Finite Group Databasebeta

← #10828159

Subgroups — #10828159

Order 768.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#548C211yesmaximal, characteristic
2C22×#835C211yesmaximal, characteristic, complemented
3C2×#5033C211yesmaximal, characteristic, complemented
4#82373C211yesmaximal, characteristic, complemented
5#72135C212yesmaximal, complemented
6#72132C211yesmaximal, characteristic, complemented
7#71722C212yesmaximal, complemented
8#71695C212yesmaximal, complemented
9#71681C211yesmaximal, characteristic, complemented
10#71628C212yesmaximal, complemented
11#71610C211yesmaximal, characteristic, complemented
12#24035—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C3×#548—24no—
16C3×#548—24no—
17C2×C3×#124—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×#935—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C2×#835—24no—
30C2×#835C2212yes—
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#835C2212yes—
34C2×#832C2212yes—
35C2×#832C2212yes—
36#5212C2211yescharacteristic
37#5212C2212yes—
38C2×#830C2211yescharacteristic
39C2×#830C2212yes—
40#5033—24no—
41#4981—21no—
42C2×#807C2212yes—
43#4953—21no—
44#4943—21no—
45#4928C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#548D311yescharacteristic
49C2×#480—31no—
50C22×#103—31no—
51#2486—31no—
52#2042—32no—
53#2036—32no—
54#1893—31no—
55#1869—31no—
56#1863—32no—
57#1788—32no—
58#1787—31no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C22×C3×Q8—21no—
64C22×C3×Q8—22no—
65C22×C3×Q8—24no—
66C22×C3×Q8—42no—
67C22×C3×Q8D414yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C3×#124—28no—
80C3×#124—22no—
81C3×#124—22no—
82C3×C22⋊Q8—28no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×#121—28no—
86C3×#121—22no—
87C3×#121D414yes—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×D4—28no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#217—21no—
106C2×#217—22no—
107#935—44no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—22no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2312yes—
113C2×Dic3⋊C4—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—42no—
116C22×C3⋊C8C2312yes—
117C22×C3⋊C8C2312yes—
118#835—24no—
119#835—24no—
120#835—24no—
121#835—42no—
122#832—24no—
123#832—42no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126C3⋊#95—21no—
127#830—22no—
128#830—42no—
129#807—24no—
130C22×#41C2×D311yescharacteristic
131C23×Q8C2×D311yescharacteristic
132C23×C8—31no—
133#548—24no—
134#548—24no—
135C2×#124—21no—
136C2×#124C2×D311yescharacteristic
137C2×#123—61no—
138C2×C22⋊Q8—21no—
139C2×C22⋊Q8C2×D311yescharacteristic
140C2×#121—21no—
141C2×#121C2×D311yescharacteristic
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4C2×D311yescharacteristic
145C22×C4⋊C4—31no—
146#480—64no—
147C2×#107—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×#103—64no—
153C2×#102—32no—
154C2×#102—32no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—32no—
157#398—61no—
158#397—61no—
159#389—32no—
160#381—61no—
161#380—31no—
162#380—32no—
163#340—32no—
164#336—32no—
165C24×C3—21no—
166C24×C3C2×D411yescharacteristic
167C24×C3C2×D411yescharacteristic
168C3×#41—48no—
169C3×#41—44no—
170C3×#41—44no—
171C2×C3×Q8—24no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—28no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—44no—
179C2×C3×Q8—44no—
180C2×C3×Q8—48no—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8C2×D412yes—
183C2×C3×Q8#4114yes—
184C2×C3×D4—24no—
185C2×C3×D4—22no—
186C2×C3×D4—24no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—44no—
193C2×C3×D4—42no—
194C2×C3×D4—42no—
195C2×C3×D4C2×D412yes—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—44no—
208C22×C3×C4—44no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4C2×D412yes—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—22no—
232C3×C42—24no—
233C3×C42—21no—
234C3×C42—21no—
235#217—44no—
236#217—42no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—42no—
242Dic3⋊C4—44no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—44no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—24no—
253C2×#41—42no—
254C22×Q8—21no—
255C22×Q8—22no—
256C22×Q8—24no—
257C22×Q8—42no—
258C22×Q8#6714yes—
259C22×Q8C22×D311yescharacteristic
260C22×Q8C22×D311yescharacteristic
261C22×D4—21no—
262C22×D4—21no—
263C22×D4#6711yescharacteristic
264C22×D4C22×D311yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—61no—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C22×C8—32no—
271C22×C8—32no—
272C22×C8—62no—
273C22×C8—122no—
274#124—28no—
275#124—22no—
276#124—22no—
277#123—124no—
278C22⋊Q8—28no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281#121—28no—
