Finite Group Databasebeta

← #11235117

Subgroups — #11235117

Order 768.

1726 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2399C211yesmaximal, characteristic, complemented
2#82476C212yesmaximal
3#82462C212yesmaximal, complemented
4#77990C214yesmaximal, complemented
5#77791C214yesmaximal, complemented
6#77267C214yesmaximal, complemented
7#77184C214yesmaximal, complemented
8#73807C211yesmaximal, characteristic, complemented
9#73800C211yesmaximal, characteristic, complemented
10#73694C212yesmaximal, complemented
11#73692C212yesmaximal, complemented
12#73645C211yesmaximal, characteristic, complemented
13#73645C211yesmaximal, characteristic, complemented
14#73391C211yesmaximal, characteristic, complemented
15#73389C211yesmaximal, characteristic
16#45430—31nomaximal, Sylow
17C2×C3×C4×Q8C2212yes—
18C2×#947C2214yes—
19C2×#898C2214yes—
20C2×#897C2214yes—
21C2×#874C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C3×#398C2212yes—
26C3×#398C2212yes—
27C3×#395C2212yes—
28C3×#395C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2212yes—
32C3×#376C2211yescharacteristic
33#5229C2214yes—
34#4994C2214yes—
35#4992C2214yes—
36#4991C2214yes—
37#4991C2214yes—
38#4990C2214yes—
39#4990C2214yes—
40#4990C2214yes—
41#4988C2214yes—
42#4987C2214yes—
43#4980C2214yes—
44#4980C2214yes—
45Dic3×C4⋊C4C2214yes—
46#4939C2212yes—
47#4939C2212yes—
48#4939C2212yes—
49#4939C2212yes—
50#4939C2211yescharacteristic
51#4939C2211yescharacteristic
52#4936C2211yescharacteristic
53#4936C2212yes—
54#4936C2211yescharacteristic
55#4935C2212yes—
56C4×C4⋊Dic3C2212yes—
57C4×C4⋊Dic3C2212yes—
58C4×C4⋊Dic3C2212yes—
59#4928C2212yes—
60#4677C2212yes—
61#4677C2212yes—
62#4675C2214yes—
63#4674C2214yes—
64#4673C2212yes—
65#4673C2212yes—
66#4670C2214yes—
67#4669C2214yes—
68#4666C2212yes—
69#4666C2212yes—
70#4665C2214yes—
71#4665C2214yes—
72#4664C2212yes—
73#4664C2212yes—
74#4654C2214yes—
75#4646C2214yes—
76#2740—34no—
77#2700—34no—
78#2563—31no—
79#2560—34no—
80#2548—32no—
81#2542—31no—
82#2521—34no—
83#2513—32no—
84#2512—32no—
85#2399D311yescharacteristic, complemented
86#2357—31no—
87#2357—31no—
88#2355—32no—
89#2331—31no—
90#2328—31no—
91C22×C3×Q8C2312yes—
92C22×D12C2311yescharacteristic
93C22×D12C2311yescharacteristic
94C22×D12C2314yes—
95C22×C4×D3C2312yes—
96C22×C4×D3C2312yes—
97C22×Dic6C2312yes—
98C3×C4×Q8—28no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2312yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111#947—48no—
112C2×(C2×D3)⋊C4C2312yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2314yes—
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⋊Dic3C2312yes—
127C2×C4⋊Dic3C2314yes—
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×C4×Dic3C2314yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2314yes—
135#898—48no—
136#897—48no—
137#874—24no—
138C4×D12—24no—
139C4×D12—24no—
140#869—24no—
141C3×#95C2312yes—
142C3×#95C2312yes—
143C3×#95C2311yescharacteristic
144C3×#95C2311yescharacteristic
145C3×#95C2311yescharacteristic
146C3×#95C2311yescharacteristic
147C3×#95C2312yes—
148C3×#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C2×C4⋊Q8—31no—
159C2×#127—31no—
160C2×#126—34no—
161C2×#124—34no—
162C2×C22⋊Q8—34no—
163C2×C4×Q8C2×D312yes—
164C2×C4×D4—31no—
165C2×C4×D4—31no—
166#399—32no—
167#399—32no—
168#398C2×D312yes—
169#398C2×D312yes—
170#397—34no—
171#397—34no—
172#397—34no—
173#396—34no—
