Finite Group Databasebeta

← #11232824

Subgroups — #11232824

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×#2548C211yesmaximal, characteristic, complemented
2#83188C211yesmaximal, characteristic
3#83172C211yesmaximal, characteristic, complemented
4#82974C212yesmaximal, complemented
5#82955C212yesmaximal, complemented
6#82460C211yesmaximal, characteristic, complemented
7#82455C211yesmaximal, characteristic, complemented
8#77148C214yesmaximal, complemented
9#76953C214yesmaximal, complemented
10#76409C214yesmaximal, complemented
11#76354C214yesmaximal, complemented
12#76338C212yesmaximal, complemented
13#76317C212yesmaximal, complemented
14#73671C211yesmaximal, characteristic, complemented
15#73645C211yesmaximal, characteristic, complemented
16#45312—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×Q8×Dic3C2212yes—
22C2×#940C2212yes—
23C2×#938C2214yes—
24C2×D4×Dic3C2212yes—
25C2×#886C2214yes—
26C2×#879C2214yes—
27C2×C4×D12C2211yescharacteristic
28C3×#394C2212yes—
29C3×#394C2212yes—
30C3×#390C2212yes—
31C3×#390C2214yes—
32C3×C4×C4⋊C4C2211yescharacteristic
33#5229C2211yescharacteristic
34#5223C2212yes—
35#5223C2212yes—
36#5221C2212yes—
37#5221C2212yes—
38#5220C2214yes—
39#5220C2214yes—
40#4990C2214yes—
41#4989C2214yes—
42#4988C2211yescharacteristic
43#4986C2212yes—
44#4984C2212yes—
45#4984C2214yes—
46#4982C2212yes—
47#4975C2211yescharacteristic
48#4961C2214yes—
49#4961C2214yes—
50#4961C2214yes—
51#4961C2214yes—
52#4958C2214yes—
53#4957C2214yes—
54#4957C2214yes—
55#4952C2214yes—
56#4951C2214yes—
57#4951C2214yes—
58#4950C2214yes—
59#4948C2214yes—
60#4944C2214yes—
61Dic3×C22⋊C4C2214yes—
62#4940C2214yes—
63#4939C2212yes—
64#4936C2212yes—
65#4936C2211yescharacteristic
66C4×C4⋊Dic3C2212yes—
67C4×C4⋊Dic3C2212yes—
68C4×C4⋊Dic3C2212yes—
69C4×C4⋊Dic3C2211yescharacteristic
70#4932C2211yescharacteristic
71C4×Dic3⋊C4C2212yes—
72#4666C2212yes—
73#4664C2212yes—
74#4661C2212yes—
75#4660C2212yes—
76#2753—34no—
77#2701—34no—
78#2559—34no—
79#2554—34no—
80#2548—32no—
81#2548D311yescharacteristic, complemented
82#2541—31no—
83#2512—32no—
84#2445—32no—
85#2402—31no—
86#2367—32no—
87#2357—31no—
88#2351—31no—
89#2328—31no—
90#2326—31no—
91C22×C3×Q8C2311yescharacteristic
92C22×C3×D4C2312yes—
93C22×C3×D4C2312yes—
94C22×C3×D4C2312yes—
95C22×#67C2314yes—
96C22×#67C2314yes—
97C23×Dic3C2314yes—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C3×#127—24no—
101C3×#124—28no—
102C3×C4×Q8—24no—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C42C2312yes—
111C2×C3×C42C2312yes—
112C2×C3×C42C2311yescharacteristic
113C2×#217C2314yes—
114C2×#217C2314yes—
115C2×#217C2314yes—
116C2×#217C2314yes—
117C2×#217C2314yes—
118#947—24no—
119Q8×Dic3—44no—
120#940—44no—
121#938—48no—
122D4×Dic3—44no—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2314yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2314yes—
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143#886—48no—
144#879—48no—
145C4×D12—24no—
146C3×#95C2311yescharacteristic
147C3×#95C2311yescharacteristic
148C3⋊#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2312yes—
151C3⋊#95C2312yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2312yes—
158C3⋊#95C2311yescharacteristic
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C2×#127—32no—
162C2×#127C2×D311yescharacteristic
163C2×#126—34no—
164C2×#124—31no—
165C2×#124C2×D312yes—
166C2×#123—34no—
167C2×#121—34no—
168C2×C4×Q8—32no—
169C2×C4×Q8C2×D311yescharacteristic
170C2×C4×D4—32no—
171C2×C4×D4—31no—
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#394C2×D312yes—
