Finite Group Databasebeta

← #11225496

Subgroups — #11225496

Order 768.

1662 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2567C211yesmaximal, characteristic, complemented
2#83259C211yesmaximal, characteristic
3#83228C211yesmaximal, characteristic, complemented
4#82463C211yesmaximal, characteristic, complemented
5#82455C211yesmaximal, characteristic
6#77265C214yesmaximal, complemented
7#77180C214yesmaximal, complemented
8#76555C212yesmaximal
9#76522C212yesmaximal, complemented
10#76158C214yesmaximal, complemented
11#76097C214yesmaximal, complemented
12#74333C212yesmaximal
13#74323C212yesmaximal, complemented
14#73669C211yesmaximal, characteristic, complemented
15#73645C211yesmaximal, characteristic
16#44979—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×#946C2214yes—
21C2×#898C2214yes—
22C2×#887C2214yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C3×#395C2212yes—
26C3×#395C2212yes—
27C3×#391C2212yes—
28C3×#389C2214yes—
29C3×#384C2212yes—
30C3×C4×C4⋊C4C2211yescharacteristic
31#5229C2212yes—
32#5229C2211yescharacteristic
33#5229C2214yes—
34#4994C2214yes—
35#4993C2212yes—
36#4993C2212yes—
37#4992C2214yes—
38#4990C2214yes—
39#4990C2214yes—
40#4989C2214yes—
41#4988C2212yes—
42#4988C2211yescharacteristic
43#4984C2212yes—
44#4982C2212yes—
45#4980C2214yes—
46#4980C2214yes—
47#4979C2212yes—
48#4976C2214yes—
49#4976C2214yes—
50#4975C2211yescharacteristic
51#4975C2212yes—
52Dic3×C4⋊C4C2212yes—
53Dic3×C4⋊C4C2212yes—
54#4972C2214yes—
55#4940C2214yes—
56#4939C2212yes—
57#4937C2212yes—
58#4936C2211yescharacteristic
59C4×C4⋊Dic3C2211yescharacteristic
60C4×C4⋊Dic3C2212yes—
61#4932C2211yescharacteristic
62#4932C2212yes—
63C4×Dic3⋊C4C2212yes—
64#4675C2214yes—
65#4673C2214yes—
66#4670C2214yes—
67#4669C2214yes—
68#4666C2212yes—
69#4664C2212yes—
70#4659C2214yes—
71#4658C2214yes—
72#4657C2212yes—
73#4656C2212yes—
74#4655C2214yes—
75#4652C2214yes—
76#2777—34no—
77#2689—34no—
78#2604—34no—
79#2592—32no—
80#2567D311yescharacteristic, complemented
81#2542—31no—
82#2512—32no—
83#2511—32no—
84#2465—34no—
85#2399—31no—
86#2357—31no—
87#2356—31no—
88Q8×C4⋊C4—32no—
89#2331—31no—
90#2326—31no—
91C22×C3×Q8C2311yescharacteristic
92C22×C3×Q8C2312yes—
93C22×D12C2311yescharacteristic
94C22×C4×D3C2312yes—
95C22×Dic6C2312yes—
96C3×C4⋊Q8—24no—
97C3×C4×Q8—24no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2311yescharacteristic
108#947—24no—
109#946—48no—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2314yes—
122C2×Dic3⋊C4C2312yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2312yes—
125C2×Dic3⋊C4C2312yes—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2314yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2311yescharacteristic
135#898—48no—
136#887—48no—
137C4×D12—24no—
138C4×Dic6—28no—
139C3×#95C2314yes—
140C3×#95C2311yescharacteristic
141C3×#95C2311yescharacteristic
142C3×#95C2314yes—
143C3⋊#95C2314yes—
144C3⋊#95C2314yes—
145C3⋊#95C2314yes—
146C3⋊#95C2312yes—
147C3⋊#95C2314yes—
148C3⋊#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2314yes—
158C2×C4⋊Q8C2×D311yescharacteristic
159C2×#126—34no—
160C2×#124—31no—
161C2×C22⋊Q8—34no—
162C2×C4×Q8—34no—
163C2×C4×Q8—32no—
164C2×C4×Q8C2×D311yescharacteristic
165C2×C4×D4—31no—
166#400—34no—
167#400—34no—
168#399—32no—