174#395C2×D312yes—
175#395C2×D312yes—
176#394—34no—
177#394—34no—
178#393—32no—
179#393—32no—
180#391—34no—
181#391—34no—
182#390—34no—
183#390—32no—
184#390—31no—
185#390—31no—
186#390—32no—
187#390—32no—
188#390—32no—
189#389—32no—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#387—34no—
196#387—34no—
197#387—34no—
198#386—34no—
199#386—34no—
200#384—32no—
201#382—32no—
202#382—31no—
203#382—31no—
204#381—32no—
205#381—32no—
206#380—34no—
207C4×C4⋊C4—34no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—32no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4C2×D311yescharacteristic
212C4×C4⋊C4C2×D311yescharacteristic
213C4×C4⋊C4C2×D312yes—
214C4×C22⋊C4—32no—
215C4×C22⋊C4—32no—
216#376C2×D311yescharacteristic
217C23×D3C2412yes—
218C23×D3C2412yes—
219C2×C3×Q8—24no—
220C2×C3×Q8—24no—
221C2×C3×Q8—42no—
222C22×C3×C4C2412yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3C2412yes—
232C22×Dic3C2414yes—
233C22×Dic3C2412yes—
234C22×Dic3C2414yes—
235C2×D12—22no—
236C2×D12—22no—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12—22no—
240C2×D12—22no—
241C2×D12—28no—
242C2×D12—44no—
243C2×D12—44no—
244C2×C4×D3—24no—
245C2×C4×D3—24no—
246C2×C4×D3—44no—
247C2×C4×D3—44no—
248C2×Dic6—24no—
249C2×Dic6—44no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—42no—
257C3×C42—24no—
258C3×C42—24no—
259C3×C42—22no—
260C3×C42#4114yes—
261C3×C42#4114yes—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—28no—
265(C2×D3)⋊C4—28no—
266(C2×D3)⋊C4—44no—
267(C2×D3)⋊C4—44no—
268(C2×D3)⋊C4—44no—
269(C2×D3)⋊C4—44no—
270(C2×D3)⋊C4—44no—
271(C2×D3)⋊C4—44no—
272(C2×D3)⋊C4—44no—
273(C2×D3)⋊C4—44no—
274C4⋊Dic3—24no—
275C4⋊Dic3—24no—
276C4⋊Dic3—22no—
277C4⋊Dic3—22no—
278C4⋊Dic3—28no—
279C4⋊Dic3—24no—
280C4⋊Dic3—44no—
281Dic3⋊C4—44no—
282C4×Dic3—44no—
283C4×Dic3—44no—
284C4×Dic3—44no—
285C22×Q8—32no—
286C22×Q8C22×D312yes—
287C22×D4—31no—
288C22×D4—31no—
289C22×D4—34no—
290C23×C4—32no—
291C23×C4—32no—
292C4⋊Q8—64no—
293#127—64no—
294#126—128no—
295#124—128no—
296C22⋊Q8—128no—
297C4×Q8—28no—
298C4×D4—64no—
299C4×D4—64no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4C22×D312yes—
309C2×C4⋊C4C22×D312yes—
310C2×C4⋊C4C22×D312yes—
311C2×C4⋊C4C22×D312yes—
312C2×C4⋊C4C22×D312yes—
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D312yes—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—32no—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—34no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4—34no—
327C2×C42—34no—
328C2×C42—34no—
329C2×C42—34no—
330C2×C42C22×D312yes—
331C2×C42C22×D312yes—
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335#95—32no—
336#95—32no—
337#95—34no—
338#95—34no—
339#95—34no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95—34no—
344#95C22×D312yes—
345#95C22×D312yes—
346#95C22×D311yescharacteristic
347#95C22×D311yescharacteristic
348#95C22×D311yescharacteristic
349#95C22×D311yescharacteristic
350#95C22×D312yes—
351#95C22×D312yes—
352C23×C3C2511yescharacteristic
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—24no—
358C22×D3—24no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—24no—
362C22×D3—24no—
363C3×Q8—48no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4C2×#4114yes—
383C2×C3×C4C2×#4112yes—
384C2×C3×C4C2×#4112yes—
385C2×C3×C4C2×#4112yes—
386C2×C3×C4C2×#4112yes—
387C2×C3×C4H(5,𝔽2)14yes—
388C2×C3×C42−1+412yes—
