Finite Group Databasebeta

← #10943849

Subgroups — #10943849

Order 768.

3926 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#41C211yesmaximal, characteristic
2C2×#5239C214yesmaximal, complemented
3C2×#5228C211yesmaximal, characteristic, complemented
4C2×#5218C211yesmaximal, characteristic, complemented
5#84031C211yesmaximal, characteristic, complemented
6#72135C214yesmaximal, complemented
7#72132C211yesmaximal, characteristic, complemented
8#72128C211yesmaximal, characteristic, complemented
9#72126C211yesmaximal, characteristic, complemented
10#32883—31nomaximal, Sylow
11C22×C3×#41—216no—
12C22×C3×#41—24no—
13C22×C3×#41—24no—
14C22×C3×#41—22no—
15C22×C3×#41C2214yes—
16C23×C3×Q8C2211yescharacteristic
17C23×C3×D4—21no—
18C23×C3×D4C2211yescharacteristic
19C24×C3×C4—21no—
20C24×C3×C4C2211yescharacteristic
21C22×#216C2212yescomplemented
22C22×#214C2212yescomplemented
23C2×#924C2211yescharacteristic
24C2×#924C2212yescomplemented
25#5247—21no—
26#5239—216no—
27#5228—24no—
28C2×#835C2212yescomplemented
29C2×#835C2214yescomplemented
30#5218—24no—
31C2×#832C2212yescomplemented
32C2×#832C2214yescomplemented
33#5212C2211yescharacteristic
34#5212C2212yescomplemented
35#5210—22no—
36#5210—21no—
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C2×#830C2214yescomplemented
40C23×#41D311yescharacteristic
41C2×#452—34no—
42C2×#451—31no—
43C2×#450—31no—
44#2848—31no—
45#1788—34no—
46#1787—31no—
47#1786—31no—
48#1779—31no—
49C25×C3—21no—
50C25×C3D411yescharacteristic
51C2×C3×#41—264no—
52C2×C3×#41—264no—
53C2×C3×#41—216no—
54C2×C3×#41—216no—
55C2×C3×#41—216no—
56C2×C3×#41—28no—
57C2×C3×#41—28no—
58C2×C3×#41—28no—
59C2×C3×#41—28no—
60C2×C3×#41—28no—
61C2×C3×#41D4116yes—
62C22×C3×Q8—28no—
63C22×C3×Q8—22no—
64C22×C3×Q8—21no—
65C22×C3×Q8D414yes—
66C22×C3×Q8C2312yes—
67C22×C3×D4—216no—
68C22×C3×D4—28no—
69C22×C3×D4—24no—
70C22×C3×D4—28no—
71C22×C3×D4—21no—
72C22×C3×D4—22no—
73C22×C3×D4D414yes—
74C22×C3×D4C2312yes—
75C23×C3×C4—216no—
76C23×C3×C4—28no—
77C23×C3×C4—21no—
78C23×C3×C4—24no—
79C23×C3×C4—22no—
80C23×C3×C4—22no—
81C23×C3×C4—28no—
82C23×C3×C4D411yescharacteristic
83C23×C3×C4D412yes—
84C23×C3×C4C2311yescharacteristic
85C23×C3×C4C2312yes—
86C22×Dic6C2311yescharacteristic
87C2×#217—21no—
88C2×#217—22no—
89C2×#217—21no—
90C2×#217—22no—
91C2×#216—24no—
92C2×#216—28no—
93C2×#214—24no—
94C2×#214—28no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97#924—22no—
98#924—22no—
99#924—28no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—22no—
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104#835—24no—
105#835—28no—
106#832—24no—
107#832—28no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#830—22no—
111#830—22no—
112#830—28no—
113C22×#41—216no—
114C22×#41—24no—
115C22×#41—24no—
116C22×#41—22no—
117C22×#41C2×D314yes—
118C23×Q8C2×D311yescharacteristic
119C23×D4—21no—
120C23×D4C2×D311yescharacteristic
121C24×C4—21no—
122C24×C4C2×D311yescharacteristic
123C22×Q16—32no—
124C22×SD16—32no—
125C2×C22⋊Q8—31no—
126C2×C22⋊Q8—32no—
127#452—616no—
128#451—64no—
129#450—64no—
130C2×#103—32no—
131C2×#103—34no—
132C2×#102—32no—
133C2×#102—34no—
134C2×C22⋊C8—31no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8—34no—
137#385—61no—
138#385—62no—
139#380—31no—
140#380—32no—
141#379—61no—
142C24×C3—28no—
143C24×C3—24no—
144C24×C3—24no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—42no—
150C24×C3—44no—
151C24×C3—42no—
152C24×C3—42no—
153C24×C3C2×D412yes—
154C24×C3C2×D411yescharacteristic
155C3×#41—2128no—
156C3×#41—232no—
157C3×#41—232no—
158C3×#41—232no—
159C3×#41—232no—
160C2×C3×Q8—216no—
161C2×C3×Q8—28no—
162C2×C3×Q8—216no—
163C2×C3×Q8—24no—
164C2×C3×Q8—24no—
