Finite Group Databasebeta

← #77108

Subgroups — #77108

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×#895C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic, complemented
5C2×#882C211yesmaximal, characteristic, complemented
6C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
7C3×#394C211yesmaximal, characteristic, complemented
8#5223C211yesmaximal, characteristic, complemented
9#4989C211yesmaximal, characteristic, complemented
10#4960C211yesmaximal, characteristic, complemented
11#4959C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4955C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2410—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C24×D3C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic, complemented
23C22×#67C2211yescharacteristic, complemented
24C2×D3×D4—24no—
25C2×D3×D4—24no—
26C2×D3×D4—24no—
27C22×D12C2211yescharacteristic
28C22×D12C2211yescharacteristic
29C22×D12C2211yescharacteristic, complemented
30C22×D12C2211yescharacteristic, complemented
31C22×C4×D3C2211yescharacteristic
32C22×C4×D3C2211yescharacteristic, complemented
33C22×C4×D3C2211yescharacteristic, complemented
34C2×C3×C4⋊C4C2211yescharacteristic, complemented
35C2×C3×C22⋊C4C2211yescharacteristic, complemented
36C2×C3×C22⋊C4C2211yescharacteristic, complemented
37C2×C3×C22⋊C4C2211yescharacteristic, complemented
38C2×C3×C22⋊C4C2211yescharacteristic, complemented
39C2×#217C2211yescharacteristic, complemented
40C2×#217C2211yescharacteristic, complemented
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
49C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51#895—24no—
52#884—24no—
53#882—24no—
54#882—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#394D311yescharacteristic, complemented
68#392—31no—
69#392—31no—
70#386—31no—
71#386—31no—
72#386—31no—
73#385—31no—
74#379—31no—
75C24×C3C2311yescharacteristic
76C24×C3C2311yescharacteristic
77C23×D3—22no—
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—41no—
87C23×D3—41no—
88C23×D3—41no—
89C23×D3—41no—
90C23×D3D414yescomplemented
91C23×D3C2311yescharacteristic
92C23×D3C2311yescharacteristic
93C23×D3C2311yescharacteristic
94C23×D3C2311yescharacteristic
95C2×C3×D4—22no—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2311yescharacteristic
103C2×#67—22no—
104C2×#67—22no—
105C2×#67—22no—
106C2×#67—22no—
107C2×#67—22no—
108C2×#67—22no—
109C2×#67—22no—
110C2×#67—22no—
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—41no—
114C2×#67D414yescomplemented
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119D3×D4—232no—
120C2×D12—22no—
121C2×D12—22no—
122C2×D12—22no—
123C2×D12—22no—
124C2×D12—22no—
125C2×D12—22no—
126C2×D12—22no—
127C2×D12—22no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—41no—
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×C4⋊C4—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—22no—
146#217—22no—
147#217—41no—
148(C2×D3)⋊C4—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—22no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4D414yescomplemented
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—31no—
167C22×D4—31no—
168C22×D4—64no—
169C22×D4—64no—
170C22×D4—64no—
171C22×D4C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic, complemented
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—64no—
193C2×C22⋊C4C2×D311yescharacteristic, complemented
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×C3C2×D412yes—
214C23×C3C2×D412yes—
215C23×C3C2411yescharacteristic
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—21no—
220C22×D3—21no—
221C22×D3—21no—
222C22×D3—21no—
223C22×D3—21no—
224C22×D3—28no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—21no—
228C22×D3—21no—
229C22×D3—21no—
230C22×D3—21no—
231C22×D3—28no—
232C22×D3—28no—
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—21no—
238C22×D3—21no—
239C22×D3—21no—
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—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
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×C4—21no—
272C2×C3×C4C2×D412yes—
273C2×C3×C4C2×D412yescomplemented
274#67—28no—
275#67—28no—
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×Dic3C2×D412yes—
289C2×Dic3C2×D412yescomplemented
290D12—28no—
291D12—44no—
292D12—44no—
293D12—44no—
294C4×D3—28no—
295C4×D3—44no—
296C4×D3—44no—
297C24—31no—
298C24—31no—
299C24—31no—
300C24—34no—
301C24—31no—
302C24—62no—
303C24—62no—
304C24—62no—
305C24—62no—
306C24—62no—
307C24—62no—
308C24—62no—
309C24—62no—
310C24—62no—
311C24—121no—
312C24—121no—
313C24—121no—
314C24—121no—