183#394C2×D312yes—
184#392—32no—
185#392—32no—
186#391—31no—
187#390—32no—
188#390—32no—
189#390—31no—
190#390—34no—
191#390—34no—
192#390—32no—
193#390C2×D312yes—
194#390C2×D314yes—
195#389—32no—
196#389—34no—
197#388—34no—
198#388—34no—
199#388—34no—
200#388—34no—
201#388—34no—
202#386—34no—
203#386—34no—
204#386—34no—
205#385—34no—
206#385—34no—
207#384—31no—
208#382—31no—
209#382—32no—
210#381—32no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4—32no—
216C4×C4⋊C4C2×D311yescharacteristic
217C4×C22⋊C4—34no—
218C4×C22⋊C4—32no—
219#376—31no—
220C24×C3C2414yes—
221C23×D3C2412yes—
222C2×C3×Q8—22no—
223C2×C3×Q8—22no—
224C2×C3×Q8#4114yes—
225C2×C3×D4—24no—
226C2×C3×D4—24no—
227C2×C3×D4—24no—
228C2×C3×D4—24no—
229C2×C3×D4—24no—
230C2×C3×D4—44no—
231C2×C3×D4—44no—
232C22×C3×C4C2412yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C2×#67—28no—
240C2×#67—44no—
241C2×#67—44no—
242C2×#67—44no—
243C2×#67—44no—
244C2×#67—44no—
245C22×Dic3—44no—
246C22×Dic3—44no—
247C22×Dic3—44no—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C22×Dic3C2414yes—
252C22×Dic3C2412yes—
253C22×Dic3C2412yes—
254C2×D12—22no—
255C2×D12—22no—
256C2×D12—22no—
257C2×C4×D3—24no—
258C2×C4×D3—44no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4—22no—
261C3×C4⋊C4—42no—
262C3×C22⋊C4—28no—
263C3×C22⋊C4—28no—
264C3×C22⋊C4—44no—
265C3×C22⋊C4—44no—
266C3×C42—24no—
267C3×C42—24no—
268C3×C42#4114yes—
269#217—28no—
270#217—44no—
271#217—44no—
272#217—44no—
273#217—44no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—28no—
276(C2×D3)⋊C4—44no—
277(C2×D3)⋊C4—44no—
278(C2×D3)⋊C4—44no—
279C4⋊Dic3—22no—
280C4⋊Dic3—24no—
281C4⋊Dic3—28no—
282C4⋊Dic3—24no—
283C4⋊Dic3—24no—
284C4⋊Dic3—42no—
285C4⋊Dic3—44no—
286Dic3⋊C4—24no—
287Dic3⋊C4—44no—
288Dic3⋊C4—42no—
289C4×Dic3—22no—
290C4×Dic3—22no—
291C4×Dic3—24no—
292C4×Dic3—44no—
293C4×Dic3—44no—
294C22×Q8C22×D311yescharacteristic
295C22×D4—34no—
296C22×D4—34no—
297C22×D4—31no—
298C22×D4C22×D312yes—
299C22×D4C22×D312yes—
300C22×D4C22×D312yes—
301C23×C4—34no—
302C23×C4—32no—
303#127—24no—
304#127—124no—
305#126—128no—
306#124—28no—
307#124—64no—
308#123—128no—
309#121—128no—
310C4×Q8—24no—
311C4×Q8—124no—
312C4×D4—64no—
313C4×D4—124no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—31no—
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×D311yescharacteristic
325C2×C4⋊C4C22×D312yes—
326C2×C4⋊C4C22×D312yes—
327C2×C22⋊C4—34no—
328C2×C22⋊C4—32no—
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⋊C4C22×D314yes—
338C2×C22⋊C4C22×D314yes—
339C2×C22⋊C4C22×D314yes—
340C2×C22⋊C4C22×D314yes—
341C2×C42—31no—
342C2×C42—32no—
343C2×C42—31no—
344C2×C42—34no—
345C2×C42—34no—
346C2×C42C22×D312yes—
347C2×C42C22×D312yes—
348C2×C42C22×D311yescharacteristic
349#95—32no—
350#95—32no—
351#95—32no—
352#95—32no—
353#95—34no—
354#95—31no—
355#95—34no—
356#95—32no—
357#95—31no—
358#95—32no—
359#95—32no—
360#95—32no—
361#95—34no—
362#95C22×D311yescharacteristic
363#95C22×D311yescharacteristic
364C23×C3—24no—
365C23×C3—24no—
366C23×C3—24no—
367C23×C3—24no—
368C23×C3—24no—
369C23×C3—24no—
370C23×C3—24no—
371C23×C3C2511yescharacteristic
372C22×D3—22no—
373C22×D3—22no—
374C22×D3—24no—
375C22×D3—22no—
376C22×D3—24no—
377C3×Q8—28no—
378C3×D4—48no—
379C3×D4—84no—
380C3×D4—84no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4C2×#4112yes—
394C2×C3×C4C2×#4114yes—
395C2×C3×C4C2×#4112yes—
396C2×C3×C4C2×#4112yes—
397C2×C3×C4C2×#4112yes—
398C2×C3×C4H(5,𝔽2)12yes—
399#67—88no—
400#67—88no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—24no—
417C2×Dic3—88no—