169#399—32no—
170#398—32no—
171#398—32no—
172#397—34no—
173#397—34no—
174#396—34no—
175#395—32no—
176#395—32no—
177#395C2×D312yes—
178#395C2×D312yes—
179#393—34no—
180#391—31no—
181#391—34no—
182#391—32no—
183#391C2×D312yes—
184#390—31no—
185#390—32no—
186#390—32no—
187#389—32no—
188#389—34no—
189#389—34no—
190#389—34no—
191#389C2×D314yes—
192#388—34no—
193#388—34no—
194#388—34no—
195#388—34no—
196#388—34no—
197#387—34no—
198#387—34no—
199#387—34no—
200#387—34no—
201#385—34no—
202#384—31no—
203#384—32no—
204#384C2×D312yes—
205#383—32no—
206#382—31no—
207#381—32no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—32no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4C2×D311yescharacteristic
214C4×C22⋊C4—32no—
215#376—31no—
216#376—32no—
217C23×D3C2412yes—
218C2×C3×Q8—22no—
219C2×C3×Q8—22no—
220C2×C3×Q8—24no—
221C2×C3×Q8—22no—
222C2×C3×Q8—44no—
223C22×C3×C4C2412yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2414yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3C2414yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12—22no—
240C2×C4×D3—24no—
241C2×C4×D3—44no—
242C2×Dic6—24no—
243C2×Dic6—24no—
244C2×Dic6—24no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—44no—
251C3×C4⋊C4—44no—
252C3×C42—24no—
253C3×C42—24no—
254C3×C42#4114yes—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—28no—
257(C2×D3)⋊C4—28no—
258(C2×D3)⋊C4—28no—
259(C2×D3)⋊C4—44no—
260(C2×D3)⋊C4—44no—
261(C2×D3)⋊C4—44no—
262C4⋊Dic3—22no—
263C4⋊Dic3—24no—
264C4⋊Dic3—24no—
265C4⋊Dic3—24no—
266C4⋊Dic3—44no—
267Dic3⋊C4—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—28no—
271Dic3⋊C4—24no—
272Dic3⋊C4—44no—
273Dic3⋊C4—44no—
274C4×Dic3—24no—
275C4×Dic3—22no—
276C4×Dic3—28no—
277C4×Dic3—44no—
278C4×Dic3—44no—
279C4×Dic3#4114yes—
280C22×Q8—32no—
281C22×Q8C22×D311yescharacteristic
282C22×Q8C22×D312yes—
283C22×D4—31no—
284C23×C4—32no—
285C4⋊Q8—24no—
286#126—128no—
287#124—64no—
288C22⋊Q8—128no—
289C4×Q8—24no—
290C4×Q8—68no—
291C4×Q8—128no—
292C4×D4—64no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—34no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D312yes—
307C2×C4⋊C4C22×D312yes—
308C2×C4⋊C4C22×D312yes—
309C2×C4⋊C4C22×D312yes—
310C2×C4⋊C4C22×D314yes—
311C2×C4⋊C4C22×D314yes—
312C2×C22⋊C4—32no—
313C2×C22⋊C4—34no—
314C2×C22⋊C4—34no—
315C2×C22⋊C4—34no—
316C2×C22⋊C4—34no—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—34no—
319C2×C42—31no—
320C2×C42—34no—
321C2×C42—32no—
322C2×C42—31no—
323C2×C42—34no—
324C2×C42—34no—
325C2×C42C22×D312yes—
326C2×C42C22×D312yes—
327C2×C42C22×D311yescharacteristic
328#95—31no—
329#95—32no—
330#95—32no—
331#95—32no—
332#95—34no—
333#95—32no—
334#95—34no—
335#95—31no—
336#95—34no—
337#95—32no—
338#95—34no—
339#95—32no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95C22×D311yescharacteristic
344#95C22×D311yescharacteristic
345#95C22×D314yes—
346#95C22×D314yes—
347C23×C3C2511yescharacteristic
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—24no—
351C22×D3—24no—
352C22×D3—22no—
353C3×Q8—44no—
354C3×Q8—84no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4C2×#4112yes—
371C2×C3×C4C2×#4112yes—
372C2×C3×C4C2×#4112yes—
373C2×C3×C4C2×#4112yes—
374C2×C3×C42−1+414yes—
375C2×C3×C42−1+412yes—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3C2×#4114yes—
