Finite Group Databasebeta

← #10822283

Subgroups — #10822283

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
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C23×#214C211yesmaximal, characteristic, complemented
4C2×#5233C211yesmaximal, characteristic, complemented
5#84044C211yesmaximal, characteristic, complemented
6C2×#5120C211yesmaximal, characteristic, complemented
7#72057C214yesmaximal, complemented
8#72045C214yesmaximal, complemented
9#71980C211yesmaximal, characteristic, complemented
10#23424—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)—24no—
16C2×C3×AGL(1,ℤ/8)—24no—
17C22×C3×SD16C2212yes—
18C22×C3×D8C2212yes—
19C22×C3×M16C2211yescharacteristic
20#5825—24no—
21#5825—24no—
22C2×#944C2212yescomplemented
23C2×#936C2212yescomplemented
24C2×#935—22no—
25C22×#214—24no—
26C22×#214—24no—
27C22×#214—24no—
28C22×#214C2214yes—
29C2×#926C2211yescharacteristic
30C2×#926C2211yescharacteristic
31C2×#924—22no—
32C23×C3⋊C8C2211yescharacteristic
33#5247—22no—
34#5233—24no—
35C2×#835C2212yescomplemented
36#5221—22no—
37#5221C2214yes—
38C2×#832C2212yescomplemented
39#5120—24no—
40C2×#816C2214yescomplemented
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)D311yescharacteristic
43C23×SD16—31no—
44C2×#536—31no—
45C2×#419—31no—
46C2×#416—31no—
47#2309—31no—
48#2013—34no—
49#2012—34no—
50#1729—31no—
51C25×C3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
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×D4C2311yescharacteristic
68C22×C3×D4C2312yes—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—21no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—22no—
78C22×Dic6D414yes—
79C22×Dic6C2312yes—
80C3×AGL(1,ℤ/8)—232no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×D8—24no—
85C2×C3×D8—24no—
86C2×C3×D8—24no—
87C2×C3×M16—22no—
88C2×C3×M16—22no—
89C2×C3×M16D414yes—
90C22×C3×C8C2312yes—
91C2×#217—24no—
92C2×#217—21no—
93C2×#217—28no—
94C2×#217D412yes—
95#944—24no—
96#944—24no—
97#936—24no—
98#936—24no—
99#935—216no—
100C2×#214—232no—
101C2×#214—28no—
102C2×#214—28no—
103C2×#214—28no—
104#926—22no—
105#926—22no—
106#926—22no—
107#926—22no—
108C2×C4⋊Dic3D412yes—
109C2×C4⋊Dic3C2312yes—
110#924—216no—
111C2×Dic3⋊C4—24no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2312yes—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—22no—
116C22×C3⋊C8D414yes—
117C22×C3⋊C8C2312yes—
118#835—24no—
119#832—24no—
120#816—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×D4C2×D311yescharacteristic
124C2×AGL(1,ℤ/8)—24no—
125C2×AGL(1,ℤ/8)—24no—
126C2×AGL(1,ℤ/8)—24no—
127C22×SD16—34no—
128C22×SD16—64no—
129C22×SD16—64no—
130C22×SD16—64no—
131C22×SD16C2×D312yes—
132C22×D8C2×D312yes—
133C22×M16C2×D311yescharacteristic
134C23×C8—31no—
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—32no—
146C2×#103—34no—
147C2×#102—32no—
148#390—34no—
149#390—62no—
150#379—62no—
151#336—34no—
152C24×C3—22no—
153C24×C3—21no—
154C24×C3—24no—
155C24×C3—42no—
156C24×C3—42no—
157C24×C3—84no—
158C24×C3C2×D411yescharacteristic
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—28no—
171C2×C3×D4—28no—
172C2×C3×D4—24no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—42no—
189C2×C3×D4C2×D414yes—
190C2×C3×D4C2×D412yes—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3C2×D414yes—
209C2×Dic6—28no—
210C2×Dic6—28no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—24no—
216C2×Dic6—22no—
217C2×Dic6—44no—
218C2×Dic6—44no—
219C2×Dic6—42no—
220C2×Dic6—42no—
221C2×Dic6C2×D414yes—
222C3×SD16—216no—
223C3×D8—216no—
224C3×M16—28no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8C2×D414yes—
228#217—216no—
229#217—24no—
230#217—42no—
231#217—88no—
232#214—232no—
233C4⋊Dic3—24no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236Dic3⋊C4—216no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—42no—
240C2×C3⋊C8—28no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8C2×D414yes—
244C25—21no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41#6714yes—
251C22×Q8—34no—
252C22×Q8—32no—
253C22×Q8—61no—
254C22×Q8—62no—
255C22×Q8—62no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8C22×D311yescharacteristic
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4#6714yes—
268C22×D4C22×D311yescharacteristic
269C22×D4C22×D312yes—
270C23×C4—21no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273AGL(1,ℤ/8)—232no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—68no—
278C2×SD16—68no—
279C2×SD16—632no—
280C2×SD16—68no—
281C2×D8—24no—
282C2×D8—24no—
283C2×D8—24no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16#6714yes—
