Finite Group Databasebeta

← #10822347

Subgroups — #10822347

Order 768.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C22×#936C211yesmaximal, characteristic, complemented
4#84086C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6C2×#5120C211yesmaximal, characteristic, complemented
7#72058C214yesmaximal, complemented
8#72045C214yesmaximal, complemented
9#71979C211yesmaximal, characteristic, complemented
10#23431—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
20C22×#217—22no—
21C2×#951—24no—
22C2×#951—24no—
23C2×#951—24no—
24C22×#216C2212yescomplemented
25C2×#936—24no—
26C2×#936—24no—
27C2×#936—22no—
28C2×#936C2214yes—
29C22×#214C2212yescomplemented
30C2×#926C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5229C2212yescomplemented
34#5221C2212yescomplemented
35#5220—22no—
36#5215—24no—
37C2×#832C2214yes—
38C4×#217—22no—
39#5120—24no—
40C2×#816C2214yescomplemented
41#4550C2214yescomplemented
42C22×#137—31no—
43C22×AGL(1,ℤ/8)D311yescharacteristic
44C22×#124—31no—
45C2×#419—31no—
46C2×#418—31no—
47#2297—31no—
48#2015—34no—
49#2012—34no—
50#1728—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—
72C23×Dic3D412yes—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6D414yes—
80C22×Dic6C2312yes—
81C3×AGL(1,ℤ/8)—232no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×SD16—24no—
85C2×C3×D8—24no—
86C2×C3×D8—24no—
87C2×C3×D8—24no—
88C2×C3×M16—22no—
89C2×C3×M16—22no—
90C2×C3×M16D414yes—
91C22×C3×C8C2312yes—
92C2×#217—216no—
93C2×#217—28no—
94C2×#217—21no—
95C2×#217—24no—
96#951—232no—
97C2×#216—24no—
98C2×#216—24no—
99C2×#216—24no—
100#936—216no—
101#936—28no—
102#936—28no—
103C2×#214—24no—
104C2×#214—24no—
105C2×#214—24no—
106#926—22no—
107#926—22no—
108C2×C4⋊Dic3C2312yes—
109C2×C4×Dic3—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D412yes—
113C2×C4×Dic3C2312yes—
114C2×#208—22no—
115C2×#208—22no—
116C2×#208D414yes—
117C22×C3⋊C8C2312yes—
118#832—28no—
119C3⋊#95—24no—
120#816—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×D4C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C2×AGL(1,ℤ/8)—24no—
128C2×AGL(1,ℤ/8)—24no—
129C2×AGL(1,ℤ/8)—24no—
130C22×Q16—32no—
131C22×SD16—32no—
132C22×SD16C2×D312yes—
133C22×D8C2×D312yes—
134C22×M16—31no—
135C22×M16C2×D311yescharacteristic
136C2×#124—34no—
137C2×#124—62no—
138C2×#124—64no—
139C2×#124—64no—
140C2×#117—31no—
141C22×C22⋊C4—62no—
142C22×C42—31no—
143#419—64no—
144#418—64no—
145C2×#103—34no—
146C2×#102—34no—
147#391—32no—
148#390—32no—
149#386—62no—
150C4×C22⋊C4—62no—
151#336—34no—
152C24×C3—24no—
153C24×C3—22no—
154C24×C3—21no—
155C24×C3—48no—
156C24×C3—42no—
157C24×C3—42no—
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—24no—
172C2×C3×D4—28no—
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4C2×D412yes—
190C2×C3×D4C2×D414yes—
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—24no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—44no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3C2×D414yes—
211C2×Dic6—28no—
212C2×Dic6—28no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—44no—
220C2×Dic6—44no—
221C2×Dic6—42no—
222C2×Dic6—42no—
223C2×Dic6C2×D414yes—
224C3×SD16—216no—
225C3×D8—216no—
226C3×M16—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8C2×D414yes—
230#217—232no—
231#217—48no—
232#217—42no—
233#216—216no—
234#214—216no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4×Dic3—24no—
238C4×Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—42no—
241#208—28no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8C2×D414yes—
245C25—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41#6714yes—
252C22×Q8—34no—
253C22×Q8—32no—
254C22×Q8—61no—
255C22×Q8—62no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8C22×D311yescharacteristic
261C22×D4—21no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—21no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4#6714yes—
269C22×D4C22×D312yes—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—32no—
273C23×C4—61no—
274C23×C4C22×D311yescharacteristic
275#137—632no—
276AGL(1,ℤ/8)—232no—
277C2×Q16—64no—
278C2×Q16—64no—
279C2×Q16—64no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—64no—
284C2×SD16—64no—
285C2×SD16—64no—
286C2×D8—24no—
287C2×D8—24no—
288C2×D8—24no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—34no—
292C2×M16—62no—
293C2×M16—62no—
294C2×M16#6714yes—
295C22×C8—32no—
296C22×C8C22×D312yes—
297#124—68no—
298#124—616no—
299#124—68no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—32no—
303C2×C22⋊C4—68no—
304C2×C22⋊C4—616no—
305C2×C22⋊C4—64no—
306C2×C22⋊C4—61no—
307C2×C42—32no—
308C2×C42—32no—
309C2×C42—62no—
310C2×C42—62no—
311C2×C42—61no—
312#103—68no—
313#102—68no—
314#95—64no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—416no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—44no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—88no—
