Finite Group Databasebeta

← #10828764

Subgroups — #10828764

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
2C3×#2487C211yesmaximal, characteristic, complemented
3C2×#4968C211yesmaximal, characteristic, complemented
4C22×#808C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic, complemented
6#69626C212yesmaximal, complemented
7#69605C212yesmaximal, complemented
8#69524C212yesmaximal, complemented
9#69515C211yesmaximal, characteristic, complemented
10#69505C212yesmaximal, complemented
11#69501C211yesmaximal, characteristic, complemented
12#24089—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21#5555—24no—
22#5555—24no—
23C2×#897—21no—
24C2×#886—21no—
25C2×#886—21no—
26C2×C4×D12C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C3×#400—21no—
29C3×#399—21no—
30C3×#389C2212yes—
31C3×#386—21no—
32C3×#385—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37#4968—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808—24no—
41C2×#808C2212yes—
42C2×#808C2212yes—
43C2×#808C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#547—31no—
49C2×#481—31no—
50C22×#106—31no—
51#2487D311yescharacteristic, complemented
52#2058—32no—
53#2034—32no—
54#1893—31no—
55#1869—31no—
56#1864—32no—
57#1853—32no—
58#1823—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—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—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87#930—22no—
88#930—22no—
89#930—22no—
90#930D414yes—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3D414yes—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C22×C3⋊C8—22no—
105C22×C3⋊C8—24no—
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2312yes—
108#897—28no—
109#886—28no—
110#886—28no—
111C4×D12—22no—
112C4×D12—22no—
113#869—22no—
114#869—22no—
115C3×#95—22no—
116C3×#95—21no—
117C3×#95—22no—
118#830—22no—
119#830—24no—
120#809—24no—
121#809—24no—
122#808—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×D4C2×D311yescharacteristic
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—31no—
148C2×C22⋊C8—32no—
149#400—21no—
150#399—21no—
151#389C2×D312yes—
152#386—21no—
153#385—21no—
154#380C2×D311yescharacteristic
155#380C2×D312yes—
156#340—32no—
157#340—32no—
158#340—32no—
159#336—32no—
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×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—42no—
175C22×C3×C4—82no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C2×#67—24no—
184C2×#67—24no—
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—42no—
188C2×#67—42no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—42no—
195C22×Dic3—42no—
196C22×Dic3—42no—
197C22×Dic3—42no—
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—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—82no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—82no—
220C3×C42—21no—
221C3×C42—21no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225C4⋊Dic3—24no—
226C4⋊Dic3—24no—
227C4⋊Dic3—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3C2×D412yes—
232C4⋊Dic3C2×D412yes—
233C4⋊Dic3C2×D412yes—
234C2×C3⋊C8—28no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
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—61no—
247C22×D4—61no—
248C22×D4#6711yescharacteristic
249C23×C4—21no—
250C23×C4—61no—
251C23×C4—61no—
252C23×C4—61no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C22×C8—32no—
256C22×C8—32no—
257C22×C8—64no—
258C22×C8—62no—
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—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—34no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4C22×D312yes—
288C2×C4⋊C4C22×D312yes—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4#6711yescharacteristic
295C2×C22⋊C4#6711yescharacteristic
296C2×C42C22×D311yescharacteristic
297#106—64no—
298#106—64no—
299#106—64no—
300#102—64no—
301#102—64no—
302C22⋊C8—64no—
303C22⋊C8—62no—
304#95—22no—
305#95—21no—
306#95—22no—
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—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4#12012yes—
345C2×C3×C4C2×SD1612yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4112yes—
349C2×C3×C42−1+412yes—
350#67—48no—
351#67—48no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—48no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—48no—
363D12—24no—
364D12—41no—
365D12—41no—
366C4×D3—48no—
367Dic6—48no—
368C3⋊C8—24no—
369C3⋊C8—42no—
370C24—61no—
371C24C2×#6711yescharacteristic
372C24C2×#6711yescharacteristic
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#21514yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—82no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4D3×D412yes—
413C22×C4D3×D412yes—
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C23×D311yescharacteristic
419C2×C8—34no—
420C2×C8—68no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8—62no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—32no—
432C4⋊C4—32no—
433C4⋊C4—32no—
434C4⋊C4—42no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—64no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—82no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—42no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—82no—
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—82no—
476C3×C4—82no—
477C3×C4#52511yescharacteristic
478C3×C4#53611yescharacteristic
479C3×C4#54712yes—
480Dic3—42no—
481Dic3—42no—
482Dic3—42no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—62no—
490C23—121no—
491C23—121no—
492C23—122no—
493C23—122no—
494C23#93712yes—
495C23C2×#21512yes—
496C23C2×#21512yes—
497C23#94912yes—
498C23C22×#6711yescharacteristic
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—24no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—84no—
529C2×C4—128no—
530C2×C4—128no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—128no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4#93712yes—
539C2×C4C2×#21512yes—
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4D3×#4112yes—
543C2×C4#101012yes—
544C8—64no—
545C8—122no—
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—42no—
561C22—41no—
562C22—41no—
563C22—122no—
564C22—122no—
565C22—122no—
566C22—242no—
567C22—242no—
568C22#504614yes—
569C22C2×#93711yescharacteristic
570C22C22×#21511yescharacteristic
571C22C2×#94911yescharacteristic
572C22#582412yes—
573C22#583612yes—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—82no—
579C4—82no—
580C4—122no—
581C4—122no—
582C4—122no—
583C4#564811yescharacteristic
584C4#565512yes—
585C4#565611yescharacteristic
586C3#2408911yescharacteristic, complemented, Sylow
587C2—24no—
588C2—42no—
589C2—242no—
590C2C2×#504611yescharacteristic
591C2#8284111yescharacteristic
592C2#8339011yescharacteristic
593C2#8360812yes—
594C2#8362212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2487C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#2487D3
hypocenterC3#24089
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C3×#2487C2semidirect
3. C2×#4968C2semidirect
4. C22×#808C2semidirect
5. #71610C2semidirect
6. #69626C2semidirect
7. #69605C2semidirect
8. #69524C2semidirect
9. #69515C2semidirect
10. #69505C2semidirect
11. #69501C2semidirect
51. #2487D3semidirect
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×#55552
C3×#24872
C2×#49682
C22×#8082
#716102
#696262
#696052
#695242
#695152
#695052
#695012
#240893