Finite Group Databasebeta

← #10822996

Subgroups — #10822996

Order 768.

1454 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#5237C211yesmaximal, characteristic, complemented
3#84072C211yesmaximal, characteristic, complemented
4C22×#833C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6#83750C211yesmaximal, characteristic, complemented
7#72053C214yesmaximal, complemented
8#72042C214yesmaximal, complemented
9#71998C211yesmaximal, characteristic, complemented
10#23521—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)—24no—
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)—22no—
16C2×C3×AGL(1,ℤ/8)—22no—
17C22×C3×SD16C2212yes—
18C22×C3×D8C2212yes—
19C22×C3×M16C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5237—24no—
25#5233—22no—
26#5232—22no—
27C2×#836C2212yescomplemented
28#5225C2212yescomplemented
29C2×#833—24no—
30C2×#833C2214yes—
31C2×#833C2212yescomplemented
32#5215—22no—
33#5215C2214yes—
34#5215C2212yescomplemented
35C2×#832—22no—
36#5119—22no—
37#5118—22no—
38#5102C2214yescomplemented
39#4552C2214yescomplemented
40C22×AGL(1,ℤ/8)D311yescharacteristic
41#2905—31no—
42#2896—31no—
43C2×#418—31no—
44C2×#413—31no—
45C22×#100—31no—
46#2046—34no—
47#2045—34no—
48#1738—31no—
49C25×C3—21no—
50C2×C3×#41—22no—
51C2×C3×#41—21no—
52C2×C3×#41—21no—
53C2×C3×#41—22no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4D414yes—
69C22×C3×D4C2311yescharacteristic
70C22×C3×D4C2312yes—
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C3×AGL(1,ℤ/8)—216no—
74C3×AGL(1,ℤ/8)—216no—
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×D8—24no—
79C2×C3×D8—24no—
80C2×C3×D8—24no—
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C2×C3×M16D412yes—
84C2×C3×M16D412yes—
85C22×C3×C8C2312yes—
86C2×#217—21no—
87#926—21no—
88#926—21no—
89#926D412yes—
90#926C2312yes—
91C2×C4⋊Dic3D412yes—
92C2×C4×Dic3—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—22no—
101C2×#208—22no—
102C2×#208C2312yes—
103C2×#208C2312yes—
104C2×#208C2312yes—
105C22×C3⋊C8—21no—
106C22×C3⋊C8—21no—
107C22×C3⋊C8—21no—
108#836—24no—
109#835—24no—
110#833—28no—
111#833—24no—
112#832—28no—
113#832—24no—
114Dic3⋊C8—28no—
115C22×#41C2×D311yescharacteristic
116C23×D4C2×D311yescharacteristic
117C2×AGL(1,ℤ/8)—24no—
118C2×AGL(1,ℤ/8)—24no—
119C2×AGL(1,ℤ/8)—22no—
120C2×AGL(1,ℤ/8)—22no—
121C22×SD16C2×D312yes—
122C22×D8C2×D312yes—
123C22×M16—31no—
124C22×M16—31no—
125C22×M16—31no—
126C22×M16C2×D311yescharacteristic
127C2×#117—31no—
128#424—34no—
129#424—62no—
130#423—62no—
131#419—32no—
132#418—34no—
133#418—32no—
134#418—62no—
135#417—62no—
136#416—62no—
137C2×#102—62no—
138#413—64no—
139C2×#101—32no—
140C2×#100—34no—
141C2×#100—32no—
142C2×#100—64no—
143#341—34no—
144C24×C3—24no—
145C24×C3—22no—
146C24×C3—21no—
147C24×C3—41no—
148C24×C3—41no—
149C24×C3—42no—
150C24×C3—84no—
151C24×C3#4111yescharacteristic
152C3×#41—24no—
153C3×#41—24no—
154C3×#41—24no—
155C3×#41—24no—
156C3×#41—42no—
157C3×#41—42no—
158C3×#41—42no—
159C3×#41—42no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8C2×D412yes—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C2×C3×D4—42no—
185C2×C3×D4—41no—
186C2×C3×D4—41no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4C2×D412yes—
193C2×C3×D4C2×D414yes—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—41no—
197C22×C3×C4—41no—
198C22×C3×C4—42no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4#4111yescharacteristic
