Finite Group Databasebeta

← #10824351

Subgroups — #10824351

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
2#82451C211yesmaximal, characteristic, complemented
3C2×C4×#215C211yesmaximal, characteristic, complemented
4C4×#940C211yesmaximal, characteristic
5#72056C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8#70585C211yesmaximal, characteristic, complemented
9#70567C211yesmaximal, characteristic, complemented
10#70280C211yesmaximal, characteristic, complemented
11#70233C211yesmaximal, characteristic, complemented
12#70193C211yesmaximal, characteristic, complemented
13#70177C211yesmaximal, characteristic
14C4×#821C211yesmaximal, characteristic, complemented
15C4×Dic3⋊C8C211yesmaximal, characteristic, complemented
16#23678—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#940C2211yescharacteristic
22C2×C4×#67—21no—
23C2×C4×#67—21no—
24C2×C4×D12C2211yescharacteristic
25C3×C4×SD16—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30#5229C2211yescharacteristic
31#5229C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C4×#215—24no—
41#4988C2211yescharacteristic
42#4984C2211yescharacteristic
43#4984C2211yescharacteristic
44#4980—21no—
45#4980—21no—
46#4975C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4957—21no—
50#4957—21no—
51C4×C4⋊Dic3—21no—
52C4×C4⋊Dic3C2211yescharacteristic
53#4932—21no—
54C42×Dic3C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×SD16D311yescharacteristic, complemented
64C4×#127—31no—
65#2295—31no—
66#2084—31no—
67#2072—31no—
68#2065—31no—
69#2065—31no—
70#2055—31no—
71#2055—31no—
72#2041—31no—
73#2014—31no—
74C4×C4⋊C8—31no—
75C4×#102—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×#67—21no—
81C22×#67—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×Q8—22no—
90C3×C4×Q8D414yescomplemented
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#215—22no—
102C2×#215—22no—
103C2×#215—22no—
104#940—22no—
105#940—22no—
106C4×#67—44no—
107C4×#67—44no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3D411yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C4×D12—22no—
128C4×D12—22no—
129C3×#106—22no—
130C3×#103—22no—
131C3×#102—22no—
132C3×C4×C8—22no—
133#835—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#821—22no—
144#821—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147#809—22no—
148#808—22no—
149C4×C3⋊C8—22no—
150C22×SD16—31no—
151C22×SD16C2×D311yescharacteristic
152C2×#127—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155C2×C4×D4—61no—
156C2×C4×D4—61no—
157C2×C4×D4C2×D311yescharacteristic
158C4×SD16—24no—
159C4×SD16—64no—
160C2×#106—31no—
161C2×#106C2×D311yescharacteristic
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102C2×D311yescharacteristic
172C2×C4×C8—31no—
173C2×C4×C8C2×D311yescharacteristic
174#391—31no—
175#391—31no—
176#390—31no—
177#390—31no—
178#390—61no—
179#390—61no—
180#389—31no—
181#389—31no—
182#387—61no—
183#387—61no—
184#384—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—61no—
187#376—61no—
188C43—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#336—31no—
194#336—31no—
195C24×C3—21no—
196C23×D3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4#4112yes—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C2×#67—24no—
219C2×#67—24no—
220C2×#67—42no—
221C2×#67—42no—
222C2×#67—42no—
223C2×#67—42no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12#4112yes—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C3×SD16—28no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8#4112yes—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D412yes—
252C3×C22⋊C4—42no—
253C3×C42—24no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256#217—42no—
257#217—42no—
258#215—28no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—42no—
269C4⋊Dic3—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—22no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—22no—
278C4×Dic3—41no—
279C4×Dic3—42no—
280C4×Dic3SD1614yescomplemented
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8#4112yes—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C22×D4—61no—
290C22×D4—61no—
291C22×D4C22×D311yescharacteristic
292C23×C4—21no—
293C23×C4—61no—
294C2×SD16—22no—
295C2×SD16—22no—
296C2×SD16—22no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—62no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304#127—62no—
305#127—62no—
306C4×Q8—22no—
307C4×Q8#6714yescomplemented
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—62no—
311C4×D4—62no—
312C4×D4—124no—
313C4×D4—124no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—21no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#106—22no—
340#106—62no—
341C4⋊C8—62no—
342C4⋊C8—62no—
343#103—22no—
344#103—62no—
345#102—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351C4×C8—22no—
