Finite Group Databasebeta

← #10822349

Subgroups — #10822349

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×#124C211yesmaximal, characteristic, complemented
2C22×#942C211yesmaximal, characteristic, complemented
3C22×#932C211yesmaximal, characteristic
4C2×#5225C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6#82070C211yesmaximal, characteristic, complemented
7#71729C214yesmaximal, complemented
8#71728C214yesmaximal, complemented
9#71527C211yesmaximal, characteristic, complemented
10#23431—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C2×C3×#124—24no—
15C2×C3×#124—24no—
16C2×C3×#124—22no—
17C2×C3×#124C2214yes—
18C22×C3×C22⋊C4—22no—
19C22×C3×C42C2211yescharacteristic
20C22×#216C2212yescomplemented
21C2×#942—24no—
22C2×#942—24no—
23C2×#942—24no—
24C22×#215C2212yescomplemented
25C22×#214C2212yes—
26C2×#932—24no—
27C2×#932—24no—
28C2×#932—24no—
29C22×#213C2212yes—
30C2×#926C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5225—24no—
34C2×#835C2214yescomplemented
35#5215—24no—
36C2×#832C2214yes—
37#5213—22no—
38#4939C2212yescomplemented
39C4×(C2×D3)⋊C4—22no—
40#4929C2212yescomplemented
41#4524C2214yescomplemented
42C22×#137—31no—
43C22×AGL(1,ℤ/8)—31no—
44C22×#124D311yescharacteristic, complemented
45C2×#419—31no—
46C2×#418—31no—
47#2297—31no—
48#2015—34no—
49#2012—34no—
50#1728—31no—
51C25×C3—21no—
52C22×C3×Q8—22no—
53C22×C3×Q8—21no—
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×C4D412yes—
70C23×C3×C4C2311yescharacteristic
71C22×#67—21no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235D414yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C3×#124—216no—
82C3×#124—28no—
83C3×#124—28no—
84C2×C3×C22⋊C4—216no—
85C2×C3×C22⋊C4—28no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42D412yes—
89C2×C3×C42C2312yes—
90C2×#217—21no—
91C2×#216—24no—
92C2×#216—24no—
93C2×#216—24no—
94#942—232no—
95C2×#215—24no—
96C2×#215—24no—
97C2×#215—24no—
98C2×#214—24no—
99C2×#214—24no—
100C2×#214—24no—
101#932—232no—
102C2×#213—24no—
103C2×#213—24no—
104C2×#213—24no—
105C2×(C2×D3)⋊C4—24no—
106#926—22no—
107#926—22no—
108C2×C4⋊Dic3C2312yes—
109C2×C4×Dic3—21no—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208—22no—
113C2×#208—22no—
114C2×#208D414yes—
115C2×#208D414yes—
116C22×C3⋊C8C2312yes—
117C22×C3⋊C8C2312yes—
118#835—28no—
119#832—28no—
120C3⋊#95—24no—
121C22×#41—31no—
122C23×Q8C2×D311yescharacteristic
123C23×D4C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C2×AGL(1,ℤ/8)—64no—
128C2×AGL(1,ℤ/8)—64no—
129C2×AGL(1,ℤ/8)—64no—
130C22×Q16—32no—
131C22×SD16—32no—
132C22×SD16—32no—
133C22×D8—32no—
134C22×M16—31no—
135C22×M16—31no—
136C2×#124—24no—
137C2×#124—24no—
138C2×#124—22no—
139C2×#124C2×D314yes—
140C2×#117—31no—
141C22×C22⋊C4—22no—
142C22×C42C2×D311yescharacteristic
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—48no—
155C24×C3—42no—
156C24×C3—42no—
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—42no—
169C2×C3×Q8—44no—
170C2×C3×Q8—42no—
171C2×C3×Q8C2×D414yes—
172C2×C3×D4—28no—
173C2×C3×D4—24no—
174C2×C3×D4—28no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—44no—
181C2×C3×D4—42no—
182C2×C3×D4—44no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—44no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—42no—
202C2×#67—42no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205#235—28no—
206#235—28no—
207#235—44no—
208#235—44no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12C2×D412yes—
215C2×C4×D3—24no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6C2×D412yes—
224C3×C22⋊C4—232no—
225C3×C42—24no—
226C3×C42—22no—
227C3×C42—22no—
228#217—42no—
229#216—216no—
230#215—216no—
231#214—216no—
232#213—216no—
233(C2×D3)⋊C4—48no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4×Dic3—42no—
237#208—28no—
238#208—28no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8C2×D414yes—
244C2×C3⋊C8C2×D414yes—
245C25—21no—
246C2×#41—34no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C22×Q8—22no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—31no—
259C22×Q8#6714yes—
260C22×Q8C22×D312yes—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—21no—
267C22×D4—31no—
268C22×D4—61no—
269C22×D4#6714yes—
270C22×D4C22×D312yes—
271C23×C4—21no—
272C23×C4—61no—
273C23×C4#6712yes—
274C23×C4C22×D311yescharacteristic
275#137—632no—
276AGL(1,ℤ/8)—632no—
277C2×Q16—64no—
278C2×Q16—64no—
279C2×Q16—64no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C2×SD16—64no—
286C2×D8—64no—
287C2×D8—64no—
288C2×D8—64no—
289C2×M16—34no—
290C2×M16—34no—
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C2×M16—62no—
295C22×C8—32no—
296C22×C8—32no—
297#124—216no—
298#124—28no—
299#124—28no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—32no—
303C2×C22⋊C4—216no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—64no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—61no—
310C2×C42#6712yes—
311C2×C42C22×D312yes—
312#103—68no—
313#102—68no—
314#95—64no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—416no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—48no—
