Finite Group Databasebeta

← #76755

Subgroups — #76755

Order 384.

1246 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×D3×D4C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×#937C211yesmaximal, characteristic, complemented
4C2×#937C211yesmaximal, characteristic, complemented
5C2×#884C211yesmaximal, characteristic, complemented
6C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
7C3×#392C211yesmaximal, characteristic, complemented
8#5223C211yesmaximal, characteristic, complemented
9#5222C211yesmaximal, characteristic, complemented
10#4961C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4955C211yesmaximal, characteristic, complemented
14#4668C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2410—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C24×D3C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic, complemented
21C22×C3×D4C2211yescharacteristic, complemented
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic, complemented
26C22×#67C2211yescharacteristic, complemented
27C2×D3×D4—24no—
28C2×D3×D4—24no—
29C2×D3×D4—24no—
30C22×D12C2211yescharacteristic
31C22×C4×D3C2211yescharacteristic
32C22×C4×D3C2211yescharacteristic, complemented
33C22×C4×D3C2211yescharacteristic, complemented
34C2×C3×C22⋊C4C2211yescharacteristic, complemented
35C2×C3×C22⋊C4C2211yescharacteristic, complemented
36C2×C3×C22⋊C4C2211yescharacteristic, complemented
37C2×#217C2211yescharacteristic, complemented
38C2×#217C2211yescharacteristic, complemented
39C2×#217C2211yescharacteristic, complemented
40C2×#217C2211yescharacteristic, complemented
41C2×#217C2211yescharacteristic, complemented
42#938—24no—
43#937—24no—
44#937—24no—
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×(C2×D3)⋊C4C2211yescharacteristic
48C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
49C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
50C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
51C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
52C2×C4⋊Dic3C2211yescharacteristic, complemented
53C2×Dic3⋊C4C2211yescharacteristic, complemented
54#884—24no—
55D3×C22⋊C4—24no—
56C3×#95C2211yescharacteristic, complemented
57C3⋊#95C2211yescharacteristic, complemented
58C3⋊#95C2211yescharacteristic, complemented
59C3⋊#95C2211yescharacteristic, complemented
60C23×D4—31no—
61C2×#121—31no—
62C2×#121—31no—
63C2×#120—31no—
64C2×#120—31no—
65C22×C22⋊C4—31no—
66#394—31no—
67#394—31no—
68#392—31no—
69#392D311yescharacteristic, complemented
70#386—31no—
71#386—31no—
72#386—31no—
73#385—31no—
74#379—31no—
75C24×C3C2311yescharacteristic
76C24×C3C2311yescharacteristic
77C24×C3C2311yescharacteristic
78C23×D3—22no—
79C23×D3—22no—
80C23×D3—22no—
81C23×D3—22no—
82C23×D3—22no—
83C23×D3—22no—
84C23×D3—22no—
85C23×D3—22no—
86C23×D3—22no—
87C23×D3—41no—
88C23×D3—41no—
89C23×D3—41no—
90C23×D3—41no—
91C23×D3D414yescomplemented
92C23×D3C2311yescharacteristic
93C23×D3C2311yescharacteristic
94C23×D3C2311yescharacteristic
95C2×C3×D4—22no—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—22no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67—22no—
109C2×#67—22no—
110C2×#67—22no—
111C2×#67—22no—
112C2×#67—22no—
113C2×#67—22no—
114C2×#67—22no—
115C2×#67—22no—
116C2×#67—22no—
117C2×#67—22no—
118C2×#67—22no—
119C2×#67—22no—
120C2×#67—22no—
121C2×#67—41no—
122C2×#67D414yescomplemented
123C22×Dic3C2311yescharacteristic
124C22×Dic3C2311yescharacteristic
125C22×Dic3C2311yescharacteristic
126C22×Dic3C2311yescharacteristic
127C22×Dic3C2311yescharacteristic
128D3×D4—232no—
129C2×D12—22no—
130C2×D12—22no—
131C2×D12D414yescomplemented
132C2×C4×D3—22no—
133C2×C4×D3—22no—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C2×C4×D3—22no—
137C2×C4×D3—22no—
138C2×C4×D3—22no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—41no—
144#217—22no—
