Finite Group Databasebeta

← #10799110

Subgroups — #10799110

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic
2C2×C3×#418C211yesmaximal, characteristic, complemented
3C2×#5221C211yesmaximal, characteristic, complemented
4C2×#5133C211yesmaximal, characteristic, complemented
5C2×Dic3×M16C211yesmaximal, characteristic, complemented
6#82228C211yesmaximal, characteristic, complemented
7C2×#4995C211yesmaximal, characteristic, complemented
8#70192C212yesmaximal, complemented
9#70191C212yesmaximal, complemented
10#70190C212yesmaximal, complemented
11#70189C212yesmaximal, complemented
12#21236—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#217—22no—
18C22×#214C2212yes—
19C2×#932—24no—
20C2×#932—24no—
21C2×#932C418yescomplemented
22C22×#213C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C22×#208C2211yescharacteristic
26C3×#418—24no—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29#5221—22no—
30#5221C2212yes—
31#5221C2212yes—
32#5133—24no—
33Dic3×M16—24no—
34C2×#821C2212yescomplemented
35C2×#816C2212yescomplemented
36C2×#815C2212yescomplemented
37C2×C8×Dic3C2212yescomplemented
38#4995—24no—
39#4988C2211yescharacteristic, complemented
40#4988C2211yescharacteristic, complemented
41#4973C2211yescharacteristic, complemented
42#4973C2211yescharacteristic, complemented
43C2×#808C2212yescomplemented
44C2×#807C2212yescomplemented
45#4958—22no—
46C4×(C2×D3)⋊C4—22no—
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#417—31no—
50C2×#418D311yescharacteristic, complemented
51C2×C4×M16—31no—
52#2297—31no—
53C2×#390—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C25×C3—21no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4C2×C414yes—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C22×#67—21no—
69C23×Dic3—21no—
70C23×Dic3D412yes—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235C2×C414yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217—24no—
91C2×#217—24no—
92C2×#217—48no—
93C2×#214—24no—
94C2×#214—24no—
95C2×#214C2×C418yescomplemented
96#932—232no—
97C2×#213—24no—
98C2×#213—24no—
99C2×#213C2×C418yescomplemented
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C2×C4×Dic3—22no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3D412yes—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2×C414yes—
112C2×#208D414yes—
113C2×#208D414yes—
114C22×C3⋊C8C2312yes—
115C3×#102—24no—
116C3×#102—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#821—24no—
120#816—24no—
121#815—24no—
122C8×Dic3—24no—
123#808—24no—
124#807—24no—
125C22×#41—31no—
126C23×D4C2×D311yescharacteristic
127C2×AGL(1,ℤ/8)—38no—
128C2×AGL(1,ℤ/8)—64no—
129C2×AGL(1,ℤ/8)—64no—
130C22×SD16—32no—
131C22×D8—32no—
132C22×M16—31no—
133C22×M16C2×D311yescharacteristic
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4—31no—
136C22×C22⋊C4—62no—
137C22×C42—31no—
138#428—64no—
139C2×#107—32no—
140C2×#106—32no—
141#418—24no—
142#417—64no—
143C2×#103—32no—
144C2×#102—32no—
145C2×#102C2×D312yes—
146C2×#102C2×D312yes—
147C4×M16—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—31no—
151#391—31no—
152#390—32no—
153#390—32no—
154#390—31no—
155#390—31no—
156#390—62no—
157#386—62no—
158C4×C22⋊C4—62no—
159C24×C3—24no—
160C24×C3—22no—
161C24×C3—44no—
162C24×C3—44no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C2×C3×D4—28no—
168C2×C3×D4—28no—
169C2×C3×D4—24no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4—42no—
179C2×C3×D4C22×C414yes—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4112yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—24no—
191C2×#67—42no—
192C2×#67—42no—
193C22×Dic3—24no—
194C22×Dic3—21no—
195C22×Dic3—24no—
196C22×Dic3—22no—
197C22×Dic3—42no—
198C22×Dic3—44no—
199C22×Dic3—42no—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D412yes—
202#235—28no—
