Finite Group Databasebeta

← #10828766

Subgroups — #10828766

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×#5555C211yesmaximal, characteristic
2C22×#817C211yesmaximal, characteristic, complemented
3#82374C211yesmaximal, characteristic, complemented
4C2×#4735C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69935C212yesmaximal, complemented
8#69928C212yesmaximal, complemented
9#69924C211yesmaximal, characteristic, complemented
10#68671C212yesmaximal, complemented
11#68633C212yesmaximal, complemented
12#24089—31nomaximal, Sylow
13C22×#237—21no—
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21#5555—24no—
22#5555—24no—
23C2×#897—21no—
24C2×#886—21no—
25C2×#886—21no—
26C2×C4×D12C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2212yes—
30#5220—21no—
31#5212C2211yescharacteristic
32#5212C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#817—24no—
36C2×#817C2212yes—
37C2×#817C2212yes—
38C2×#817C2212yes—
39#4986—21no—
40#4949—21no—
41#4946—21no—
42#4928C2212yes—
43#4735—24no—
44C3×#336C2212yes—
45#4551C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#547—31no—
49C2×#481—31no—
50C22×#106—31no—
51#2487—31no—
52#2058—32no—
53#2034—32no—
54#1893—31no—
55#1869D311yescharacteristic, complemented
56#1864—32no—
57#1853—32no—
58#1823—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C2×#237—28no—
68C2×#237—44no—
69C2×#237—44no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—21no—
73C22×C3×C8—22no—
74C22×C3×C8—24no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84C2×#217—22no—
85#930—22no—
86#930—22no—
87#930—22no—
88#930D414yes—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3D414yes—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—22no—
108#897—28no—
109#886—28no—
110#886—28no—
111C4×D12—22no—
112C4×D12—22no—
113#869—22no—
114#869—22no—
115C3×C22⋊C8—22no—
116C3×C22⋊C8—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#821—24no—
121#821—24no—
122#817—24no—
123#817—24no—
124#817—24no—
125C22×#41—61no—
126C23×C8C2×D311yescharacteristic
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×D4C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—31no—
140#481—64no—
141C2×#106—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8C2×D311yescharacteristic
148C2×C22⋊C8C2×D312yes—
149#400—61no—
150#399—61no—
151#389—32no—
152#386—61no—
153#385—61no—
154#380—31no—
155#380—32no—
156#340—32no—
157#340—32no—
158#340—32no—
159#336C2×D312yes—
160C24×C3C2×D411yescharacteristic
161C24×C3C2×D411yescharacteristic
162C23×D3—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4SD1614yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C2×#67—42no—
184C2×#67—42no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—42no—
193C22×Dic3—42no—
194C22×Dic3—42no—
195C22×Dic3—42no—
196C22×Dic3—42no—
197C22×Dic3—82no—
198#237—432no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—21no—
202C2×D12C2×D412yes—
203C2×C4×D3—24no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×Dic6—24no—
207C2×Dic6—42no—
208C2×Dic6—42no—
209C2×C3×C8—28no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8#4114yescomplemented
216C3×C4⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C42—21no—
220C3×C42—21no—
221#217—42no—
222#217—82no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226C4⋊Dic3—24no—
227C4⋊Dic3—24no—
228C4⋊Dic3—24no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—42no—
237C4⋊Dic3C2×D412yes—
238C4⋊Dic3C2×D412yes—
239C4⋊Dic3C2×D412yes—
240Dic3⋊C4—82no—
241C2×#41—68no—
242C2×#41—124no—
243C2×#41—124no—
244C22×Q8—61no—
245C22×D4—31no—
246C22×D4—61no—
247C22×D4—61no—
248C22×D4D1211yescharacteristic
249C23×C4—61no—
250C23×C4—61no—
251C23×C4—61no—
252C23×C4—61no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C22×C8—22no—
256C22×C8—24no—
257C22×C8C22×D312yes—
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—22no—
269C4×D4—22no—
270C4×D4—62no—
271C4×D4—62no—
272C2×C4⋊C4—34no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4D1211yescharacteristic
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4D1211yescharacteristic
295C2×C22⋊C4D1211yescharacteristic
296C2×C42C22×D311yescharacteristic
297#106—64no—
298#106—64no—
299#106—64no—
300#102—64no—
301#102—64no—
302C22⋊C8—22no—
303C22⋊C8—24no—
304#95—62no—
305#95—61no—
306#95—62no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3#12012yes—
314C23×C3C2×SD1612yes—
315C23×C3C2×SD1612yes—
316C23×C3AGL(1,ℤ/8)12yes—
317C23×C3C22×D411yescharacteristic
