Finite Group Databasebeta

← #10828794

Subgroups — #10828794

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic
2C2×#5034C211yesmaximal, characteristic, complemented
3#82374C211yesmaximal, characteristic, complemented
4C22×#807C211yesmaximal, characteristic, complemented
5#71739C212yesmaximal, complemented
6#71681C211yesmaximal, characteristic, complemented
7#71679C212yesmaximal, complemented
8#71610C211yesmaximal, characteristic, complemented
9#71606C212yesmaximal, complemented
10#71576C212yesmaximal, complemented
11#71554C211yesmaximal, characteristic, complemented
12#24092—31nomaximal, Sylow
13C3×#553—24no—
14C3×#553—24no—
15C2×C3×#126—21no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×#123—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5220—21no—
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5212C2211yescharacteristic
32#5212C2212yes—
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35#5034—24no—
36#4986—21no—
37C2×#808C2212yes—
38C2×#807—24no—
39C2×#807C2212yes—
40C2×#807C2212yes—
41#4949—21no—
42#4946—21no—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#553D311yescharacteristic
49C2×#481—31no—
50C22×#107—31no—
51#2487—31no—
52#2059—32no—
53#2033—32no—
54#1893—31no—
55#1869—31no—
56#1862—32no—
57#1854—32no—
58#1824—31no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×Dic3—21no—
68C3×#126—28no—
69C3×#125—22no—
70C3×#125—22no—
71C3×#123—28no—
72C3×#121—28no—
73C3×#121—22no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121D414yes—
77C3×C4×D4—28no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D414yes—
90C2×C3×C4⋊C4C2311yescharacteristic
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×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2312yes—
108C2×Dic3⋊C4—22no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8—24no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#832—24no—
114#832—24no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—21no—
118#830—22no—
119#830—24no—
120#808—24no—
121#807—24no—
122#807—24no—
123C23×C8—31no—
124#553—24no—
125#553—24no—
126C2×#126—21no—
127C2×#125C2×D311yescharacteristic
128C2×#123—21no—
129C2×#121—21no—
130C2×#121C2×D311yescharacteristic
131C2×#121C2×D311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4C2×D311yescharacteristic
134C2×C4×D4C2×D311yescharacteristic
135C2×#117C2×D311yescharacteristic
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#481—64no—
139C2×#107—32no—
140C2×#107—32no—
141C2×#107—64no—
142C2×#106—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—32no—
147#400—61no—
148#399—61no—
149#389—32no—
150#386—61no—
151#385—61no—
152#380—31no—
153#380—32no—
154#340—32no—
155#340—32no—
156#340—32no—
157#336—32no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×D4—22no—
162C2×C3×D4—24no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—82no—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
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—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4C2×D412yes—
205C3×C4⋊C4C2×D412yes—
206C3×C4⋊C4C2×D412yes—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—22no—
213C3×C42—24no—
214C3×C42—21no—
215C3×C42—21no—
216#217—42no—
217#217—82no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—42no—
223Dic3⋊C4—82no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8#4114yes—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4#6711yescharacteristic
234C22×D4C22×D311yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—61no—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240C22×C8—32no—
241C22×C8—32no—
242C22×C8—64no—
243C22×C8—62no—
244#126—28no—
245#125—22no—
246#125—22no—
247#123—28no—
248#121—28no—
249#121—22no—
250#121—22no—
251#121—22no—
252#121#6714yes—
253C4×D4—28no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258#117—22no—
259#117—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4#6711yescharacteristic
271C2×C4⋊C4#6714yes—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4#6711yescharacteristic
282C2×C22⋊C4#6711yescharacteristic
283C2×C42—21no—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#107—64no—
287#106—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—64no—
291C22⋊C8—62no—
292#95—62no—
293#95—61no—
294#95—62no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3#12012yes—
308C23×C3#13512yes—
309C23×C3#13712yes—
310C23×C3C22×D411yescharacteristic
311C3×D4—24no—
312C3×D4—44no—
313C3×D4—48no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—84no—
339C2×C3×C4—84no—
340C2×C3×C4#12012yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C2×#4112yes—
344C2×C3×C42−1+412yes—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—84no—
352C3⋊C8—24no—
353C3⋊C8—42no—
354C24—21no—
355C24C2×#6711yescharacteristic
356C24C2×#6711yescharacteristic
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—82no—
368C2×D4C2×#6712yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—122no—
385C22×C4—242no—
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C2×#6711yescharacteristic
390C22×C4C2×#6712yes—
391C22×C4C2×#6712yes—
392C22×C4C23×D311yescharacteristic
393C2×C8—34no—
394C2×C8—68no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—122no—
412C4⋊C4—242no—
413C4⋊C4C2×#6712yes—
414C4⋊C4C2×#6712yes—
415C4⋊C4C2×#6712yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—122no—
423C22⋊C4—242no—
424C42—24no—
425C42—21no—
426C42—21no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—41no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3#45214yes—
442C22×C3C2×#12011yescharacteristic
443C22×C3#54712yes—
444C22×C3C2×#13511yescharacteristic
445C22×C3C2×#13711yescharacteristic
446C22×C3#57612yes—
447C3×C4—22no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4#52511yescharacteristic
455C3×C4#53611yescharacteristic
456C3×C4#54712yes—
457Dic3—82no—
458Dic3—82no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23#95012yes—
472C23#95112yes—
473C23#95312yes—
474C23C22×#6711yescharacteristic
475D4—24no—
476D4—44no—
477D4—48no—
478D4—41no—
479D4—41no—
480D4—84no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—244no—
511C2×C4#95312yes—
512C2×C4D3×#4112yes—
513C2×C4#101012yes—
514C2×C4C22×#6711yescharacteristic
515C2×C4C22×#6711yescharacteristic
516C8—64no—
517C8—122no—
518C2×C3—24no—
519C2×C3—81no—
520C2×C3—82no—
521C2×C3#240611yescharacteristic
522C2×C3C2×#45211yescharacteristic
523C2×C3#300811yescharacteristic
524C2×C3#333912yes—
525C2×C3#334612yes—
526C22—22no—
527C22—22no—
528C22—24no—
529C22—24no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22#523914yes—
541C22C2×#95011yescharacteristic
542C22C2×#95111yescharacteristic
543C22#582412yes—
544C22#583512yes—
545C22C2×#95311yescharacteristic
546C4—22no—
547C4—41no—
548C4—42no—
549C4—41no—
550C4—41no—
551C4—42no—
552C4—42no—
553C4—242no—
554C4—242no—
555C4#582412yes—
556C4#582511yescharacteristic
557C4#582811yescharacteristic
558C3#2409211yescharacteristic, complemented, Sylow
559C2—24no—
560C2—81no—
561C2—82no—
562C2#8360712yes—
563C2#8360912yes—
564C2C2×#523911yescharacteristic
565C2#8409011yescharacteristic
566C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#553C2
socleC23×C3C22×D4
hypercenterC2×#553D3
hypocenterC3#24092
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
2. C2×#5034C2semidirect
3. #82374C2semidirect
4. C22×#807C2semidirect
5. #71739C2semidirect
6. #71681C2semidirect
7. #71679C2semidirect
8. #71610C2semidirect
9. #71606C2semidirect
10. #71576C2semidirect
11. #71554C2semidirect
558. C3#24092semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5532
C2×#50342
#823742
C22×#8072
#717392
#716812
#716792
#716102
#716062
#715762
#715542
#240923