Finite Group Databasebeta

← #10828763

Subgroups — #10828763

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5507C211yesmaximal, characteristic
2C22×#808C211yesmaximal, characteristic, complemented
3C2×#4735C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic, complemented
5C3×#1823C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic, complemented
7#69515C211yesmaximal, characteristic, complemented
8#68895C212yesmaximal, complemented
9#68875C212yesmaximal, complemented
10#68827C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#24089—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×#929—21no—
17C2×C4×#67C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C2×#896—21no—
21#5507—24no—
22#5507—24no—
23C2×#895C2211yescharacteristic
24C2×#894—21no—
25C2×#893C2211yescharacteristic
26C2×D3×C4⋊C4C2211yescharacteristic
27C2×#888C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C2×#821C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2212yes—
33#4987C2212yes—
34#4984C2212yes—
35C2×#809C2212yes—
36C2×#808—24no—
37C2×#808C2212yes—
38C2×#808C2212yes—
39#4958—21no—
40#4950—21no—
41#4735—24no—
42#4677—21no—
43#4659—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#547—31no—
49C2×#481—31no—
50C22×#106—31no—
51#2487—31no—
52#2058—32no—
53#2034—32no—
54#1893—31no—
55#1869—31no—
56#1864—32no—
57#1853—32no—
58#1823D311yescharacteristic, complemented
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67D411yescharacteristic
63C23×Dic3—21no—
64C2×#235—28no—
65C2×#235—44no—
66C2×#235—44no—
67C22×D12—21no—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3—21no—
71C22×C4×D3C2311yescharacteristic
72C22×Dic6—21no—
73C22×C3×C8C2312yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D414yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×#217D411yescharacteristic
82#930—22no—
83#930D414yes—
84#929—28no—
85C4×#67—22no—
86C4×#67—22no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4D411yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8—22no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8C2312yes—
105#896—28no—
106#895—22no—
107#895—22no—
108#894—28no—
109#893—22no—
110#893—22no—
111D3×C4⋊C4—22no—
112D3×C4⋊C4—22no—
113#888—22no—
114#888—22no—
115C3×C22⋊C8—22no—
116C3×#95—21no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#821—24no—
120(C2×D3)⋊C8—24no—
121#817—24no—
122#809—24no—
123#808—24no—
124#808—24no—
125C22×#41—61no—
126C23×C8—31no—
127#547—64no—
128#547—64no—
129C2×C4⋊Q8—31no—
130C2×#123—61no—
131C2×#123—61no—
132C2×C22⋊Q8—61no—
133C2×#121—31no—
134C2×#121—31no—
135C2×C4×D4—31no—
136C2×C4×D4—31no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140#481—64no—
141C2×#106—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8—32no—
148C2×C22⋊C8C2×D311yescharacteristic
149#400—61no—
150#399—61no—
151#389—32no—
152#386—61no—
153#385—61no—
154#380—32no—
155#380C2×D311yescharacteristic
156#340—32no—
157#340C2×D312yes—
158#340C2×D312yes—
159#336—32no—
160C24×C3C2×D411yescharacteristic
161C23×D3—21no—
162C23×D3C2×D411yescharacteristic
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—24no—
178C2×#67—42no—
179C2×#67—42no—
180C2×#67SD1614yes—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—42no—
184C22×Dic3—82no—
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188#235—432no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—24no—
192C2×D12—21no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12C2×D412yes—
196C2×C4×D3—24no—
197C2×C4×D3—24no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×Dic6—24no—
207C2×Dic6—42no—
208C2×Dic6—42no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4C2×D412yes—
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—42no—
220#217—22no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—82no—
226C4⋊Dic3—21no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—42no—
230C4⋊Dic3—82no—
231C4⋊Dic3C2×D412yes—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234Dic3⋊C4—22no—
235C4×Dic3—21no—
236C4×Dic3—21no—
237C2×C3⋊C8—28no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8#4114yescomplemented
241C2×#41—68no—
242C2×#41—124no—
243C2×#41—124no—
244C22×Q8—61no—
245C22×D4—31no—
246C22×D4—31no—
247C22×D4—61no—
248C22×D4—61no—
249C23×C4—21no—
250C23×C4—31no—
251C23×C4—61no—
252C23×C4—61no—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C22×C8—32no—
256C22×C8—64no—
257C22×C8—62no—
258C22×C8C22×D312yes—
259C4⋊Q8—62no—
260C4⋊Q8—62no—
261#123—68no—
262#123—68no—
263C22⋊Q8—68no—
264#121—34no—
265#121—62no—
266#121—62no—
267#121—62no—
268C4×D4—62no—
269C4×D4—62no—
270C4×D4—62no—
271C4×D4—62no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4D1214yes—
286C2×C4⋊C4C22×D312yes—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—62no—
296C2×C42—31no—
297#106—64no—
298#106—64no—
299#106—64no—
300#102—64no—