318C22×D3—22no—
319C22×D3—42no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—42no—
323C3×D4—28no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4#12012yes—
338C2×C3×C4C2×SD1612yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4112yes—
342C2×C3×C42−1+412yes—
343#67—48no—
344#67—48no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—48no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—48no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—84no—
363D12—24no—
364D12—41no—
365D12—41no—
366C4×D3—48no—
367Dic6—48no—
368C3×C8—24no—
369C3×C8—42no—
370C24—61no—
371C24C2×D1211yescharacteristic
372C24C2×D1211yescharacteristic
373#41—1232no—
374C2×Q8—64no—
375C2×Q8—122no—
376C2×Q8—122no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—32no—
380C2×D4—64no—
381C2×D4—64no—
382C2×D4—61no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—122no—
386C2×D4—122no—
387C2×D4—122no—
388C2×D4—122no—
389C2×D4#20414yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—64no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—122no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—242no—
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4D3×D412yes—
417C22×C4D3×D412yes—
418C22×C4C23×D311yescharacteristic
419C2×C8—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#23714yescomplemented
426C4⋊C4—22no—
427C4⋊C4—32no—
428C4⋊C4—32no—
429C4⋊C4—32no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—62no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—122no—
441C4⋊C4—242no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—64no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—122no—
448C22⋊C4—242no—
449C42—21no—
450C42—21no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—42no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3#45014yes—
461C22×C3C2×#12011yescharacteristic
462C22×C3#54712yes—
463C22×C3C22×SD1611yescharacteristic
464C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
465C22×C3#57712yes—
466C2×D3—42no—
467C2×D3—42no—
468C2×D3—42no—
469C2×D3—82no—
470C2×D3—82no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4#52511yescharacteristic
476C3×C4#53611yescharacteristic
477C3×C4#54712yes—
478Dic3—42no—
479Dic3—42no—
480Dic3—42no—
481Dic3—82no—
482Dic3—82no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—62no—
490C23—121no—
491C23—121no—
492C23—122no—
493C23—122no—
494C23#88212yes—
495C23C2×#20412yes—
496C23C2×#20412yes—
497C23#90812yes—
498C23C22×D1211yescharacteristic
499Q8—128no—
500D4—28no—
501D4—64no—
502D4—128no—
503D4—128no—
504D4—121no—
505D4—121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—128no—
523C2×C4—128no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—128no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—244no—
538C2×C4#88212yes—
539C2×C4C2×#20412yes—
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×#23712yes—
543C2×C4#101012yes—
544C8—24no—
545C8—42no—
546C2×C3—24no—
547C2×C3—42no—
548C2×C3#240611yescharacteristic
549C2×C3C2×#45011yescharacteristic
550C2×C3#301011yescharacteristic
551C2×C3#333512yes—
552C2×C3#334512yes—
553D3—82no—
554C22—24no—
555C22—22no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—24no—
560C22—41no—
561C22—42no—
562C22—41no—
563C22—122no—
564C22—122no—
565C22—122no—
566C22—242no—
567C22—242no—
568C22#473114yes—
569C22C2×#88211yescharacteristic
570C22#555512yes—
571C22C22×#20411yescharacteristic
572C22C2×#90811yescharacteristic
573C22#575112yes—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—122no—
579C4—122no—
580C4—122no—
581C4—242no—
582C4—242no—
583C4#558611yescharacteristic
584C4#561112yes—
585C4#564811yescharacteristic
586C3#2408911yescharacteristic, complemented, Sylow
587C2—24no—
588C2—42no—
589C2—242no—
590C2#7468711yescharacteristic
591C2C2×#473111yescharacteristic
592C2#7867811yescharacteristic
593C2#7941212yes—
594C2#7947112yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#1869D3
hypocenterC3#24089
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#817C2semidirect
3. #82374C2semidirect
4. C2×#4735C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69935C2semidirect
8. #69928C2semidirect
9. #69924C2semidirect
10. #68671C2semidirect
11. #68633C2semidirect
55. #1869D3semidirect
215. C2×C3×C8#41semidirect
425. C2×C8#237semidirect
586. C3#24089semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C22×#8172
#823742
C2×#47352
C3×#18692
#719732
#699352
#699282
#699242
#686712
#686332
#240893