Finite Group Databasebeta

← #10822351

Subgroups — #10822351

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×#947C211yesmaximal, characteristic, complemented
2C22×#942C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic
4C2×C3×#419C211yesmaximal, characteristic, complemented
5#82228C211yesmaximal, characteristic, complemented
6C2×#5001C211yesmaximal, characteristic, complemented
7#71979C211yesmaximal, characteristic, complemented
8#70278C214yesmaximal, complemented
9#70277C214yesmaximal, complemented
10#23431—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×D12C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#947—24no—
17C2×#947—24no—
18C2×#947—22no—
19C2×#947C2214yes—
20C22×#216C2212yescomplemented
21C2×#942—24no—
22C2×#942—24no—
23C2×#942—24no—
24C22×#215C2212yescomplemented
25C22×(C2×D3)⋊C4—22no—
26C22×C4×Dic3C2211yescharacteristic
27C22×#208C2211yescharacteristic
28C2×#908—24no—
29C2×#908—24no—
30C2×#908—24no—
31C22×D24C2212yes—
32C22×#204C2212yes—
33C3×#419—24no—
34C2×C3×#103C2214yescomplemented
35#5001—24no—
36#4988C2212yescomplemented
37#4973C2212yescomplemented
38C2×#809C2214yes—
39#4958—22no—
40C4×(C2×D3)⋊C4—22no—
41#4550C2214yescomplemented
42C22×#137—31no—
43C22×AGL(1,ℤ/8)—31no—
44C22×#124—31no—
45C2×#419D311yescharacteristic, complemented
46C2×#418—31no—
47#2297—31no—
48#2015—34no—
49#2012—34no—
50#1728—31no—
51C24×D3—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—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C23×Dic3D412yes—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235D414yes—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—22no—
74C22×D12—22no—
75C22×D12—21no—
76C22×D12D414yes—
77C22×D12C2312yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C2×C3×M16D414yes—
84C22×C3×C8C2312yes—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90#947—216no—
91#947—28no—
92#947—28no—
93C2×#216—24no—
94C2×#216—24no—
95C2×#216—24no—
96#942—232no—
97C2×#215—24no—
98C2×#215—24no—
99C2×#215—24no—
100C2×(C2×D3)⋊C4—216no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—24no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3D412yes—
106C2×C4×Dic3C2312yes—
107C2×#208—22no—
108C2×#208—22no—
109C2×#208D414yes—
110C22×C3⋊C8C2312yes—
111#908—232no—
112C2×D24—24no—
113C2×D24—24no—
114C2×D24—24no—
115C2×#204—24no—
116C2×#204—24no—
117C2×#204—24no—
118C3×#103—28no—
119C3⋊#95—24no—
120#809—28no—
121C22×#41—31no—
122C23×Q8C2×D311yescharacteristic
123C23×D4—31no—
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×M16C2×D311yescharacteristic
136C2×#124—34no—
137C2×#124—62no—
138C2×#124—64no—
139C2×#124—64no—
140C2×#117C2×D311yescharacteristic
141C22×C22⋊C4—62no—
142C22×C42—31no—
143#419—24no—
144#418—64no—
145C2×#103C2×D314yescomplemented
146C2×#102—34no—
147#391—32no—
148#390—32no—
149#386—62no—
150C4×C22⋊C4—62no—
151#336—34no—
152C24×C3C2×D411yescharacteristic
153C23×D3—24no—
154C23×D3—21no—
155C23×D3—22no—
156C23×D3—48no—
157C23×D3—42no—
158C23×D3—42no—
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—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4#4112yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—42no—
187C2×#67—42no—
188C22×Dic3—24no—
189C22×Dic3—21no—
190C22×Dic3—44no—
191C22×Dic3C2×D414yes—
192#235—28no—
193#235—28no—
194#235—44no—
195#235—44no—
196C2×D12—28no—
197C2×D12—28no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—22no—
206C2×D12—24no—
207C2×D12—22no—
208C2×D12—21no—
209C2×D12—44no—
210C2×D12—42no—
211C2×D12—44no—
212C2×D12—42no—
213C2×D12C2×D414yes—
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×M16—28no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8C2×D414yes—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C22⋊C4—42no—
231C3×C42—42no—
232#216—216no—
233#215—216no—
234(C2×D3)⋊C4—232no—
235(C2×D3)⋊C4—48no—
236C4×Dic3—24no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239#208—28no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×D414yes—
243D24—216no—
244#204—216no—
245C25—61no—
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×Q8D1214yes—
260C22×Q8C22×D312yes—
261C22×D4—34no—
262C22×D4—32no—
263C22×D4—31no—
264C22×D4—61no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4—61no—
270C22×D4—62no—
271C23×C4—21no—
272C23×C4—32no—
273C23×C4—61no—
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—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—22no—
301#117—22no—
302C2×C4⋊C4C22×D312yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—68no—
305C2×C22⋊C4—616no—
306C2×C22⋊C4—64no—
307C2×C42—21no—
308C2×C42—32no—
309C2×C42—32no—
310C2×C42—62no—
311C2×C42—62no—
312#103—28no—
313#102—68no—
314#95—64no—
315C23×C3—24no—
316C23×C3#12012yes—
317C23×C3#12112yes—
318C23×C3#12712yes—
319C23×C3C22×D411yescharacteristic
320C22×D3—24no—
321C22×D3—22no—
322C22×D3—416no—
323C22×D3—41no—
324C22×D3—42no—
325C22×D3—41no—
326C22×D3—42no—
327C22×D3—44no—
328C22×D3—48no—
329C22×D3—48no—
330C22×D3—88no—
331C22×D3—88no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—48no—
