Finite Group Databasebeta

← #10827377

Subgroups — #10827377

Order 768.

1500 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5514C211yesmaximal, characteristic, complemented
2C22×#809C211yesmaximal, characteristic, complemented
3C2×#4732C211yesmaximal, characteristic, complemented
4#74016C211yesmaximal, characteristic, complemented
5C3×#1824C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic, complemented
7#69515C211yesmaximal, characteristic, complemented
8#68899C212yesmaximal, complemented
9#68897C212yesmaximal, complemented
10#68886C212yesmaximal, complemented
11#68881C212yesmaximal, complemented
12#23963—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2211yescharacteristic
17C2×C4×#67—21no—
18C2×C4×#67C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×D24C2212yes—
21#5514—24no—
22#5514—24no—
23C2×#898—21no—
24C2×#895C2212yes—
25C2×#895C2211yescharacteristic, complemented
26C2×#894—21no—
27C2×#893C2211yescharacteristic, complemented
28C2×#892C2211yescharacteristic, complemented
29C2×#890C2211yescharacteristic, complemented
30C2×#882—21no—
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#380C2211yescharacteristic
33C2×#821C2212yes—
34C2×(C2×D3)⋊C8C2212yescomplemented
35#4988C2212yes—
36C2×#809—24no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yescomplemented
40#4955—21no—
41#4732—24no—
42#4672—21no—
43#4671—21no—
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C2×#553—31no—
49C2×#447—31no—
50C22×#102—31no—
51#2408—31no—
52#2025—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1875—32no—
57#1869—31no—
58#1824D311yescharacteristic, complemented
59C24×D3—21no—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67D411yescharacteristic
64C22×D12—22no—
65C22×D12—24no—
66C22×D12—21no—
67C22×D12—22no—
68C22×D12—21no—
69C22×D12—22no—
70C22×D12—42no—
71C22×D12C2311yescharacteristic
72C22×D12C2312yes—
73C22×C4×D3—21no—
74C22×C4×D3C2311yescharacteristic
75C22×C3×C8C2312yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4D414yescomplemented
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×#217D411yescharacteristic
84#930—28no—
85#930—22no—
86#930D414yescomplemented
87C4×#67—28no—
88C4×#67—22no—
89C4×#67—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4D411yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8—22no—
105C22×C3⋊C8—42no—
106C22×C3⋊C8C2312yes—
107C2×D24—24no—
108C2×D24—24no—
109C2×D24—42no—
110#898—28no—
111#895—22no—
112#895—22no—
113#895—24no—
114#895—42no—
115#894—28no—
116#893—22no—
117#893—22no—
118#892—22no—
119#892—22no—
120#890—22no—
121#890—22no—
122#882—44no—
123C3×C22⋊C8—22no—
124C3×#95—21no—
125#821—24no—
126(C2×D3)⋊C8—42no—
127#809—24no—
128#809—24no—
129#809—42no—
130C23×D4—31no—
131C22×D8—32no—
132C23×C8—31no—
133#553—64no—
134#553—64no—
135C2×#126—61no—
136C2×#125—31no—
137C2×#123—61no—
138C2×#121—31no—
139C2×#121—31no—
140C2×#121—32no—
141C2×#121—61no—
142C2×#120—61no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×#117—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148#447—64no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—64no—
154C2×C22⋊C8—32no—
155C2×C22⋊C8C2×D311yescharacteristic
156#394—61no—
157#392—61no—
158#390—32no—
159#380C2×D311yescharacteristic
160#379—61no—
161#340—32no—
162#340C2×D312yescomplemented
163#340C2×D312yescomplemented
164#336—32no—
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C23×D3—22no—
168C23×D3—44no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—82no—
173C23×D3C2×D411yescharacteristic
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—82no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—22no—
186C2×#67—24no—
187C2×#67—22no—
188C2×#67—22no—
189C2×#67—42no—
190C2×#67—42no—