203#235—28no—
204#235—44no—
205#235—44no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C22×C412yes—
212C2×C4×D3—24no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C22×C412yes—
221C3×M16—28no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8C2×D414yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C22⋊C4—42no—
230C3×C42—42no—
231#217—48no—
232#217—48no—
233#214—216no—
234#213—216no—
235(C2×D3)⋊C4—44no—
236(C2×D3)⋊C4—44no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—42no—
240C4×Dic3—24no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245#208C22⋊C4116yescomplemented
246C2×C3⋊C8C22×C414yes—
247C2×C3⋊C8C2×D414yes—
248C2×C3⋊C8C2×D414yes—
249C25—21no—
250C2×#41—34no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—62no—
256C22×Q8—31no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—31no—
264C22×D4—61no—
265C22×D4C4×D314yes—
266C22×D4C22×D312yes—
267C23×C4—32no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271AGL(1,ℤ/8)—632no—
272C2×SD16—38no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×D8—38no—
276C2×D8—64no—
277C2×D8—64no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—34no—
281C2×M16—34no—
282C2×M16—34no—
283C2×M16#6714yes—
284C22×C8—32no—
285C22×C8C22×D312yes—
286#117—22no—
287#117—22no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—64no—
299C2×C22⋊C4—128no—
300C2×C42—21no—
301C2×C42—32no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—62no—
305C2×C42—62no—
306#107—64no—
307#106—64no—
308#103—64no—
309#102—24no—
310#102—24no—
311#102—64no—
312#97—64no—
313C4×C8—64no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—24no—
319C23×C3—416no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—44no—
323C23×C3—44no—
324C23×C3—48no—
325C23×C3—84no—
326C23×C3—84no—
327C23×C3—88no—
328C23×C3C4×D412yes—
329C23×C3#12112yes—
330C23×C3#12712yes—
331C23×C3C22×D411yescharacteristic
332C22×D3—22no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—42no—
336C22×D3—42no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—48no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4C2×C22⋊C414yes—
347C2×C3×C4C4×D412yes—
348C2×C3×C4C4×D414yes—
349C2×C3×C4#12112yes—
350C2×C3×C4#12114yes—
351C2×C3×C4#12414yes—
352C2×C3×C4#12712yes—
353C2×C3×C4C23×C411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359#67—48no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—48no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—44no—
371C2×Dic3—84no—
372C2×Dic3AGL(1,ℤ/8)14yescomplemented
373D12—22no—
374D12—22no—
375D12—41no—
376D12—41no—
377C4×D3—48no—
378Dic6—22no—
379Dic6—22no—
380Dic6—41no—
381Dic6—41no—
382C3×C8—24no—
383C3⋊C8C2×C22⋊C418yescomplemented
384C24—24no—
385C24—22no—
386C24—44no—
387C24—44no—
388C24—42no—
389C24—42no—
390C24—61no—
391C24D3×D411yescharacteristic
392#41—68no—
393#41—68no—
394#41—124no—
395#41—124no—
396C2×Q8—32no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—32no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—64no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—122no—
422C2×D4—122no—
423C2×D4C2×C4×D314yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—61no—
431C22×C4—64no—
432C22×C4—62no—
433C22×C4—122no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—124no—
437C22×C4—122no—
438C22×C4C2×C4×D312yes—
439C22×C4#23512yes—
440C22×C4D3×D412yes—
441C22×C4D3×D412yes—
442C22×C4D3×D411yescharacteristic
443C22×C4#23712yes—
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446SD16—616no—
447D8—616no—
448M16—28no—
449M16—316no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—34no—
453C2×C8—34no—
454C2×C8—34no—
455C2×C8C2×#6714yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—122no—
463C22⋊C4—42no—
464C22⋊C4—124no—
465C22⋊C4—128no—
466C22⋊C4—124no—
467C22⋊C4—128no—
468C42—42no—
