Finite Group Databasebeta

← #10720176

Subgroups — #10720176

Order 768.

2634 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2C22×#926C211yesmaximal, characteristic, complemented
3C2×#5220C212yesmaximal
4C22×C3⋊#95C211yesmaximal, characteristic, complemented
5#71565C212yesmaximal, complemented
6#71564C212yesmaximal, complemented
7#71546C212yesmaximal, complemented
8#71545C212yesmaximal, complemented
9#71544C212yesmaximal, complemented
10#14135—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×Dic3C2212yes—
14C2×C3×#123—24no—
15C2×C3×#123—24no—
16C2×C3×#123C418yescomplemented
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×#217C2214yes—
23C22×#217C2212yes—
24C2×#926—24no—
25C2×#926—28no—
26C22×C4⋊Dic3C2212yes—
27C22×C4×Dic3C2212yes—
28#5220—28no—
29C2×C3⋊#95—24no—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2214yes—
34C2×C3⋊#95C2214yes—
35C22×#123D311yescharacteristic
36C22×#117—31no—
37C2×#386—32no—
38C22×#95—31no—
39#1441—32no—
40#1439—32no—
41#1437—32no—
42#1436—32no—
43C22⋊#95—32no—
44C25×C3C2311yescharacteristic
45C25×C3C2311yescharacteristic
46C22×C3×D4—24no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—24no—
50C22×C3×D4C2×C414yes—
51C23×C3×C4—24no—
52C23×C3×C4—24no—
53C23×C3×C4C2×C414yes—
54C23×C3×C4D414yes—
55C23×C3×C4D414yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—24no—
62C23×Dic3—28no—
63C23×Dic3—24no—
64C23×Dic3—24no—
65C23×Dic3—44no—
66C23×Dic3C2314yes—
67C23×Dic3C2314yes—
68C3×#123—232no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4C2×C414yescomplemented
74C2×C3×C4⋊C4C2×C414yescomplemented
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4C2×C414yes—
82C2×C3×C22⋊C4C2×C414yescomplemented
83C2×C3×C22⋊C4C2×C414yescomplemented
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—28no—
87C2×#217—28no—
88C2×#217—44no—
89C2×#217D418yes—
90#926—232no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—28no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—28no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—28no—
98C3⋊#95—28no—
99C3⋊#95—44no—
100C23×D4C2×D311yescharacteristic
101C24×C4—32no—
102C24×C4C2×D311yescharacteristic
103C2×#123—24no—
104C2×#123—24no—
105C2×#123Dic318yescomplemented
106C2×#117—68no—
107C2×#117—64no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C22⋊C4—32no—
112C22×C22⋊C4—34no—
113C22×C22⋊C4C2×D311yescharacteristic
114C22×C22⋊C4C2×D311yescharacteristic
115C22×C22⋊C4C2×D311yescharacteristic
116C22×C42—32no—
117#386—68no—
118C2×#95—34no—
119C2×#95—32no—
120C2×#95—34no—
121C2×#95—32no—
122C2×#95—34no—
123C2×#95—64no—
124C24×C3—24no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—21no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3C22×C412yes—
135C24×C3C22×C412yes—
136C24×C3C2×D414yes—
137C24×C3C2×D412yes—
138C24×C3C2×D412yes—
139C24×C3#4114yes—
140C24×C3C2411yescharacteristic
141C2×C3×D4—28no—
142C2×C3×D4—216no—
143C2×C3×D4—44no—
144C2×C3×D4—48no—
145C2×C3×D4—44no—
146C22×C3×C4—216no—
147C22×C3×C4—28no—
148C22×C3×C4—216no—
149C22×C3×C4—28no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4C22⋊C4116yescomplemented
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×Dic3—216no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—416no—
189C22×Dic3—416no—
190C22×Dic3—48no—
191C22×Dic3—48no—
192C22×Dic3C2×D418yes—
193C22×Dic3#4118yes—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C22⋊C4—28no—
199#217—216no—
200#217—416no—
201C4⋊Dic3—216no—
202C4×Dic3—216no—
203C25C22×D311yescharacteristic
204C25C22×D311yescharacteristic
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4C2×Dic314yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—34no—
213C23×C4—34no—
214C23×C4—68no—
215C23×C4—64no—
216C23×C4—64no—
217C23×C4—64no—
218C23×C4—124no—
219C23×C4C2×Dic314yes—
220C23×C4#6714yes—
221C23×C4#6714yes—
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227#123—232no—
228#117—632no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—68no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4C2×Dic314yescomplemented
236C2×C4⋊C4C2×Dic314yescomplemented
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—38no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—68no—
247C2×C22⋊C4—68no—
248C2×C22⋊C4—124no—
249C2×C22⋊C4C2×Dic314yes—
250C2×C22⋊C4C2×Dic314yescomplemented
251C2×C22⋊C4C2×Dic314yescomplemented
252C2×C42—68no—
253C2×C42—64no—
254#95—68no—
255#95—68no—
256#95—64no—
257#95—64no—
258#95—124no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—28no—
267C23×C3—28no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—44no—
283C23×C3—44no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3C2×C22⋊C414yes—
287C23×C3C4×D418yes—
288C23×C3#12014yes—
289C23×C3#12118yes—
290C23×C3#12314yes—
291C23×C3C23×C411yescharacteristic
292C23×C3C22×D412yes—
293C23×C3C22×D411yescharacteristic
294C23×C3C22×D411yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3C2×#4114yes—
