Finite Group Databasebeta

← #11069522

Subgroups — #11069522

Order 768.

2168 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5602C212yesmaximal
2C3×C4×#121C211yesmaximal, characteristic
3#82373C212yesmaximal, complemented
4#82373C212yesmaximal, complemented
5D4×C4⋊Dic3C212yesmaximal, complemented
6#82076C211yesmaximal, characteristic, complemented
7C2×#4928C211yesmaximal, characteristic, complemented
8#76398C212yesmaximal, complemented
9#76127C212yesmaximal, complemented
10#75671C214yesmaximal, complemented
11#74694C214yesmaximal, complemented
12#73953C212yesmaximal, complemented
13#73682C212yesmaximal, complemented
14#73512C212yesmaximal, complemented
15#73493C212yesmaximal, complemented
16#39376—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C2×#935C2214yes—
22C2×#935C2214yes—
23C2×D4×Dic3C2214yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2212yes—
26C22×C4⋊Dic3C2211yescharacteristic
27C2×#924C2212yes—
28C22×Dic3⋊C4C2212yes—
29#5602—216no—
30C2×#889C2214yes—
31C2×#879C2214yes—
32C2×#878C2214yes—
33C2×#869C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#388C2212yes—
37C3×#386C2212yes—
38C3×#380C2211yescharacteristic
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×C4×C22⋊C4C2212yes—
41#5212C2212yes—
42#5212C2212yes—
43#5212C2212yes—
44#5212C2212yes—
45#5212C2212yes—
46C2×C3⋊#95C2212yes—
47#4984C2214yes—
48#4981C2214yes—
49#4981C2214yes—
50#4981C2214yes—
51#4953C2214yes—
52#4953C2214yes—
53#4953C2214yes—
54#4950C2214yes—
55#4950C2214yes—
56#4950C2214yes—
57#4943C2214yes—
58#4943C2214yes—
59#4935C2212yes—
60#4935C2211yescharacteristic
61C4×C4⋊Dic3C2212yes—
62#4929C2212yes—
63#4928C2212yes—
64#4928C2212yes—
65#4928C2212yes—
66#4928C2212yes—
67#4928C2212yes—
68#4928C2212yes—
69#4928C2212yes—
70#4928C2212yes—
71#4661C2214yes—
72#4659C2214yes—
73#4659C2214yes—
74#4656C2214yes—
75#4653C2214yes—
76#4653C2214yes—
77C2×#539—32no—
78#2857—31no—
79#2726—32no—
80#2724—32no—
81#2689—34no—
82#2627—32no—
83#2506—32no—
84#2496—32no—
85#2486—32no—
86#2486—32no—
87#2440—32no—
88#2419—34no—
89D4×C4⋊C4—32no—
90C4×#121D311yescharacteristic
91C2×#389—31no—
92C22×C3×D4C2312yes—
93C22×C3×D4C2311yescharacteristic
94C23×C3×C4C2312yes—
95C23×C3×C4C2311yescharacteristic
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2311yescharacteristic
98C23×Dic3C2312yes—
99C23×Dic3C2314yes—
100C23×Dic3C2312yes—
101C22×Dic6C2312yes—
102C3×#121D418yes—
103C3×C4×D4—28no—
104C3×C4×D4—42no—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C42—42no—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2312yes—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×#217C2314yes—
121C2×#217C2314yes—
122C2×#217C2314yes—
123C2×#217C2314yes—
124#935—216no—
125#935—216no—
126D4×Dic3—216no—
127C2×C4⋊Dic3—24no—
128C2×C4⋊Dic3—44no—
129C2×C4⋊Dic3—42no—
130C2×C4⋊Dic3C2312yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2314yes—
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2312yes—
142#924—28no—
143C2×Dic3⋊C4—28no—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2314yes—
149C2×C4×Dic3C2314yes—
