Finite Group Databasebeta

← #11232828

Subgroups — #11232828

Order 768.

1898 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2357C211yesmaximal, characteristic, complemented
2#82378C211yesmaximal, characteristic
3#82328C211yesmaximal, characteristic, complemented
4#76352C212yesmaximal, complemented
5#76318C212yesmaximal, complemented
6#75883C214yesmaximal, complemented
7#75733C214yesmaximal, complemented
8#74157C211yesmaximal, characteristic, complemented
9#74153C211yesmaximal, characteristic, complemented
10#74099C214yesmaximal, complemented
11#73712C211yesmaximal, characteristic, complemented
12#73693C212yesmaximal, complemented
13#73671C211yesmaximal, characteristic, complemented
14#73654C212yesmaximal, complemented
15#73603C214yesmaximal, complemented
16#45312—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2212yes—
19C2×#936C2211yescharacteristic
20C2×#930C2214yes—
21C2×#893C2212yes—
22C2×#887C2212yes—
23C2×#883C2214yes—
24C2×#876C2214yes—
25C2×#875C2212yes—
26C2×#874C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#390C2212yes—
29C3×#388C2214yes—
30C3×#382C2211yescharacteristic
31C3×#381C2212yes—
32C3×C4×C4⋊C4C2212yes—
33C3×C4×C4⋊C4C2211yescharacteristic
34C3×C4×C22⋊C4C2212yes—
35#5221C2211yescharacteristic
36#5210C2214yes—
37#4988C2212yes—
38#4988C2214yes—
39#4988C2212yes—
40#4986C2212yes—
41#4982C2212yes—
42#4975C2211yescharacteristic
43#4973C2211yescharacteristic
44#4972C2212yes—
45#4961C2214yes—
46#4960C2212yes—
47#4960C2212yes—
48#4959C2214yes—
49#4959C2214yes—
50#4959C2214yes—
51#4957C2214yes—
52#4957C2214yes—
53#4956C2214yes—
54#4944C2214yes—
55#4940C2214yes—
56#4940C2214yes—
57#4939C2212yes—
58C4×(C2×D3)⋊C4C2214yes—
59C4×C4⋊Dic3C2212yes—
60C4×C4⋊Dic3C2211yescharacteristic
61#4932C2211yescharacteristic
62C4×Dic3⋊C4C2212yes—
63#4676C2214yes—
64#4675C2214yes—
65#4672C2212yes—
66#4672C2212yes—
67#4670C2214yes—
68#4668C2214yes—
69#4665C2214yes—
70#4665C2214yes—
71#4661C2212yes—
72#4660C2212yes—
73#4650C2212yes—
74#4649C2212yes—
75#4647C2214yes—
76#2753—34no—
77#2701—34no—
78#2559—34no—
79#2554—34no—
80#2548—32no—
81#2548—31no—
82#2541—31no—
83#2512—32no—
84#2445—32no—
85#2402—31no—
86#2367—32no—
87#2357D311yescharacteristic, complemented
88#2351—31no—
89#2328—31no—
90#2326—31no—
91C22×C3×D4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C22×#67C2314yes—
94C22×#67C2314yes—
95C22×D12C2312yes—
96C22×D12C2312yes—
97C22×D12C2312yes—
98C22×C4×D3C2314yes—
99C22×Dic6C2311yescharacteristic
100C3×C4×D4—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C42C2312yes—
108C2×C3×C42C2314yes—
109C2×C3×C42C2311yescharacteristic
110C2×#217C2314yes—
111C2×#217C2314yes—
112#940—44no—
113#936—24no—
114#930—48no—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2312yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2312yes—
133C2×Dic3⋊C4C2314yes—
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2314yes—
139#893—44no—
140#887—44no—
141#883—48no—
142#876—48no—
143#875—28no—
144#874—24no—
145C4×Dic6—24no—
146C3×#95C2312yes—
147C3×#95C2312yes—
148C3×#95C2311yescharacteristic
149C3×#95C2311yescharacteristic
150C3×#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2311yescharacteristic
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C2×#127—32no—
162C2×#127—31no—
163C2×#126—34no—
164C2×#124—31no—
165C2×#124—32no—
166C2×#123—34no—
167C2×#121—34no—
168C2×C4×Q8—32no—
169C2×C4×Q8—31no—
170C2×C4×D4—32no—
171C2×C4×D4C2×D311yescharacteristic
172#400—32no—
173#400—32no—
174#398—32no—
175#398—32no—
176#397—34no—
177#396—34no—
178#394—34no—
179#394—34no—
180#394—34no—
181#394—34no—
182#394—32no—
183#394—32no—
184#392—32no—
185#392—32no—
186#391—31no—
187#390—31no—
188#390—32no—
189#390—32no—
