Finite Group Databasebeta

← #11232946

Subgroups — #11232946

Order 768.

1486 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2568C211yesmaximal, characteristic, complemented
2#83260C211yesmaximal, characteristic
3#83228C211yesmaximal, characteristic, complemented
4#82460C211yesmaximal, characteristic, complemented
5#82455C211yesmaximal, characteristic
6#78376C214yesmaximal, complemented
7#78171C214yesmaximal, complemented
8#76575C214yesmaximal, complemented
9#76531C214yesmaximal, complemented
10#76507C212yesmaximal
11#76490C212yesmaximal, complemented
12#74186C212yesmaximal, complemented
13#74172C212yesmaximal
14#73671C211yesmaximal, characteristic, complemented
15#73645C211yesmaximal, characteristic
16#45318—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2212yes—
21C2×C4×D12C2211yescharacteristic
22C3×#400C2212yes—
23C3×#400C2212yes—
24C3×#391C2214yes—
25C3×#391C2212yes—
26C3×#382C2212yes—
27C3×C4×C4⋊C4C2211yescharacteristic
28#5231C2212yes—
29#5231C2212yes—
30#5229C2211yescharacteristic
31#5229C2214yes—
32#4990C2214yes—
33#4989C2214yes—
34#4988C2211yescharacteristic
35#4988C2212yes—
36#4987C2214yes—
37#4986C2212yes—
38#4984C2214yes—
39#4984C2212yes—
40#4984C2212yes—
41#4982C2212yes—
42#4980C2214yes—
43#4977C2212yes—
44#4975C2211yescharacteristic
45Dic3×C4⋊C4C2212yes—
46#4940C2214yes—
47#4939C2212yes—
48#4936C2211yescharacteristic
49#4936C2212yes—
50C4×C4⋊Dic3C2211yescharacteristic
51C4×C4⋊Dic3C2212yes—
52C4×C4⋊Dic3C2212yes—
53C4×C4⋊Dic3C2212yes—
54#4932C2211yescharacteristic
55C4×Dic3⋊C4C2212yes—
56#4677C2214yes—
57#4675C2214yes—
58#4674C2214yes—
59#4674C2214yes—
60#4670C2214yes—
61#4670C2214yes—
62#4670C2214yes—
63#4669C2214yes—
64#4666C2212yes—
65#4664C2212yes—
66#4661C2212yes—
67#4661C2214yes—
68#4660C2214yes—
69#4660C2212yes—
70#4655C2214yes—
71#4654C2214yes—
72#4654C2214yes—
73#4653C2214yes—
74#4651C2214yes—
75#4648C2214yes—
76#2754—34no—
77#2738—34no—
78#2570—32no—
79#2568D311yescharacteristic, complemented
80#2563—31no—
81#2532—34no—
82#2530—34no—
83#2513—32no—
84#2448—32no—
85#2402—31no—
86#2357—31no—
87#2356—32no—
88#2351—31no—
89#2331—31no—
90#2326—31no—
91C22×C3×Q8C2311yescharacteristic
92C22×C3×Q8C2312yes—
93C22×D12C2311yescharacteristic
94C22×C4×D3C2312yes—
95C3×C4⋊Q8—24no—
96C3×C4×Q8—24no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C42C2312yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105#947—24no—
106Q8×Dic3—28no—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2312yes—
109C2×(C2×D3)⋊C4C2314yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2312yes—
117C2×C4⋊Dic3C2314yes—
118C2×C4⋊Dic3C2314yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2312yes—
122C2×Dic3⋊C4C2312yes—
123C2×Dic3⋊C4C2312yes—
124C2×Dic3⋊C4C2314yes—
125C2×C4×Dic3C2314yes—
126C2×C4×Dic3C2311yescharacteristic
127C2×C4×Dic3C2312yes—
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2314yes—
130C2×C4×Dic3C2314yes—
131C4×D12—24no—
132C3×#95C2314yes—
133C3×#95C2314yes—
134C3×#95C2314yes—
135C3×#95C2311yescharacteristic
136C3×#95C2314yes—
137C3×#95C2311yescharacteristic
138C3⋊#95C2312yes—
139C3⋊#95C2314yes—
140C3⋊#95C2314yes—
141C3⋊#95C2312yes—
142C3⋊#95C2314yes—
143C3⋊#95C2314yes—
144C3⋊#95C2314yes—
145C3⋊#95C2312yes—
146C3⋊#95C2314yes—
147C3⋊#95C2312yes—
148C3⋊#95C2314yes—
149C3⋊#95C2312yes—
150C3⋊#95C2312yes—
151C3⋊#95C2312yes—
152C3⋊#95C2311yescharacteristic
153C3⋊#95C2312yes—
