Finite Group Databasebeta

← #10827351

Subgroups — #10827351

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5507C211yesmaximal, characteristic
2C22×#810C211yesmaximal, characteristic, complemented
3C2×#4738C211yesmaximal, characteristic, complemented
4#74017C211yesmaximal, characteristic, complemented
5C3×#1823C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic, complemented
7#69515C211yesmaximal, characteristic, complemented
8#68910C212yesmaximal, complemented
9#68896C212yesmaximal, complemented
10#68887C212yesmaximal, complemented
11#68878C212yesmaximal, complemented
12#23961—31nomaximal, Sylow
13C22×#235—21no—
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#929—21no—
18C2×C4×#67C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×#204C2212yes—
21C2×#897C2212yes—
22C2×#896—21no—
23#5507—24no—
24#5507—24no—
25C2×#895C2211yescharacteristic
26C2×#894—21no—
27C2×#893C2211yescharacteristic
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×#888C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33C2×#816C2212yes—
34#4973C2212yes—
35C2×#810—24no—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39#4958—21no—
40#4950—21no—
41#4738—24no—
42#4673—21no—
43#4656—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#547—31no—
49C2×#448—31no—
50C22×#103—31no—
51#2409—31no—
52#2024—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1872—32no—
57#1869—31no—
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—22no—
73C22×Dic6—24no—
74C22×Dic6—21no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—42no—
78C22×Dic6C2312yes—
79C22×C3×C8C2312yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4D414yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×#217D411yescharacteristic
88#930—22no—
89#930D414yes—
90#929—28no—
91C4×#67—22no—
92C4×#67—22no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4D411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8—22no—
105C22×C3⋊C8—42no—
106C22×C3⋊C8C2312yes—
107C2×#204—24no—
108C2×#204—24no—
109C2×#204—42no—
110#897—24no—
111#897—42no—
112#896—28no—
113#895—22no—
114#895—22no—
115#894—28no—
116#893—22no—
117#893—22no—
118D3×C4⋊C4—22no—
119D3×C4⋊C4—22no—
120#888—22no—
121#888—22no—
122C3×C22⋊C8—22no—
123C3×#95—21no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126(C2×D3)⋊C8—42no—
127#816—24no—
128#810—24no—
129#810—24no—
130#809—42no—
131C22×#41—61no—
132C23×Q8—31no—
133C22×SD16—32no—
134C23×C8—31no—
135#547—64no—
136#547—64no—
137C2×C4⋊Q8—31no—
138C2×#123—61no—
139C2×#123—61no—
140C2×C22⋊Q8—32no—
141C2×C22⋊Q8—61no—
142C2×#121—31no—
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4—31no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148#448—64no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×#103—64no—
153C2×#102—32no—
154C2×C22⋊C8—32no—
155C2×C22⋊C8C2×D311yescharacteristic
156#395—61no—
157#393—61no—
158#391—32no—
159#386—61no—
160#385—61no—
161#380C2×D311yescharacteristic
162#340—32no—
163#340C2×D312yes—
164#340C2×D312yes—
165#336—32no—
166C24×C3C2×D411yescharacteristic
167C23×D3—21no—
168C23×D3C2×D411yescharacteristic
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C2×#67—22no—
182C2×#67—22no—
183C2×#67—24no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67SD1614yescomplemented
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—44no—
190C22×Dic3—42no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194#235—432no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—24no—
198C2×D12—21no—
199C2×D12—42no—
200C2×D12—42no—
201C2×D12C2×D412yes—
202C2×C4×D3—24no—
203C2×C4×D3—24no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×Dic6—28no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—48no—
220C2×Dic6—44no—
221C2×Dic6—42no—
222C2×Dic6—44no—
223C2×Dic6—42no—
224C2×Dic6—42no—
225C2×Dic6#4114yescomplemented
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4C2×D412yes—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—42no—
237#217—22no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—44no—
243C4⋊Dic3—21no—
244C4⋊Dic3—44no—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248Dic3⋊C4—22no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—44no—
255#204—48no—
256C2×#41—68no—
257C2×#41—124no—
258C2×#41—124no—
259C22×Q8—32no—
260C22×Q8—61no—
261C22×Q8—64no—
262C22×Q8—61no—
263C22×Q8—62no—
264C22×Q8—62no—
265C22×Q8—122no—
266C22×D4—31no—
267C22×D4—31no—
268C22×D4—61no—
269C22×D4—61no—
270C23×C4—21no—
271C23×C4—31no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276C2×SD16—64no—
277C2×SD16—64no—
278C2×SD16—122no—
279C22×C8—32no—
280C22×C8—62no—
281C22×C8—122no—
282C22×C8C22×D312yes—
283C4⋊Q8—62no—
284C4⋊Q8—62no—
285#123—68no—
286#123—68no—
287C22⋊Q8—68no—
288C22⋊Q8—64no—
289C22⋊Q8—122no—
290#121—34no—
291#121—62no—
292#121—62no—
293#121—62no—
294C4×D4—62no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—62no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—62no—
307C2×C4⋊C4D1214yes—
308C2×C4⋊C4C22×D312yes—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—31no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319#103—64no—