190#390—34no—
191#390—34no—
192#390—34no—
193#390—32no—
194#390C2×D312yes—
195#389—32no—
196#389—34no—
197#388—34no—
198#388—34no—
199#388—34no—
200#388—34no—
201#388C2×D314yes—
202#386—34no—
203#386—34no—
204#386—34no—
205#385—34no—
206#385—34no—
207#384—31no—
208#382—32no—
209#382C2×D311yescharacteristic
210#381C2×D312yes—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4C2×D312yes—
216C4×C4⋊C4C2×D311yescharacteristic
217C4×C22⋊C4—34no—
218C4×C22⋊C4C2×D312yes—
219#376—31no—
220C24×C3C2412yes—
221C23×D3C2414yes—
222C2×C3×D4—22no—
223C2×C3×D4—22no—
224C2×C3×D4—22no—
225C22×C3×C4—24no—
226C22×C3×C4—44no—
227C22×C3×C4C2412yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2414yes—
230C22×C3×C4C2412yes—
231C22×C3×C4C2411yescharacteristic
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C2×#67—28no—
235C2×#67—44no—
236C2×#67—44no—
237C2×#67—44no—
238C2×#67—44no—
239C2×#67—44no—
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242C22×Dic3C2412yes—
243C22×Dic3C2412yes—
244C22×Dic3C2412yes—
245C22×Dic3C2412yes—
246C2×D12—24no—
247C2×D12—24no—
248C2×D12—24no—
249C2×D12—24no—
250C2×D12—24no—
251C2×D12—44no—
252C2×D12—44no—
253C2×C4×D3—44no—
254C2×C4×D3—44no—
255C2×C4×D3—44no—
256C2×Dic6—22no—
257C2×Dic6—22no—
258C2×Dic6#4114yes—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—42no—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—44no—
264C3×C22⋊C4—44no—
265C3×C42—24no—
266C3×C42—44no—
267C3×C42#4114yes—
268#217—28no—
269#217—44no—
270(C2×D3)⋊C4—28no—
271(C2×D3)⋊C4—28no—
272(C2×D3)⋊C4—28no—
273(C2×D3)⋊C4—44no—
274(C2×D3)⋊C4—44no—
275(C2×D3)⋊C4—44no—
276(C2×D3)⋊C4—44no—
277(C2×D3)⋊C4—44no—
278(C2×D3)⋊C4—44no—
279C4⋊Dic3—22no—
280C4⋊Dic3—24no—
281C4⋊Dic3—24no—
282C4⋊Dic3—42no—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—28no—
286Dic3⋊C4—44no—
287Dic3⋊C4—44no—
288Dic3⋊C4—42no—
289C4×Dic3—24no—
290C4×Dic3—22no—
291C4×Dic3—24no—
292C4×Dic3—22no—
293C4×Dic3—44no—
294C22×Q8—31no—
295C22×D4—34no—
296C22×D4—34no—
297C22×D4—32no—
298C22×D4—32no—
299C22×D4—32no—
300C22×D4C22×D311yescharacteristic
301C23×C4—34no—
302C23×C4C22×D312yes—
303#127—64no—
304#127—124no—
305#126—128no—
306#124—68no—
307#124—64no—
308#123—128no—
309#121—128no—
310C4×Q8—64no—
311C4×Q8—124no—
312C4×D4—24no—
313C4×D4—124no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D312yes—
327C2×C22⋊C4—34no—
328C2×C22⋊C4—34no—
329C2×C22⋊C4—34no—
330C2×C22⋊C4—34no—
331C2×C22⋊C4—34no—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4—34no—
336C2×C22⋊C4—34no—
337C2×C22⋊C4—34no—
338C2×C22⋊C4C22×D312yes—
339C2×C22⋊C4C22×D314yes—
340C2×C22⋊C4C22×D314yes—
341C2×C42—31no—
342C2×C42—32no—
343C2×C42—32no—
344C2×C42—31no—
345C2×C42—34no—
346C2×C42C22×D312yes—
347C2×C42C22×D314yes—
348C2×C42C22×D311yescharacteristic
349#95—32no—
350#95—32no—
351#95—32no—
352#95—34no—
353#95—31no—
354#95—31no—
355#95—32no—
356#95—32no—
357#95—32no—
358#95—34no—
359#95C22×D311yescharacteristic
360#95C22×D312yes—
361#95C22×D312yes—
362#95C22×D311yescharacteristic
363#95C22×D314yes—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—24no—
367C23×C3—22no—
368C23×C3—24no—
369C23×C3C2511yescharacteristic
370C22×D3—24no—
371C22×D3—24no—
372C22×D3—24no—
373C22×D3—24no—
374C22×D3—24no—
375C22×D3—24no—
376C22×D3—24no—
377C3×D4—44no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—22no—
391C2×C3×C4—84no—
392C2×C3×C4C2×#4112yes—
393C2×C3×C4C2×#4112yes—
394C2×C3×C4C2×#4112yes—
395C2×C3×C4C2×#4112yes—
396C2×C3×C4H(5,𝔽2)12yes—
397#67—88no—
398#67—88no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3C2×#4114yes—
415C2×Dic3H(5,𝔽2)14yes—
416D12—48no—
417D12—84no—
418D12—84no—
419C4×D3—88no—
420Dic6—28no—
421C24—34no—
422C24C23×D312yes—
