Finite Group Databasebeta

← #10827346

Subgroups — #10827346

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×#5611C211yesmaximal, characteristic
2C2×C3×#448C211yesmaximal, characteristic, complemented
3#84068C211yesmaximal, characteristic, complemented
4C22×#835C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#72047C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#70268C212yesmaximal, complemented
9#70231C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic, complemented
11#69924C211yesmaximal, characteristic, complemented
12#23961—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#237—21no—
15C22×C3×SD16C2212yes—
16C2×C3×#121C2211yescharacteristic
17C2×#946C2212yes—
18C2×#938C2211yescharacteristic
19C2×#935—21no—
20C2×D4×Dic3C2211yescharacteristic
21C2×C4×#67C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24#5611—24no—
25#5611—24no—
26C2×D3×C4⋊C4C2211yescharacteristic
27C2×#888C2211yescharacteristic
28C2×#883—21no—
29C2×#878—21no—
30C3×#448—24no—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33#5231—21no—
34#5230—21no—
35#5229C2212yes—
36C2×#835—24no—
37C2×#835C2212yes—
38C2×#835C2212yes—
39C2×#832C2212yes—
40#5213—21no—
41#5212C2211yescharacteristic
42#5210—21no—
43C2×(C2×D3)⋊C8C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#547—31no—
49C2×#448D311yescharacteristic, complemented
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#1823—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4—21no—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67D411yescharacteristic
71C23×Dic3—21no—
72C23×Dic3—21no—
73C2×#237—28no—
74C2×#237—44no—
75C2×#237—44no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—21no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—42no—
82C22×C3×C8C2312yes—
83C3×#121—22no—
84C3×#121D414yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217D411yescharacteristic
91#946—24no—
92#946—42no—
93#938—22no—
94#938—22no—
95#935—28no—
96D4×Dic3—22no—
97D4×Dic3—22no—
98C4×#67—22no—
99C4×#67—22no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3D414yes—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—22no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8—22no—
114C22×C3⋊C8—42no—
115C22×C3⋊C8C2312yes—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—22no—
118#888—22no—
119#888—22no—
120#883—28no—
121#878—28no—
122C3×#103—24no—
123C3×C22⋊C8—22no—
124#835—24no—
125#835—24no—
126#832—42no—
127C3⋊#95—21no—
128C3⋊#95—22no—
129C3⋊#95—22no—
130(C2×D3)⋊C8—42no—
131C22×#41—61no—
132C23×Q8C2×D311yescharacteristic
133C22×SD16C2×D312yes—
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×#121C2×D311yescharacteristic
144C2×C4×D4—31no—
145C2×C4×D4—31no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4—31no—
148#448—24no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—64no—
152C2×#103C2×D312yes—
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#380—31no—
162#340—32no—
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3—21no—
167C24×C3C2×D411yescharacteristic
168C23×D3C2×D411yescharacteristic
169C2×C3×Q8—22no—
170C2×C3×Q8—28no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—42no—
179C2×C3×Q8#4114yescomplemented
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4C2×D412yes—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—44no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—22no—
198C2×#67—22no—
199C2×#67—24no—
200C2×#67—42no—
201C2×#67—42no—
202C2×#67SD1614yescomplemented
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—22no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215#237—432no—
216C2×C4×D3—24no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×Dic6—24no—
224C2×Dic6—42no—
225C2×Dic6—42no—
226C3×SD16—48no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2×D412yes—
232C3×C22⋊C4—24no—
233#217—24no—
234#217—24no—
235#217—22no—
236#217—42no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—44no—
239C4⋊Dic3—24no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—22no—
243C4⋊Dic3—22no—
244C4⋊Dic3C2×D412yes—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248Dic3⋊C4—22no—
249Dic3⋊C4—44no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—44no—
256C2×#41—68no—
257C2×#41—124no—
258C2×#41—124no—
259C22×Q8—21no—
260C22×Q8—24no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—42no—
264C22×Q8—61no—
265C22×Q8C22×D312yes—
266C22×D4—21no—
267C22×D4—31no—
268C22×D4—61no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—31no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—42no—
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—22no—
291#121—62no—
292#121—62no—
293#121#6714yes—
294C4×D4—62no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—62no—
298C2×C4⋊C4—34no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
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—62no—
317C2×C22⋊C4—61no—