191C22×Dic3—21no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×D12—28no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12—22no—
204C2×D12—24no—
205C2×D12—48no—
206C2×D12—44no—
207C2×D12—44no—
208C2×D12—42no—
209C2×D12—44no—
210C2×D12—44no—
211C2×D12—82no—
212C2×D12—82no—
213C2×D12C2×D412yes—
214C2×D12#4114yes—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—24no—
220C2×C4×D3—82no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4C2×D412yes—
230C3×C4⋊C4C2×D412yes—
231C3×C22⋊C4—42no—
232#217—22no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—44no—
239(C2×D3)⋊C4—44no—
240(C2×D3)⋊C4—82no—
241C4⋊Dic3—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3C2×D412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—22no—
246C4×Dic3—24no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—44no—
253D24—48no—
254C25—61no—
255C22×D4—31no—
256C22×D4—31no—
257C22×D4—32no—
258C22×D4—64no—
259C22×D4—61no—
260C22×D4—62no—
261C22×D4—61no—
262C22×D4—62no—
263C22×D4—62no—
264C22×D4—61no—
265C22×D4—122no—
266C23×C4—21no—
267C23×C4—31no—
268C23×C4—61no—
269C23×C4C22×D311yescharacteristic
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—122no—
273C22×C8—32no—
274C22×C8—62no—
275C22×C8—122no—
276C22×C8C22×D312yes—
277#126—68no—
278#125—62no—
279#125—62no—
280#123—68no—
281#121—34no—
282#121—62no—
283#121—68no—
284#121—62no—
285#121—62no—
286#121—64no—
287#121—122no—
288#120—124no—
289C4×D4—62no—
290C4×D4—68no—
291C4×D4—62no—
292C4×D4—62no—
293C4×D4—62no—
294#117—62no—
295#117—62no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4D1214yescomplemented
303C2×C4⋊C4C22×D312yes—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—62no—
316C2×C42—31no—
317C2×C42—61no—
318#102—64no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—22no—
323C22⋊C8—122no—
324#95—21no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3#12012yes—
328C23×C3#13512yes—
329C23×C3AGL(1,ℤ/8)12yes—
330C23×C3C22×D411yescharacteristic
331C22×D3—21no—
332C22×D3—22no—
333C22×D3—24no—
334C22×D3—22no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—22no—
338C22×D3—48no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—41no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—41no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—88no—
350C22×D3—88no—
351C22×D3—84no—
352C22×D3—84no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—84no—
365C2×C3×C4#12012yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4112yes—
369C2×C3×C4H(5,𝔽2)12yes—
370#67—44no—
371#67—48no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382D12—24no—
383D12—24no—
384D12—24no—
385D12—48no—
386D12—41no—
387D12—41no—
388D12—42no—
389D12—42no—
390D12—84no—
391D12—88no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—42no—
395C4×D3—84no—
396C3×C8—42no—
397C3⋊C8—42no—
398C24—31no—
399C24—61no—
400C24—62no—
401C24—124no—
402C24—122no—
403C24—122no—
404C24—122no—
405C24—242no—
406C24D3×D411yescharacteristic
407C2×D4—32no—
408C2×D4—34no—
409C2×D4—68no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—61no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—128no—
424C2×D4—124no—
425C2×D4—124no—
426C2×D4—124no—
427C2×D4—124no—
428C2×D4—122no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4—242no—
432C2×D4—242no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—82no—
449C22×C4—242no—
450C22×C4C2×D1212yes—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D412yes—
453C22×C4C23×D311yescharacteristic
454D8—128no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—124no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—32no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4C2×D1212yes—
473C4⋊C4C2×D1212yes—
474C22⋊C4—42no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C22⋊C4—64no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—64no—