206C22×C3×C4#4112yes—
207C22×C3×C4#4112yes—
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—22no—
210C3×SD16—216no—
211C3×D8—216no—
212C3×M16—24no—
213C3×M16—24no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8C2×D414yes—
217#217—22no—
218#217—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3C2×D414yes—
221C4×Dic3—22no—
222C4×Dic3—22no—
223#208—22no—
224#208—22no—
225#208—22no—
226#208—22no—
227#208—22no—
228#208—22no—
229#208—42no—
230#208—42no—
231#208—42no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—42no—
236C2×C3⋊C8—42no—
237C2×C3⋊C8—42no—
238C25—21no—
239C2×#41—22no—
240C2×#41—21no—
241C2×#41—21no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41#6714yes—
247C22×Q8C22×D311yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4#6714yes—
258C22×D4C22×D311yescharacteristic
259C22×D4C22×D312yes—
260C23×C4—21no—
261C23×C4C22×D311yescharacteristic
262AGL(1,ℤ/8)—216no—
263AGL(1,ℤ/8)—216no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—24no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—32no—
273C2×M16—32no—
274C2×M16—32no—
275C2×M16—61no—
276C2×M16—61no—
277C2×M16—61no—
278C2×M16—62no—
279C2×M16—62no—
280C2×M16—62no—
281C2×M16—62no—
282C2×M16—62no—
283C2×M16—62no—
284C2×M16#6712yes—
285C2×M16#6712yes—
286C22×C8—61no—
287C22×C8—61no—
288C22×C8—61no—
289C22×C8C22×D312yes—
290#117—32no—
291#117—32no—
292#117—61no—
293#117—61no—
294C2×C4⋊C4—32no—
295C2×C22⋊C4—61no—
296C2×C42—61no—
297C4⋊C8—68no—
298#103—64no—
299#102—68no—
300#102—64no—
301#101—64no—
302#100—68no—
303#100—64no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3—41no—
310C23×C3—42no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—82no—
314C23×C3—84no—
315C23×C3—84no—
316C23×C3—84no—
317C23×C3—88no—
318C23×C3—88no—
319C23×C3#12111yescharacteristic
320C23×C3#12111yescharacteristic
321C23×C3#12312yes—
322C23×C3#12312yes—
323C23×C3C2×#4111yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—41no—
327C3×Q8—41no—
328C3×D4—22no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—44no—
333C3×D4—44no—
334C3×D4—42no—
335C3×D4—41no—
336C3×D4—42no—
337C3×D4—41no—
338C3×D4—44no—
339C3×D4—44no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—44no—
344C2×C3×C4—44no—
345C2×C3×C4#12012yes—
346C2×C3×C4#12012yes—
347C2×C3×C4#12111yescharacteristic
348C2×C3×C4#12111yescharacteristic
349C2×C3×C4#12112yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12112yes—
352C2×C3×C4#12112yes—
353C2×C3×C4#12312yes—
354C2×C3×C4#12312yes—
355C2×C3×C4#12412yes—
356C2×C3×C4#12412yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—24no—
364C2×Dic3—82no—
365C3×C8—24no—
366C3⋊C8—42no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C24—24no—
370C24—21no—
371C24—22no—
372C24—42no—
373C24—41no—
374C24—41no—
375C24—84no—
376C24#23711yescharacteristic
377#41—24no—
378#41—24no—
379#41—24no—
380#41—24no—
381#41—42no—
382#41—42no—
383#41—42no—
384#41—42no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8C2×#6712yes—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×#6712yes—
418C2×D4C2×#6714yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—62no—
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4#23711yescharacteristic
428C22×C4#23712yes—
429C22×C4#23712yes—
430C22×C4C2×#6712yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C23×D311yescharacteristic
435SD16—216no—
436D8—216no—
437M16—24no—
438M16—24no—
439M16—62no—