145#217—22no—
146#217—22no—
147#217—22no—
148#217—22no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—22no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4D414yescomplemented
156C4⋊Dic3—22no—
157Dic3⋊C4—22no—
158C25—31no—
159C25—31no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—31no—
163C22×D4—31no—
164C22×D4—31no—
165C22×D4—31no—
166C22×D4—64no—
167C22×D4—64no—
168C22×D4—64no—
169C22×D4C2×D311yescharacteristic
170C22×D4C2×D311yescharacteristic, complemented
171C22×D4C2×D311yescharacteristic, complemented
172C23×C4—31no—
173C23×C4—31no—
174C23×C4—31no—
175#121—64no—
176#121—64no—
177#120—64no—
178#120—64no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—31no—
193C2×C22⋊C4—64no—
194C2×C22⋊C4C2×D311yescharacteristic, complemented
195C2×C22⋊C4C2×D311yescharacteristic, complemented
196C2×C22⋊C4C2×D311yescharacteristic, complemented
197#95—31no—
198#95—31no—
199#95—31no—
200#95C2×D311yescharacteristic, complemented
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3C2×D412yes—
221C23×C3C2×D412yes—
222C23×C3C2411yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—21no—
228C22×D3—21no—
229C22×D3—21no—
230C22×D3—21no—
231C22×D3—28no—
232C22×D3—21no—
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—21no—
238C22×D3—28no—
239C22×D3—28no—
240C22×D3—44no—
241C22×D3—44no—
242C22×D3—44no—
243C22×D3—44no—
244C22×D3—44no—
245C22×D3—44no—
246C22×D3—44no—
247C22×D3—44no—
248C22×D3—44no—
249C22×D3—44no—
250C22×D3—44no—
251C22×D3C2×D412yes—
252C22×D3C2×D412yes—
253C22×D3C2×D412yes—
254C22×D3C2×D412yes—
255C22×D3C2×D412yes—
256C22×D3C2×D412yes—
257C22×D3#4112yes—
258C3×D4—28no—
259C3×D4—44no—
260C3×D4—44no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4C2×D412yes—
272C2×C3×C4C2×D412yescomplemented
273#67—28no—
274#67—28no—
275#67—44no—
276#67—44no—
277#67—44no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3C2×D412yes—
292C2×Dic3C2×D412yescomplemented
293D12—28no—
294C4×D3—28no—
295C4×D3—44no—
296C4×D3—44no—
297C24—31no—
298C24—31no—
299C24—31no—
300C24—34no—
301C24—62no—
302C24—62no—
303C24—62no—
304C24—62no—
305C24—62no—
306C24—62no—
307C24—62no—
308C24—62no—
309C24—62no—
310C24—121no—
311C24—121no—
312C24—121no—
313C24—121no—
314C24C22×D311yescharacteristic
315C24C22×D311yescharacteristic
316C24C22×D311yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—34no—
324C2×D4—34no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—632no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—121no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4C22×D311yescharacteristic
360C22×C4C22×D311yescharacteristic
361C22×C4C22×D311yescharacteristic
362C22×C4C22×D311yescharacteristic
363C4⋊C4—62no—
364C4⋊C4—62no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—34no—
368C22⋊C4—41no—
369C22⋊C4—62no—
370C22⋊C4—62no—
371C22⋊C4—62no—
372C22⋊C4—62no—
373C22⋊C4—62no—
374C22⋊C4—62no—
375C22⋊C4—62no—
376C22⋊C4—62no—
377C22⋊C4—62no—
378C22⋊C4—62no—
379C22⋊C4—121no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3C22×D411yescharacteristic
402C22×C3C22×D411yescharacteristic
403C22×C3C22×D411yescharacteristic
404C22×C3C22×D411yescharacteristic
405C22×C3C2×#4111yescharacteristic
406C22×C3H(5,𝔽2)11yescharacteristic
407C22×C3H(5,𝔽2)11yescharacteristic
408C2×D3—22no—
409C2×D3—22no—
410C2×D3—22no—
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—22no—
416C2×D3—22no—
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—416no—
424C2×D3—416no—
425C2×D3—41no—
426C2×D3—41no—
427C2×D3—41no—
428C2×D3—41no—
429C2×D3#12014yescomplemented
430C2×D3#12114yescomplemented
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—41no—
434C3×C4—41no—
435Dic3—22no—
436Dic3—22no—
437Dic3—41no—
438Dic3—41no—
439Dic3—41no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—32no—
460C23—32no—
461C23—32no—
