Finite Group Databasebeta

← #10720175

Subgroups — #10720175

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×#929C211yesmaximal, characteristic
2C22×#926C211yesmaximal, characteristic, complemented
3C22×C3×#95C211yesmaximal, characteristic, complemented
4C2×#5213C212yesmaximal
5#72127C212yesmaximal, complemented
6#69427C212yesmaximal, complemented
7#69410C212yesmaximal, complemented
8#69408C212yesmaximal, complemented
9#69405C212yesmaximal, complemented
10#14135—31nomaximal, Sylow
11C24×C3×C4C2212yes—
12C24×C3×C4C2211yescharacteristic
13C23×#67C2211yescharacteristic
14C22×#217C2211yescharacteristic
15C22×#217C2212yes—
16C2×#929—24no—
17C2×#929—24no—
18C2×#929C418yescomplemented
19C22×(C2×D3)⋊C4C2214yes—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C2×#926—24no—
23C2×#926—28no—
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×C4×Dic3C2212yes—
28C2×C3×#95—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31#5213—28no—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2214yes—
34C2×C3⋊#95C2214yes—
35C22×#123—31no—
36C22×#117—31no—
37C2×#386—32no—
38C22×#95D311yescharacteristic, complemented
39#1441—32no—
40#1439—32no—
41#1437—32no—
42#1436—32no—
43C22⋊#95—32no—
44C25×C3C2311yescharacteristic
45C24×D3C2311yescharacteristic
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—28no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—44no—
53C23×C3×C4C2×C414yes—
54C23×C3×C4D414yes—
55C23×C3×C4D414yes—
56C23×C3×C4C2314yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67—24no—
60C22×#67—22no—
61C22×#67—22no—
62C22×#67—24no—
63C22×#67C2×C414yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2314yes—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217C2×C414yes—
73C2×#217D418yes—
74#929—232no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—28no—
80C2×(C2×D3)⋊C4—28no—
81C2×(C2×D3)⋊C4—44no—
82C2×(C2×D3)⋊C4C2×C414yescomplemented
83C2×(C2×D3)⋊C4C2×C414yescomplemented
84#926—232no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—28no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4C2×C414yescomplemented
92C2×Dic3⋊C4C2×C414yescomplemented
93C2×C4×Dic3—24no—
94C2×C4×Dic3—28no—
95C3×#95—24no—
96C3×#95—24no—
97C3⋊#95—28no—
98C3⋊#95—28no—
99C3⋊#95—44no—
100C23×D4—31no—
101C24×C4C2×D312yes—
102C24×C4C2×D311yescharacteristic
103C2×#123—38no—
104C2×#123—64no—
105C2×#123—64no—
106C2×#117—68no—
107C2×#117—64no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—32no—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—31no—
113C22×C22⋊C4—31no—
114C22×C22⋊C4—31no—
115C22×C22⋊C4—34no—
116C22×C42—32no—
117#386—68no—
118C2×#95—24no—
119C2×#95—34no—
120C2×#95—34no—
121C2×#95—34no—
122C2×#95C2×D312yes—
123C2×#95C2×D312yes—
124C24×C3—24no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3C22×C412yes—
128C24×C3C2×D412yes—
129C24×C3C2×D414yes—
130C24×C3C2×D412yes—
131C24×C3#4114yes—
132C24×C3C2411yescharacteristic
133C23×D3—21no—
134C23×D3—22no—
135C23×D3—21no—
136C23×D3—24no—
137C23×D3—22no—
138C23×D3—22no—
139C23×D3—22no—
140C23×D3C22×C412yes—
141C22×C3×C4—216no—
142C22×C3×C4—28no—
143C22×C3×C4—216no—
144C22×C3×C4—28no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—216no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—416no—
157C22×C3×C4—416no—
158C22×C3×C4—48no—
159C22×C3×C4—48no—
160C22×C3×C4C22⋊C4116yescomplemented
161C22×C3×C4C22×C412yes—
162C22×C3×C4C22×C412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4#4118yes—
168C2×#67—28no—
169C2×#67—216no—
170C2×#67—44no—
171C2×#67—48no—
172C2×#67—44no—