440M16—62no—
441M16—62no—
442M16—62no—
443M16—62no—
444M16—62no—
445M16—122no—
446M16—122no—
447M16—122no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—62no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8—122no—
454C2×C8—122no—
455C2×C8—122no—
456C2×C8C2×#6714yes—
457C4⋊C4—34no—
458C4⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C42—62no—
462C42—62no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—82no—
471C22×C3—84no—
472C22×C3—82no—
473C22×C3—88no—
474C22×C3—84no—
475C22×C3—88no—
476C22×C3—82no—
477C22×C3—84no—
478C22×C3#39412yes—
479C22×C3#39412yes—
480C22×C3C2×#12111yescharacteristic
481C22×C3C2×#12311yescharacteristic
482C22×C3C2×#12311yescharacteristic
483C3×C4—42no—
484C3×C4#39412yes—
485C3×C4#39412yes—
486C3×C4C2×#12011yescharacteristic
487C3×C4C2×#12111yescharacteristic
488C3×C4C2×#12111yescharacteristic
489C3×C4C2×#12411yescharacteristic
490Dic3—44no—
491C23—22no—
492C23—24no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—84no—
501C23—82no—
502C23—84no—
503C23—84no—
504C23—88no—
505C23—88no—
506C23#93512yes—
507C23#93512yes—
508C23#93811yescharacteristic
509C23#93811yescharacteristic
510C23C2×#23711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515D4—24no—
516D4—22no—
517D4—22no—
518D4—24no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—42no—
524D4—41no—
525D4—42no—
526D4—41no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—64no—
533C2×C4—242no—
534C2×C4#93512yes—
535C2×C4#93512yes—
536C2×C4#93612yes—
537C2×C4#93612yes—
538C2×C4#93811yescharacteristic
539C2×C4#93811yescharacteristic
540C2×C4#93912yes—
541C2×C4#93912yes—
542C2×C4#93912yes—
543C2×C4#93912yes—
544C2×C4#95312yes—
545C2×C4#95312yes—
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×#23711yescharacteristic
548C2×C4C2×#23711yescharacteristic
549C2×C4C22×#6711yescharacteristic
550C2×C4C22×#6711yescharacteristic
551C2×C4C22×#6711yescharacteristic
552C8—24no—
553C8—122no—
554C8—122no—
555C8—122no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—22no—
559C2×C3—22no—
560C2×C3—82no—
561C2×C3—84no—
562C2×C3C2×#39411yescharacteristic
563C2×C3#306812yes—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—84no—
572C22—82no—
573C22—84no—
574C22—88no—
575C22—88no—
576C22—82no—
577C22—84no—
578C22—82no—
579C22#522212yes—
580C22#522212yes—
581C22C2×#93511yescharacteristic
582C22C2×#93511yescharacteristic
583C22C2×#93811yescharacteristic
584C4—42no—
585C4—124no—
586C4#522212yes—
587C4#522212yes—
588C4C2×#93611yescharacteristic
589C4C2×#93911yescharacteristic
590C4C2×#93911yescharacteristic
591C4C2×#95311yescharacteristic
592C3#2352111yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—22no—
596C2—22no—
597C2—84no—
598C2—82no—
599C2#8396312yes—
600C2C2×#522211yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC22×C3C2×#121
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23521
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C2×#5237C2semidirect
3. #84072C2semidirect
4. C22×#833C2semidirect
5. C2×#5215C2semidirect
6. #83750C2semidirect
7. #72053C2semidirect
8. #72042C2semidirect
9. #71998C2semidirect
27. C2×#836C22semidirect
28. #5225C22semidirect
31. C2×#833C22semidirect
34. #5215C22semidirect
38. #5102C22semidirect
39. #4552C22semidirect
592. C3#23521semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
C2×#52372
#840722
C22×#8332
C2×#52152
#837502
#720532
#720422
#719982
#235213