Finite Group Databasebeta

← #10944195

Subgroups — #10944195

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#5242C211yesmaximal, characteristic, complemented
3C2×#5241C211yesmaximal, characteristic
4#82527C212yesmaximal, complemented
5#82381C211yesmaximal, characteristic, complemented
6#71667C212yesmaximal, complemented
7#71666C212yesmaximal, complemented
8#71650C211yesmaximal, characteristic, complemented
9#71649C211yesmaximal, characteristic, complemented
10#71646C212yesmaximal, complemented
11#71617C211yesmaximal, characteristic, complemented
12#32924—31nomaximal, Sylow
13C2×C3×H(5,𝔽2)C2211yescharacteristic
14C3×#520—22no—
15C3×#520C2212yes—
16C3×#518—22no—
17C3×#518—22no—
18C3×#518C2211yescharacteristic
19C3×C4×#41—22no—
20C3×C4×#41C2212yes—
21C2×C3×C4×D4—22no—
22C2×C3×C4×D4—22no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×#951C2212yes—
25C2×#949C2212yes—
26C2×#940C2211yescharacteristic
27C2×#936C2211yescharacteristic
28C2×#875C2211yescharacteristic, complemented
29#5242—24no—
30#5241—24no—
31C2×#837C2212yes—
32C2×#837C2212yes—
33C2×#833C2211yescharacteristic
34C2×#833C2211yescharacteristic
35#5213—22no—
36#5028—22no—
37#5028—22no—
38#5016—22no—
39#5016—22no—
40#5004C2212yescomplemented
41#5001C2212yescomplemented
42#4988C2211yescharacteristic, complemented
43#4973C2211yescharacteristic, complemented
44#4961—22no—
45C4×C4⋊Dic3C2211yescharacteristic, complemented
46#4924C2212yescomplemented
47#4532C2212yescomplemented
48#4532C2212yescomplemented
49#4530C2212yescomplemented
50C4×H(5,𝔽2)D311yescharacteristic, complemented
51#3040—32no—
52C2×#455—31no—
53C2×#453—31no—
54#2544—31no—
55#1887—31no—
56#1885—31no—
57#1883—32no—
58#1882—32no—
59#1806—32no—
60#1800—31no—
61C3×H(5,𝔽2)—24no—
62C3×H(5,𝔽2)—22no—
63C3×H(5,𝔽2)D414yes—
64C2×C3×#41—22no—
65C2×C3×#41C2312yes—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—22no—
70C23×C3×C4D412yes—
71C22×#67—22no—
72C22×D12C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C3×C4×Q8—22no—
75C3×C4×Q8D412yescomplemented
76C3×C4×Q8D412yescomplemented
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4D412yescomplemented
90C3×C4×D4D412yescomplemented
91C3×#117—24no—
92C3×#117—22no—
93C3×#117—22no—
94C3×#117C2312yes—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—24no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4D412yes—
102C2×C3×C42—22no—
103C2×C3×C42—22no—
104C2×C3×C42C2311yescharacteristic
105C2×#217—22no—
106#951—24no—
107#951—24no—
108#949—24no—
109#949—24no—
110C2×#216—22no—
111C2×#215—22no—
112#940—22no—
113#940—22no—
114#936—22no—
115#936—22no—
116C2×#214—22no—
117C2×#213—22no—
118C2×(C2×D3)⋊C4—21no—
119C2×(C2×D3)⋊C4—21no—
120C2×(C2×D3)⋊C4—22no—
121C2×C4⋊Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C2×#208C2312yes—
125C2×#208C2312yes—
126#875—22no—
127#875—22no—
128#837—24no—
129#837—24no—
130#833—22no—
131#833—22no—
132C3⋊#95—21no—
133C3⋊#95—22no—
134#810—22no—
135#810—22no—
136#809—22no—
137#809—22no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×H(5,𝔽2)C2×D311yescharacteristic
141C2×#137—32no—
142C2×AGL(1,ℤ/8)—32no—
143C2×#127—31no—
144C2×#124—31no—
145C2×#124—31no—
146#520—22no—
147#520C2×D312yes—
148#518—22no—
149#518—22no—
150#518C2×D311yescharacteristic
151C4×#41—22no—
152C4×#41C2×D312yes—
153C2×C4×D4—22no—
154C2×C4×D4—22no—
155C2×C4×D4C2×D311yescharacteristic
156#464—62no—
157#464—62no—
158#463—62no—
159#463—62no—
160#455—64no—
161#453—64no—
162#423—32no—
163C2×#104—32no—
164C2×#104—32no—
165#419—32no—
166#418—32no—
167C2×#100—31no—
168C2×#100—31no—
169#394—62no—
170#391—31no—
171#390—31no—
172#386—62no—
173C4×C4⋊C4—31no—
174#343—32no—
175#339—32no—
176#339—32no—
177C24×C3—22no—