327C23×C3—88no—
328C23×C3#12012yes—
329C23×C3#12112yes—
330C23×C3#12712yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—24no—
339C3×D4—24no—
340C3×D4—48no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—48no—
346C2×C3×C4—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—48no—
350C2×C3×C4#12014yes—
351C2×C3×C4#12012yes—
352C2×C3×C4#12014yes—
353C2×C3×C4#12114yes—
354C2×C3×C4#12114yes—
355C2×C3×C4#12112yes—
356C2×C3×C4#12712yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—48no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—48no—
371C2×Dic3—42no—
372C2×Dic3—44no—
373Dic6—24no—
374Dic6—24no—
375Dic6—42no—
376Dic6—42no—
377Dic6—48no—
378C3×C8—24no—
379C3⋊C8—24no—
380C24—24no—
381C24—21no—
382C24—22no—
383C24—48no—
384C24—42no—
385C24—42no—
386C24D3×D411yescharacteristic
387#41—28no—
388#41—28no—
389#41—44no—
390#41—44no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—34no—
397C2×Q8—64no—
398C2×Q8—62no—
399C2×Q8—68no—
400C2×Q8—68no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—124no—
406C2×Q8—122no—
407C2×Q8—122no—
408C2×Q8—124no—
409C2×Q8C2×#6712yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4C2×#6714yes—
431C2×D4C2×#6712yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—34no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—124no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4D3×D412yes—
445C22×C4D3×D411yescharacteristic
446C22×C4D3×D412yes—
447C22×C4#23712yes—
448C22×C4C2×#6712yes—
449C22×C4C2×#6712yes—
450C22×C4C2×#6712yes—
451C22×C4C23×D311yescharacteristic
452Q16—616no—
453SD16—216no—
454SD16—616no—
455D8—216no—
456M16—28no—
457M16—68no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—34no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8C2×#6714yes—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C22⋊C4—632no—
467C22⋊C4—128no—
468C22⋊C4—122no—
469C42—64no—
470C42—62no—
471C42—62no—
472C42—122no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—48no—
479C22×C3—44no—
480C22×C3—816no—
481C22×C3—82no—
482C22×C3—84no—
483C22×C3—82no—
484C22×C3—84no—
485C22×C3#39214yes—
486C22×C3C2×#12011yescharacteristic
487C22×C3C2×#12111yescharacteristic
488C22×C3C2×#12711yescharacteristic
489C22×C3#57514yes—
490C3×C4—42no—
491C3×C4#39214yes—
492C3×C4C2×#12011yescharacteristic
493C3×C4C2×#12011yescharacteristic
494C3×C4C2×#12111yescharacteristic
495C3×C4C2×#12111yescharacteristic
496Dic3—44no—
497Dic3—44no—
498Dic3—82no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—416no—
503C23—48no—
504C23—44no—
505C23—48no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—88no—
511C23—88no—
512C23#93712yes—
513C23#93912yes—
514C23#94012yes—
515C23C2×D3×D411yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—64no—
521Q8—64no—
522Q8—128no—
523Q8—122no—
524Q8—122no—
525D4—22no—
526D4—24no—
527D4—22no—
528D4—24no—
529D4—48no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—41no—
534D4—41no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—48no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—128no—
544C2×C4—128no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—124no—
548C2×C4—122no—
549C2×C4#93712yes—
550C2×C4#93714yes—
551C2×C4#93814yes—
552C2×C4#93912yes—
553C2×C4#93914yes—
554C2×C4#94012yes—
555C2×C4#95314yes—
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C22×#6711yescharacteristic
560C2×C4C22×#6711yescharacteristic
561C2×C4C22×#6711yescharacteristic
562C8—24no—
563C8—64no—
564C2×C3—24no—
565C2×C3—82no—
566C2×C3—84no—
567C2×C3C2×#39211yescharacteristic
568C2×C3#300512yes—
569C2×C3#304012yes—
570C2×C3#315312yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—48no—
578C22—816no—
579C22—84no—
580C22—84no—
581C22—82no—
582C22—82no—
583C22#522314yes—
584C22#577314yes—
585C22C2×#93711yescharacteristic
586C22C2×#93911yescharacteristic
587C22C2×#94011yescharacteristic
588C4—42no—
589C4—124no—
590C4—124no—
591C4—242no—
592C4#522314yes—
593C4C2×#93711yescharacteristic
594C4C2×#93811yescharacteristic
595C4C2×#93911yescharacteristic
596C4C2×#95311yescharacteristic
597C3#2343111yescharacteristic, complemented, Sylow
598C2—24no—
599C2—84no—
600C2—82no—
601C2#8393312yes—
602C2#8393812yes—
603C2#8394512yes—
604C2C2×#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#23431
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. C22×#951C2semidirect
3. C22×#936C2semidirect
4. #84086C2semidirect
5. C2×#5215C2semidirect
6. C2×#5120C2semidirect
7. #72058C2semidirect
8. #72045C2semidirect
9. #71979C2semidirect
24. C22×#216C22semidirect
29. C22×#214C22semidirect
33. #5229C22semidirect
34. #5221C22semidirect
40. C2×#816C22semidirect
41. #4550C22semidirect
597. C3#23431semidirect

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
C22×#9512
C22×#9362
#840862
C2×#52152
C2×#51202
#720582
#720452
#719792
#234313