392C2×Dic3H(5,𝔽2)14yes—
393D12—44no—
394C4×D3—84no—
395Dic6—216no—
396C24—32no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—64no—
403C2×Q8—64no—
404C2×Q8—64no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—34no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—64no—
415C22×C4—124no—
416C22×C4C23×D312yes—
417C22×C4C23×D312yes—
418C22×C4C23×D314yes—
419C22×C4C23×D312yes—
420C22×C4C23×D311yescharacteristic
421C22×C4C23×D311yescharacteristic
422C22×C4C23×D312yes—
423C22×C4C23×D311yescharacteristic
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—68no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—62no—
438C4⋊C4—64no—
439C4⋊C4—64no—
440C4⋊C4—124no—
441C4⋊C4—124no—
442C4⋊C4—124no—
443C22⋊C4—68no—
444C22⋊C4—68no—
445C22⋊C4—64no—
446C22⋊C4—68no—
447C22⋊C4—124no—
448C22⋊C4—124no—
449C22⋊C4—124no—
450C42—24no—
451C42—24no—
452C42—34no—
453C42—68no—
454C42—62no—
455C42—64no—
456C42—124no—
457C42—124no—
458C42#23714yes—
459C22×C3C22×#4111yescharacteristic
460C22×C3C22×#4111yescharacteristic
461C22×C3C2×H(5,𝔽2)12yes—
462C22×C3C2×2−1+411yescharacteristic
463C22×C3C2×2−1+412yes—
464C2×D3—42no—
465C2×D3—44no—
466C2×D3—42no—
467C2×D3—42no—
468C2×D3—44no—
469C3×C4—24no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4#55614yes—
477Dic3—24no—
478Dic3—24no—
479Dic3—42no—
480Dic3—42no—
481Dic3—44no—
482Dic3—42no—
483C23—62no—
484C23—64no—
485C23—62no—
486C23—62no—
487C23—64no—
488C23C24×D311yescharacteristic
489Q8—44no—
490Q8—616no—
491Q8—84no—
492D4—124no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—34no—
509C2×C4—34no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—244no—
526C2×C4C2×#23712yes—
527C2×C4C2×#23712yes—
528C2×C4C2×#23712yes—
529C2×C4#100714yes—
530C2×C4#100712yes—
531C2×C4D3×#4112yes—
532C2×C3C2×#55611yescharacteristic
533C2×C3#349212yes—
534C2×C3#350811yescharacteristic
535C2×C3#350911yescharacteristic
536C2×C3#355012yes—
537D3—82no—
538C22—124no—
539C22—122no—
540C22—124no—
541C22—122no—
542C22—122no—
543C22C22×#23711yescharacteristic
544C22C2×#100711yescharacteristic
545C22C2×#100712yes—
546C22C2×D3×#4111yescharacteristic
547C22D3×H(5,𝔽2)12yes—
548C4—22no—
549C4—24no—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—64no—
556C4—64no—
557C4—122no—
558C4—124no—
559C4—122no—
560C4—122no—
561C4#564114yes—
562C3#4497911yescharacteristic, complemented, Sylow
563C2—242no—
564C2C2×#564111yescharacteristic
565C2#8620411yescharacteristic
566C2#8631911yescharacteristic
567C2#8694712yes—
568C2#8704412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2567C2
prime radicalC22×D12C23
hypercenter#2567D3
hypocenterC3#44979
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2567

Complements

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

NH ≅ G/Ntype
1. C3×#2567C2semidirect
3. #83228C2semidirect
4. #82463C2semidirect
6. #77265C2semidirect
7. #77180C2semidirect
9. #76522C2semidirect
10. #76158C2semidirect
11. #76097C2semidirect
13. #74323C2semidirect
14. #73669C2semidirect
80. #2567D3semidirect
562. C3#44979semidirect

Maximal subgroups

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

subgroupindex
C3×#25672
#832592
#832282
#824632
#824552
#772652
#771802
#765552
#765222
#761582
#760972
#743332
#743232
#736692
#736452
#449793