Finite Group Databasebeta

← #10799109

Subgroups — #10799109

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×#908C211yesmaximal, characteristic
2C2×C3×C4×M16C211yesmaximal, characteristic, complemented
3C2×#5133C211yesmaximal, characteristic, complemented
4C2×#5131C211yesmaximal, characteristic, complemented
5C2×#5121C211yesmaximal, characteristic, complemented
6#82070C211yesmaximal, characteristic, complemented
7C2×#4939C211yesmaximal, characteristic, complemented
8#69826C212yesmaximal, complemented
9#69825C212yesmaximal, complemented
10#69776C212yesmaximal, complemented
11#69774C212yesmaximal, complemented
12#21236—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×(C2×D3)⋊C4—22no—
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#908—24no—
22C2×#908—24no—
23C2×#908C418yescomplemented
24C22×D24C2212yes—
25C22×#204C2212yes—
26C3×C4×M16—24no—
27C2×C3×#97C2212yescomplemented
28C2×C3×C4×C8C2212yescomplemented
29#5213—22no—
30#5133—24no—
31#5131—24no—
32#5121—24no—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#821C2212yescomplemented
36C2×#818C2212yescomplemented
37C2×#817C2212yescomplemented
38C2×#816C2212yescomplemented
39#4939—22no—
40#4939C2212yes—
41#4939C2212yes—
42#4939C2211yescharacteristic, complemented
43#4939C2211yescharacteristic, complemented
44C4×(C2×D3)⋊C4—22no—
45#4929C2211yescharacteristic, complemented
46#4929C2211yescharacteristic, complemented
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#417—31no—
50C2×#418—31no—
51C2×C4×M16D311yescharacteristic, complemented
52#2297—31no—
53C2×#390—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C24×D3—21no—
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235C2×C414yes—
69C22×D12—22no—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12C2×C414yes—
76C22×D12C2312yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16C2×C414yes—
83C2×C3×M16D414yes—
84C2×C3×M16D414yes—
85C2×C3×M16D414yes—
86C22×C3×C8C2312yes—
87C22×C3×C8C2312yes—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42D412yes—
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—48no—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2312yes—
107C2×C4×Dic3—21no—
108#908—232no—
109C2×D24—24no—
110C2×D24—24no—
111C2×D24C2×C418yescomplemented
112C2×#204—24no—
113C2×#204—24no—
114C2×#204C2×C418yescomplemented
115C3×#97—24no—
116C3×C4×C8—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#821—24no—
120#821—24no—
121#821—24no—
122#818—24no—
123#817—24no—
124#816—24no—
125C22×#41—31no—
126C23×D4—31no—
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×M16C2×D311yescharacteristic
133C22×M16C2×D311yescharacteristic
134C2×#117—31no—
135C22×C4⋊C4—31no—
136C22×C22⋊C4—62no—
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—24no—
148C2×#97C2×D312yescomplemented
149C2×C4×C8C2×D312yescomplemented
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×C3C2×D411yescharacteristic
160C23×D3—24no—
161C23×D3—21no—
162C23×D3—22no—
163C23×D3—44no—
164C23×D3—44no—
165C23×D3—42no—
166C23×D3—42no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—42no—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—44no—
188C22×Dic3—42no—
189#235—28no—
190#235—28no—
191#235—44no—
192#235—44no—
193C2×D12—28no—
194C2×D12—28no—
195C2×D12—22no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—22no—
205C2×D12—21no—
206C2×D12—44no—
207C2×D12—42no—
208C2×D12—44no—
209C2×D12—42no—
210C2×D12C22×C414yes—
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—
222C3×M16C22⋊C4116yescomplemented
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8C22×C414yes—
226C2×C3×C8C2×D414yes—
227C2×C3×C8C2×D414yes—
228C2×C3×C8C2×D414yes—
229C3×C42—24no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234#217—42no—