173C22×Dic3—22no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3C22×C412yes—
191C22×Dic3C22×C412yes—
192C22×Dic3C22×C412yes—
193C22×Dic3C2×D418yes—
194#217—28no—
195#217—216no—
196(C2×D3)⋊C4—28no—
197(C2×D3)⋊C4—28no—
198(C2×D3)⋊C4—416no—
199C4⋊Dic3—216no—
200Dic3⋊C4—28no—
201Dic3⋊C4—28no—
202C4×Dic3—216no—
203C25—31no—
204C25C22×D311yescharacteristic
205C22×D4—34no—
206C22×D4—62no—
207C22×D4—64no—
208C22×D4—64no—
209C22×D4—62no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—34no—
219C23×C4—31no—
220C23×C4—44no—
221C23×C4C4×D314yes—
222C23×C4D1214yes—
223C23×C4#6714yes—
224C23×C4C22×D314yes—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227#123—632no—
228#117—632no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—68no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C22⋊C4—34no—
238C2×C22⋊C4—38no—
239C2×C22⋊C4—34no—
240C2×C22⋊C4—34no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—64no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—68no—
250C2×C22⋊C4—68no—
251C2×C22⋊C4—124no—
252C2×C42—68no—
253C2×C42—64no—
254#95—24no—
255#95—24no—
256#95—68no—
257#95—68no—
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×C3C2×C22⋊C414yes—
271C23×C3C4×D418yes—
272C23×C3#12014yes—
273C23×C3#12118yes—
274C23×C3#12314yes—
275C23×C3C23×C411yescharacteristic
276C23×C3C22×D411yescharacteristic
277C23×C3C22×D412yes—
278C23×C3C22×D411yescharacteristic
279C23×C3C2×#4114yes—
280C23×C3C2×#4112yes—
281C23×C3H(5,𝔽2)12yes—
282C23×C32−1+412yes—
283C22×D3—22no—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—41no—
288C22×D3—42no—
289C22×D3—41no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—42no—
295C22×D3—44no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3—44no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—232no—
308C2×C3×C4—28no—
309C2×C3×C4—28no—
310C2×C3×C4—28no—
311C2×C3×C4—432no—
312C2×C3×C4—44no—
313C2×C3×C4—44no—
314C2×C3×C4—44no—
315C2×C3×C4—44no—
316C2×C3×C4C2×C22⋊C414yes—
317C2×C3×C4C2×C22⋊C414yes—
318#67—416no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—28no—
326C2×Dic3—28no—
327C2×Dic3—28no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—44no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3—44no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—32no—
344C24—61no—
345C24—64no—
346C24—62no—
347C24—62no—
348C24—62no—
349C24—62no—
350C24—61no—
351C24C2×C4×D312yes—
352C24C2×D1212yes—
353C24D3×D414yes—
354C24#23714yes—
355C24C2×#6712yes—
356C24C23×D311yescharacteristic
357C2×D4—616no—
358C2×D4—68no—
359C2×D4—124no—
360C2×D4—124no—
361C2×D4—128no—
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—216no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—32no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—38no—
381C22×C4—416no—
382C22×C4—416no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—64no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—64no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—64no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4(C2×D3)⋊C4116yescomplemented
403C22×C4C2×C4×D312yes—
404C22×C4C2×C4×D312yes—
405C22×C4C2×D1212yes—
406C22×C4C2×D1212yes—
407C22×C4#23518yes—
408C22×C4C2×#6712yes—
409C22×C4C2×#6712yes—
410C4⋊C4—616no—
411C4⋊C4—68no—
412C4⋊C4—68no—
413C22⋊C4—616no—
414C22⋊C4—68no—
415C22⋊C4—68no—
416C22⋊C4—68no—
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#38618yes—
427C22×C3C22×C22⋊C411yescharacteristic
428C22×C3C2×C4×D412yes—
429C22×C3C4×#4112yes—
430C22×C3#52012yes—
431C22×C3C2×#12011yescharacteristic
432C22×C3C2×#12112yes—
