Finite Group Databasebeta

← #10967698

Subgroups — #10967698

Order 768.

1834 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2552C211yesmaximal, characteristic, complemented
2#83058C212yesmaximal
3Dic3×#125C212yesmaximal, complemented
4#77817C214yesmaximal, complemented
5#77739C214yesmaximal
6#76223C212yesmaximal
7#75997C212yesmaximal
8#75967C212yesmaximal, complemented
9#75966C212yesmaximal, complemented
10#74336C211yesmaximal, characteristic, complemented
11#74327C212yesmaximal
12#74320C212yesmaximal, complemented
13#74318C211yesmaximal, characteristic
14#73725C211yesmaximal, characteristic, complemented
15#73715C211yesmaximal, characteristic, complemented
16C4×#892C211yesmaximal, characteristic
17C4×#892C211yesmaximal, characteristic
18#34978—31nomaximal, Sylow
19C2×C3×#125C2212yes—
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#892C2211yescharacteristic
23C2×#892C2211yescharacteristic
24C2×#892C2212yes—
25C2×#890C2212yes—
26C2×#888C2212yes—
27C2×#887C2214yes—
28C2×C42×D3C2211yescharacteristic
29C2×#869C2212yes—
30C3×#398C2214yes—
31C3×#389C2212yes—
32C3×#389C2212yes—
33C3×#384C2211yescharacteristic
34C3×#384C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C4⋊C4C2211yescharacteristic
37#4994C2214yes—
38#4994C2214yes—
39#4993C2214yes—
40#4987C2212yes—
41#4987C2212yes—
42#4984C2212yes—
43#4984C2212yes—
44#4982C2214yes—
45#4980C2214yes—
46#4980C2214yes—
47#4978C2214yes—
48#4977C2212yes—
49#4977C2212yes—
50#4976C2214yes—
51#4975C2212yes—
52Dic3×C4⋊C4C2212yes—
53Dic3×C4⋊C4C2212yes—
54Dic3×C4⋊C4C2211yescharacteristic
55Dic3×C4⋊C4C2214yes—
56Dic3×C4⋊C4C2214yes—
57Dic3×C4⋊C4C2211yescharacteristic
58Dic3×C4⋊C4C2211yescharacteristic
59Dic3×C4⋊C4C2211yescharacteristic
60#4973C2214yes—
61#4972C2212yes—
62#4972C2212yes—
63#4972C2214yes—
64C4×(C2×D3)⋊C4C2212yes—
65C4×(C2×D3)⋊C4C2212yes—
66#4936C2212yes—
67#4935C2211yescharacteristic
68C4×C4⋊Dic3C2211yescharacteristic
69C4×C4⋊Dic3C2211yescharacteristic
70C42×Dic3C2212yes—
71C42×Dic3C2212yes—
72#4928C2212yes—
73#4928C2212yes—
74#4673C2214yes—
75#4666C2212yes—
76#4666C2212yes—
77#4660C2214yes—
78#4656C2214yes—
79#4653C2214yes—
80#4649C2214yes—
81#4648C2212yes—
82#4648C2212yes—
83#4646C2214yes—
84#2612—34no—
85#2571—31no—
86#2552—31no—
87#2552D311yescharacteristic, complemented
88#2551—31no—
89#2441—32no—
90#2441—32no—
91#2439—32no—
92#2439—32no—
93#2396—34no—
94C4×C4⋊Q8—32no—
95C4×C4⋊Q8—32no—
96C4×#125—32no—
97C4×#125—32no—
98#2285—31no—
99C4×#117—31no—
100C4×#117—31no—
101C22×C4×D3C2311yescharacteristic
102C22×C4×D3C2311yescharacteristic
103C22×C4×D3C2311yescharacteristic
104C22×Dic6C2314yes—
105C3×#125—28no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C42C2312yes—
117C2×C3×C42C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2314yes—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2312yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2311yescharacteristic
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2312yes—
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2311yescharacteristic
151#892—24no—
152#892—24no—