154C3⋊#95C2311yescharacteristic
155C2×C4⋊Q8C2×D311yescharacteristic
156C2×#124—31no—
157C2×C4×Q8—32no—
158C2×C4×Q8C2×D311yescharacteristic
159C2×C4×D4—31no—
160#400—32no—
161#400—34no—
162#400—32no—
163#400C2×D312yes—
164#400C2×D312yes—
165#399—34no—
166#398—32no—
167#398—34no—
168#398—32no—
169#397—34no—
170#397—34no—
171#397—34no—
172#393—32no—
173#393—32no—
174#391—31no—
175#391—34no—
176#391C2×D312yes—
177#391C2×D314yes—
178#390—32no—
179#390—31no—
180#390—32no—
181#389—32no—
182#389—32no—
183#389—34no—
184#388—34no—
185#388—34no—
186#388—34no—
187#388—34no—
188#388—34no—
189#388—34no—
190#387—34no—
191#387—34no—
192#387—34no—
193#387—34no—
194#387—34no—
195#385—34no—
196#384—31no—
197#383—34no—
198#382—32no—
199#382—31no—
200#382—32no—
201#382C2×D312yes—
202#381—32no—
203#380—34no—
204C4×C4⋊C4—32no—
205C4×C4⋊C4—31no—
206C4×C4⋊C4—32no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—32no—
210C4×C4⋊C4C2×D311yescharacteristic
211C4×C22⋊C4—32no—
212#376—31no—
213#376—34no—
214C23×D3C2412yes—
215C2×C3×Q8—22no—
216C2×C3×Q8—24no—
217C2×C3×Q8—22no—
218C2×C3×Q8—44no—
219C2×C3×Q8#4114yes—
220C22×C3×C4C2414yes—
221C22×C3×C4C2412yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×Dic3C2412yes—
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2414yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C2×D12—22no—
235C2×D12—22no—
236C2×D12—22no—
237C2×C4×D3—24no—
238C2×C4×D3—44no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—42no—
244C3×C42—24no—
245C3×C42—24no—
246C3×C42#4114yes—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—28no—
249(C2×D3)⋊C4—44no—
250(C2×D3)⋊C4—44no—
251(C2×D3)⋊C4—44no—
252(C2×D3)⋊C4—44no—
253(C2×D3)⋊C4—44no—
254C4⋊Dic3—22no—
255C4⋊Dic3—28no—
256C4⋊Dic3—24no—
257C4⋊Dic3—44no—
258C4⋊Dic3—44no—
259C4⋊Dic3—42no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—44no—
264Dic3⋊C4—42no—
265C4×Dic3—28no—
266C4×Dic3—24no—
267C4×Dic3—22no—
268C4×Dic3—22no—
269C4×Dic3—44no—
270C4×Dic3—44no—
271C22×Q8C22×D311yescharacteristic
272C22×Q8C22×D312yes—
273C22×D4—31no—
274C23×C4—32no—
275C4⋊Q8—24no—
276#124—64no—
277C4×Q8—24no—
278C4×Q8—68no—
279C4×D4—64no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—34no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—34no—
290C2×C4⋊C4—34no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D312yes—
293C2×C4⋊C4C22×D312yes—
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D312yes—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—34no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—34no—
300C2×C22⋊C4—34no—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—34no—
303C2×C42—34no—
304C2×C42—32no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—34no—
308C2×C42—34no—
309C2×C42C22×D312yes—
310C2×C42C22×D312yes—
311C2×C42C22×D311yescharacteristic
312#95—32no—
313#95—34no—
314#95—34no—
315#95—32no—
316#95—32no—
317#95—32no—
318#95—31no—
319#95—34no—
320#95—32no—
321#95—31no—
322#95—32no—
323#95—34no—
324#95—32no—
325#95—32no—
326#95—34no—
327#95—34no—
328#95—34no—
329#95C22×D314yes—
330#95C22×D311yescharacteristic
331#95C22×D311yescharacteristic
332#95C22×D314yes—
333#95C22×D314yes—
334#95C22×D314yes—
335C23×C3C2511yescharacteristic
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—24no—
340C22×D3—24no—
341C3×Q8—28no—
342C3×Q8—84no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4C2×#4112yes—
359C2×C3×C4C2×#4112yes—