315C24C22×D311yescharacteristic
316C24C22×D311yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—34no—
321C2×D4—34no—
322C2×D4—62no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—632no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C2×D4—121no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4C22×D311yescharacteristic
359C22×C4C22×D311yescharacteristic
360C22×C4C22×D311yescharacteristic
361C22×C4C22×D311yescharacteristic
362C22×C4C22×D311yescharacteristic
363C4⋊C4—22no—
364C4⋊C4—62no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—34no—
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—121no—
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×C3C22×D411yescharacteristic
395C22×C3C22×D411yescharacteristic
396C22×C3C22×D411yescharacteristic
397C22×C3C22×D411yescharacteristic
398C22×C3C2×#4111yescharacteristic
399C22×C3H(5,𝔽2)11yescharacteristic
400C22×C3H(5,𝔽2)11yescharacteristic
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—22no—
407C2×D3—22no—
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—416no—
417C2×D3—416no—
418C2×D3—41no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—41no—
422C2×D3—41no—
423C2×D3—41no—
424C2×D3—41no—
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—
435C3×C4—41no—
436Dic3—22no—
437Dic3—22no—
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—32no—
453C23—32no—
454C23—32no—
455C23—32no—
456C23—32no—
457C23—32no—
458C23—32no—
459C23—68no—
460C23—68no—
461C23—68no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—61no—
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—68no—
499D4—68no—
500D4—68no—
501D4—124no—
502D4—124no—
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—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—68no—
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×C3C2×#12011yescharacteristic
539C2×C3C2×#12111yescharacteristic
540C2×C3#53411yescharacteristic
541C2×C3D4211yescharacteristic
542C2×C3D4211yescharacteristic
543C2×C3#54611yescharacteristic
544C2×C3#54611yescharacteristic
545D3—24no—
546D3—24no—
547D3—42no—
548D3—81no—
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—62no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—62no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—62no—
577C22—62no—
578C22—62no—
579C22—62no—
580C22—1216no—
581C22—1216no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22—121no—
593C22C2×D3×D411yescharacteristic
594C22C2×D3×D411yescharacteristic
595C22C2×D3×D411yescharacteristic
596C22C2×D3×D411yescharacteristic
597C22D3×#4111yescharacteristic
598C22#100911yescharacteristic
599C22#100911yescharacteristic
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—62no—
606C4—62no—
607C4—121no—
608C4—121no—
609C3#241011yescharacteristic, complemented, Sylow
610C2—42no—
611C2—42no—
612C2—64no—
613C2—64no—
614C2—122no—
615C2—241no—
616C2D3×#12011yescharacteristic
617C2D3×#12111yescharacteristic
618C2#560911yescharacteristic
619C2#560911yescharacteristic
620C2#565311yescharacteristic
621C2#565411yescharacteristic
622C2#567611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2410
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#895C2semidirect
3. C2×#884C2semidirect
4. C2×#882C2semidirect
5. C2×#882C2semidirect
6. C2×D3×C22⋊C4C2semidirect
7. C3×#394C2semidirect
8. #5223C2semidirect
9. #4989C2semidirect
10. #4960C2semidirect
11. #4959C2semidirect
12. #4956C2semidirect
13. #4955C2semidirect
14. #4665C2semidirect
15. #4665C2semidirect
22. C22×#67C22semidirect
23. C22×#67C22semidirect
29. C22×D12C22semidirect
30. C22×D12C22semidirect
32. C22×C4×D3C22semidirect
33. C22×C4×D3C22semidirect
34. C2×C3×C4⋊C4C22semidirect
35. C2×C3×C22⋊C4C22semidirect
36. C2×C3×C22⋊C4C22semidirect
37. C2×C3×C22⋊C4C22semidirect
38. C2×C3×C22⋊C4C22semidirect
39. C2×#217C22semidirect
40. C2×#217C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×(C2×D3)⋊C4C22semidirect
49. C2×(C2×D3)⋊C4C22semidirect
50. C2×Dic3⋊C4C22semidirect
56. C3×#95C22semidirect
57. C3⋊#95C22semidirect
58. C3⋊#95C22semidirect
59. C3⋊#95C22semidirect
67. #394D3semidirect
90. C23×D3D4semidirect
114. C2×#67D4semidirect
131. C2×D12D4semidirect
156. (C2×D3)⋊C4D4semidirect
180. C2×C4⋊C4C2×D3semidirect
193. C2×C22⋊C4C2×D3semidirect
194. C2×C22⋊C4C2×D3semidirect
195. C2×C22⋊C4C2×D3semidirect
196. C2×C22⋊C4C2×D3semidirect
200. #95C2×D3semidirect
273. C2×C3×C4C2×D4semidirect
289. 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×#8952
C2×#8842
C2×#8822
C2×D3×C22⋊C42
C3×#3942
#52232
#49892
#49602
#49592
#49562
#49552
#46652
#24103