320#103—64no—
321#103—64no—
322#102—122no—
323C22⋊C8—22no—
324C22⋊C8—122no—
325#95—21no—
326#95—62no—
327#95—62no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3#12012yes—
331C23×C3C2×SD1612yes—
332C23×C3AGL(1,ℤ/8)12yes—
333C23×C3C22×D411yescharacteristic
334C22×D3—21no—
335C22×D3—22no—
336C22×D3—21no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—42no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—42no—
343C22×D3C2×SD1612yes—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—48no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4#12012yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C2×#4112yes—
362C2×C3×C4H(5,𝔽2)12yes—
363#67—28no—
364#67—48no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—22no—
373C2×Dic3—48no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3C2×SD1612yes—
379D12—24no—
380D12—48no—
381D12—41no—
382D12—41no—
383C4×D3—48no—
384C4×D3—48no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—42no—
388Dic6—24no—
389Dic6—24no—
390Dic6—48no—
391Dic6—48no—
392Dic6—42no—
393Dic6—42no—
394C3×C8—42no—
395C3⋊C8—42no—
396C24—31no—
397C24—61no—
398C24D3×D411yescharacteristic
399#41—1232no—
400C2×Q8—34no—
401C2×Q8—64no—
402C2×Q8—68no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—124no—
409C2×Q8—128no—
410C2×Q8—124no—
411C2×Q8—122no—
412C2×Q8—122no—
413C2×Q8—122no—
414C2×D4—34no—
415C2×D4—32no—
416C2×D4—64no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—64no—
420C2×D4—61no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4—122no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—124no—
451C22×C4—122no—
452C22×C4C2×D1212yes—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D412yes—
455C22×C4C23×D311yescharacteristic
456SD16—128no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—124no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—32no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—62no—
473C4⋊C4—61no—
474C4⋊C4—124no—
475C4⋊C4C2×D1212yes—
476C4⋊C4C2×D1212yes—
477C22⋊C4—42no—
478C22⋊C4—64no—
479C22⋊C4—64no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—64no—
483C22⋊C4—124no—
484C42—61no—
485C42—61no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3#45014yes—
491C22×C3C2×#12011yescharacteristic
492C22×C3#54612yes—
493C22×C3C22×SD1611yescharacteristic
494C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
495C22×C3#57812yes—
496C2×D3—22no—
497C2×D3—24no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—82no—
505C2×D3—82no—
506C3×C4—22no—
507C3×C4—42no—
508C3×C4—42no—
509C3×C4—82no—
510C3×C4#52511yescharacteristic
511C3×C4#53511yescharacteristic
512C3×C4#54612yes—
513Dic3—22no—
514Dic3—44no—
515Dic3—41no—
516Dic3—42no—
517Dic3—41no—
518C23—22no—
519C23—22no—
520C23—32no—
521C23—62no—
522C23—61no—
523C23—61no—
524C23—62no—
525C23—62no—
526C23—122no—
527C23—121no—
528C23—121no—
529C23—122no—
530C23#88212yes—
531C23#91112yes—
532C23D3×SD1612yes—
533C23C2×D3×D411yescharacteristic
534Q8—64no—
535Q8—64no—
536Q8—128no—
537Q8—128no—
538Q8—122no—
539Q8—122no—
540D4—68no—
541D4—64no—
542D4—128no—
543D4—128no—
544D4—121no—
545D4—121no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—64no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—128no—
570C2×C4—128no—
571C2×C4—128no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4#88212yes—
580C2×C4C22×D1211yescharacteristic
581C2×C4C2×D3×D411yescharacteristic
582C2×C4D3×#4112yes—
583C2×C4#100912yes—
584C8—42no—
585C8—122no—
586C2×C3—24no—
587C2×C3#240411yescharacteristic
588C2×C3C2×#45011yescharacteristic
589C2×C3#301111yescharacteristic
590C2×C3#329812yes—
591C2×C3#330812yes—
592D3—42no—
593D3—82no—
594C22—24no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—64no—
599C22—62no—
600C22—121no—
601C22—121no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—242no—
607C22—242no—
608C22#477614yes—
609C22C2×#88211yescharacteristic
610C22#565412yes—
611C22C2×#91111yescharacteristic
612C22C2×D3×SD1611yescharacteristic
613C22#577912yes—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—62no—
618C4—82no—
619C4—121no—
620C4—121no—
621C4—124no—
622C4—122no—
623C4#552411yescharacteristic
624C4#552511yescharacteristic
625C4#565412yes—
626C3#2396111yescharacteristic, complemented, Sylow
627C2—24no—
628C2—122no—
629C2—242no—
630C2#7412511yescharacteristic
631C2C2×#477611yescharacteristic
632C2#7896111yescharacteristic
633C2#8130612yes—
634C2#8145012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC2×#5507C2
hypercenter#1823D3
hypocenterC3#23961
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×#810C2semidirect
3. C2×#4738C2semidirect
4. #74017C2semidirect
5. C3×#1823C2semidirect
6. #69952C2semidirect
7. #69515C2semidirect
8. #68910C2semidirect
9. #68896C2semidirect
10. #68887C2semidirect
11. #68878C2semidirect
58. #1823D3semidirect
186. C2×#67SD16semidirect
225. C2×Dic6#41semidirect
626. C3#23961semidirect

Maximal subgroups

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

subgroupindex
C2×#55072
C22×#8102
C2×#47382
#740172
C3×#18232
#699522
#695152
#689102
#688962
#688872
#688782
#239613