360C2×C3×C4C2×#4112yes—
361C2×C3×C4C2×#4114yes—
362C2×C3×C4C2×#4112yes—
363C2×C3×C42−1+412yes—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381D12—44no—
382C4×D3—84no—
383C24—32no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8#23914yes—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—34no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—64no—
399C22×C4—124no—
400C22×C4C23×D314yes—
401C22×C4C23×D312yes—
402C22×C4C23×D312yes—
403C22×C4C23×D312yes—
404C22×C4C23×D311yescharacteristic
405C22×C4C23×D311yescharacteristic
406C22×C4C23×D312yes—
407C22×C4C23×D311yescharacteristic
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—42no—
413C4⋊C4—62no—
414C4⋊C4—68no—
415C4⋊C4—64no—
416C4⋊C4—122no—
417C4⋊C4—122no—
418C4⋊C4—124no—
419C4⋊C4—122no—
420C4⋊C4—124no—
421C4⋊C4—122no—
422C4⋊C4—122no—
423C4⋊C4—124no—
424C22⋊C4—64no—
425C22⋊C4—68no—
426C22⋊C4—124no—
427C22⋊C4—124no—
428C22⋊C4—124no—
429C22⋊C4—124no—
430C22⋊C4—124no—
431C42—24no—
432C42—24no—
433C42—68no—
434C42—64no—
435C42—62no—
436C42—62no—
437C42—124no—
438C42—124no—
439C42#23714yes—
440C22×C3C22×#4111yescharacteristic
441C22×C3C22×#4111yescharacteristic
442C22×C3C2×2−1+411yescharacteristic
443C22×C3#58512yes—
444C22×C3#58512yes—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—44no—
449C2×D3—44no—
450C3×C4—24no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—44no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4#55214yes—
458Dic3—42no—
459Dic3—42no—
460Dic3—44no—
461Dic3—42no—
462Dic3—42no—
463Dic3—42no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—64no—
468C23—64no—
469C23C24×D311yescharacteristic
470Q8—28no—
471Q8—84no—
472D4—124no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—64no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—244no—
506C2×C4C2×#23712yes—
507C2×C4C2×#23712yes—
508C2×C4C2×#23712yes—
509C2×C4C2×#23914yes—
510C2×C4C2×#23912yes—
511C2×C4#101012yes—
512C2×C3C2×#55211yescharacteristic
513C2×C3#351011yescharacteristic
514C2×C3#351011yescharacteristic
515C2×C3#352412yes—
516C2×C3#356512yes—
517D3—82no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—124no—
522C22—124no—
523C22C22×#23711yescharacteristic
524C22C22×#23911yescharacteristic
525C22C2×#101011yescharacteristic
526C22#596412yes—
527C22#596812yes—
528C4—22no—
529C4—22no—
530C4—24no—
531C4—44no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—122no—
536C4—122no—
537C4—122no—
538C4—122no—
539C4—124no—
540C4—122no—
541C4#564214yes—
542C3#4531811yescharacteristic, complemented, Sylow
543C2—242no—
544C2C2×#564211yescharacteristic
545C2#8606511yescharacteristic
546C2#8634411yescharacteristic
547C2#8700012yes—
548C2#8702612yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2568C2
prime radicalC22×D12C23
hypercenter#2568D3
hypocenterC3#45318
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2568

Complements

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

NH ≅ G/Ntype
1. C3×#2568C2semidirect
3. #83228C2semidirect
4. #82460C2semidirect
6. #78376C2semidirect
7. #78171C2semidirect
8. #76575C2semidirect
9. #76531C2semidirect
11. #76490C2semidirect
12. #74186C2semidirect
14. #73671C2semidirect
79. #2568D3semidirect
542. C3#45318semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
#832602
#832282
#824602
#824552
#783762
#781712
#765752
#765312
#765072
#764902
#741862
#741722
#736712
#736452
#453183