150#889—216no—
151#879—216no—
152#878—216no—
153#869—42no—
154C3×#95C2312yes—
155C3×#95C2312yes—
156C3×#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167#539—616no—
168C2×C4⋊Q8—31no—
169C2×#126—34no—
170C2×#125—34no—
171C2×#123—34no—
172C2×#123—34no—
173C2×C22⋊Q8—32no—
174C2×C22⋊Q8—34no—
175C2×#121C2×D311yescharacteristic
176C2×C4×D4—34no—
177C2×C4×D4C2×D312yes—
178C2×C4×D4C2×D311yescharacteristic
179C22×C4⋊C4—32no—
180C22×C4⋊C4—31no—
181C22×C4⋊C4—31no—
182C22×C4⋊C4—32no—
183C22×C42C2×D311yescharacteristic
184#400—34no—
185#400—34no—
186#398—34no—
187#398—34no—
188#398—34no—
189#398—34no—
190#397—34no—
191#397—34no—
192#397—34no—
193#395—34no—
194#391—32no—
195#390C2×D311yescharacteristic
196#389—34no—
197#389—32no—
198#389—32no—
199#389—32no—
200#389—32no—
201#389—32no—
202#389—32no—
203#389—32no—
204#389—32no—
205#389C2×D311yescharacteristic
206#388C2×D312yes—
207#387—34no—
208#387—34no—
209#386C2×D312yes—
210#385—34no—
211#385—34no—
212#385—34no—
213#384—31no—
214#384—32no—
215#381—34no—
216#381—34no—
217#380—32no—
218#380—32no—
219#380—32no—
220#380—32no—
221#380—32no—
222#380C2×D311yescharacteristic
223C4×C4⋊C4—32no—
224C4×C4⋊C4C2×D311yescharacteristic
225C4×C22⋊C4C2×D312yes—
226C2×#95—32no—
227C24×C3C2412yes—
228C24×C3C2411yescharacteristic
229C2×C3×D4—24no—
230C2×C3×D4—24no—
231C2×C3×D4—41no—
232C2×C3×D4—41no—
233C2×C3×D4C2×D418yes—
234C2×C3×D4C2×D414yes—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—24no—
239C22×C3×C4—24no—
240C22×C3×C4—22no—
241C22×C3×C4—24no—
242C22×C3×C4—41no—
243C22×C3×C4—41no—
244C22×C3×C4—41no—
245C22×C3×C4C2×D414yes—
246C22×C3×C4#4114yes—
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2412yes—
249C22×C3×C4C2412yes—
250C22×C3×C4C2412yes—
251C22×C3×C4C2411yescharacteristic
252C22×C3×C4C2411yescharacteristic
253C22×C3×C4C2411yescharacteristic
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2411yescharacteristic
256C22×Dic3—24no—
257C22×Dic3—24no—
258C22×Dic3—24no—
259C22×Dic3—28no—
260C22×Dic3—24no—
261C22×Dic3—44no—
262C22×Dic3—42no—
263C22×Dic3—44no—
264C22×Dic3—42no—
265C22×Dic3C2414yes—
266C22×Dic3C2414yes—
267C22×Dic3C2414yes—
268C22×Dic3C2414yes—
269C2×Dic6—24no—
270C2×Dic6—42no—
271C2×Dic6—42no—
272C3×C4⋊C4—24no—
273C3×C4⋊C4—41no—
274C3×C4⋊C4—41no—
275C3×C4⋊C4C2×D414yes—
276C3×C22⋊C4—24no—
277C3×C22⋊C4—24no—
278C3×C22⋊C4—24no—
279C3×C22⋊C4—42no—
280C3×C22⋊C4C2×D418yes—
281C3×C42—24no—
282C3×C42—22no—
283C3×C42—22no—
284C3×C42—41no—
285C3×C42—41no—
286#217—28no—
287#217—28no—
288#217—28no—
289#217—28no—
290C4⋊Dic3—28no—
291C4⋊Dic3—24no—
292C4⋊Dic3—24no—
293C4⋊Dic3—24no—
294C4⋊Dic3—24no—
295C4⋊Dic3—24no—
296C4⋊Dic3—24no—
297C4⋊Dic3—24no—
298C4⋊Dic3—42no—
299C4⋊Dic3—44no—
300C4⋊Dic3—42no—
301C4⋊Dic3—44no—
302Dic3⋊C4—28no—
303Dic3⋊C4—28no—
304Dic3⋊C4—28no—
305Dic3⋊C4—28no—
306Dic3⋊C4—28no—
307C4×Dic3—28no—
308C22×Q8—32no—
309C22×D4C22×D312yes—
310C22×D4C22×D311yescharacteristic
311C23×C4—32no—
312C23×C4—32no—
313C23×C4—34no—
314C23×C4C22×D312yes—
315C23×C4C22×D311yescharacteristic
316C23×C4C22×D311yescharacteristic
317C23×C4C22×D311yescharacteristic