235(C2×D3)⋊C4—44no—
236(C2×D3)⋊C4—48no—
237(C2×D3)⋊C4—44no—
238(C2×D3)⋊C4—48no—
239C4⋊Dic3—22no—
240C4⋊Dic3—21no—
241C4⋊Dic3—22no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—42no—
246C4×Dic3—42no—
247D24—216no—
248#204—216no—
249C25—61no—
250C2×#41—34no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—62no—
256C22×Q8—31no—
257C22×D4—34no—
258C22×D4—31no—
259C22×D4—32no—
260C22×D4—61no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4—61no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C23×C4—61no—
268C23×C4—61no—
269C23×C4D1212yes—
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×M16C4×D314yes—
281C2×M16D1214yes—
282C2×M16D1214yes—
283C2×M16#6714yes—
284C22×C8C22×D312yes—
285C22×C8C22×D312yes—
286#117—62no—
287#117—62no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—128no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—61no—
303C2×C42D1212yes—
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—24no—
313C4×C8—24no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3C4×D412yes—
319C23×C3#12112yes—
320C23×C3#12712yes—
321C23×C3C22×D411yescharacteristic
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—416no—
325C22×D3—41no—
326C22×D3—42no—
327C22×D3—41no—
328C22×D3—42no—
329C22×D3—44no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—44no—
333C22×D3—48no—
334C22×D3—84no—
335C22×D3—84no—
336C22×D3—88no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—48no—
343C2×C3×C4C2×C22⋊C414yes—
344C2×C3×C4C4×D414yes—
345C2×C3×C4C4×D412yes—
346C2×C3×C4#12114yes—
347C2×C3×C4#12112yes—
348C2×C3×C4#12414yes—
349C2×C3×C4#12712yes—
350C2×C3×C4AGL(1,ℤ/8)14yescomplemented
351C2×C3×C4C23×C411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357#67—48no—
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—44no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—84no—
368D12—24no—
369D12—22no—
370D12—24no—
371D12—22no—
372D12—42no—
373D12—41no—
374D12—42no—
375D12—41no—
376D12—48no—
377C4×D3—48no—
378Dic6—22no—
379Dic6—22no—
380Dic6—41no—
381Dic6—41no—
382C3×C8—24no—
383C3×C8C2×C22⋊C418yescomplemented
384C24—64no—
385C24—61no—
386C24—62no—
387C24—124no—
388C24—124no—
389C24—122no—
390C24—122no—
391C24C2×D1211yescharacteristic
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—32no—
403C2×D4—34no—
404C2×D4—64no—
405C2×D4—68no—
406C2×D4—64no—
407C2×D4—68no—
408C2×D4—62no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4—62no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—62no—
418C2×D4—122no—
419C2×D4—122no—
420C2×D4—122no—
421C2×D4—124no—
422C2×D4—122no—
423C2×D4—124no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—42no—
427C22×C4—61no—
428C22×C4—64no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—122no—
433C22×C4—122no—
434C22×C4—124no—
435C22×C4—122no—
436C22×C4C2×C4×D312yes—
437C22×C4C2×D1212yes—
438C22×C4C2×D1212yes—
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1212yes—
441C22×C4C2×D1212yes—
442C22×C4#23512yes—
443C22×C4#23512yes—
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446SD16—616no—
447D8—616no—
448M16—28no—
449M16(C2×D3)⋊C4116yescomplemented
450C2×C8—22no—
451C2×C8—22no—
452C2×C8C2×C4×D314yes—
453C2×C8C2×D1214yes—
454C2×C8C2×D1214yes—
455C2×C8C2×#6714yes—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—122no—
463C22⋊C4—128no—
464C22⋊C4—124no—
465C22⋊C4—122no—
466C22⋊C4—128no—
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#39014yes—
479C22×C3C2×C4×D411yescharacteristic
480C22×C3C2×#12111yescharacteristic
481C22×C3C2×#12711yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)12yes—
483C22×C3#57512yes—
484C2×D3—44no—
485C2×D3—48no—
486C2×D3—42no—
487C2×D3—44no—
488C2×D3—816no—
489C2×D3—81no—
490C2×D3—82no—
491C2×D3—81no—