153#892—24no—
154#892—24no—
155#892—44no—
156#890—28no—
157#888—28no—
158#887—48no—
159C42×D3Q818yescomplemented
160#869—28no—
161C3×#95C2314yes—
162C3×#95C2312yes—
163C3×#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2312yes—
172C3⋊#95C2314yes—
173C3⋊#95C2314yes—
174C3⋊#95C2314yes—
175C3⋊#95C2314yes—
176C3⋊#95C2314yes—
177C2×C4⋊Q8—32no—
178C2×C4⋊Q8—32no—
179C2×#125—32no—
180C2×#125C2×D312yes—
181C2×C4×Q8—34no—
182C2×#117—31no—
183C2×#117—31no—
184C2×#117—31no—
185C2×#117—31no—
186C2×#117—32no—
187C22×C42—31no—
188#400—34no—
189#399—34no—
190#398—34no—
191#398C2×D314yes—
192#396—34no—
193#396—34no—
194#395—34no—
195#393—34no—
196#391—34no—
197#391—34no—
198#389—32no—
199#389—32no—
200#389—32no—
201#389—34no—
202#389—34no—
203#389—32no—
204#389—32no—
205#389—32no—
206#389C2×D312yes—
207#389C2×D312yes—
208#387—34no—
209#387—34no—
210#387—34no—
211#387—34no—
212#384—32no—
213#384—31no—
214#384C2×D312yes—
215#384C2×D311yescharacteristic
216#382—32no—
217#382—32no—
218#382—32no—
219#380—32no—
220#380—32no—
221C4×C4⋊C4—34no—
222C4×C4⋊C4—31no—
223C4×C4⋊C4—31no—
224C4×C4⋊C4—34no—
225C4×C4⋊C4—32no—
226C4×C4⋊C4—32no—
227C4×C4⋊C4—31no—
228C4×C4⋊C4—31no—
229C4×C4⋊C4—31no—
230C4×C4⋊C4—31no—
231C4×C4⋊C4—34no—
232C4×C4⋊C4C2×D311yescharacteristic
233C4×C4⋊C4C2×D311yescharacteristic
234C4×C22⋊C4—32no—
235C4×C22⋊C4—32no—
236C4×C22⋊C4—32no—
237C4×C22⋊C4—32no—
238#376—32no—
239#376—32no—
240C43—32no—
241C43—32no—
242C23×D3C2411yescharacteristic
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2414yes—
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2412yes—
248C22×C3×C4C2412yes—
249C22×C3×C4C2411yescharacteristic
250C22×C3×C4C2411yescharacteristic
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253C22×Dic3C2412yes—
254C22×Dic3C2411yescharacteristic
255C22×Dic3C2411yescharacteristic
256C22×Dic3C2412yes—
257C22×Dic3C2412yes—
258C22×Dic3C2411yescharacteristic
259C2×C4×D3—22no—
260C2×C4×D3—22no—
261C2×C4×D3—22no—
262C2×C4×D3—22no—
263C2×C4×D3—22no—
264C2×C4×D3—22no—
265C2×C4×D3C2×Q814yes—
266C2×C4×D3C2×Q814yes—
267C2×C4×D3C2×Q814yes—
268C2×Dic6—28no—
269C2×Dic6—28no—
270C2×Dic6—44no—
271C3×C4⋊C4—24no—
272C3×C4⋊C4—22no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—28no—
275C3×C4⋊C4—28no—
276C3×C4⋊C4—24no—
277C3×C4⋊C4—44no—
278C3×C4⋊C4—42no—
279C3×C4⋊C4#4114yes—
280C3×C4⋊C4#4114yes—
281C3×C42—24no—
282C3×C42—24no—
283C3×C42C2×Q814yes—
284(C2×D3)⋊C4—24no—
285(C2×D3)⋊C4—24no—
286(C2×D3)⋊C4—24no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—44no—
289C4⋊Dic3—24no—
290C4⋊Dic3—24no—
291C4⋊Dic3—22no—
292C4⋊Dic3—22no—
293C4⋊Dic3—28no—
294C4⋊Dic3—28no—
295C4⋊Dic3—42no—
296C4⋊Dic3#4114yes—
297C4⋊Dic3#4114yes—
298Dic3⋊C4—28no—
299Dic3⋊C4—28no—
300Dic3⋊C4—44no—
301Dic3⋊C4—44no—
302C4×Dic3—24no—
303C4×Dic3—24no—
304C4×Dic3—24no—
305C4×Dic3—24no—
306C4×Dic3—24no—
307C4×Dic3—24no—
308C4×Dic3—24no—
309C4×Dic3—24no—
310C4×Dic3—28no—
311C4×Dic3—24no—
312C4×Dic3—24no—
313C4×Dic3C2×Q814yes—
314C4×Dic3C2×Q814yes—
315C4×Dic3C2×Q814yes—
316C22×Q8—34no—
317C23×C4—31no—
318C23×C4—31no—
319C23×C4—31no—
