Finite Group Databasebeta

← #10828545

Subgroups — #10828545

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×#5535C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic, complemented
3C2×#4968C211yesmaximal, characteristic, complemented
4C2×#4967C211yesmaximal, characteristic, complemented
5#71586C211yesmaximal, characteristic, complemented
6#71586C211yesmaximal, characteristic, complemented
7#71584C211yesmaximal, characteristic, complemented
8#69627C212yesmaximal, complemented
9#69626C212yesmaximal, complemented
10#69625C212yesmaximal, complemented
11#69606C212yesmaximal, complemented
12#24069—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#930C2211yescharacteristic
18#5535—24no—
19#5535—24no—
20C2×#894—21no—
21C2×#894—21no—
22C2×#884—21no—
23C2×#884—21no—
24C2×#874C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#870C2211yescharacteristic
28C3×#389C2212yes—
29C3×#389C2212yes—
30C3×#385—21no—
31C3×#385—21no—
32C3×#384—21no—
33C3×#382—21no—
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#4968—24no—
37#4967—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#540—31no—
49C2×#481—31no—
50C2×#480—31no—
51#2317D311yescharacteristic, complemented
52#2059—32no—
53#2041—32no—
54#1875—31no—
55#1875—31no—
56#1864—32no—
57#1862—32no—
58#1818—31no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67—21no—
63C22×D12—21no—
64C22×D12—21no—
65C22×D12C2311yescharacteristic
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C22×C4×D3—21no—
69C3×#117—22no—
70C3×#117—22no—
71C3×#117—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C22⋊C4D411yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87#930—22no—
88#930—22no—
89#930D414yes—
90#930D414yes—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2312yes—
101#894—28no—
102#894—28no—
103#884—28no—
104#884—28no—
105#874—22no—
106#874—22no—
107C4×D12—22no—
108C4×D12—22no—
109C4×D12—22no—
110C4×D12—22no—
111#870—22no—
112#870—22no—
113C3×#95—22no—
114C3×#95—22no—
115#830—22no—
116#830—22no—
117#809—24no—
118#809—24no—
119#809—24no—
120#809—24no—
121#808—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—31no—
131C2×#121—61no—
132C2×#121—61no—
133C2×C4×D4—31no—
134C2×C4×D4—31no—
135C2×#117C2×D311yescharacteristic
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
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#389C2×D312yes—
149#389C2×D312yes—
150#385—21no—
151#385—21no—
152#384—21no—
153#382—21no—
154#340—32no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C23×D3—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—42no—
166C22×C3×C4—82no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—82no—
178C2×#67—82no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—24no—
188C2×D12—24no—
189C2×D12—82no—
190C2×D12—82no—
191C2×D12C2×D412yes—
192C2×D12C2×D412yes—
193C2×C4×D3—24no—
194C2×C4×D3—24no—
195C2×C4×D3—82no—
196C2×C4×D3—82no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—82no—
204C3×C4⋊C4—82no—
205C3×C4⋊C4#4114yes—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C42—24no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3C2×D412yes—
221C4⋊Dic3C2×D412yes—
222Dic3⋊C4—24no—
223Dic3⋊C4—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—31no—
231C22×D4—31no—
232C22×D4—61no—
233C22×D4—61no—
234C22×D4—61no—
235C22×D4—61no—
236C23×C4—21no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C22×C8—32no—
241C22×C8—32no—
242#127—62no—
243#127—62no—
244#125—62no—
245#125—62no—
246#123—68no—
247#123—68no—
248#121—34no—
249#121—34no—
250#121—68no—
251#121—68no—
252#121—62no—
253#121—62no—
254C4×D4—62no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—62no—
258#117—22no—
259#117—22no—
260#117—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4#6711yescharacteristic
271C2×C4⋊C4#6711yescharacteristic
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4#6711yescharacteristic
280C2×C22⋊C4#6711yescharacteristic
281C2×C42—21no—
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—22no—
293#95—22no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3#12012yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—84no—
329C2×C3×C4#12012yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4112yes—
333C2×C3×C42−1+412yes—
334#67—84no—
335#67—84no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340D12—24no—
341D12—24no—
342D12—42no—
343D12—42no—
344D12—84no—
345D12—84no—
346C4×D3—84no—
347C4×D3—84no—
348C3⋊C8—42no—
349C3⋊C8—42no—
350C24—61no—
351C24—61no—
352C24C2×#6711yescharacteristic
353C2×D4—32no—
354C2×D4—32no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—242no—
366C2×D4—242no—
367C2×D4—242no—
368C2×D4—242no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—64no—
375C22×C4—64no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—82no—
379C22×C4—242no—
380C22×C4—242no—
381C22×C4D3×D412yes—
382C22×C4D3×D412yes—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C23×D311yescharacteristic
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—32no—
401C4⋊C4—32no—
402C4⋊C4—42no—
403C4⋊C4—64no—
404C4⋊C4—64no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4#23514yes—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C42—24no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3C2×#12011yescharacteristic
428C22×C3#52511yescharacteristic
429C22×C3#53611yescharacteristic
430C22×C3#57612yes—
431C22×C3#57712yes—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C3×C4—22no—
443C3×C4—44no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—82no—
449C3×C4#54712yes—
450C3×C4#54712yes—
451Dic3—42no—
452Dic3—42no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—62no—
458C23—62no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23#93712yes—
468C23C22×#6711yescharacteristic
469D4—64no—
470D4—64no—
471D4—122no—
472D4—122no—
473D4—244no—
474D4—244no—
475D4—244no—
476D4—244no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—84no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—244no—
501C2×C4—244no—
502C2×C4#93712yes—
503C2×C4C2×#23512yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4#100712yes—
507C8—122no—
508C8—122no—
509C2×C3—42no—
510C2×C3#240611yescharacteristic
511C2×C3#300811yescharacteristic
512C2×C3#300911yescharacteristic
513C2×C3#311912yes—
514C2×C3#324112yes—
515D3—82no—
516D3—82no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—242no—
529C22—242no—
530C22—242no—
531C22—242no—
532C22#579111yescharacteristic
533C22C2×#93711yescharacteristic
534C22#581611yescharacteristic
535C22#583512yes—
536C22#583712yes—
537C4—22no—
538C4—44no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—82no—
544C4—122no—
545C4—122no—
546C4#550712yes—
547C4#550712yes—
548C3#2406911yescharacteristic, complemented, Sylow
549C2—42no—
550C2—242no—
551C2—242no—
552C2#8219211yescharacteristic
553C2#8256812yes—
554C2#8338311yescharacteristic
555C2#8340411yescharacteristic
556C2#8347712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC23×C3C22×D4
prime radicalC2×#5535C2
hypercenter#2317D3
hypocenterC3#24069
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C3×#2317C2semidirect
3. C2×#4968C2semidirect
4. C2×#4967C2semidirect
5. #71586C2semidirect
6. #71586C2semidirect
7. #71584C2semidirect
8. #69627C2semidirect
9. #69626C2semidirect
10. #69625C2semidirect
11. #69606C2semidirect
51. #2317D3semidirect
548. C3#24069semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C3×#23172
C2×#49682
C2×#49672
#715862
#715842
#696272
#696262
#696252
#696062
#240693