Finite Group Databasebeta

← #10828544

Subgroups — #10828544

Order 768.

1130 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5619C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic, complemented
3#82233C211yesmaximal, characteristic, complemented
4C2×#4735C211yesmaximal, characteristic, complemented
5C3×#1875C211yesmaximal, characteristic, complemented
6#71586C211yesmaximal, characteristic, complemented
7#70177C212yesmaximal, complemented
8#70170C212yesmaximal, complemented
9#69911C211yesmaximal, characteristic, complemented
10#68894C212yesmaximal, complemented
11#68828C212yesmaximal, complemented
12#24069—31nomaximal, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#940C2211yescharacteristic
16C2×#939—21no—
17C2×#938—21no—
18C2×D4×Dic3C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22#5619—24no—
23#5619—24no—
24C2×#893C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#886—21no—
27C2×#883—21no—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×#832C2212yes—
31C2×#830C2211yescharacteristic
32C2×#821C2212yes—
33C2×#817C2212yes—
34#5033—24no—
35#4984C2212yes—
36#4984C2212yes—
37C2×#809C2212yes—
38C2×#807C2212yes—
39#4950—21no—
40#4950—21no—
41#4936—21no—
42#4935—21no—
43#4735—24no—
44C3×#340C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#540—31no—
49C2×#481—31no—
50C2×#480—31no—
51#2317—31no—
52#2059—32no—
53#2041—32no—
54#1875—31no—
55#1875D311yescharacteristic, complemented
56#1864—32no—
57#1862—32no—
58#1818—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C22×#67—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×C3×C8C2312yes—
70C3×#121—22no—
71C3×#121D414yes—
72C3×#117—24no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42—21no—
78C2×#217—21no—
79C2×#217D411yescharacteristic
80C2×#217D411yescharacteristic
81#940—22no—
82#940—22no—
83#939—28no—
84#938—28no—
85D4×Dic3—22no—
86D4×Dic3—22no—
87#930—22no—
88#930D414yes—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91#926—22no—
92#926—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2312yes—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107#893—22no—
108#893—22no—
109#890—22no—
110#890—22no—
111#886—28no—
112#883—28no—
113C3×#102—24no—
114C3×C22⋊C8—22no—
115#832—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#821—24no—
120#817—24no—
121#809—24no—
122#807—24no—
123#540—64no—
124#540—64no—
125C2×#127—31no—
126C2×#125—31no—
127C2×#123—61no—
128C2×#123—61no—
129C2×#121—31no—
130C2×#121—61no—
131C2×#121—61no—
132C2×#121C2×D311yescharacteristic
133C2×C4×D4—31no—
134C2×C4×D4—31no—
135C2×#117—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138#481—64no—
139#480—64no—
140C2×#107—32no—
141C2×#106—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102C2×D312yes—
146C2×C22⋊C8—31no—
147C2×C22⋊C8C2×D311yescharacteristic
148#389—32no—
149#389—32no—
150#385—61no—
151#385—61no—
152#384—61no—
153#382—61no—
154#340—32no—
155#340C2×D312yes—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C23×D3—21no—
161C2×C3×D4—24no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—82no—
166C2×C3×D4C2×D412yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—24no—
178C2×#67—82no—
179C2×#67—82no—
180C2×#67—82no—
181C22×Dic3—21no—
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—42no—
185C22×Dic3—82no—
186C22×Dic3—82no—
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12C2×D412yes—
195C2×C4×D3—24no—
196C2×C4×D3—82no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4#4114yes—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C42—24no—
207#217—24no—
208#217—22no—
209#217—22no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212C4⋊Dic3—24no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—42no—
219C4⋊Dic3—82no—
220C4⋊Dic3—82no—
221C4⋊Dic3C2×D412yes—
222Dic3⋊C4—24no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×D4—21no—
231C22×D4—31no—
232C22×D4—61no—
233C22×D4—61no—
234C22×D4—61no—
235C22×D4C22×D311yescharacteristic
236C23×C4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C22×C8—32no—
241C22×C8C22×D312yes—
242#127—62no—
243#127—62no—
244#125—62no—
245#125—62no—
246#123—68no—
247#123—68no—
248#121—22no—
249#121—34no—
250#121—68no—
251#121—68no—
252#121—62no—
253#121D1214yes—
