Finite Group Databasebeta

← #10828541

Subgroups — #10828541

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×C3×#540C211yesmaximal, characteristic
2C2×#5034C211yesmaximal, characteristic, complemented
3C2×#5033C211yesmaximal, characteristic, complemented
4#82234C211yesmaximal, characteristic, complemented
5#71762C212yesmaximal, complemented
6#71739C212yesmaximal, complemented
7#71678C211yesmaximal, characteristic, complemented
8#71678C211yesmaximal, characteristic, complemented
9#71607C212yesmaximal, complemented
10#71606C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic, complemented
12#24069—31nomaximal, Sylow
13C3×#540—24no—
14C3×#540—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#123—21no—
18C2×C3×#123—21no—
19C2×C3×#121—21no—
20C2×C3×#121—21no—
21C2×C3×#121C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32C2×#830C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#5034—24no—
35#5033—24no—
36#4977—21no—
37#4975—21no—
38C2×#808C2212yes—
39C2×#807C2212yes—
40#4946—21no—
41#4946—21no—
42#4928C2212yes—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#540D311yescharacteristic
49C2×#481—31no—
50C2×#480—31no—
51#2317—31no—
52#2059—32no—
53#2041—32no—
54#1875—31no—
55#1875—31no—
56#1864—32no—
57#1862—32no—
58#1818—31no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—21no—
69C3×#127—22no—
70C3×#127—22no—
71C3×#125—22no—
72C3×#125—22no—
73C3×#123—28no—
74C3×#123—28no—
75C3×#121—28no—
76C3×#121—28no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121D414yes—
80C3×#121D414yes—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102#926—24no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2312yes—
108C2×Dic3⋊C4—22no—
109C2×Dic3⋊C4—22no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#832—24no—
114#832—24no—
115#832—24no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—22no—
120#830—22no—
121#808—24no—
122#807—24no—
123#540—24no—
124#540—24no—
125C2×#127C2×D311yescharacteristic
126C2×#125C2×D311yescharacteristic
127C2×#123—21no—
128C2×#123—21no—
129C2×#121—21no—
130C2×#121—21no—
131C2×#121C2×D311yescharacteristic
132C2×#121C2×D311yescharacteristic
133C2×C4×D4C2×D311yescharacteristic
134C2×C4×D4C2×D311yescharacteristic
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×#102—32no—
146C2×C22⋊C8—31no—
147C2×C22⋊C8—31no—
148#389—32no—
149#389—32no—
150#385—61no—
151#385—61no—
152#384—61no—
153#382—61no—
154#340—32no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—82no—
172C2×C3×D4—82no—
173C2×C3×D4—82no—
174C2×C3×D4—82no—
175C2×C3×D4C2×D412yes—
176C2×C3×D4C2×D412yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—82no—
184C22×C3×C4—82no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—42no—
196C22×Dic3—82no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C2×D412yes—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215#217—24no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220C4⋊Dic3#4114yes—
221Dic3⋊C4—82no—
222Dic3⋊C4—82no—
223C4×Dic3—24no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C22×D311yescharacteristic
235C22×D4C22×D311yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C22×C8—32no—
241C22×C8—32no—
242#127—22no—
243#127—22no—
244#125—22no—
245#125—22no—
246#123—28no—
247#123—28no—
248#121—28no—
249#121—28no—
250#121—22no—
251#121—22no—
252#121#6714yes—
253#121#6714yes—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258#117—22no—
259#117—22no—
260#117—64no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4#6711yescharacteristic
271C2×C4⋊C4#6711yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4#6711yescharacteristic
280C2×C22⋊C4#6711yescharacteristic
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284#107—64no—
285#106—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—62no—
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—22no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3#12012yes—
309C23×C3C22×D411yescharacteristic
310C3×D4—24no—
311C3×D4—24no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—84no—
315C3×D4—84no—
316C3×D4—84no—
317C3×D4—84no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—84no—
335C2×C3×C4—84no—
336C2×C3×C4#12012yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4112yes—
340C2×C3×C42−1+412yes—
341C2×Dic3—24no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—84no—
348C3⋊C8—42no—
349C3⋊C8—42no—
350C24—21no—
351C24—21no—
352C24C2×#6711yescharacteristic
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4C2×#6712yes—
368C2×D4C2×#6712yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—122no—
380C22×C4—242no—
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6712yes—
387C22×C4C2×#6712yes—
388C22×C4C23×D311yescharacteristic
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—34no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4—242no—
407C4⋊C4—242no—
408C4⋊C4C2×#6712yes—
409C4⋊C4C2×#6712yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—64no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—64no—
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—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#52511yescharacteristic
439C22×C3#53611yescharacteristic
440C22×C3#57612yes—
441C22×C3#57712yes—
442C3×C4—22no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—42no—
449C3×C4#54712yes—
450C3×C4#54712yes—
451Dic3—44no—
452Dic3—82no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23#95312yes—
468C23C22×#6711yescharacteristic
469D4—24no—
470D4—24no—
471D4—42no—
472D4—42no—
473D4—84no—
474D4—84no—
475D4—84no—
476D4—84no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—64no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—244no—
502C2×C4#95312yes—
503C2×C4D3×#4112yes—
504C2×C4#101012yes—
505C2×C4C22×#6711yescharacteristic
506C2×C4C22×#6711yescharacteristic
507C8—122no—
508C8—122no—
509C2×C3—42no—
510C2×C3—82no—
511C2×C3—82no—
512C2×C3#240611yescharacteristic
513C2×C3#300811yescharacteristic
514C2×C3#300911yescharacteristic
515C2×C3#311912yes—
516C2×C3#324112yes—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22#582511yescharacteristic
533C22#582811yescharacteristic
534C22#583512yes—
535C22#583612yes—
536C22C2×#95311yescharacteristic
537C4—22no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—124no—
545C4—242no—
546C4#582412yes—
547C4#582412yes—
548C3#2406911yescharacteristic, complemented, Sylow
549C2—42no—
550C2—82no—
551C2—82no—
552C2#8343012yes—
553C2#8343412yes—
554C2#8409011yescharacteristic
555C2#8410111yescharacteristic
556C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#540C2
socleC23×C3C22×D4
hypercenterC2×#540D3
hypocenterC3#24069
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
2. C2×#5034C2semidirect
3. C2×#5033C2semidirect
4. #82234C2semidirect
5. #71762C2semidirect
6. #71739C2semidirect
7. #71678C2semidirect
8. #71678C2semidirect
9. #71607C2semidirect
10. #71606C2semidirect
11. #71584C2semidirect
548. C3#24069semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×#50342
C2×#50332
#822342
#717622
#717392
#716782
#716072
#716062
#715842
#240693