492C2×D3—82no—
493C2×D3—82no—
494C2×D3—84no—
495C3×C4—24no—
496C3×C4—42no—
497C3×C4#39014yes—
498C3×C4C22×C22⋊C411yescharacteristic
499C3×C4C2×C4×D411yescharacteristic
500C3×C4C2×#12111yescharacteristic
501C3×C4C2×#12411yescharacteristic
502Dic3—42no—
503Dic3—81no—
504Dic3—82no—
505Dic3—81no—
506C23—22no—
507C23—22no—
508C23—62no—
509C23—64no—
510C23—1216no—
511C23—128no—
512C23—121no—
513C23—122no—
514C23—124no—
515C23—122no—
516C23—122no—
517C23—124no—
518C23—121no—
519C23—122no—
520C23—248no—
521C23—244no—
522C23—244no—
523C23C4×D1212yes—
524C23#87412yes—
525C23#93012yes—
526C23C22×D1211yescharacteristic
527Q8—62no—
528Q8—62no—
529Q8—121no—
530Q8—121no—
531D4—64no—
532D4—62no—
533D4—62no—
534D4—64no—
535D4—128no—
536D4—128no—
537D4—122no—
538D4—121no—
539D4—122no—
540D4—121no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—48no—
547C2×C4—62no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—128no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—122no—
554C2×C4—124no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—244no—
558C2×C4C4×D1212yes—
559C2×C4C4×D1214yes—
560C2×C4#87412yes—
561C2×C4#87514yes—
562C2×C4#90814yescomplemented
563C2×C4C2×(C2×D3)⋊C414yes—
564C2×C4#93012yes—
565C2×C4#93014yes—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C22×D1211yescharacteristic
568C2×C4C22×D1211yescharacteristic
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23511yescharacteristic
571C2×C4C22×#6711yescharacteristic
572C8—24no—
573C8C2×(C2×D3)⋊C418yescomplemented
574C2×C3—24no—
575C2×C3C2×#39011yescharacteristic
576C2×C3C4×AGL(1,ℤ/8)12yes—
577C2×C3#305512yes—
578C2×C3#315111yescharacteristic
579C2×C3#315311yescharacteristic
580D3—82no—
581D3—84no—
582C22—24no—
583C22—24no—
584C22—22no—
585C22—22no—
586C22—122no—
587C22—124no—
588C22—128no—
589C22—124no—
590C22—2416no—
591C22—244no—
592C22—242no—
593C22—242no—
594C22—241no—
595C22—242no—
596C22—241no—
597C22#493914yes—
598C22C2×C4×D1211yescharacteristic
599C22C2×#87411yescharacteristic
600C22C2×#90812yes—
601C22#574812yes—
602C22C2×#93011yescharacteristic
603C4—24no—
604C4—42no—
605C4—122no—
606C4—242no—
607C4—241no—
608C4—241no—
609C4#493914yes—
610C4C2×C4×D1211yescharacteristic
611C4C2×#87511yescharacteristic
612C4C22×(C2×D3)⋊C411yescharacteristic
613C4C2×#93011yescharacteristic
614C3#2123611yescharacteristic, complemented, Sylow
615C2—24no—
616C2—244no—
617C2—242no—
618C2C4×#90812yes—
619C2#7853011yescharacteristic
620C2#7853111yescharacteristic
621C2C2×#493911yescharacteristic
622C2#8377212yes—
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×M16C2
socleC23×C3C22×D4
prime radicalC22×#908C2
cosocleC22×C3×C4C24
hypercenterC2×C4×M16D3
hypocenterC3#21236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×M16C2semidirect
3. C2×#5133C2semidirect
4. C2×#5131C2semidirect
5. C2×#5121C2semidirect
6. #82070C2semidirect
7. C2×#4939C2semidirect
8. #69826C2semidirect
9. #69825C2semidirect
10. #69776C2semidirect
11. #69774C2semidirect
23. C2×#908C4semidirect
27. C2×C3×#97C22semidirect
28. C2×C3×C4×C8C22semidirect
35. C2×#821C22semidirect
36. C2×#818C22semidirect
37. C2×#817C22semidirect
38. C2×#816C22semidirect
42. #4939C22semidirect
43. #4939C22semidirect
45. #4929C22semidirect
46. #4929C22semidirect
51. C2×C4×M16D3semidirect
111. C2×D24C2×C4semidirect
114. C2×#204C2×C4semidirect
148. C2×#97C2×D3semidirect
149. C2×C4×C8C2×D3semidirect
222. C3×M16C22⋊C4semidirect
350. C2×C3×C4AGL(1,ℤ/8)semidirect
383. C3×C8C2×C22⋊C4semidirect
449. M16(C2×D3)⋊C4semidirect
562. C2×C4#908semidirect
573. C8C2×(C2×D3)⋊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×#9082
C2×C3×C4×M162
C2×#51332
C2×#51312
C2×#51212
#820702
C2×#49392
#698262
#698252
#697762
#697742
#212363