Finite Group Databasebeta

← #10799107

Subgroups — #10799107

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×#390C211yesmaximal, characteristic, complemented
3C2×#5215C211yesmaximal, characteristic, complemented
4C2×#4966C211yesmaximal, characteristic, complemented
5C2×#4964C211yesmaximal, characteristic, complemented
6#82070C211yesmaximal, characteristic, complemented
7C2×C4×#208C211yesmaximal, characteristic, complemented
8#69649C212yesmaximal, complemented
9#69648C212yesmaximal, complemented
10#69647C212yesmaximal, complemented
11#69646C212yesmaximal, complemented
12#21236—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4—22no—
17C22×C3×C42C2211yescharacteristic
18C22×#214C2212yes—
19C2×#932—24no—
20C2×#932—24no—
21C2×#932C418yescomplemented
22C22×#213C2212yes—
23C2×#926C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C22×#208C2211yescharacteristic
26C3×#390—22no—
27C3×#390C2212yes—
28C3×#390C2212yes—
29#5215—24no—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5213—22no—
33C2×#810C2212yescomplemented
34#4966—24no—
35C2×#809C2212yescomplemented
36#4964—24no—
37C2×#808C2212yescomplemented
38C2×#807C2212yescomplemented
39#4939C2211yescharacteristic, complemented
40#4939C2211yescharacteristic, complemented
41C4×(C2×D3)⋊C4—22no—
42#4929C2211yescharacteristic, complemented
43#4929C2211yescharacteristic, complemented
44C4×#208—24no—
45C2×#803C2212yescomplemented
46C2×C4×C3⋊C8C2212yescomplemented
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#417—31no—
50C2×#418—31no—
51C2×C4×M16—31no—
52#2297—31no—
53C2×#390D311yescharacteristic, complemented
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×C4—21no—
68C23×C3×C4D412yes—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
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×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—48no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42D412yes—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
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—
102#926—22no—
103#926—22no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4×Dic3—21no—
107C2×#208—22no—
108C2×#208—22no—
109C2×#208C2×C414yes—
110C2×#208D414yes—
111C2×#208D414yes—
112C2×#208D414yes—
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2312yes—
115#832—24no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#810—24no—
120#809—24no—
121#808—24no—
122#807—24no—
123#803—24no—
124C4×C3⋊C8—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×M16—31no—
134C2×#117—31no—
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C42C2×D311yescharacteristic
138#428—64no—
139C2×#107—32no—
140C2×#106—32no—
141#418—64no—
142#417—64no—
143C2×#103—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C4×M16—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—31no—
151#391—31no—
152#390—22no—
153#390—31no—
154#390—31no—
155#390C2×D312yes—
156#390C2×D312yes—
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—44no—
176C2×C3×D4—42no—
177C2×C3×D4—44no—
178C2×C3×D4—42no—
179C2×C3×D4C22×C414yes—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—42no—
184C22×C3×C4—44no—
185C22×C3×C4—42no—
186C22×C3×C4C22×C412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4#4112yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—24no—
197C2×#67—42no—
198C2×#67—42no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
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×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224C3×C22⋊C4—48no—
225C3×C22⋊C4—48no—
226C3×C42—24no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231#217—42no—
232#214—216no—
233#213—216no—
234(C2×D3)⋊C4—44no—
235(C2×D3)⋊C4—44no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4×Dic3—42no—
241#208—28no—
242#208C22⋊C4116yescomplemented
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8C22×C414yes—
246C2×C3⋊C8C2×D414yes—
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—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—31no—
264C22×D4—61no—
265C22×D4C4×D314yes—
266C22×D4C22×D312yes—
267C23×C4—21no—
268C23×C4—61no—
269C23×C4#6712yes—
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—34no—
279C2×M16—34no—
280C2×M16—34no—
281C2×M16—34no—
282C2×M16—62no—
283C2×M16—62no—
284C22×C8—32no—
285C22×C8—32no—
286#117—62no—
287#117—62no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4C22×D312yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—61no—
303C2×C42#6712yes—