481C22⋊C4—124no—
482C22⋊C4—124no—
483C22⋊C4—242no—
484C42—64no—
485C42—61no—
486C42—61no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3#44914yes—
492C22×C3C2×#12011yescharacteristic
493C22×C3#54612yes—
494C22×C3C2×#13511yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3#57612yes—
497C2×D3—41no—
498C2×D3—44no—
499C2×D3—42no—
500C2×D3—41no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—41no—
504C2×D3—42no—
505C2×D3—44no—
506C2×D3—42no—
507C2×D3—44no—
508C2×D3—816no—
509C2×D3—82no—
510C2×D3—82no—
511C2×D3—82no—
512C2×D3—82no—
513C2×D3—82no—
514C2×D3—82no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—82no—
519C3×C4#52511yescharacteristic
520C3×C4#53511yescharacteristic
521C3×C4#54612yes—
522Dic3—41no—
523Dic3—41no—
524Dic3—42no—
525Dic3—41no—
526C23—22no—
527C23—22no—
528C23—64no—
529C23—62no—
530C23—62no—
531C23—61no—
532C23—61no—
533C23—61no—
534C23—62no—
535C23—128no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—121no—
541C23—121no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—248no—
547C23—248no—
548C23—244no—
549C23—244no—
550C23#88212yes—
551C23#91412yes—
552C23#91912yes—
553C23C2×D3×D411yescharacteristic
554D4—64no—
555D4—64no—
556D4—64no—
557D4—128no—
558D4—124no—
559D4—128no—
560D4—122no—
561D4—121no—
562D4—122no—
563D4—121no—
564D4—244no—
565D4—248no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—62no—
585C2×C4—84no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—244no—
592C2×C4#88212yes—
593C2×C4C22×D1211yescharacteristic
594C2×C4C2×D3×D411yescharacteristic
595C2×C4D3×#4112yes—
596C2×C4#100912yes—
597C8—42no—
598C8—122no—
599C2×C3—24no—
600C2×C3#240411yescharacteristic
601C2×C3C2×#44911yescharacteristic
602C2×C3#300811yescharacteristic
603C2×C3#329712yes—
604C2×C3#331812yes—
605D3—81no—
606D3—82no—
607D3—84no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—122no—
613C22—124no—
614C22—122no—
615C22—121no—
616C22—121no—
617C22—124no—
618C22—122no—
619C22—122no—
620C22—124no—
621C22—121no—
622C22—122no—
623C22—2416no—
624C22—242no—
625C22—242no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22#481014yes—
631C22C2×#88211yescharacteristic
632C22#565412yes—
633C22C2×#91411yescharacteristic
634C22C2×#91911yescharacteristic
635C22#577412yes—
636C4—22no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—121no—
641C4—121no—
642C4—121no—
643C4—122no—
644C4#552411yescharacteristic
645C4#552511yescharacteristic
646C4#565412yes—
647C3#2396311yescharacteristic, complemented, Sylow
648C2—24no—
649C2—241no—
650C2—244no—
651C2—242no—
652C2#7412511yescharacteristic
653C2C2×#481011yescharacteristic
654C2#7892811yescharacteristic
655C2#8134312yes—
656C2#8148512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1824C2
socleC23×C3C22×D4
hypercenter#1824D3
hypocenterC3#23963
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
1. C2×#5514C2semidirect
2. C22×#809C2semidirect
3. C2×#4732C2semidirect
4. #74016C2semidirect
5. C3×#1824C2semidirect
6. #69952C2semidirect
7. #69515C2semidirect
8. #68899C2semidirect
9. #68897C2semidirect
10. #68886C2semidirect
11. #68881C2semidirect
25. C2×#895C22semidirect
27. C2×#893C22semidirect
28. C2×#892C22semidirect
29. C2×#890C22semidirect
34. C2×(C2×D3)⋊C8C22semidirect
39. C2×#809C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
58. #1824D3semidirect
78. C2×C3×C4⋊C4D4semidirect
86. #930D4semidirect
162. #340C2×D3semidirect
163. #340C2×D3semidirect
302. C2×C4⋊C4D12semidirect
647. C3#23963semidirect

Maximal subgroups

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

subgroupindex
C2×#55142
C22×#8092
C2×#47322
#740162
C3×#18242
#699522
#695152
#688992
#688972
#688862
#688812
#239633