433C22×C3C2×#12311yescharacteristic
434C22×C3#52512yes—
435C22×C3#53212yes—
436C22×C3#53511yescharacteristic
437C22×C3#53611yescharacteristic
438C22×C3#53712yes—
439C22×C3#54614yes—
440C22×C3#54714yes—
441C22×C3#55114yes—
442C22×C3#55214yes—
443C2×D3—42no—
444C2×D3—48no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—24no—
452C3×C4—24no—
453C3×C4—48no—
454Dic3—42no—
455Dic3—48no—
456Dic3—42no—
457Dic3—42no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—28no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—64no—
470C23—64no—
471C23—64no—
472C23—62no—
473C23—122no—
474C23—121no—
475C23—124no—
476C23—121no—
477C23—124no—
478C23—124no—
479C23—122no—
480C23—124no—
481C23—122no—
482C23—122no—
483C23—121no—
484C23—121no—
485C23#67⋊C418yes—
486C23#88214yes—
487C23#88614yes—
488C23C2×(C2×D3)⋊C414yes—
489C23#93918yes—
490C23C22×C4×D311yescharacteristic
491C23C22×D1211yescharacteristic
492C23C2×#23514yes—
493C23C2×D3×D412yes—
494C23C2×#23712yes—
495C23#100412yes—
496C23#100712yes—
497C23C22×#6711yescharacteristic
498D4—1216no—
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—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—432no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—68no—
519C2×C4—68no—
520C2×C4—64no—
521C2×C4—68no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—122no—
526C2×C4—124no—
527C2×C4—121no—
528C2×C4—124no—
529C2×C4—121no—
530C2×C4—124no—
531C2×C4—122no—
532C2×C4—124no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4C2×(C2×D3)⋊C414yes—
538C2×C4C2×(C2×D3)⋊C414yes—
539C2×C3—28no—
540C2×C3C2×#38611yescharacteristic
541C2×C3#229711yescharacteristic
542C2×C3#231111yescharacteristic
543C2×C3#235914yes—
544C2×C3#241112yes—
545C2×C3#241312yes—
546C2×C3#243012yes—
547C2×C3#249812yes—
548D3—82no—
549C22—24no—
550C22—216no—
551C22—28no—
552C22—24no—
553C22—28no—
554C22—28no—
555C22—28no—
556C22—122no—
557C22—128no—
558C22—122no—
559C22—122no—
560C22—242no—
561C22—242no—
562C22—242no—
563C22—242no—
564C22#495818yes—
565C22C4×#23512yes—
566C22#547712yes—
567C22C2×#67⋊C412yes—
568C22C2×#88211yescharacteristic
569C22#549014yes—
570C22C2×#88611yescharacteristic
571C22#549514yes—
572C22#550412yes—
573C22#550714yes—
574C22#551012yes—
575C22#551314yes—
576C22C22×(C2×D3)⋊C411yescharacteristic
577C22#579112yes—
578C22C2×#93912yes—
579C22#580511yescharacteristic
580C22#581611yescharacteristic
581C4—24no—
582C4—24no—
583C4—48no—
584C4—122no—
585C4—122no—
586C4—128no—
587C4—122no—
588C3#1413511yescharacteristic, complemented, Sylow
589C2—28no—
590C2—242no—
591C2#7333714yes—
592C2#7335412yes—
593C2#7337612yes—
594C2#8207011yescharacteristic
595C2C2×#495811yescharacteristic
596C2#8210612yes—
597C2#8217012yes—
598C2#8218611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×#95C2
socle, cosocleC24×C3C24
prime radicalC23×#67C22
hypercenterC22×#95D3
hypocenterC3#14135
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
2. C22×#926C2semidirect
3. C22×C3×#95C2semidirect
5. #72127C2semidirect
6. #69427C2semidirect
7. #69410C2semidirect
8. #69408C2semidirect
9. #69405C2semidirect
18. C2×#929C4semidirect
38. C22×#95D3semidirect
82. C2×(C2×D3)⋊C4C2×C4semidirect
83. C2×(C2×D3)⋊C4C2×C4semidirect
91. C2×Dic3⋊C4C2×C4semidirect
92. C2×Dic3⋊C4C2×C4semidirect
160. C22×C3×C4C22⋊C4semidirect
402. C22×C4(C2×D3)⋊C4semidirect
588. C3#14135semidirect

Maximal subgroups

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

subgroupindex
C22×#9292
C22×#9262
C22×C3×#952
C2×#52132
#721272
#694272
#694102
#694082
#694052
#141353