320C4⋊Q8—68no—
321C4⋊Q8—68no—
322#125—28no—
323#125—68no—
324C4×Q8—128no—
325#117—64no—
326#117—64no—
327#117—64no—
328#117—64no—
329#117—124no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—34no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D312yes—
346C2×C4⋊C4C22×D312yes—
347C2×C4⋊C4C22×D314yes—
348C2×C4⋊C4C22×D314yes—
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D314yes—
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D311yescharacteristic
353C2×C22⋊C4—32no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—34no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—38no—
362C2×C42—34no—
363C2×C42—32no—
364C2×C42—32no—
365C2×C42—32no—
366C2×C42—32no—
367C2×C42—32no—
368C2×C42—32no—
369C2×C42—32no—
370C2×C42—32no—
371C2×C42—32no—
372C2×C42—32no—
373C2×C42C22×D312yes—
374C2×C42C22×D312yes—
375C2×C42C22×D311yescharacteristic
376#95—32no—
377#95—32no—
378#95—32no—
379#95—32no—
380#95—32no—
381#95—32no—
382#95—34no—
383#95—34no—
384#95—34no—
385#95—34no—
386#95—34no—
387#95—34no—
388#95—34no—
389#95C22×D312yes—
390#95C22×D312yes—
391#95C22×D314yes—
392C23×C3C2511yescharacteristic
393C22×D3—21no—
394C22×D3—21no—
395C22×D3—21no—
396C22×D3—21no—
397C22×D3—21no—
398C22×D3—21no—
399C22×D3C22×Q812yes—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—24no—
412C2×C3×C4—22no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4C22×Q812yes—
416C2×C3×C4C22×Q812yes—
417C2×C3×C4C22×Q812yes—
418C2×C3×C4C2×#4114yes—
419C2×C3×C4C2×#4112yes—
420C2×C3×C4C2×#4114yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C4C2×#4112yes—
423C2×C3×C4C2×#4112yes—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—24no—
437C2×Dic3—24no—
438C2×Dic3—22no—
439C2×Dic3—24no—
440C2×Dic3—22no—
441C2×Dic3C22×Q812yes—
442C2×Dic3C22×Q812yes—
443C2×Dic3C22×Q812yes—
444C2×Dic3C2×#4114yes—
445C2×Dic3C2×#4114yes—
446C2×Dic3H(5,𝔽2)12yes—
447C2×Dic32−1+412yes—
448C4×D3—28no—
449C4×D3—28no—
450C4×D3—28no—
451Dic6—416no—
452C24—31no—
453C2×Q8—68no—
454C2×Q8—68no—
455C2×Q8—124no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—34no—
462C22×C4—34no—
463C22×C4—34no—
464C22×C4—32no—
465C22×C4—34no—
466C22×C4—32no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4C23×D312yes—
474C22×C4C23×D312yes—
475C22×C4C23×D314yes—
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D312yes—
479C22×C4C23×D311yescharacteristic
480C22×C4C23×D311yescharacteristic
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—34no—
488C4⋊C4—34no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—68no—
492C4⋊C4—68no—
493C4⋊C4—68no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—68no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—122no—
500C4⋊C4—124no—
501C4⋊C4—124no—
502C4⋊C4#23714yes—
503C4⋊C4#23914yes—
504C22⋊C4—64no—
505C22⋊C4—64no—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—124no—
509C42—24no—
510C42—24no—
511C42—34no—
512C42—34no—
513C42—34no—
514C42—64no—
515C42—64no—
516C42—68no—
517C42—64no—
518C42—64no—
519C42—64no—
520C42—64no—
521C42—64no—
522C42—64no—
523C42—64no—
524C42—64no—
525C42D3×Q814yes—