423C2×Q8—34no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—68no—
430C2×D4—64no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—124no—
436C2×D4—124no—
437C2×D4—124no—
438C2×D4—124no—
439C2×D4—124no—
440C2×D4—124no—
441C2×D4—124no—
442C22×C4—24no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—44no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—124no—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D312yes—
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D311yescharacteristic
459C22×C4C23×D311yescharacteristic
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—64no—
464C4⋊C4—64no—
465C4⋊C4—68no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—62no—
469C4⋊C4—124no—
470C4⋊C4—124no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C22⋊C4—24no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—68no—
477C22⋊C4—68no—
478C22⋊C4—68no—
479C22⋊C4—68no—
480C22⋊C4—124no—
481C22⋊C4—124no—
482C22⋊C4—124no—
483C22⋊C4—124no—
484C22⋊C4—124no—
485C22⋊C4—124no—
486C22⋊C4—124no—
487C42—24no—
488C42—44no—
489C42—64no—
490C42—62no—
491C42—62no—
492C42—64no—
493C42—124no—
494C42#23514yes—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—44no—
500C22×C3C22×#4111yescharacteristic
501C22×C3C22×#4111yescharacteristic
502C22×C3C2×H(5,𝔽2)11yescharacteristic
503C22×C3C2×H(5,𝔽2)12yes—
504C22×C3#58512yes—
505C2×D3—44no—
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—44no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—44no—
518C3×C4#55314yes—
519Dic3—24no—
520Dic3—24no—
521Dic3—42no—
522Dic3—42no—
523Dic3—42no—
524Dic3—42no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—64no—
531C23—64no—
532C23—64no—
533C23—64no—
534C23—64no—
535C23—64no—
536C23—64no—
537C23C24×D311yescharacteristic
538Q8—68no—
539D4—44no—
540D4—128no—
541D4—244no—
542D4—244no—
543D4—248no—
544D4—248no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—34no—
559C2×C4—34no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—84no—
576C2×C4—248no—
577C2×C4C2×#23512yes—
578C2×C4C2×#23512yes—
579C2×C4C2×#23512yes—
580C2×C4D3×#4112yes—
581C2×C4#100912yes—
582C2×C3—82no—
583C2×C3C2×#55311yescharacteristic
584C2×C3#350811yescharacteristic
585C2×C3#351011yescharacteristic
586C2×C3#351712yes—
587C2×C3#355812yes—
588D3—84no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—42no—
594C22—124no—
595C22—124no—
596C22—124no—
597C22—124no—
598C22—124no—
599C22—124no—
600C22—124no—
601C22C22×#23511yescharacteristic
602C22C2×D3×#4111yescharacteristic
603C22C2×#100911yescharacteristic
604C22#596412yes—
605C22D3×H(5,𝔽2)12yes—
606C4—22no—
607C4—22no—
608C4—44no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—64no—
613C4—64no—
614C4—122no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4#553014yes—
619C3#4531211yescharacteristic, complemented, Sylow
620C2—82no—
621C2—244no—
622C2C2×#553011yescharacteristic
623C2#8584911yescharacteristic
624C2#8586211yescharacteristic
625C2#8688612yes—
626C2#8706912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2357C2
prime radical#82378C2
hypercenter#2357D3
hypocenterC3#45312
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. C3×#2357C2semidirect
3. #82328C2semidirect
4. #76352C2semidirect
5. #76318C2semidirect
6. #75883C2semidirect
7. #75733C2semidirect
8. #74157C2semidirect
9. #74153C2semidirect
10. #74099C2semidirect
11. #73712C2semidirect
12. #73693C2semidirect
13. #73671C2semidirect
14. #73654C2semidirect
15. #73603C2semidirect
87. #2357D3semidirect
619. C3#45312semidirect

Maximal subgroups

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

subgroupindex
C3×#23572
#823782
#823282
#763522
#763182
#758832
#757332
#741572
#741532
#740992
#737122
#736932
#736712
#736542
#736032
#453123