418C2×Dic3H(5,𝔽2)14yes—
419D12—44no—
420C4×D3—84no—
421C24—32no—
422C24C23×D314yes—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8#23714yes—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—68no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—124no—
438C2×D4—124no—
439C2×D4—124no—
440C2×D4—124no—
441C2×D4—124no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—32no—
448C22×C4—64no—
449C22×C4—124no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—124no—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D312yes—
456C22×C4C23×D311yescharacteristic
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D312yes—
459C22×C4C23×D311yescharacteristic
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—64no—
464C4⋊C4—64no—
465C4⋊C4—68no—
466C4⋊C4—64no—
467C4⋊C4—62no—
468C4⋊C4—64no—
469C4⋊C4—124no—
470C4⋊C4—124no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—68no—
478C22⋊C4—64no—
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—24no—
489C42—62no—
490C42—62no—
491C42—64no—
492C42—124no—
493C42—124no—
494C42#23714yes—
495C22×C3—44no—
496C22×C3—44no—
497C22×C3—44no—
498C22×C3—44no—
499C22×C3—44no—
500C22×C3—44no—
501C22×C3—44no—
502C22×C3C22×#4111yescharacteristic
503C22×C3C22×#4111yescharacteristic
504C22×C3C2×H(5,𝔽2)11yescharacteristic
505C22×C3C2×H(5,𝔽2)12yes—
506C22×C3#58512yes—
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—44no—
511C2×D3—44no—
512C3×C4—24no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4#55314yes—
519Dic3—24no—
520Dic3—42no—
521Dic3—42no—
522Dic3—44no—
523Dic3—42no—
524Dic3—42no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—64no—
536C23—64no—
537C23C24×D311yescharacteristic
538Q8—28no—
539D4—48no—
540D4—84no—
541D4—84no—
542D4—124no—
543D4—248no—
544D4—248no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—34no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—244no—
575C2×C4—248no—
576C2×C4C2×#23712yes—
577C2×C4C2×#23712yes—
578C2×C4C2×#23714yes—
579C2×C4C2×#23712yes—
580C2×C4C2×#23712yes—
581C2×C4#100912yes—
582C2×C3—84no—
583C2×C3C2×#55311yescharacteristic
584C2×C3#350811yescharacteristic
585C2×C3#351011yescharacteristic
586C2×C3#351712yes—
587C2×C3#355812yes—
588D3—82no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—124no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—124no—
601C22C22×#23711yescharacteristic
602C22C22×#23711yescharacteristic
603C22C2×#100911yescharacteristic
604C22#596412yes—
605C22D3×H(5,𝔽2)12yes—
606C4—22no—
607C4—22no—
608C4—24no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—64no—
613C4—122no—
614C4—124no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4#561814yes—
619C3#4531211yescharacteristic, complemented, Sylow
620C2—84no—
621C2—242no—
622C2C2×#561811yescharacteristic
623C2#8606511yescharacteristic
624C2#8631911yescharacteristic
625C2#8677212yes—
626C2#8706912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2548C2
prime radical#83188C2
hypercenter#2548D3
hypocenterC3#45312
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C3×#2548C2semidirect
3. #83172C2semidirect
4. #82974C2semidirect
5. #82955C2semidirect
6. #82460C2semidirect
7. #82455C2semidirect
8. #77148C2semidirect
9. #76953C2semidirect
10. #76409C2semidirect
11. #76354C2semidirect
12. #76338C2semidirect
13. #76317C2semidirect
14. #73671C2semidirect
15. #73645C2semidirect
81. #2548D3semidirect
619. C3#45312semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#831882
#831722
#829742
#829552
#824602
#824552
#771482
#769532
#764092
#763542
#763382
#763172
#736712
#736452
#453123