165C2×C3×Q8—22no—
166C2×C3×Q8C2×D418yes—
167C2×C3×Q8C2×D414yes—
168C2×C3×D4—264no—
169C2×C3×D4—216no—
170C2×C3×D4—216no—
171C2×C3×D4—28no—
172C2×C3×D4—216no—
173C2×C3×D4—216no—
174C2×C3×D4—216no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4C2×D418yes—
179C2×C3×D4C2×D414yes—
180C22×C3×C4—28no—
181C22×C3×C4—264no—
182C22×C3×C4—216no—
183C22×C3×C4—216no—
184C22×C3×C4—216no—
185C22×C3×C4—216no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—216no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—22no—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D418yes—
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—22no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×Dic6—21no—
208C2×Dic6—24no—
209C2×Dic6—21no—
210#217—44no—
211#217—44no—
212#217—42no—
213#217—42no—
214#216—48no—
215#214—48no—
216C4⋊Dic3—21no—
217C4⋊Dic3—24no—
218C4⋊Dic3—21no—
219Dic3⋊C4—44no—
220Dic3⋊C4—42no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—24no—
223C2×C3⋊C8—24no—
224C2×C3⋊C8—22no—
225C25—21no—
226C25#6711yescharacteristic
227C2×#41—264no—
228C2×#41—264no—
229C2×#41—216no—
230C2×#41—216no—
231C2×#41—216no—
232C2×#41—28no—
233C2×#41—28no—
234C2×#41—28no—
235C2×#41—28no—
236C2×#41—28no—
237C2×#41#67116yes—
238C22×Q8—28no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—31no—
242C22×Q8#6714yes—
243C22×Q8C22×D312yes—
244C22×D4—216no—
245C22×D4—28no—
246C22×D4—24no—
247C22×D4—28no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4#6714yes—
251C22×D4C22×D312yes—
252C23×C4—216no—
253C23×C4—28no—
254C23×C4—21no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—28no—
259C23×C4#6711yescharacteristic
260C23×C4#6712yes—
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D312yes—
263C2×Q16—64no—
264C2×Q16—68no—
265C2×SD16—64no—
266C2×SD16—68no—
267C22×C8—32no—
268C22×C8—32no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271C22⋊Q8—68no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280#103—64no—
281#103—68no—
282#102—64no—
283#102—68no—
284C22⋊C8—62no—
285C22⋊C8—62no—
286C22⋊C8—68no—
287#95—62no—
288#95—61no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—28no—
292C23×C3—216no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3—28no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3—416no—
304C23×C3—44no—
305C23×C3—48no—
306C23×C3—416no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—48no—
312C23×C3#12014yes—
313C23×C3#12012yes—
314C23×C3#13512yes—
315C23×C3#13712yes—
316C23×C3C22×D411yescharacteristic
317C3×Q8—216no—
318C3×Q8—24no—
319C3×Q8—28no—
320C3×Q8#12018yes—
321C3×D4—264no—
322C3×D4—216no—
323C3×D4—28no—
324C3×D4—24no—
325C3×D4#12018yes—
326C2×C3×C4—28no—
327C2×C3×C4—216no—
328C2×C3×C4—264no—
329C2×C3×C4—216no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—28no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—28no—
338C2×C3×C4—28no—
339C2×C3×C4#12014yes—
340C2×C3×C4#12012yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D414yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×Dic3—42no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351Dic6—42no—
352Dic6—42no—
353C3⋊C8—42no—
354C3⋊C8—42no—
355C24—28no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—42no—
363C24—44no—
364C24—42no—
365C24—42no—
366C24C2×#6712yes—
367C24C2×#6711yescharacteristic
368#41—2128no—
369#41—232no—
370#41—232no—
371#41—232no—
372#41—232no—
373C2×Q8—216no—
374C2×Q8—28no—
375C2×Q8—216no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—61no—
380C2×Q8—64no—
381C2×Q8—61no—
382C2×Q8C2×#6718yes—
383C2×Q8C2×#6714yes—
384C2×D4—264no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—28no—