526C22×C3C23×Q811yescharacteristic
527C22×C3C22×#4111yescharacteristic
528C22×C3C22×#4111yescharacteristic
529C22×C3C22×#4111yescharacteristic
530C22×C3C22×#4111yescharacteristic
531C22×C3C2×H(5,𝔽2)11yescharacteristic
532C22×C3C2×2−1+411yescharacteristic
533C2×D3—22no—
534C2×D3—22no—
535C2×D3—22no—
536C2×D3—41no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—41no—
540C3×C4—22no—
541C3×C4—24no—
542C3×C4—24no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—44no—
546C3×C4#53314yes—
547C3×C4#53314yes—
548Dic3—22no—
549Dic3—22no—
550Dic3—24no—
551Dic3—24no—
552Dic3—44no—
553Dic3—42no—
554Dic3—42no—
555Dic3#54414yes—
556C23—32no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23C24×D311yescharacteristic
564Q8—1216no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—32no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—32no—
585C2×C4—34no—
586C2×C4—34no—
587C2×C4—68no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—68no—
597C2×C4—68no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4C2×#23714yes—
608C2×C4C2×#23712yes—
609C2×C4C2×D3×Q812yes—
610C2×C4C2×D3×Q812yes—
611C2×C4C2×D3×Q812yes—
612C2×C4C2×#23914yes—
613C2×C4C2×#23912yes—
614C2×C4D3×#4112yes—
615C2×C4D3×#4112yes—
616C2×C3C2×#53311yescharacteristic
617C2×C3C2×#53311yescharacteristic
618C2×C3C2×#54411yescharacteristic
619C2×C3#348111yescharacteristic
620C2×C3#348211yescharacteristic
621C2×C3#348211yescharacteristic
622C2×C3#348311yescharacteristic
623D3—42no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—121no—
628C22—121no—
629C22—121no—
630C22—121no—
631C22C22×#23711yescharacteristic
632C22C22×D3×Q811yescharacteristic
633C22C22×#23911yescharacteristic
634C22C2×D3×#4111yescharacteristic
635C22C2×D3×#4111yescharacteristic
636C22D3×H(5,𝔽2)11yescharacteristic
637C22D3×2−1+411yescharacteristic
638C4—22no—
639C4—24no—
640C4—24no—
641C4—34no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—62no—
646C4—64no—
647C4—62no—
648C4—64no—
649C4—124no—
650C4—122no—
651C4—122no—
652C4#568814yes—
653C4#572814yes—
654C3#3497811yescharacteristic, complemented, Sylow
655C2—122no—
656C2C2×#568811yescharacteristic
657C2C2×#572811yescharacteristic
658C2D3×#54411yescharacteristic
659C2#8667611yescharacteristic
660C2#8676811yescharacteristic
661C2#8692411yescharacteristic
662C2#8701711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2552C2
prime radicalC23×D3C24
hypercenter#2552D3
hypocenterC3#34978
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2552

Complements

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

NH ≅ G/Ntype
1. C3×#2552C2semidirect
3. Dic3×#125C2semidirect
4. #77817C2semidirect
8. #75967C2semidirect
9. #75966C2semidirect
10. #74336C2semidirect
12. #74320C2semidirect
14. #73725C2semidirect
15. #73715C2semidirect
87. #2552D3semidirect
159. C42×D3Q8semidirect
654. C3#34978semidirect

Maximal subgroups

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

subgroupindex
C3×#25522
#830582
Dic3×#1252
#778172
#777392
#762232
#759972
#759672
#759662
#743362
#743272
#743202
#743182
#737252
#737152
C4×#8922
#349783