Finite Group Databasebeta

← #10822352

Subgroups — #10822352

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×#936C211yesmaximal, characteristic, complemented
2C22×#932C211yesmaximal, characteristic
3C22×#909C211yesmaximal, characteristic, complemented
4C2×C3×#418C211yesmaximal, characteristic, complemented
5#82228C211yesmaximal, characteristic, complemented
6C2×#5004C211yesmaximal, characteristic, complemented
7#71979C211yesmaximal, characteristic, complemented
8#70215C214yesmaximal, complemented
9#70214C214yesmaximal, complemented
10#23431—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#217—22no—
17C2×#936—24no—
18C2×#936—24no—
19C2×#936—22no—
20C2×#936C2214yes—
21C22×#214C2212yes—
22C2×#932—24no—
23C2×#932—24no—
24C2×#932—24no—
25C22×#213C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C22×#208C2211yescharacteristic
28C2×#909—24no—
29C2×#909—24no—
30C2×#909—24no—
31C22×Dic12C2212yescomplemented
32C22×#204C2212yescomplemented
33C3×#418—24no—
34C2×C3×#102C2214yes—
35#5004—24no—
36#4988C2212yescomplemented
37#4973C2212yescomplemented
38C2×#810C2214yescomplemented
39#4958—22no—
40C4×(C2×D3)⋊C4—22no—
41#4550C2214yescomplemented
42C22×#137—31no—
43C22×AGL(1,ℤ/8)—31no—
44C22×#124—31no—
45C2×#419—31no—
46C2×#418D311yescharacteristic, complemented
47#2297—31no—
48#2015—34no—
49#2012—34no—
50#1728—31no—
51C25×C3—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4D414yes—
59C22×C3×D4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C23×Dic3D412yes—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235D414yes—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6D414yes—
79C22×Dic6C2312yes—
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—
90C2×#217—216no—
91C2×#217—28no—
92#936—216no—
93#936—28no—
94#936—28no—
95C2×#214—24no—
96C2×#214—24no—
97C2×#214—24no—
98#932—232no—
99C2×#213—24no—
100C2×#213—24no—
101C2×#213—24no—
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#909—232no—
112C2×Dic12—24no—
113C2×Dic12—24no—
114C2×Dic12—24no—
115C2×#204—24no—
116C2×#204—24no—
117C2×#204—24no—
118C3×#102—28no—
119C3⋊#95—24no—
120#810—28no—
121C22×#41—31no—
122C23×Q8—31no—
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×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—64no—
144#418—24no—
145C2×#103—34no—
146C2×#102C2×D314yes—
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×D4—28no—
160C2×C3×D4—24no—
161C2×C3×D4—28no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—44no—
168C2×C3×D4—44no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—22no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4#4112yes—
180C22×C3×C4C2411yescharacteristic
181C2×#67—24no—
182C2×#67—42no—
183C2×#67—42no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—44no—
189C22×Dic3—42no—
190C22×Dic3—42no—
191C22×Dic3C2×D414yes—
192#235—28no—
193#235—28no—
194#235—44no—
195#235—44no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12C2×D412yes—
202C2×C4×D3—24no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×Dic6—28no—
206C2×Dic6—28no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—22no—
215C2×Dic6—24no—
216C2×Dic6—22no—
217C2×Dic6—21no—
218C2×Dic6—44no—
219C2×Dic6—42no—
220C2×Dic6—44no—
221C2×Dic6—42no—
222C2×Dic6C2×D414yes—
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#217—232no—
233#214—216no—
234#213—216no—
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—
243Dic12—216no—
244#204—216no—
245C25—21no—
246C2×#41—34no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C22×Q8—34no—
253C22×Q8—32no—
254C22×Q8—31no—
255C22×Q8—62no—
256C22×Q8—61no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—21no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—31no—
268C22×D4—61no—
269C22×D4D1214yes—
270C22×D4C22×D312yes—
271C23×C4—32no—
272C23×C4—61no—
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—68no—
313#102—28no—
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×D4—24no—