297C23×C3H(5,𝔽2)12yes—
298C23×C32−1+412yes—
299C3×D4—416no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—232no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4C2×C22⋊C414yes—
324C2×C3×C4C2×C22⋊C414yes—
325C2×Dic3—28no—
326C2×Dic3—28no—
327C2×Dic3—28no—
328C2×Dic3—28no—
329C2×Dic3—28no—
330C2×Dic3—28no—
331C2×Dic3—432no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C2×Dic3—44no—
335C2×Dic3—44no—
336C2×Dic3—44no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3—44no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24D3×D414yes—
351C24#23714yes—
352C24C22×Dic312yes—
353C24C22×Dic312yes—
354C24C2×#6712yes—
355C24C2×#6712yes—
356C24C23×D311yescharacteristic
357C2×D4—216no—
358C2×D4—28no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—38no—
382C22×C4—38no—
383C22×C4—616no—
384C22×C4—64no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—64no—
392C22×C4—64no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—1216no—
397C22×C4—1216no—
398C22×C4—128no—
399C22×C4—128no—
400C22×C4#217116yescomplemented
401C22×C4C22×Dic312yes—
402C22×C4C22×Dic312yes—
403C22×C4C22×Dic312yes—
404C22×C4C22×Dic312yes—
405C22×C4C22×Dic312yes—
406C22×C4C2×#6712yes—
407C22×C4C2×#6712yes—
408C22×C4C2×#6712yes—
409C22×C4C2×#6712yes—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—616no—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—616no—
417C22⋊C4—1216no—
418C42—616no—
419C22×C3—24no—
420C22×C3—216no—
421C22×C3—28no—
422C22×C3—28no—
423C22×C3—28no—
424C22×C3—28no—
425C22×C3—24no—
426C22×C3—42no—
427C22×C3—48no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3#38618yes—
435C22×C3C22×C22⋊C411yescharacteristic
436C22×C3C2×C4×D412yes—
437C22×C3C4×#4112yes—
438C22×C3#52012yes—
439C22×C3C2×#12011yescharacteristic
440C22×C3C2×#12112yes—
441C22×C3C2×#12311yescharacteristic
442C22×C3#52512yes—
443C22×C3#53212yes—
444C22×C3#53511yescharacteristic
445C22×C3#53611yescharacteristic
446C22×C3#53712yes—
447C22×C3#54614yes—
448C22×C3#54714yes—
449C22×C3#55114yes—
450C22×C3#55214yes—
451C3×C4—24no—
452C3×C4—24no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456Dic3—48no—
457Dic3—48no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23D4×Dic318yes—
486C23#93514yes—
487C23#93714yes—
488C23#93918yes—
489C23C2×#21714yes—
490C23C2×D3×D412yes—
491C23C2×#23712yes—
492C23D3×#4114yes—
493C23#100912yes—
494C23#101012yes—
495C23C23×Dic311yescharacteristic
496C23C22×#6711yescharacteristic
497C23C22×#6711yescharacteristic
498D4—416no—
499C2×C4—232no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—68no—
523C2×C4—68no—
524C2×C4—68no—
525C2×C4—68no—
526C2×C4—68no—
527C2×C4—68no—
528C2×C4—1232no—
529C2×C4—124no—
530C2×C4—124no—
531C2×C4—124no—
532C2×C4—124no—
533C2×C4—124no—
534C2×C4—124no—
535C2×C4—124no—
536C2×C4—124no—
537C2×C4C2×#21714yes—
538C2×C4C2×#21714yes—
539C2×C3—28no—
540C2×C3—82no—
541C2×C3C2×#38611yescharacteristic
542C2×C3#229711yescharacteristic
543C2×C3#231111yescharacteristic
544C2×C3#235914yes—
545C2×C3#241112yes—
546C2×C3#241312yes—
547C2×C3#243012yes—
548C2×C3#249812yes—
549C22—24no—
550C22—24no—
551C22—216no—
552C22—28no—
553C22—28no—
554C22—28no—
555C22—28no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—48no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22#522018yes—
565C22#565014yes—
566C22#565414yes—
567C22#565514yes—
568C22#565814yes—
569C22C2×D4×Dic312yes—
570C22C2×#93511yescharacteristic
571C22C2×#93711yescharacteristic
572C22C2×#93912yes—
573C22#582511yescharacteristic
574C22Dic3×#4112yes—
575C22#582712yes—
576C22#582812yes—
577C22#583011yescharacteristic
578C22#583212yes—
579C22#583312yes—
580C22C22×#21711yescharacteristic
581C4—24no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—128no—
587C4—128no—
588C3#1413511yescharacteristic, complemented, Sylow
589C2—28no—
590C2—82no—
591C2#8283814yes—
592C2#8284312yes—
593C2#8284812yes—
594C2#8285812yes—
595C2#8289412yes—
596C2C2×#522011yescharacteristic
597C2#8408611yescharacteristic
598C2#8408711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×#123C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC22×#123D3
hypocenterC3#14135
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C22×#926C2semidirect
4. C22×C3⋊#95C2semidirect
5. #71565C2semidirect
6. #71564C2semidirect
7. #71546C2semidirect
8. #71545C2semidirect
9. #71544C2semidirect
16. C2×C3×#123C4semidirect
73. C2×C3×C4⋊C4C2×C4semidirect
74. C2×C3×C4⋊C4C2×C4semidirect
82. C2×C3×C22⋊C4C2×C4semidirect
83. C2×C3×C22⋊C4C2×C4semidirect
105. C2×#123Dic3semidirect
165. C22×C3×C4C22⋊C4semidirect
235. C2×C4⋊C4C2×Dic3semidirect
236. C2×C4⋊C4C2×Dic3semidirect
250. C2×C22⋊C4C2×Dic3semidirect
251. C2×C22⋊C4C2×Dic3semidirect
400. C22×C4#217semidirect
588. C3#14135semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C22×#9262
C2×#52202
C22×C3⋊#952
#715652
#715642
#715462
#715452
#715442
#141353