301#102—64no—
302C22⋊C8—22no—
303C22⋊C8—64no—
304#95—21no—
305#95—62no—
306#95—62no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3#12012yes—
310C23×C3C2×SD1612yes—
311C23×C3AGL(1,ℤ/8)12yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—21no—
314C22×D3—22no—
315C22×D3—21no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—42no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3C2×SD1612yes—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—48no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12012yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C2×#4112yes—
341C2×C3×C42−1+412yes—
342#67—28no—
343#67—48no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—84no—
357C2×Dic3C2×SD1612yes—
358D12—24no—
359D12—48no—
360D12—41no—
361D12—41no—
362C4×D3—48no—
363C4×D3—48no—
364C4×D3—41no—
365C4×D3—41no—
366C4×D3—42no—
367Dic6—48no—
368C3×C8—42no—
369C3⋊C8—24no—
370C24—31no—
371C24—61no—
372C24D3×D411yescharacteristic
373#41—1232no—
374C2×Q8—64no—
375C2×Q8—122no—
376C2×Q8—122no—
377C2×D4—34no—
378C2×D4—32no—
379C2×D4—64no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—64no—
383C2×D4—61no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—122no—
387C2×D4—122no—
388C2×D4—122no—
389C2×D4—122no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—242no—
415C22×C4C2×D1212yes—
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D412yes—
418C22×C4C23×D311yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—34no—
423C2×C8—68no—
424C2×C8—62no—
425C2×C8—62no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—32no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—62no—
435C4⋊C4—61no—
436C4⋊C4—62no—
437C4⋊C4—62no—
438C4⋊C4—122no—
439C4⋊C4—242no—
440C4⋊C4C2×D1212yes—
441C4⋊C4C2×D1212yes—
442C22⋊C4—42no—
443C22⋊C4—64no—
444C22⋊C4—64no—
445C22⋊C4—62no—
446C22⋊C4—62no—
447C22⋊C4—64no—
448C22⋊C4—242no—
449C42—61no—
450C42—61no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3#45014yes—
456C22×C3C2×#12011yescharacteristic
457C22×C3#54712yes—
458C22×C3C22×SD1611yescharacteristic
459C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
460C22×C3#57712yes—
461C2×D3—24no—
462C2×D3—22no—
463C2×D3—42no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—42no—
467C2×D3—42no—
468C2×D3—42no—
469C2×D3—82no—
470C2×D3—82no—
471C3×C4—22no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—82no—
475C3×C4#52511yescharacteristic
476C3×C4#53611yescharacteristic
477C3×C4#54712yes—
478Dic3—22no—
479Dic3—41no—
480Dic3—42no—
481Dic3—41no—
482Dic3—82no—
483C23—22no—
484C23—22no—
485C23—32no—
486C23—62no—
487C23—61no—
488C23—61no—
489C23—62no—
490C23—62no—
491C23—122no—
492C23—121no—
493C23—121no—
494C23—122no—
495C23#88212yes—
496C23#91112yes—
497C23D3×SD1612yes—
498C23C2×D3×D411yescharacteristic
499Q8—128no—
500D4—68no—
501D4—64no—
502D4—128no—
503D4—128no—
504D4—121no—
505D4—121no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—32no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—128no—
529C2×C4—128no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—244no—
539C2×C4#88212yes—
540C2×C4C22×D1211yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4D3×#4112yes—
543C2×C4#101012yes—
544C8—42no—
545C8—64no—
546C2×C3—24no—
547C2×C3#240611yescharacteristic
548C2×C3C2×#45011yescharacteristic
549C2×C3#301011yescharacteristic
550C2×C3#333512yes—
551C2×C3#334512yes—
552D3—42no—
553D3—82no—
554C22—24no—
555C22—22no—
556C22—22no—
557C22—24no—
558C22—64no—
559C22—62no—
560C22—121no—
561C22—121no—
562C22—122no—
563C22—122no—
564C22—122no—
565C22—122no—
566C22—242no—
567C22—242no—
568C22#477614yes—
569C22C2×#88211yescharacteristic
570C22#565512yes—
571C22C2×#91111yescharacteristic
572C22C2×D3×SD1611yescharacteristic
573C22#577812yes—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—62no—
578C4—82no—
579C4—121no—
580C4—121no—
581C4—122no—
582C4—242no—
583C4#552411yescharacteristic
584C4#552611yescharacteristic
585C4#565512yes—
586C3#2408911yescharacteristic, complemented, Sylow
587C2—24no—
588C2—122no—
589C2—242no—
590C2#7406311yescharacteristic
591C2C2×#477611yescharacteristic
592C2#7896011yescharacteristic
593C2#8129412yes—
594C2#8143512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC2×#5507C2
hypercenter#1823D3
hypocenterC3#24089
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. C22×#808C2semidirect
3. C2×#4735C2semidirect
4. #74028C2semidirect
5. C3×#1823C2semidirect
6. #69952C2semidirect
7. #69515C2semidirect
8. #68895C2semidirect
9. #68875C2semidirect
10. #68827C2semidirect
11. #68823C2semidirect
58. #1823D3semidirect
240. C2×C3⋊C8#41semidirect
586. C3#24089semidirect

Maximal subgroups

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

subgroupindex
C2×#55072
C22×#8082
C2×#47352
#740282
C3×#18232
#699522
#695152
#688952
#688752
#688272
#688232
#240893