318C2×C42—31no—
319#103—24no—
320#103—64no—
321#103—64no—
322#102—122no—
323C22⋊C8—22no—
324C22⋊C8—122no—
325#95—62no—
326#95—62no—
327#95—61no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—42no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—42no—
335C23×C3#12012yes—
336C23×C3C2×SD1612yes—
337C23×C3AGL(1,ℤ/8)12yes—
338C23×C3C22×D411yescharacteristic
339C22×D3—21no—
340C22×D3—22no—
341C22×D3—21no—
342C22×D3—22no—
343C22×D3C2×SD1612yes—
344C3×Q8—24no—
345C3×Q8—24no—
346C3×Q8—48no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×D4—24no—
350C3×D4—48no—
351C3×D4—41no—
352C3×D4—41no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—48no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4#12012yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4112yes—
368C2×C3×C4H(5,𝔽2)12yes—
369#67—28no—
370#67—48no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—48no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—48no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3C2×SD1612yes—
389C4×D3—48no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—42no—
393Dic6—48no—
394C3×C8—42no—
395C3⋊C8—42no—
396C24—21no—
397C24—31no—
398C24D3×D411yescharacteristic
399#41—1232no—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—64no—
411C2×Q8—122no—
412C2×Q8—122no—
413C2×Q8#23714yescomplemented
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—34no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—64no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4C2×#6712yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4D3×D411yescharacteristic
453C22×C4D3×D412yes—
454C22×C4C2×#6712yes—
455C22×C4C23×D311yescharacteristic
456SD16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—124no—
464C4⋊C4—21no—
465C4⋊C4—32no—
466C4⋊C4—32no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4—124no—
476C4⋊C4C2×#6712yes—
477C22⋊C4—24no—
478C22⋊C4—64no—
479C22⋊C4—64no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—124no—
483C22⋊C4—122no—
484C42—61no—
485C42—61no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3#45014yes—
496C22×C3C2×#12011yescharacteristic
497C22×C3#54612yes—
498C22×C3C22×SD1611yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57812yes—
501C2×D3—22no—
502C2×D3—24no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—42no—
506C3×C4—22no—
507C3×C4—44no—
508C3×C4—42no—
509C3×C4#52511yescharacteristic
510C3×C4#53511yescharacteristic
511C3×C4#54612yes—
512Dic3—22no—
513Dic3—41no—
514Dic3—42no—
515Dic3—42no—
516Dic3—41no—
517Dic3—82no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—32no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—61no—
527C23—62no—
528C23—61no—
529C23—62no—
530C23D3×SD1612yes—
531C23#91412yes—
532C23#93712yes—
533C23C2×D3×D411yescharacteristic
534Q8—24no—
535Q8—24no—
536Q8—48no—
537Q8—42no—
538Q8—42no—
539Q8—128no—
540D4—24no—
541D4—48no—
542D4—41no—
543D4—41no—
544D4—68no—
545D4—128no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—32no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—128no—
567C2×C4—128no—
568C2×C4—128no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4#93712yes—
580C2×C4C2×D3×D411yescharacteristic
581C2×C4C2×#23712yes—
582C2×C4#100412yes—
583C2×C4C22×#6711yescharacteristic
584C8—42no—
585C8—122no—
586C2×C3—24no—
587C2×C3—82no—
588C2×C3#240411yescharacteristic
589C2×C3C2×#45011yescharacteristic
590C2×C3#301111yescharacteristic
591C2×C3#329812yes—
592C2×C3#330812yes—
593D3—42no—
594C22—22no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—62no—
602C22—64no—
603C22—82no—
604C22—82no—
605C22—121no—
606C22—121no—
607C22—122no—
608C22#516914yes—
609C22#559112yes—
610C22C2×D3×SD1611yescharacteristic
611C22C2×#91411yescharacteristic
612C22#577912yes—
613C22C2×#93711yescharacteristic
614C4—22no—
615C4—42no—
616C4—44no—
617C4—62no—
618C4—121no—
619C4—122no—
620C4—122no—
621C4—121no—
622C4—242no—
623C4#558611yescharacteristic
624C4#559211yescharacteristic
625C4#580612yes—
626C3#2396111yescharacteristic, complemented, Sylow
627C2—24no—
628C2—82no—
629C2—122no—
630C2#8004912yes—
631C2#8005112yes—
632C2#8258111yescharacteristic
633C2C2×#516911yescharacteristic
634C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radicalC2×#5611C2
hypercenterC2×#448D3
hypocenterC3#23961
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C2×C3×#448C2semidirect
3. #84068C2semidirect
4. C22×#835C2semidirect
5. #72054C2semidirect
6. #72047C2semidirect
7. #71681C2semidirect
8. #70268C2semidirect
9. #70231C2semidirect
10. #69952C2semidirect
11. #69924C2semidirect
49. C2×#448D3semidirect
179. C2×C3×Q8#41semidirect
202. C2×#67SD16semidirect
413. C2×Q8#237semidirect
626. C3#23961semidirect

Maximal subgroups

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

subgroupindex
C2×#56112
C2×C3×#4482
#840682
C22×#8352
#720542
#720472
#716812
#702682
#702312
#699522
#699242
#239613