304C2×C42C22×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306#107—64no—
307#106—64no—
308#103—64no—
309#102—64no—
310#102—64no—
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—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—48no—
348C2×C3×C4—42no—
349C2×C3×C4—44no—
350C2×C3×C4—84no—
351C2×C3×C4C2×C22⋊C414yes—
352C2×C3×C4C4×D414yes—
353C2×C3×C4C4×D412yes—
354C2×C3×C4#12114yes—
355C2×C3×C4#12112yes—
356C2×C3×C4#12414yes—
357C2×C3×C4#12712yes—
358C2×C3×C4AGL(1,ℤ/8)14yescomplemented
359C2×C3×C4C23×C411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365#67—48no—
366C2×Dic3—22no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—42no—
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—
391C24C2×#6711yescharacteristic
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—28no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—32no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
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—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—61no—
431C22×C4—64no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4C2×C4×D312yes—
437C22×C4#23512yes—
438C22×C4D3×D412yes—
439C22×C4D3×D412yes—
440C22×C4D3×D412yes—
441C22×C4#23712yes—
442C22×C4C2×#6712yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6711yescharacteristic
445C22×C4C23×D311yescharacteristic
446SD16—616no—
447D8—616no—
448M16—316no—
449M16—68no—
450C2×C8—34no—
451C2×C8—34no—
452C2×C8—34no—
453C2×C8—34no—
454C2×C8—62no—
455C2×C8—62no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—124no—
466C22⋊C4—122no—
467C22⋊C4—124no—
468C42—24no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—122no—
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—24no—
496C3×C4—42no—
497C3×C4—82no—
498C3×C4#39014yes—
499C3×C4C22×C22⋊C411yescharacteristic
500C3×C4C2×C4×D411yescharacteristic
501C3×C4C2×#12111yescharacteristic
502C3×C4C2×#12411yescharacteristic
503Dic3—42no—
504Dic3—81no—
505Dic3—81no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—416no—
510C23—48no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—62no—
516C23—88no—
517C23—84no—
518C23—84no—
519C23—122no—
520C23—121no—
521C23—121no—
522C23—122no—
523C23C4×#6712yes—
524C23#93912yes—
525C23#94012yes—
526C23C22×#6711yescharacteristic
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—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—62no—
550C2×C4—84no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—128no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—122no—
558C2×C4D3×C22⋊C414yes—
559C2×C4#67⋊C414yes—
560C2×C4#88414yes—
561C2×C4#88514yes—
562C2×C4C4×#6712yes—
563C2×C4#93214yescomplemented
564C2×C4#93912yes—
565C2×C4#94012yes—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C8—38no—
573C8—64no—
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×C4×#6711yescharacteristic
599C22C2×#93212yes—
600C22C2×#93911yescharacteristic
601C22C2×#94011yescharacteristic
602C22#582212yes—
603C4—24no—
604C4—42no—
605C4—82no—
606C4—122no—
607C4—241no—
608C4—241no—
609C4#495714yes—
610C4C2×D3×C22⋊C411yescharacteristic
611C4C2×#67⋊C411yescharacteristic
612C4C2×#88411yescharacteristic
613C4C2×#88511yescharacteristic
614C3#2123611yescharacteristic, complemented, Sylow
615C2—24no—
616C2—84no—
617C2—242no—
618C2C2×#495711yescharacteristic
619C2C4×#93212yes—
620C2#8262112yes—
621C2#8294311yescharacteristic
622C2#8316311yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#390C2
socleC23×C3C22×D4
prime radicalC22×#932C2
cosocleC22×C3×C4C24
hypercenterC2×#390D3
hypocenterC3#21236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×C3×#390C2semidirect
3. C2×#5215C2semidirect
4. C2×#4966C2semidirect
5. C2×#4964C2semidirect
6. #82070C2semidirect
7. C2×C4×#208C2semidirect
8. #69649C2semidirect
9. #69648C2semidirect
10. #69647C2semidirect
11. #69646C2semidirect
21. C2×#932C4semidirect
33. C2×#810C22semidirect
35. C2×#809C22semidirect
37. C2×#808C22semidirect
38. C2×#807C22semidirect
39. #4939C22semidirect
40. #4939C22semidirect
42. #4929C22semidirect
43. #4929C22semidirect
45. C2×#803C22semidirect
46. C2×C4×C3⋊C8C22semidirect
53. C2×#390D3semidirect
95. C2×#214C2×C4semidirect
99. C2×#213C2×C4semidirect
242. #208C22⋊C4semidirect
358. C2×C3×C4AGL(1,ℤ/8)semidirect
383. C3⋊C8C2×C22⋊C4semidirect
563. C2×C4#932semidirect
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×#3902
C2×#52152
C2×#49662
C2×#49642
#820702
C2×C4×#2082
#696492
#696482
#696472
#696462
#212363