352C4×C8—62no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3C22×D411yescharacteristic
368C22×D3—22no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C3×Q8—22no—
376C3×Q8—22no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—41no—
382C3×D4—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—84no—
395C2×C3×C4#12112yes—
396C2×C3×C4#12112yes—
397C2×C3×C4C2×SD1612yes—
398C2×C3×C4AGL(1,ℤ/8)12yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405#67—48no—
406#67—48no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3C2×SD1612yes—
426C2×Dic3C2×SD1612yes—
427C2×Dic3#13512yes—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433C3×C8—22no—
434C3×C8—41no—
435C3⋊C8—22no—
436C3⋊C8—41no—
437C24—21no—
438C24—61no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×#6712yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—32no—
451C2×D4—64no—
452C2×D4—64no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—122no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—122no—
462C2×D4#23512yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—122no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482SD16—28no—
483SD16—68no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8#23512yes—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4—122no—
510C4⋊C4C2×#6712yes—
511C22⋊C4—42no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C42—24no—
518C42—34no—
519C42—62no—
520C42—62no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—61no—
525C42—121no—
526C42—122no—
527C42#23712yes—
528C42C2×#6712yes—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3C2×#12111yescharacteristic
537C22×C3#52511yescharacteristic
538C22×C3#53211yescharacteristic
539C22×C3C22×SD1611yescharacteristic
540C22×C3C2×#13511yescharacteristic
541C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
542C22×C3#57511yescharacteristic
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—42no—
553C3×C4—81no—
554C3×C4#46014yes—
555C3×C4#52511yescharacteristic
556C3×C4#52511yescharacteristic
557C3×C4#52911yescharacteristic
558C3×C4#52911yescharacteristic
559Dic3—22no—
560Dic3—22no—
561Dic3—22no—
562Dic3—41no—
563Dic3—81no—
564Dic3—81no—
565Dic3—81no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—62no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23C2×D3×D411yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—41no—
584Q8—41no—
585D4—22no—
586D4—22no—
587D4—41no—
588D4—41no—
589D4—62no—
590D4—62no—
591D4—128no—
592D4—128no—
593D4—121no—
594D4—121no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—84no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—244no—
629C2×C4D3×SD1612yes—
630C2×C4#91412yes—
631C2×C4#93912yes—
632C2×C4#93912yes—
633C2×C4C2×#23511yescharacteristic
634C2×C4C2×#23511yescharacteristic
635C2×C4C2×#23711yescharacteristic
636C2×C4D3×#4111yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4C22×#6711yescharacteristic
639C8—22no—
640C8—41no—
641C8—62no—
642C8—121no—
643C2×C3—82no—
644C2×C3#239211yescharacteristic
645C2×C3C2×#46011yescharacteristic
646C2×C3#303511yescharacteristic
647C2×C3#310111yescharacteristic
648C2×C3#310511yescharacteristic
649C2×C3#316711yescharacteristic
650C2×C3#317111yescharacteristic
651D3—82no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—241no—
663C22—241no—
664C22—241no—
665C22—241no—
666C22#554711yescharacteristic
667C22#554811yescharacteristic
668C22C2×D3×SD1611yescharacteristic
669C22C2×#91411yescharacteristic
670C22D3×#13511yescharacteristic
671C22#576811yescharacteristic
672C22C2×#93911yescharacteristic
673C4—22no—
674C4—42no—
675C4—42no—
676C4—62no—
677C4—62no—
678C4—62no—
679C4—81no—
680C4—121no—
681C4—241no—
682C4—241no—
683C4—241no—
684C4#516314yes—
685C4#553811yescharacteristic
686C4#555711yescharacteristic
687C4#579111yescharacteristic
688C4#582811yescharacteristic
689C3#2367811yescharacteristic, complemented, Sylow
690C2—82no—
691C2—242no—
692C2#7965111yescharacteristic
693C2#7965411yescharacteristic
694C2#7966211yescharacteristic
695C2#7966411yescharacteristic
696C2#8234511yescharacteristic
697C2C2×#516311yescharacteristic
698C2#8390011yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenterC2×C4×SD16D3
hypocenterC3#23678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SD16C2semidirect
2. #82451C2semidirect
3. C2×C4×#215C2semidirect
6. #72051C2semidirect
7. C4×#832C2semidirect
8. #70585C2semidirect
9. #70567C2semidirect
10. #70280C2semidirect
11. #70233C2semidirect
12. #70193C2semidirect
14. C4×#821C2semidirect
15. C4×Dic3⋊C8C2semidirect
63. C2×C4×SD16D3semidirect
90. C3×C4×Q8D4semidirect
280. C4×Dic3SD16semidirect
307. C4×Q8#67semidirect
689. C3#23678semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SD162
#824512
C2×C4×#2152
C4×#9402
#720562
#720512
C4×#8322
#705852
#705672
#702802
#702332
#701932
#701772
C4×#8212
C4×Dic3⋊C82
#236783