178C24×C3C2×D412yes—
179C23×D3—21no—
180C3×#41—28no—
181C3×#41—28no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41C2×D414yes—
185C3×#41#4114yes—
186C2×C3×Q8C2×D412yes—
187C2×C3×D4—24no—
188C2×C3×D4—28no—
189C2×C3×D4—28no—
190C2×C3×D4—28no—
191C2×C3×D4—22no—
192C2×C3×D4—28no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4C2×D412yes—
196C2×C3×D4C2×D412yes—
197C2×C3×D4#4112yes—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—24no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—82no—
208C22×C3×C4—84no—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—44no—
214C2×#67—44no—
215C2×#67—82no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C2×D12—22no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—22no—
223C2×Dic6—22no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—22no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—24no—
239C3×C42—24no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42C2×D412yes—
244C3×C42C2×D412yes—
245#217—44no—
246#216—44no—
247#215—44no—
248#214—44no—
249#213—44no—
250(C2×D3)⋊C4—41no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—41no—
253(C2×D3)⋊C4—82no—
254C4⋊Dic3—42no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259#208—22no—
260#208—22no—
261#208—22no—
262#208—22no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—22no—
265H(5,𝔽2)—24no—
266H(5,𝔽2)—22no—
267H(5,𝔽2)D1214yes—
268C2×#41—22no—
269C2×#41C22×D312yes—
270C22×Q8—31no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—31no—
274C22×D4—62no—
275C22×D4C22×D311yescharacteristic
276C23×C4—22no—
277C23×C4#6712yes—
278#137—64no—
279#137—64no—
280AGL(1,ℤ/8)—64no—
281AGL(1,ℤ/8)—64no—
282C2×Q16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×D8—62no—
286C2×M16—32no—
287C2×M16—32no—
288#127—62no—
289#127—62no—
290#124—62no—
291#124—62no—
292#124—62no—
293#124—62no—
294C4×Q8—22no—
295C4×Q8#6712yescomplemented
296C4×Q8#6712yescomplemented
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4#6712yescomplemented
310C4×D4#6712yescomplemented
311#117—24no—
312#117—22no—
313#117—22no—
314#117C22×D312yes—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—62no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4#6712yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42C22×D311yescharacteristic
332C4⋊C8—62no—
333C4⋊C8—62no—
334#104—64no—
335#104—64no—
336#103—62no—
337#103—62no—
338#102—62no—
339#102—62no—
340#100—62no—
341#100—62no—
342#95—61no—
343#95—62no—
344C23×C3—28no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—82no—
351C23×C3—82no—
352C23×C3#12114yes—
353C23×C3C22×D411yescharacteristic
354C22×D3—42no—
355C22×D3—42no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C3×Q8—22no—
360C3×Q8#12114yes—
361C3×D4—216no—
362C3×D4—24no—
363C3×D4—24no—
364C3×D4—24no—
365C3×D4—22no—
366C3×D4—24no—
367C3×D4#12114yes—
368C2×C3×C4—28no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—44no—
378C2×C3×C4—44no—
379C2×C3×C4—42no—
380C2×C3×C4—82no—
381C2×C3×C4—82no—
382C2×C3×C4—82no—
383C2×C3×C4—84no—
384C2×C3×C4—82no—
385C2×C3×C4#12011yescharacteristic
386C2×C3×C4#12012yes—
387C2×C3×C4#12011yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D412yes—
390C2×C3×C4C2×#4112yes—
391C2×C3×C4C2×#4111yescharacteristic
392#67—88no—
393C2×Dic3—22no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402D12—42no—
403D12—42no—
404Dic6—42no—
405Dic6—42no—
406C3⋊C8—42no—
407C3⋊C8—42no—
408C24—22no—
409C24—61no—
410C24C2×#6712yes—
411#41—28no—
412#41—28no—
413#41—22no—
414#41—22no—
415#41C2×D1214yes—
416#41#23514yes—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8C2×#6712yes—
422C2×D4—24no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—22no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—124no—
