Finite Group Databasebeta

← #10822284

Subgroups — #10822284

Order 768.

2098 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
1C23×C3×SD16C211yesmaximal, characteristic, complemented
2C2×#5816C211yesmaximal, characteristic, complemented
3C22×#932C211yesmaximal, characteristic
4C2×#5225C211yesmaximal, characteristic, complemented
5#84044C211yesmaximal, characteristic, complemented
6C2×#5113C211yesmaximal, characteristic, complemented
7#72057C214yesmaximal, complemented
8#72055C214yesmaximal, complemented
9#71965C211yesmaximal, characteristic, complemented
10#23424—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C22×C3×SD16—24no—
15C22×C3×SD16—24no—
16C22×C3×SD16—24no—
17C22×C3×SD16C2214yes—
18C23×C3×C8C2211yescharacteristic
19#5816—24no—
20#5816—24no—
21C2×#947C2212yescomplemented
22C2×#946—22no—
23C2×#944C2212yescomplemented
24C22×#214C2212yes—
25C2×#932—24no—
26C2×#932—24no—
27C2×#932—24no—
28C22×#213C2212yes—
29C2×#929—22no—
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5247—22no—
34#5225—24no—
35C2×#835C2214yescomplemented
36#5221—22no—
37#5221C2214yes—
38#5113—24no—
39C2×#821C2212yescomplemented
40C2×#816C2212yescomplemented
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C23×SD16D311yescharacteristic, complemented
44C2×#536—31no—
45C2×#419—31no—
46C2×#416—31no—
47#2309—31no—
48#2013—34no—
49#2012—34no—
50#1729—31no—
51C25×C3—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8D414yes—
59C22×C3×Q8C2312yes—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4D414yes—
67C22×C3×D4C2312yes—
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235D414yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×SD16—232no—
81C2×C3×SD16—28no—
82C2×C3×SD16—28no—
83C2×C3×SD16—28no—
84C22×C3×C8—22no—
85C22×C3×C8—22no—
86C22×C3×C8D414yes—
87C22×C3×C8C2312yes—
88C2×#217—21no—
89C2×#217—28no—
90C2×#217D412yes—
91#947—24no—
92#947—24no—
93#946—216no—
94#944—24no—
95#944—24no—
96C2×#214—24no—
97C2×#214—24no—
98C2×#214—24no—
99#932—232no—
100C2×#213—24no—
101C2×#213—24no—
102C2×#213—24no—
103#929—216no—
104C2×(C2×D3)⋊C4—24no—
105#926—22no—
106#926—22no—
107#926—22no—
108#926—22no—
109C2×C4⋊Dic3D412yes—
110C2×C4⋊Dic3C2312yes—
111C2×Dic3⋊C4—24no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2312yes—
114C2×#208—22no—
115C2×#208—22no—
116C2×#208D414yes—
117C22×C3⋊C8C2312yes—
118#835—28no—
119#821—24no—
120#816—24no—
121C22×#41—31no—
122C23×Q8C2×D311yescharacteristic
123C23×D4C2×D311yescharacteristic
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—64no—
126C2×AGL(1,ℤ/8)—64no—
127C22×SD16—24no—
128C22×SD16—24no—
129C22×SD16—24no—
130C22×SD16—32no—
131C22×SD16C2×D314yes—
132C22×D8—32no—
133C22×M16—31no—
134C23×C8C2×D311yescharacteristic
135#536—64no—
136#536—64no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—62no—
141C2×#117—31no—
142C2×#117—31no—
143#419—64no—
144#416—64no—
145C2×#103—34no—
146C2×#103—32no—
147C2×#102—32no—
148#390—34no—
149#390—62no—
150#379—62no—
151#336—34no—
152C24×C3—22no—
153C24×C3—24no—
154C24×C3—42no—
155C24×C3—42no—
156C24×C3—84no—
157C24×C3C2×D411yescharacteristic
158C23×D3—21no—
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—44no—
168C2×C3×Q8—44no—
169C2×C3×Q8—42no—
170C2×C3×Q8—42no—
171C2×C3×Q8C2×D414yes—
172C2×C3×D4—28no—
173C2×C3×D4—28no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—44no—
182C2×C3×D4—44no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—42no—
199C2×#67—42no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3C2×D414yes—
203#235—28no—
204#235—28no—
205#235—44no—
206#235—44no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×D412yes—
213C2×C4×D3—24no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C2×D412yes—
222C3×SD16—232no—
223C2×C3×C8—28no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8C2×D414yes—
227#217—24no—
228#217—42no—
229#217—88no—
230#214—216no—
231#213—216no—
232(C2×D3)⋊C4—216no—
233C4⋊Dic3—24no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236Dic3⋊C4—216no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—42no—
240#208—28no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8C2×D414yes—
244C25—21no—
245C2×#41—34no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—31no—
258C22×Q8#6714yes—
259C22×Q8C22×D312yes—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4#6714yes—
269C22×D4C22×D312yes—
270C23×C4—21no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273AGL(1,ℤ/8)—632no—
274C2×SD16—232no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—64no—
279C2×SD16—64no—
280C2×SD16—64no—
281C2×D8—64no—
282C2×D8—64no—
283C2×D8—64no—
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—32no—
290C22×C8#6714yes—
291C22×C8C22×D312yes—
292C4⋊Q8—64no—
293C4⋊Q8—64no—
294#124—64no—
295#124—64no—
296#123—616no—
297C22⋊Q8—616no—
298#117—62no—
299#117—62no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—64no—
305C2×C22⋊C4—32no—
306C2×C22⋊C4—64no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—68no—
309C2×C42—32no—
310C2×C42—61no—
311#103—68no—
312#103—64no—
313#102—64no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—48no—
319C23×C3—48no—
320C23×C3—88no—
321C23×C3—88no—
322C23×C3—88no—
323C23×C3#12012yes—
324C23×C3#12112yes—