389C2×Dic3—24no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399D12—44no—
400D12—44no—
401D12—88no—
402C4×D3—84no—
403C4×D3—84no—
404Dic6—84no—
405C24—32no—
406C24—32no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—42no—
410C2×Q8—64no—
411C2×Q8—124no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—68no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—124no—
420C2×D4—124no—
421C22×C4—32no—
422C22×C4—34no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—124no—
428C22×C4—124no—
429C22×C4C23×D312yes—
430C22×C4C23×D312yes—
431C22×C4C23×D312yes—
432C22×C4C23×D312yes—
433C22×C4C23×D312yes—
434C22×C4C23×D311yescharacteristic
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—62no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—68no—
451C4⋊C4—124no—
452C4⋊C4—124no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—68no—
456C22⋊C4—68no—
457C22⋊C4—124no—
458C22⋊C4—124no—
459C22⋊C4—124no—
460C22⋊C4—124no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C22⋊C4—124no—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—124no—
469C42—124no—
470C42—124no—
471C42#23714yes—
472C42#23914yes—
473C22×C3C22×#4111yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C2×H(5,𝔽2)12yes—
476C22×C3C2×2−1+411yescharacteristic
477C22×C3#58512yes—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—44no—
487C2×D3—44no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4#55614yes—
497Dic3—44no—
498Dic3—42no—
499Dic3—42no—
500Dic3—44no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—62no—
505C23—62no—
506C23—62no—
507C23—64no—
508C23—64no—
509C23—64no—
510C23—64no—
511C23C24×D311yescharacteristic
512Q8—48no—
513Q8—244no—
514D4—124no—
515D4—124no—
516D4—248no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—244no—
546C2×C4—244no—
547C2×C4C2×#23714yes—
548C2×C4C2×#23712yes—
549C2×C4C2×#23912yes—
550C2×C4C2×#23912yes—
551C2×C4C2×#23912yes—
552C2×C4#100712yes—
553C2×C4#100914yes—
554C2×C3C2×#55611yescharacteristic
555C2×C3#349412yes—
556C2×C3#350811yescharacteristic
557C2×C3#351011yescharacteristic
558C2×C3#355412yes—
559D3—82no—
560D3—82no—
561C22—124no—
562C22—124no—
563C22—124no—
564C22—124no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22C22×#23711yescharacteristic
572C22C22×#23911yescharacteristic
573C22C2×#100711yescharacteristic
574C22C2×#100912yes—
575C22#596412yes—
576C4—24no—
577C4—24no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—124no—
585C4—122no—
586C4—124no—
587C4—122no—
588C4#573214yes—
589C3#4543011yescharacteristic, complemented, Sylow
590C2—242no—
591C2—242no—
592C2#8602712yes—
593C2C2×#573211yescharacteristic
594C2#8626111yescharacteristic
595C2#8626311yescharacteristic
596C2#8628912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2399C2
prime radicalC2×#874C22
hypercenter#2399D3
hypocenterC3#45430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. C3×#2399C2semidirect
3. #82462C2semidirect
4. #77990C2semidirect
5. #77791C2semidirect
6. #77267C2semidirect
7. #77184C2semidirect
8. #73807C2semidirect
9. #73800C2semidirect
10. #73694C2semidirect
11. #73692C2semidirect
12. #73645C2semidirect
13. #73645C2semidirect
14. #73391C2semidirect
85. #2399D3semidirect
589. C3#45430semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#824762
#824622
#779902
#777912
#772672
#771842
#738072
#738002
#736942
#736922
#736452
#733912
#733892
#454303