333C3×D4—24no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—44no—
339C2×C3×C4#12014yes—
340C2×C3×C4#12012yes—
341C2×C3×C4#12014yes—
342C2×C3×C4#12114yes—
343C2×C3×C4#12112yes—
344C2×C3×C4#12114yes—
345C2×C3×C4#12712yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352#67—48no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—48no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—48no—
364D12—22no—
365D12—22no—
366D12—41no—
367D12—41no—
368C4×D3—48no—
369Dic6—24no—
370Dic6—22no—
371Dic6—22no—
372Dic6—24no—
373Dic6—42no—
374Dic6—42no—
375Dic6—41no—
376Dic6—41no—
377Dic6—48no—
378C3×C8—24no—
379C3⋊C8—24no—
380C24—24no—
381C24—22no—
382C24—48no—
383C24—42no—
384C24—42no—
385C24—61no—
386C24D3×D411yescharacteristic
387#41—68no—
388#41—68no—
389#41—124no—
390#41—124no—
391C2×Q8—34no—
392C2×Q8—32no—
393C2×Q8—64no—
394C2×Q8—62no—
395C2×Q8—61no—
396C2×Q8—68no—
397C2×Q8—68no—
398C2×Q8—61no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—61no—
402C2×Q8—62no—
403C2×Q8—61no—
404C2×Q8—62no—
405C2×Q8—61no—
406C2×Q8—124no—
407C2×Q8—122no—
408C2×Q8—124no—
409C2×Q8—122no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—32no—
419C2×D4—42no—
420C2×D4—44no—
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×D1214yes—
432C22×C4—22no—
433C22×C4—34no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—64no—
439C22×C4—124no—
440C22×C4—122no—
441C22×C4—122no—
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—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—82no—
491C3×C4#39214yes—
492C3×C4C2×#12011yescharacteristic
493C3×C4C2×#12011yescharacteristic
494C3×C4C2×#12111yescharacteristic
495C3×C4C2×#12111yescharacteristic
496Dic3—44no—
497Dic3—42no—
498Dic3—44no—
499C23—24no—
500C23—24no—
501C23—416no—
502C23—48no—
503C23—42no—
504C23—48no—
505C23—42no—
506C23—62no—
507C23—88no—
508C23—88no—
509C23—121no—
510C23—124no—
511C23—121no—
512C23#88212yes—
513C23#88412yes—
514C23#94012yes—
515C23C2×D3×D411yescharacteristic
516Q8—62no—
517Q8—64no—
518Q8—64no—
519Q8—62no—
520Q8—128no—
521Q8—122no—
522Q8—121no—
523Q8—122no—
524Q8—121no—
525D4—24no—
526D4—24no—
527D4—48no—
528D4—42no—
529D4—42no—
530D4—62no—
531D4—62no—
532D4—128no—
533D4—121no—
534D4—121no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—128no—
543C2×C4—128no—
544C2×C4—121no—
545C2×C4—122no—
546C2×C4—121no—
547C2×C4—122no—
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×C3—84no—
566C2×C3C2×#39211yescharacteristic
567C2×C3#300512yes—
568C2×C3#304012yes—
569C2×C3#315312yes—
570D3—82no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—48no—
575C22—44no—
576C22—816no—
577C22—84no—
578C22—84no—
579C22—124no—
580C22—122no—
581C22—242no—
582C22—242no—
583C22#496014yes—
584C22C2×#88211yescharacteristic
585C22C2×#88411yescharacteristic
586C22#577314yes—
587C22C2×#94011yescharacteristic
588C4—82no—
589C4—124no—
590C4—122no—
591C4—124no—
592C4#496014yes—
593C4C2×#88211yescharacteristic
594C4C2×#88411yescharacteristic
595C4C2×#93011yescharacteristic
596C4C2×#93711yescharacteristic
597C3#2343111yescharacteristic, complemented, Sylow
598C2—24no—
599C2—84no—
600C2—242no—
601C2#7892712yes—
602C2#7894312yes—
603C2C2×#496011yescharacteristic
604C2#8393812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenterC2×#418D3
hypocenterC3#23431
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
1. C22×#936C2semidirect
3. C22×#909C2semidirect
4. C2×C3×#418C2semidirect
5. #82228C2semidirect
6. C2×#5004C2semidirect
7. #71979C2semidirect
8. #70215C2semidirect
9. #70214C2semidirect
31. C22×Dic12C22semidirect
32. C22×#204C22semidirect
36. #4988C22semidirect
37. #4973C22semidirect
38. C2×#810C22semidirect
41. #4550C22semidirect
46. C2×#418D3semidirect
597. C3#23431semidirect

Maximal subgroups

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

subgroupindex
C22×#9362
C22×#9322
C22×#9092
C2×C3×#4182
#822282
C2×#50042
#719792
#702152
#702142
#234313