462C23—32no—
463C23—32no—
464C23—32no—
465C23—32no—
466C23—68no—
467C23—68no—
468C23—68no—
469C23—61no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23—61no—
474C23—61no—
475C23—61no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—124no—
484C23—124no—
485C23—124no—
486C23—124no—
487C23—124no—
488C23—124no—
489C23—124no—
490C23—124no—
491C23—124no—
492C23—124no—
493C23—124no—
494C23D3×D412yes—
495C23D3×D412yes—
496C23C23×D311yescharacteristic
497D4—28no—
498D4—44no—
499D4—44no—
500D4—68no—
501D4—68no—
502D4—68no—
503D4—124no—
504D4—124no—
505D4—124no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—32no—
517C2×C4—32no—
518C2×C4—68no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—124no—
533C2×C4—124no—
534C2×C4D3×D412yes—
535C2×C4D3×D412yescomplemented
536C2×C3—42no—
537C2×C3—42no—
538C2×C3—81no—
539C2×C3C2×#12011yescharacteristic
540C2×C3C2×#12111yescharacteristic
541C2×C3#53411yescharacteristic
542C2×C3D4211yescharacteristic
543C2×C3D4211yescharacteristic
544C2×C3#54611yescharacteristic
545C2×C3#54611yescharacteristic
546D3—24no—
547D3—24no—
548D3—42no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—34no—
556C22—34no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—62no—
577C22—62no—
578C22—62no—
579C22—62no—
580C22—62no—
581C22—62no—
582C22—62no—
583C22—62no—
584C22—62no—
585C22—62no—
586C22—62no—
587C22—1216no—
588C22—1216no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22—121no—
593C22C2×D3×D411yescharacteristic
594C22C2×D3×D411yescharacteristic
595C22C2×D3×D411yescharacteristic
596C22C2×D3×D411yescharacteristic
597C22#100411yescharacteristic
598C22#100411yescharacteristic
599C22D3×#4111yescharacteristic
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—62no—
605C4—62no—
606C4—121no—
607C4—121no—
608C4—121no—
609C3#241011yescharacteristic, complemented, Sylow
610C2—42no—
611C2—42no—
612C2—64no—
613C2—64no—
614C2—81no—
615C2—122no—
616C2D3×#12011yescharacteristic
617C2#558911yescharacteristic
618C2#559011yescharacteristic
619C2D3×#12111yescharacteristic
620C2#561011yescharacteristic
621C2#561211yescharacteristic
622C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2410
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#938C2semidirect
3. C2×#937C2semidirect
4. C2×#937C2semidirect
5. C2×#884C2semidirect
6. C2×D3×C22⋊C4C2semidirect
7. C3×#392C2semidirect
8. #5223C2semidirect
9. #5222C2semidirect
10. #4961C2semidirect
11. #4956C2semidirect
12. #4956C2semidirect
13. #4955C2semidirect
14. #4668C2semidirect
15. #4665C2semidirect
20. C22×C3×D4C22semidirect
21. C22×C3×D4C22semidirect
25. C22×#67C22semidirect
26. C22×#67C22semidirect
32. C22×C4×D3C22semidirect
33. C22×C4×D3C22semidirect
34. C2×C3×C22⋊C4C22semidirect
35. C2×C3×C22⋊C4C22semidirect
36. C2×C3×C22⋊C4C22semidirect
37. C2×#217C22semidirect
38. C2×#217C22semidirect
39. C2×#217C22semidirect
40. C2×#217C22semidirect
41. C2×#217C22semidirect
48. C2×(C2×D3)⋊C4C22semidirect
49. C2×(C2×D3)⋊C4C22semidirect
50. C2×(C2×D3)⋊C4C22semidirect
51. C2×(C2×D3)⋊C4C22semidirect
52. C2×C4⋊Dic3C22semidirect
53. C2×Dic3⋊C4C22semidirect
56. C3×#95C22semidirect
57. C3⋊#95C22semidirect
58. C3⋊#95C22semidirect
59. C3⋊#95C22semidirect
69. #392D3semidirect
91. C23×D3D4semidirect
122. C2×#67D4semidirect
131. C2×D12D4semidirect
155. (C2×D3)⋊C4D4semidirect
170. C22×D4C2×D3semidirect
171. C22×D4C2×D3semidirect
194. C2×C22⋊C4C2×D3semidirect
195. C2×C22⋊C4C2×D3semidirect
196. C2×C22⋊C4C2×D3semidirect
200. #95C2×D3semidirect
272. C2×C3×C4C2×D4semidirect
292. C2×Dic3C2×D4semidirect
429. C2×D3#120semidirect
430. C2×D3#121semidirect
535. C2×C4D3×D4semidirect
609. C3#2410semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#9382
C2×#9372
C2×#8842
C2×D3×C22⋊C42
C3×#3922
#52232
#52222
#49612
#49562
#49552
#46682
#46652
#24103