318C4⋊Q8—122no—
319#126—616no—
320#125—616no—
321#123—616no—
322#123—616no—
323C22⋊Q8—68no—
324C22⋊Q8—616no—
325#121D1218yes—
326C4×D4—28no—
327C4×D4—42no—
328C4×D4—616no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4—34no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—68no—
347C2×C4⋊C4—64no—
348C2×C4⋊C4—122no—
349C2×C4⋊C4—124no—
350C2×C4⋊C4C22×D312yes—
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D311yescharacteristic
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C22⋊C4—34no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4C22×D312yes—
359C2×C22⋊C4C22×D312yes—
360C2×C22⋊C4C22×D312yes—
361C2×C22⋊C4C22×D312yes—
362C2×C22⋊C4C22×D312yes—
363C2×C42—34no—
364C2×C42—42no—
365C2×C42C22×D312yes—
366C2×C42C22×D311yescharacteristic
367C2×C42C22×D311yescharacteristic
368C2×C42C22×D311yescharacteristic
369C2×C42C22×D311yescharacteristic
370#95—34no—
371#95—34no—
372#95—34no—
373#95—34no—
374#95—34no—
375#95—34no—
376#95—34no—
377#95—34no—
378#95—34no—
379#95—34no—
380#95C22×D312yes—
381#95C22×D312yes—
382#95C22×D312yes—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—21no—
387C23×C3—22no—
388C23×C3—22no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3C22×D414yes—
394C23×C3C22×D412yes—
395C23×C3C2×#4112yes—
396C23×C32−1+412yes—
397C23×C3C2511yescharacteristic
398C3×D4—216no—
399C3×D4—44no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—28no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—22no—
420C2×C3×C4—44no—
421C2×C3×C4—44no—
422C2×C3×C4—48no—
423C2×C3×C4C22×D414yes—
424C2×C3×C4C22×D412yes—
425C2×C3×C4C22×D412yes—
426C2×C3×C4C2×#4112yes—
427C2×C3×C4C2×#4112yes—
428C2×C3×C4H(5,𝔽2)12yes—
429C2×C3×C42−1+412yes—
430C2×C3×C42−1+412yes—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—24no—
435C2×Dic3—24no—
436C2×Dic3—24no—
437C2×Dic3—24no—
438C2×Dic3—24no—
439C2×Dic3—24no—
440C2×Dic3—24no—
441C2×Dic3—24no—
442C2×Dic3—24no—
443C2×Dic3—48no—
444C2×Dic3—88no—
445C2×Dic3—84no—
446Dic6—84no—
447C24C23×D312yes—
448C24C23×D311yescharacteristic
449C2×Q8—64no—
450C2×Q8—122no—
451C2×Q8—122no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4C2×D1218yes—
457C2×D4C2×D1214yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—34no—
466C22×C4—34no—
467C22×C4—34no—
468C22×C4—34no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—64no—
473C22×C4—64no—
474C22×C4—64no—
475C22×C4—64no—
476C22×C4—68no—
477C22×C4—122no—
478C22×C4—124no—
479C22×C4—122no—
480C22×C4—124no—
481C22×C4C2×D1214yes—
482C22×C4#23714yes—
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D312yes—
485C22×C4C23×D312yes—
486C22×C4C23×D312yes—
487C22×C4C23×D311yescharacteristic
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D311yescharacteristic
490C22×C4C23×D311yescharacteristic
491C22×C4C23×D311yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—68no—
496C4⋊C4—68no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—68no—
500C4⋊C4—64no—
501C4⋊C4—68no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—122no—
509C4⋊C4—124no—
510C4⋊C4—122no—
511C4⋊C4—124no—
