C22—122no—
645C22—124no—
646C22—124no—
647C22—122no—
648C22—124no—
649C22—124no—
650C22—124no—
651C22—122no—
652C22—124no—
653C22C22×#23511yescharacteristic
654C22C2×D3×#4111yescharacteristic
655C22C2×#100912yes—
656C22C2×#100911yescharacteristic
657C22D3×H(5,𝔽2)12yes—
658C4—22no—
659C4—22no—
660C4—24no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—64no—
665C4—64no—
666C4—122no—
667C4—122no—
668C4—122no—
669C4#553214yes—
670C3#4543711yescharacteristic, complemented, Sylow
671C2—82no—
672C2—244no—
673C2—242no—
674C2C2×#553211yescharacteristic
675C2#8584811yescharacteristic
676C2#8586211yescharacteristic
677C2#8691612yes—
678C2#8705112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2357C2
hypercenter#2357D3
hypocenterC3#45437
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. C3×#2357C2semidirect
2. #82378C2semidirect
3. #82329C2semidirect
4. #76353C2semidirect
5. #76348C2semidirect
6. #75924C2semidirect
7. #75597C2semidirect
8. #74160C2semidirect
9. #74153C2semidirect
10. #74097C2semidirect
11. #73799C2semidirect
12. #73712C2semidirect
13. #73669C2semidirect
14. #73646C2semidirect
15. #73600C2semidirect
88. #2357D3semidirect
670. C3#45437semidirect

Maximal subgroups

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

subgroupindex
C3×#23572
#823782
#823292
#763532
#763482
#759242
#755972
#741602
#741532
#740972
#737992
#737122
#736692
#736462
#736002
#454373