282#121—22no—
283#121#6714yes—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×D4—28no—
287C4×D4—22no—
288C4×D4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4#6711yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—62no—
305C2×C22⋊C4#6711yescharacteristic
306C2×C22⋊C4#6711yescharacteristic
307C2×C42—21no—
308C2×C42C22×D311yescharacteristic
309#107—64no—
310#103—64no—
311#103—64no—
312#103—64no—
313#103—122no—
314#102—64no—
315#102—122no—
316C22⋊C8—62no—
317C22⋊C8—122no—
318#95—62no—
319#95—61no—
320#95—62no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—42no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—42no—
333C23×C3#12012yes—
334C23×C3#13512yes—
335C23×C3AGL(1,ℤ/8)12yes—
336C23×C3C22×D411yescharacteristic
337C3×Q8—22no—
338C3×Q8—24no—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—48no—
342C3×Q8—42no—
343C3×Q8#12314yescomplemented
344C3×D4—24no—
345C3×D4—48no—
346C3×D4—44no—
347C3×D4—48no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—48no—
366C2×C3×C4—48no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4#12012yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4112yes—
377C2×C3×C4H(5,𝔽2)12yes—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—84no—
385C3⋊C8—42no—
386C3⋊C8—42no—
387C24—21no—
388C24C2×#6711yescharacteristic
389C24C2×#6711yescharacteristic
390#41—48no—
391#41—44no—
392#41—44no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—48no—
403C2×Q8#23714yes—
404C2×Q8C2×#6712yes—
405C2×Q8C2×#6712yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×#6712yes—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—122no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6712yes—
441C22×C4C2×#6712yes—
442C22×C4C23×D311yescharacteristic
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—124no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—62no—
457C4⋊C4—62no—
458C4⋊C4—124no—
459C4⋊C4—122no—
460C4⋊C4C2×#6712yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—124no—
468C22⋊C4—122no—
469C42—24no—
470C42—21no—
471C42—21no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—42no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—42no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3#44914yes—
487C22×C3C2×#12011yescharacteristic
488C22×C3#54612yes—
489C22×C3C2×#13511yescharacteristic
490C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
491C22×C3#57812yes—
492C3×C4—22no—
493C3×C4—24no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—42no—
498C3×C4—41no—
499C3×C4C2×#12311yescharacteristic
500C3×C4#53411yescharacteristic
501C3×C4#54612yes—
502Dic3—82no—
503Dic3—82no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—21no—
511C23—22no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23#94912yes—
517C23#95012yes—
518C23#95312yes—
519C23C22×#6711yescharacteristic
520Q8—22no—
521Q8—24no—
522Q8—22no—
523Q8—22no—
524Q8—48no—
525Q8—42no—
526Q8#93514yescomplemented
527D4—24no—
528D4—48no—
529D4—44no—
530D4—48no—
531D4—41no—
532D4—41no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—244no—
563C2×C4#95312yes—
564C2×C4C2×#23712yes—
565C2×C4#100412yes—
566C2×C4C22×#6711yescharacteristic
567C2×C4C22×#6711yescharacteristic
568C8—122no—
569C8—122no—
570C2×C3—24no—
571C2×C3—81no—
572C2×C3—82no—
573C2×C3#240511yescharacteristic
574C2×C3C2×#44911yescharacteristic
575C2×C3#301111yescharacteristic
576C2×C3#330012yes—
577C2×C3#330812yes—
578C22—24no—
579C22—22no—
580C22—24no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22#523814yes—
593C22#580612yes—
594C22C2×#94911yescharacteristic
595C22C2×#95011yescharacteristic
596C22#583812yes—
597C22C2×#95311yescharacteristic
598C4—22no—
599C4—24no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—242no—
606C4—242no—
607C4C2×#93511yescharacteristic
608C4#580411yescharacteristic
609C4#580612yes—
610C3#2403511yescharacteristic, complemented, Sylow
611C2—24no—
612C2—81no—
613C2—82no—
614C2#8353412yes—
615C2#8353612yes—
616C2#8404711yescharacteristic
617C2C2×#523811yescharacteristic
618C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#548C2
socleC23×C3C22×D4
hypercenterC2×#548D3
hypocenterC3#24035
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
2. C22×#835C2semidirect
3. C2×#5033C2semidirect
4. #82373C2semidirect
5. #72135C2semidirect
6. #72132C2semidirect
7. #71722C2semidirect
8. #71695C2semidirect
9. #71681C2semidirect
10. #71628C2semidirect
11. #71610C2semidirect
343. C3×Q8#123semidirect
526. Q8#935semidirect
610. C3#24035semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5482
C22×#8352
C2×#50332
#823732
#721352
#721322
#717222
#716952
#716812
#716282
#716102
#240353