337C2×C3×C4—24no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—48no—
341C2×C3×C4—42no—
342C2×C3×C4—44no—
343C2×C3×C4#12014yes—
344C2×C3×C4#12012yes—
345C2×C3×C4#12014yes—
346C2×C3×C4#12114yes—
347C2×C3×C4#12112yes—
348C2×C3×C4#12114yes—
349C2×C3×C4#12712yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356#67—48no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—48no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364D12—24no—
365D12—22no—
366D12—22no—
367D12—24no—
368D12—42no—
369D12—42no—
370D12—41no—
371D12—41no—
372D12—48no—
373C4×D3—48no—
374Dic6—22no—
375Dic6—22no—
376Dic6—41no—
377Dic6—41no—
378C3×C8—24no—
379C3⋊C8—24no—
380C24—64no—
381C24—61no—
382C24—62no—
383C24—128no—
384C24—122no—
385C24—122no—
386C24D3×D411yescharacteristic
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×D1214yes—
410C2×D4—34no—
411C2×D4—32no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—68no—
415C2×D4—68no—
416C2×D4—61no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—61no—
422C2×D4—62no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—62no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4—124no—
429C2×D4—122no—
430C2×D4—124no—
431C2×D4—122no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—34no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—64no—
439C22×C4—61no—
440C22×C4—64no—
441C22×C4—124no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4C2×D1212yes—
445C22×C4#23512yes—
446C22×C4D3×D411yescharacteristic
447C22×C4D3×D412yes—
448C22×C4D3×D412yes—
449C22×C4D3×D412yes—
450C22×C4C2×#6712yes—
451C22×C4C23×D311yescharacteristic
452Q16—616no—
453SD16—616no—
454SD16—616no—
455D8—616no—
456M16—28no—
457M16—68no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—34no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8C2×#6714yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C22⋊C4—42no—
467C22⋊C4—632no—
468C22⋊C4—128no—
469C42—42no—
470C42—64no—
471C42—62no—
472C42—62no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3#39214yes—
477C22×C3C2×#12011yescharacteristic
478C22×C3C2×#12111yescharacteristic
479C22×C3C2×#12711yescharacteristic
480C22×C3#57514yes—
481C2×D3—42no—
482C2×D3—44no—
483C2×D3—48no—
484C2×D3—44no—
485C2×D3—816no—
486C2×D3—82no—
487C2×D3—84no—
488C2×D3—82no—
489C2×D3—84no—
490C3×C4—44no—
491C3×C4—82no—
492C3×C4#39214yes—
493C3×C4C2×#12011yescharacteristic
494C3×C4C2×#12011yescharacteristic
495C3×C4C2×#12111yescharacteristic
496C3×C4C2×#12111yescharacteristic
497Dic3—42no—
498Dic3—44no—
499C23—24no—
500C23—64no—
501C23—62no—
502C23—1216no—
503C23—128no—
504C23—121no—
505C23—122no—
506C23—124no—
507C23—128no—
508C23—121no—
509C23—122no—
510C23—248no—
511C23—248no—
512C23#88212yes—
513C23#88412yes—
514C23#94012yes—
515C23C2×D3×D411yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521Q8—62no—
522Q8—62no—
523Q8—121no—
524Q8—121no—
525D4—62no—
526D4—64no—
527D4—62no—
528D4—64no—
529D4—128no—
530D4—128no—
531D4—122no—
532D4—121no—
533D4—122no—
534D4—121no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—128no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—128no—
549C2×C4#88212yes—
550C2×C4#88214yes—
551C2×C4#88412yes—
552C2×C4#88414yes—
553C2×C4#93014yes—
554C2×C4#93714yes—
555C2×C4#94012yes—
556C2×C4C22×D1211yescharacteristic
557C2×C4C2×#23511yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C22×#6711yescharacteristic
562C8—24no—
563C8—64no—
564C2×C3—24no—
565C2×C3C2×#39211yescharacteristic
566C2×C3#300512yes—
567C2×C3#304012yes—
568C2×C3#315312yes—
569D3—82no—
570D3—84no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—128no—
575C22—124no—
576C22—122no—
577C22—124no—
578C22—2416no—
579C22—244no—
580C22—242no—
581C22—242no—
582C22—244no—
583C22#496014yes—
584C22C2×#88211yescharacteristic
585C22C2×#88411yescharacteristic
586C22#577714yes—
587C22C2×#94011yescharacteristic
588C4—44no—
589C4—82no—
590C4—124no—
591C4—122no—
592C4#496014yes—
593C4C2×#88211yescharacteristic
594C4C2×#88411yescharacteristic
595C4C2×#93011yescharacteristic
596C4C2×#93711yescharacteristic
597C3#2343111yescharacteristic, complemented, Sylow
598C2—24no—
599C2—244no—
600C2—242no—
601C2#7895912yes—
602C2#7897512yes—
603C2C2×#496011yescharacteristic
604C2#8397312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#908C2
hypercenterC2×#419D3
hypocenterC3#23431
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C22×#947C2semidirect
2. C22×#942C2semidirect
4. C2×C3×#419C2semidirect
5. #82228C2semidirect
6. C2×#5001C2semidirect
7. #71979C2semidirect
8. #70278C2semidirect
9. #70277C2semidirect
20. C22×#216C22semidirect
24. C22×#215C22semidirect
34. C2×C3×#103C22semidirect
36. #4988C22semidirect
37. #4973C22semidirect
41. #4550C22semidirect
45. C2×#419D3semidirect
145. C2×#103C2×D3semidirect
597. C3#23431semidirect

Maximal subgroups

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

subgroupindex
C22×#9472
C22×#9422
C22×#9082
C2×C3×#4192
#822282
C2×#50012
#719792
#702782
#702772
#234313