469C42—64no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42—61no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—48no—
479C22×C3—44no—
480C22×C3—816no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—84no—
484C22×C3#39014yes—
485C22×C3C2×C4×D411yescharacteristic
486C22×C3C2×#12111yescharacteristic
487C22×C3C2×#12711yescharacteristic
488C22×C3C2×AGL(1,ℤ/8)12yes—
489C22×C3#57512yes—
490C2×D3—44no—
491C2×D3—42no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—82no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#39014yes—
498C3×C4C22×C22⋊C411yescharacteristic
499C3×C4C2×C4×D411yescharacteristic
500C3×C4C2×#12111yescharacteristic
501C3×C4C2×#12411yescharacteristic
502Dic3—24no—
503Dic3—42no—
504Dic3—42no—
505Dic3—82no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—416no—
510C23—48no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—62no—
516C23—88no—
517C23—84no—
518C23—84no—
519C23—121no—
520C23—122no—
521C23—122no—
522C23—121no—
523C23#67⋊C412yes—
524C23#88412yes—
525C23#94012yes—
526C23C2×D3×D411yescharacteristic
527Q8—62no—
528Q8—62no—
529Q8—121no—
530Q8—121no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—42no—
535D4—42no—
536D4—62no—
537D4—62no—
538D4—128no—
539D4—121no—
540D4—121no—
541C2×C4—34no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—128no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—128no—
556C2×C4—124no—
557C2×C4—122no—
558C2×C4—244no—
559C2×C4D3×C22⋊C414yes—
560C2×C4#67⋊C412yes—
561C2×C4#88412yes—
562C2×C4#88514yes—
563C2×C4C4×#6714yes—
564C2×C4#93914yes—
565C2×C4#94012yes—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4C22×#6711yescharacteristic
572C8—24no—
573C8—38no—
574C2×C3—24no—
575C2×C3—84no—
576C2×C3C2×#39011yescharacteristic
577C2×C3C4×AGL(1,ℤ/8)12yes—
578C2×C3#305512yes—
579C2×C3#315111yescharacteristic
580C2×C3#315311yescharacteristic
581D3—82no—
582C22—24no—
583C22—24no—
584C22—22no—
585C22—22no—
586C22—48no—
587C22—44no—
588C22—816no—
589C22—84no—
590C22—82no—
591C22—82no—
592C22—124no—
593C22—122no—
594C22—242no—
595C22—241no—
596C22—241no—
597C22#495714yes—
598C22C2×#67⋊C411yescharacteristic
599C22C2×#88411yescharacteristic
600C22D3×AGL(1,ℤ/8)12yes—
601C22#577312yes—
602C22C2×#94011yescharacteristic
603C4—64no—
604C4—81no—
605C4—81no—
606C4—122no—
607C4—122no—
608C4—242no—
609C4#495714yes—
610C4C2×D3×C22⋊C411yescharacteristic
611C4C2×#88511yescharacteristic
612C4C2×C4×#6711yescharacteristic
613C4C2×#93911yescharacteristic
614C3#2123611yescharacteristic, complemented, Sylow
615C2—24no—
616C2—84no—
617C2—242no—
618C2#7891412yes—
619C2#7893012yes—
620C2C2×#495711yescharacteristic
621C2#8393711yescharacteristic
622C2#8393811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC23×C3C22×D4
prime radicalC22×#932C2
cosocleC22×C3×C4C24
hypercenterC2×#418D3
hypocenterC3#21236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×C3×#418C2semidirect
3. C2×#5221C2semidirect
4. C2×#5133C2semidirect
5. C2×Dic3×M16C2semidirect
6. #82228C2semidirect
7. C2×#4995C2semidirect
8. #70192C2semidirect
9. #70191C2semidirect
10. #70190C2semidirect
11. #70189C2semidirect
21. C2×#932C4semidirect
34. C2×#821C22semidirect
35. C2×#816C22semidirect
36. C2×#815C22semidirect
37. C2×C8×Dic3C22semidirect
39. #4988C22semidirect
40. #4988C22semidirect
41. #4973C22semidirect
42. #4973C22semidirect
43. C2×#808C22semidirect
44. C2×#807C22semidirect
50. C2×#418D3semidirect
95. C2×#214C2×C4semidirect
99. C2×#213C2×C4semidirect
245. #208C22⋊C4semidirect
372. C2×Dic3AGL(1,ℤ/8)semidirect
383. C3⋊C8C2×C22⋊C4semidirect
614. C3#21236semidirect

Maximal subgroups

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

subgroupindex
C22×#9322
C2×C3×#4182
C2×#52212
C2×#51332
C2×Dic3×M162
#822282
C2×#49952
#701922
#701912
#701902
#701892
#212363