321C23×C3—48no—
322C23×C3—88no—
323C23×C3—88no—
324C23×C3#12012yes—
325C23×C3#12112yes—
326C23×C3#12712yes—
327C23×C3C22×D411yescharacteristic
328C22×D3—22no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—44no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—48no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—48no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—48no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—48no—
350C2×C3×C4#12014yes—
351C2×C3×C4#12014yes—
352C2×C3×C4#12012yes—
353C2×C3×C4#12112yes—
354C2×C3×C4#12114yes—
355C2×C3×C4#12114yes—
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
363#67—48no—
364C2×Dic3—22no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—42no—
368C2×Dic3—44no—
369D12—22no—
370D12—22no—
371D12—41no—
372D12—41no—
373C4×D3—48no—
374Dic6—22no—
375Dic6—22no—
376Dic6—41no—
377Dic6—41no—
378C3⋊C8—24no—
379C3⋊C8—24no—
380C24—24no—
381C24—22no—
382C24—48no—
383C24—42no—
384C24—42no—
385C24—61no—
386C24C2×#6711yescharacteristic
387#41—68no—
388#41—68no—
389#41—124no—
390#41—124no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—32no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8C2×#6714yes—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—32no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—64no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4C2×#6714yes—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—61no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4D3×D412yes—
444C22×C4D3×D412yes—
445C22×C4D3×D412yes—
446C22×C4#23712yes—
447C22×C4C2×#6714yes—
448C22×C4C2×#6712yes—
449C22×C4C2×#6712yes—
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—616no—
453SD16—616no—
454SD16—616no—
455D8—616no—
456M16—68no—
457M16—68no—
458C2×C8—34no—
459C2×C8—34no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—62no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C22⋊C4—232no—
467C22⋊C4—128no—
468C22⋊C4—122no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—122no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—44no—
477C22×C3—48no—
478C22×C3—816no—
479C22×C3—84no—
480C22×C3—84no—
481C22×C3#39214yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×#12111yescharacteristic
484C22×C3C2×#12711yescharacteristic
485C22×C3#57514yes—
486C2×D3—42no—
487C2×D3—44no—
488C2×D3—82no—
489C2×D3—82no—
490C3×C4—44no—
491C3×C4—44no—
492C3×C4#39214yes—
493C3×C4C2×#12011yescharacteristic
494C3×C4C2×#12011yescharacteristic
495C3×C4C2×#12111yescharacteristic
496C3×C4C2×#12111yescharacteristic
497Dic3—42no—
498Dic3—82no—
499C23—24no—
500C23—24no—
501C23—416no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—48no—
506C23—62no—
507C23—88no—
508C23—88no—
509C23—121no—
510C23—124no—
511C23—121no—
512C23#93812yes—
513C23#94012yes—
514C23#95312yes—
515C23C22×#6711yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521Q8—62no—
522Q8—62no—
523Q8—121no—
524Q8—121no—
525D4—24no—
526D4—24no—
527D4—48no—
528D4—42no—
529D4—42no—
530D4—62no—
531D4—62no—
532D4—128no—
533D4—121no—
534D4—121no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—62no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—128no—
547C2×C4—122no—
548C2×C4—124no—
549C2×C4#93714yes—
550C2×C4#93714yes—
551C2×C4#93812yes—
552C2×C4#93914yes—
553C2×C4#93914yes—
554C2×C4#94012yes—
555C2×C4#95312yes—
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×#23711yescharacteristic
560C2×C4C22×#6711yescharacteristic
561C2×C4C22×#6711yescharacteristic
562C8—64no—
563C8—64no—
564C2×C3—24no—
565C2×C3—84no—
566C2×C3C2×#39211yescharacteristic
567C2×C3#300512yes—
568C2×C3#304012yes—
569C2×C3#315312yes—
570D3—82no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—48no—
576C22—816no—
577C22—84no—
578C22—84no—
579C22—124no—
580C22—122no—
581C22—242no—
582C22—242no—
583C22#522314yes—
584C22C2×#93811yescharacteristic
585C22C2×#94011yescharacteristic
586C22#582214yes—
587C22C2×#95311yescharacteristic
588C4—44no—
589C4—44no—
590C4—122no—
591C4—242no—
592C4#522314yes—
593C4C2×#93711yescharacteristic
594C4C2×#93711yescharacteristic
595C4C2×#93911yescharacteristic
596C4C2×#93911yescharacteristic
597C3#2343111yescharacteristic, complemented, Sylow
598C2—24no—
599C2—84no—
600C2—242no—
601C2#8294312yes—
602C2#8294512yes—
603C2C2×#522311yescharacteristic
604C2#8408112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#124C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenterC22×#124D3
hypocenterC3#23431
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. C22×C3×#124C2semidirect
2. C22×#942C2semidirect
4. C2×#5225C2semidirect
5. C2×#5215C2semidirect
6. #82070C2semidirect
7. #71729C2semidirect
8. #71728C2semidirect
9. #71527C2semidirect
20. C22×#216C22semidirect
24. C22×#215C22semidirect
34. C2×#835C22semidirect
38. #4939C22semidirect
40. #4929C22semidirect
41. #4524C22semidirect
44. C22×#124D3semidirect
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×#1242
C22×#9422
C22×#9322
C2×#52252
C2×#52152
#820702
#717292
#717282
#715272
#234313