512C4⋊C4C2×D1214yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—68no—
518C22⋊C4—68no—
519C22⋊C4—68no—
520C22⋊C4—68no—
521C22⋊C4C2×D1218yes—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—68no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—22no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—24no—
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3#54714yes—
542C22×C3C23×D411yescharacteristic
543C22×C3C22×#4111yescharacteristic
544C22×C3C2×H(5,𝔽2)11yescharacteristic
545C22×C3C2×2−1+411yescharacteristic
546C22×C3C2×2−1+411yescharacteristic
547C22×C3C2×2−1+411yescharacteristic
548C22×C3#58511yescharacteristic
549C3×C4—22no—
550C3×C4—24no—
551C3×C4—22no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4#53914yescomplemented
558Dic3—44no—
559Dic3—44no—
560Dic3—44no—
561Dic3—44no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23C22×D1214yes—
573C23C22×D1212yes—
574C23C2×#23712yes—
575C23#101012yes—
576C23C24×D311yescharacteristic
577Q8—244no—
578D4—216no—
579D4—44no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—128no—
616C2×C4—244no—
617C2×C4—248no—
618C2×C4C22×D1214yes—
619C2×C4C22×D1212yes—
620C2×C4C22×D1212yes—
621C2×C4C2×#23712yes—
622C2×C4C2×#23712yes—
623C2×C4#100412yes—
624C2×C4#101012yes—
625C2×C4#101012yes—
626C2×C3—24no—
627C2×C3—44no—
628C2×C3C2×#53911yescharacteristic
629C2×C3C2×#54711yescharacteristic
630C2×C3#349611yescharacteristic
631C2×C3#349911yescharacteristic
632C2×C3#350711yescharacteristic
633C2×C3#353811yescharacteristic
634C2×C3#356311yescharacteristic
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22#555514yes—
649C22C23×D1211yescharacteristic
650C22C22×#23711yescharacteristic
651C22C2×#100411yescharacteristic
652C22C2×#101011yescharacteristic
653C22C2×#101011yescharacteristic
654C22C2×#101011yescharacteristic
655C22#596411yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—24no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—124no—
665C4—124no—
666C4—124no—
667C4—124no—
668C4#560214yescomplemented
669C3#3937611yescharacteristic, complemented, Sylow
670C2—24no—
671C2—44no—
672C2C2×#555511yescharacteristic
673C2#8578611yescharacteristic
674C2#8582711yescharacteristic
675C2C2×#560211yescharacteristic
676C2#8603111yescharacteristic
677C2#8614211yescharacteristic
678C2#8622411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×C4×#121C2
prime radicalC2×C3×#121C22
hypercenterC4×#121D3
hypocenterC3#39376
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
3. #82373C2semidirect
4. #82373C2semidirect
5. D4×C4⋊Dic3C2semidirect
6. #82076C2semidirect
7. C2×#4928C2semidirect
8. #76398C2semidirect
9. #76127C2semidirect
10. #75671C2semidirect
11. #74694C2semidirect
12. #73953C2semidirect
13. #73682C2semidirect
14. #73512C2semidirect
15. #73493C2semidirect
557. C3×C4#539semidirect
668. C4#5602semidirect
669. C3#39376semidirect

Maximal subgroups

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

subgroupindex
C2×#56022
C3×C4×#1212
#823732
D4×C4⋊Dic32
#820762
C2×#49282
#763982
#761272
#756712
#746942
#739532
#736822
#735122
#734932
#393763