325C23×C3#12712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—44no—
331C3×Q8—24no—
332C3×Q8—24no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—48no—
336C3×D4—24no—
337C3×D4—24no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—48no—
341C2×C3×C4—24no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—48no—
345C2×C3×C4#12014yes—
346C2×C3×C4#12012yes—
347C2×C3×C4#12014yes—
348C2×C3×C4#12114yes—
349C2×C3×C4#12114yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12712yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358#67—48no—
359C2×Dic3—24no—
360C2×Dic3—22no—
361C2×Dic3—28no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—44no—
366D12—22no—
367D12—22no—
368D12—41no—
369D12—41no—
370C4×D3—48no—
371Dic6—22no—
372Dic6—22no—
373Dic6—41no—
374Dic6—41no—
375C3×C8—24no—
376C3⋊C8—24no—
377C24—22no—
378C24—24no—
379C24—42no—
380C24—42no—
381C24—61no—
382C24—84no—
383C24D3×D411yescharacteristic
384#41—68no—
385#41—68no—
386#41—124no—
387#41—124no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—32no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8C2×#6714yes—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—32no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—64no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4C2×#6714yes—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—34no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—61no—
435C22×C4—64no—
436C22×C4—62no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4D3×D412yes—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D412yes—
442C22×C4#23712yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6712yes—
445C22×C4C2×#6712yes—
446C22×C4C23×D311yescharacteristic
447SD16—232no—
448SD16—616no—
449D8—616no—
450M16—68no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—34no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8C2×#6714yes—
458C4⋊C4—616no—
459C4⋊C4—64no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C22⋊C4—616no—
463C22⋊C4—64no—
464C22⋊C4—122no—
465C22⋊C4—248no—
466C42—62no—
467C42—62no—
468C42—122no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—44no—
473C22×C3—48no—
474C22×C3—816no—
475C22×C3—84no—
476C22×C3—84no—
477C22×C3#39214yes—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×#12111yescharacteristic
480C22×C3C2×#12711yescharacteristic
481C22×C3#57714yes—
482C2×D3—42no—
483C2×D3—44no—
484C2×D3—82no—
485C2×D3—82no—
486C3×C4—44no—
487C3×C4#39214yes—
488C3×C4C2×#12011yescharacteristic
489C3×C4C2×#12011yescharacteristic
490C3×C4C2×#12111yescharacteristic
491C3×C4C2×#12111yescharacteristic
492Dic3—42no—
493Dic3—44no—
494Dic3—82no—
495C23—24no—
496C23—24no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—62no—
502C23—88no—
503C23—88no—
504C23—88no—
505C23—124no—
506C23—121no—
507C23—121no—
508C23#93712yes—
509C23#93912yes—
510C23#94012yes—
511C23C2×D3×D411yescharacteristic
512Q8—24no—
513Q8—24no—
514Q8—48no—
515Q8—42no—
516Q8—42no—
517Q8—62no—
518Q8—62no—
519Q8—121no—
520Q8—121no—
521D4—24no—
522D4—24no—
523D4—48no—
524D4—42no—
525D4—42no—
526D4—62no—
527D4—62no—
528D4—128no—
529D4—121no—
530D4—121no—
531C2×C4—24no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—68no—
536C2×C4—64no—
537C2×C4—62no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—128no—
541C2×C4—124no—
542C2×C4—122no—
543C2×C4#93712yes—
544C2×C4#93714yes—
545C2×C4#93814yes—
546C2×C4#93912yes—
547C2×C4#93914yes—
548C2×C4#94012yes—
549C2×C4#95314yes—
550C2×C4C2×D3×D411yescharacteristic
551C2×C4C2×D3×D411yescharacteristic
552C2×C4C2×#23711yescharacteristic
553C2×C4C22×#6711yescharacteristic
554C2×C4C22×#6711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—24no—
557C8—64no—
558C2×C3—24no—
559C2×C3—84no—
560C2×C3C2×#39211yescharacteristic
561C2×C3#301012yes—
562C2×C3#304212yes—
563C2×C3#315412yes—
564D3—82no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—44no—
569C22—48no—
570C22—816no—
571C22—84no—
572C22—84no—
573C22—122no—
574C22—124no—
575C22—242no—
576C22—242no—
577C22#522314yes—
578C22#576214yes—
579C22C2×#93711yescharacteristic
580C22C2×#93911yescharacteristic
581C22C2×#94011yescharacteristic
582C4—44no—
583C4—122no—
584C4—124no—
585C4—242no—
586C4#522314yes—
587C4C2×#93711yescharacteristic
588C4C2×#93811yescharacteristic
589C4C2×#93911yescharacteristic
590C4C2×#95311yescharacteristic
591C3#2342411yescharacteristic, complemented, Sylow
592C2—24no—
593C2—84no—
594C2—242no—
595C2#8384812yes—
596C2#8385412yes—
597C2#8386012yes—
598C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×SD16C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenterC23×SD16D3
hypocenterC3#23424
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×C3×SD16C2semidirect
2. C2×#5816C2semidirect
4. C2×#5225C2semidirect
5. #84044C2semidirect
6. C2×#5113C2semidirect
7. #72057C2semidirect
8. #72055C2semidirect
9. #71965C2semidirect
21. C2×#947C22semidirect
23. C2×#944C22semidirect
35. C2×#835C22semidirect
39. C2×#821C22semidirect
40. C2×#816C22semidirect
41. #4550C22semidirect
43. C23×SD16D3semidirect
591. C3#23424semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SD162
C2×#58162
C22×#9322
C2×#52252
#840442
C2×#51132
#720572
#720552
#719652
#234243