435C2×D4—124no—
436C2×D4—242no—
437C2×D4C2×D1212yes—
438C2×D4#23512yes—
439C2×D4C2×#6712yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—82no—
453C22×C4—84no—
454C22×C4C2×#6712yes—
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—124no—
459SD16—124no—
460SD16—124no—
461D8—124no—
462M16—62no—
463M16—62no—
464M16—62no—
465M16—62no—
466C2×C8—62no—
467C2×C8—62no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—122no—
473C4⋊C4C2×#6712yes—
474C4⋊C4C2×#6712yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—124no—
482C22⋊C4—121no—
483C22⋊C4—122no—
484C22⋊C4—121no—
485C22⋊C4—242no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—61no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42C2×#6712yes—
495C42C2×#6712yes—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—82no—
506C22×C3—88no—
507C22×C3C2×#12011yescharacteristic
508C22×C3C2×#12111yescharacteristic
509C22×C3#52511yescharacteristic
510C2×D3—82no—
511C2×D3—82no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4C2×#12112yes—
523C3×C4#52512yes—
524Dic3—42no—
525Dic3—81no—
526Dic3—81no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—122no—
536C23—122no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23#93014yes—
541C23C22×#6711yescharacteristic
542Q8—22no—
543Q8—122no—
544Q8—122no—
545Q8#93014yes—
546D4—216no—
547D4—24no—
548D4—24no—
549D4—24no—
550D4—24no—
551D4—22no—
552D4—122no—
553D4—122no—
554D4—248no—
555D4#93014yes—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—62no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4#95311yescharacteristic
583C2×C4#95312yes—
584C2×C4#95311yescharacteristic
585C2×C4C22×D1211yescharacteristic
586C2×C4C2×#23512yes—
587C2×C4C2×#23511yescharacteristic
588C2×C4C22×#6712yes—
589C8—122no—
590C8—122no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—44no—
594C2×C3—44no—
595C2×C3#284711yescharacteristic
596C2×C3#301811yescharacteristic
597C2×C3#301911yescharacteristic
598D3—161no—
599C22—21no—
600C22—21no—
601C22—24no—
602C22—24no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—82no—
609C22—88no—
610C22—242no—
611C22—242no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22C2×#93011yescharacteristic
616C22#579111yescharacteristic
617C22C2×#95311yescharacteristic
618C4—24no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4C2×#93012yes—
629C4#579112yes—
630C3#3292411yescharacteristic, complemented, Sylow
631C2—21no—
632C2—21no—
633C2—44no—
634C2—44no—
635C2—481no—
636C2#8403211yescharacteristic
637C2#8410411yescharacteristic
638C2#8412111yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×H(5,𝔽2)C2
socleC22×C3C2×#120
prime radicalC2×#5241C2
hypercenterC4×H(5,𝔽2)D3
hypocenterC3#32924
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×C4×H(5,𝔽2)C2semidirect
2. C2×#5242C2semidirect
4. #82527C2semidirect
5. #82381C2semidirect
6. #71667C2semidirect
7. #71666C2semidirect
8. #71650C2semidirect
9. #71649C2semidirect
10. #71646C2semidirect
11. #71617C2semidirect
28. C2×#875C22semidirect
40. #5004C22semidirect
41. #5001C22semidirect
42. #4988C22semidirect
43. #4973C22semidirect
45. C4×C4⋊Dic3C22semidirect
46. #4924C22semidirect
47. #4532C22semidirect
48. #4532C22semidirect
49. #4530C22semidirect
50. C4×H(5,𝔽2)D3semidirect
75. C3×C4×Q8D4semidirect
76. C3×C4×Q8D4semidirect
89. C3×C4×D4D4semidirect
90. C3×C4×D4D4semidirect
295. C4×Q8#67semidirect
296. C4×Q8#67semidirect
309. C4×D4#67semidirect
310. C4×D4#67semidirect
630. C3#32924semidirect

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C2×#52422
C2×#52412
#825272
#823812
#716672
#716662
#716502
#716492
#716462
#716172
#329243