254C4×D4—62no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—62no—
258#117—24no—
259#117—62no—
260#117—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—31no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C42—21no—
282C2×C42—31no—
283C2×C42—31no—
284#107—64no—
285#106—64no—
286#102—24no—
287#102—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—22no—
291C22⋊C8—62no—
292#95—62no—
293#95—62no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3#12012yes—
304C23×C3C22×D411yescharacteristic
305C22×D3—22no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—42no—
310C3×D4—24no—
311C3×D4—42no—
312C3×D4—84no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4#12012yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4112yes—
326C2×C3×C42−1+412yes—
327#67—84no—
328#67—84no—
329#67—84no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—84no—
344C2×Dic3—84no—
345D12—24no—
346D12—42no—
347C4×D3—84no—
348C3×C8—42no—
349C3⋊C8—42no—
350C24—21no—
351C24—61no—
352C24D3×D411yescharacteristic
353C2×D4—24no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—32no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—61no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—82no—
365C2×D4—242no—
366C2×D4—242no—
367C2×D4—242no—
368C2×D4C2×D1212yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—64no—
378C22×C4—61no—
379C22×C4—64no—
380C22×C4—62no—
381C22×C4—122no—
382C22×C4—242no—
383C22×C4—242no—
384C22×C4—242no—
385C22×C4C2×D1212yes—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D412yes—
388C22×C4C23×D311yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—64no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4—242no—
407C4⋊C4—242no—
408C4⋊C4C2×D1212yes—
409C4⋊C4#23714yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4—62no—
415C22⋊C4—62no—
416C22⋊C4—64no—
417C42—24no—
418C42—61no—
419C42—61no—
420C42—61no—
421C42—61no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#52511yescharacteristic
434C22×C3#53611yescharacteristic
435C22×C3#57612yes—
436C22×C3#57712yes—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—82no—
441C2×D3—82no—
442C3×C4—22no—
443C3×C4—44no—
444C3×C4—42no—
445C3×C4#54712yes—
446C3×C4#54712yes—
447Dic3—41no—
448Dic3—41no—
449Dic3—41no—
450Dic3—41no—
451Dic3—42no—
452Dic3—82no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—62no—
463C23—121no—
464C23—122no—
465C23—122no—
466C23—121no—
467C23#88212yes—
468C23C2×D3×D411yescharacteristic
469D4—24no—
470D4—42no—
471D4—64no—
472D4—84no—
473D4—122no—
474D4—244no—
475D4—244no—
476D4—244no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—244no—
500C2×C4—244no—
501C2×C4—244no—
502C2×C4#88212yes—
503C2×C4C22×D1211yescharacteristic
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×#23712yes—
506C2×C4#101012yes—
507C8—42no—
508C8—122no—
509C2×C3—42no—
510C2×C3—82no—
511C2×C3#240611yescharacteristic
512C2×C3#300811yescharacteristic
513C2×C3#300911yescharacteristic
514C2×C3#311912yes—
515C2×C3#324112yes—
516D3—82no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—242no—
531C22—242no—
532C22C2×#88211yescharacteristic
533C22#558611yescharacteristic
534C22#564811yescharacteristic
535C22#577012yes—
536C22#577412yes—
537C4—22no—
538C4—42no—
539C4—44no—
540C4—121no—
541C4—121no—
542C4—121no—
543C4—121no—
544C4—122no—
545C4—242no—
546C4#555512yes—
547C4#561112yes—
548C3#2406911yescharacteristic, complemented, Sylow
549C2—42no—
550C2—82no—
551C2—242no—
552C2#7468711yescharacteristic
553C2#7883511yescharacteristic
554C2#7892811yescharacteristic
555C2#8007312yes—
556C2#8136012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radicalC2×#5619C2
hypercenter#1875D3
hypocenterC3#24069
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. C2×#5033C2semidirect
3. #82233C2semidirect
4. C2×#4735C2semidirect
5. C3×#1875C2semidirect
6. #71586C2semidirect
7. #70177C2semidirect
8. #70170C2semidirect
9. #69911C2semidirect
10. #68894C2semidirect
11. #68828C2semidirect
55. #1875D3semidirect
548. C3#24069semidirect

Maximal subgroups

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

subgroupindex
C2×#56192
C2×#50332
#822332
C2×#47352
C3×#18752
#715862
#701772
#701702
#699112
#688942
#688282
#240693