287C22×C8—34no—
288C22×C8—32no—
289C22×C8—62no—
290C22×C8—62no—
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—64no—
312#103—68no—
313#102—64no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—24no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—48no—
322C23×C3—44no—
323C23×C3—48no—
324C23×C3—88no—
325C23×C3—88no—
326C23×C3—88no—
327C23×C3#12012yes—
328C23×C3#12112yes—
329C23×C3#12712yes—
330C23×C3C22×D411yescharacteristic
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×D4—24no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4—48no—
340C3×D4—41no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—41no—
344C3×D4—48no—
345C2×C3×C4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—48no—
349C2×C3×C4#12014yes—
350C2×C3×C4#12012yes—
351C2×C3×C4#12014yes—
352C2×C3×C4#12114yes—
353C2×C3×C4#12114yes—
354C2×C3×C4#12112yes—
355C2×C3×C4#12712yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—28no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—48no—
368C2×Dic3—42no—
369C2×Dic3—44no—
370Dic6—24no—
371Dic6—24no—
372Dic6—42no—
373Dic6—42no—
374Dic6—48no—
375C3×C8—24no—
376C3⋊C8—24no—
377C24—21no—
378C24—22no—
379C24—24no—
380C24—42no—
381C24—42no—
382C24—84no—
383C24D3×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—34no—
394C2×Q8—64no—
395C2×Q8—62no—
396C2×Q8—68no—
397C2×Q8—68no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8—124no—
403C2×Q8—122no—
404C2×Q8—122no—
405C2×Q8—124no—
406C2×Q8C2×#6712yes—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4C2×#6712yes—
428C2×D4C2×#6714yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—34no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—64no—
435C22×C4—62no—
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—216no—
448SD16—632no—
449D8—216no—
450M16—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—34no—
454C2×C8—68no—
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—42no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—48no—
476C22×C3—816no—
477C22×C3—82no—
478C22×C3—84no—
479C22×C3—84no—
480C22×C3—82no—
481C22×C3#39214yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×#12111yescharacteristic
484C22×C3C2×#12711yescharacteristic
485C22×C3#57714yes—
486C3×C4—42no—
487C3×C4#39214yes—
488C3×C4C2×#12011yescharacteristic
489C3×C4C2×#12011yescharacteristic
490C3×C4C2×#12111yescharacteristic
491C3×C4C2×#12111yescharacteristic
492Dic3—44no—
493Dic3—44no—
494Dic3—82no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—44no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—88no—
506C23—88no—
507C23—88no—
508C23#93712yes—
509C23#93912yes—
510C23#94012yes—
511C23C2×D3×D411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—64no—
517Q8—64no—
518Q8—128no—
519Q8—122no—
520Q8—122no—
521D4—24no—
522D4—22no—
523D4—22no—
524D4—24no—
525D4—48no—
526D4—48no—
527D4—41no—
528D4—42no—
529D4—41no—
530D4—42no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—68no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—128no—
539C2×C4—122no—
540C2×C4—122no—
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—82no—
560C2×C3—84no—
561C2×C3C2×#39211yescharacteristic
562C2×C3#301012yes—
563C2×C3#304212yes—
564C2×C3#315412yes—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—44no—
570C22—48no—
571C22—44no—
572C22—816no—
573C22—82no—
574C22—84no—
575C22—82no—
576C22—84no—
577C22#522314yes—
578C22#577514yes—
579C22C2×#93711yescharacteristic
580C22C2×#93911yescharacteristic
581C22C2×#94011yescharacteristic
582C4—42no—
583C4—124no—
584C4—124no—
585C4—242no—
586C4#522314yes—
587C4C2×#93711yescharacteristic
588C4C2×#93811yescharacteristic
589C4C2×#93911yescharacteristic
590C4C2×#95311yescharacteristic
591C3#2342411yescharacteristic, complemented, Sylow
592C2—24no—
593C2—84no—
594C2—82no—
595C2#8393512yes—
596C2#8393912yes—
597C2#8394812yes—
598C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC23×C3C22×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23424
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C23×#214C2semidirect
4. C2×#5233C2semidirect
5. #84044C2semidirect
6. C2×#5120C2semidirect
7. #72057C2semidirect
8. #72045C2semidirect
9. #71980C2semidirect
22. C2×#944C22semidirect
23. C2×#936C22semidirect
35. C2×#835C22semidirect
38. C2×#832C22semidirect
40. C2×#816C22semidirect
41. #4550C22semidirect
591. C3#23424semidirect

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×#58252
C23×#2142
C2×#52332
#840442
C2×#51202
#720572
#720452
#719802
#234243