388C2×D4—216no—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4C2×#6718yes—
395C2×D4C2×#6714yes—
396C22×C4—28no—
397C22×C4—264no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4C2×#6712yes—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6712yes—
420C22×C4C2×#6712yes—
421C22×C4C2×#6718yes—
422C22×C4C23×D311yescharacteristic
423Q16—128no—
424SD16—128no—
425C2×C8—62no—
426C2×C8—64no—
427C2×C8—64no—
428C2×C8—62no—
429C4⋊C4—61no—
430C4⋊C4—64no—
431C4⋊C4—61no—
432C4⋊C4—124no—
433C4⋊C4—122no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C22⋊C4—124no—
437C22⋊C4—122no—
438C22×C3—22no—
439C22×C3—24no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—28no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—416no—
447C22×C3—432no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—42no—
451C22×C3—48no—
452C22×C3—42no—
453C22×C3—44no—
454C22×C3—42no—
455C22×C3—48no—
456C22×C3#45214yescomplemented
457C22×C3C2×#12011yescharacteristic
458C22×C3C2×#12012yes—
459C22×C3C2×#13511yescharacteristic
460C22×C3C2×#13711yescharacteristic
461C22×C3#57512yes—
462C3×C4—24no—
463C3×C4—28no—
464C3×C4—22no—
465C3×C4C2×#12012yes—
466C3×C4C2×#12012yes—
467Dic3—81no—
468Dic3—82no—
469Dic3—81no—
470C23—22no—
471C23—22no—
472C23—28no—
473C23—216no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—24no—
482C23—48no—
483C23—48no—
484C23—416no—
485C23—44no—
486C23—48no—
487C23—416no—
488C23—44no—
489C23—48no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23#95012yes—
494C23#95112yes—
495C23#95314yes—
496C23#95312yes—
497C23C22×#6711yescharacteristic
498Q8—216no—
499Q8—24no—
500Q8—28no—
501Q8—122no—
502Q8—122no—
503Q8#95318yes—
504D4—264no—
505D4—216no—
506D4—28no—
507D4—24no—
508D4#95318yes—
509C2×C4—28no—
510C2×C4—216no—
511C2×C4—264no—
512C2×C4—216no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4#95314yes—
530C2×C4#95312yes—
531C2×C4C22×#6711yescharacteristic
532C2×C4C22×#6714yes—
533C2×C4C22×#6711yescharacteristic
534C8—122no—
535C8—122no—
536C2×C3—24no—
537C2×C3—44no—
538C2×C3—48no—
539C2×C3#284611yescharacteristic
540C2×C3C2×#45211yescharacteristic
541C2×C3#300212yes—
542C2×C3#300511yescharacteristic
543C2×C3#300712yes—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—28no—
552C22—416no—
553C22—432no—
554C22—42no—
555C22—48no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—48no—
562C22#523914yescomplemented
563C22C2×#95011yescharacteristic
564C22#582212yes—
565C22C2×#95111yescharacteristic
566C22C2×#95312yes—
567C22C2×#95311yescharacteristic
568C4—24no—
569C4—28no—
570C4—22no—
571C4—242no—
572C4—241no—
573C4—241no—
574C4C2×#95312yes—
575C4C2×#95312yes—
576C3#3288311yescharacteristic, complemented, Sylow
577C2—24no—
578C2—44no—
579C2—48no—
580C2#8408012yes—
581C2#8408111yescharacteristic
582C2C2×#523911yescharacteristic
583C2#8408412yes—
584C2#8412811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC23×C3×#41C2
socleC23×C3C22×D4
hypercenterC23×#41D3
hypocenterC3#32883
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
2. C2×#5239C2semidirect
3. C2×#5228C2semidirect
4. C2×#5218C2semidirect
5. #84031C2semidirect
6. #72135C2semidirect
7. #72132C2semidirect
8. #72128C2semidirect
9. #72126C2semidirect
21. C22×#216C22semidirect
22. C22×#214C22semidirect
24. C2×#924C22semidirect
28. C2×#835C22semidirect
29. C2×#835C22semidirect
31. C2×#832C22semidirect
32. C2×#832C22semidirect
34. #5212C22semidirect
39. C2×#830C22semidirect
456. C22×C3#452semidirect
562. C22#5239semidirect
576. C3#32883semidirect

Maximal subgroups

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

subgroupindex
C23×C3×#412
C2×#52392
C2×#52282
C2×#52182
#840312
#721352
#721322
#721282
#721262
#328833