Finite Group Databasebeta

← #10822346

Subgroups — #10822346

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×#137C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic
3C22×#947C211yesmaximal, characteristic, complemented
4#84086C211yesmaximal, characteristic, complemented
5C2×#5225C211yesmaximal, characteristic, complemented
6C2×#5131C211yesmaximal, characteristic, complemented
7#72072C214yesmaximal, complemented
8#72055C214yesmaximal, complemented
9#71979C211yesmaximal, characteristic, complemented
10#23431—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C23×D12C2211yescharacteristic
14C2×C3×#137—24no—
15C2×C3×#137—24no—
16C2×C3×#137—24no—
17C22×C3×Q16C2212yescomplemented
18C22×C3×SD16C2212yescomplemented
19C22×C3×M16C2211yescharacteristic
20C2×#949—24no—
21C2×#949—24no—
22C2×#949—24no—
23C2×#947—24no—
24C2×#947—24no—
25C2×#947—22no—
26C2×#947C2214yes—
27C22×#215C2212yes—
28C22×#213C2212yes—
29C22×(C2×D3)⋊C4—22no—
30C2×#926C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5229C2212yescomplemented
34#5225—24no—
35C2×#835C2214yescomplemented
36#5221C2212yescomplemented
37#5220—22no—
38C4×#217—22no—
39#5131—24no—
40C2×#821C2214yes—
41#4550C2214yescomplemented
42C22×#137D311yescharacteristic, complemented
43C22×AGL(1,ℤ/8)—31no—
44C22×#124—31no—
45C2×#419—31no—
46C2×#418—31no—
47#2297—31no—
48#2015—34no—
49#2012—34no—
50#1728—31no—
51C24×D3—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×Q8—22no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D414yes—
65C22×C3×Q8C2312yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C23×Dic3D412yes—
73C22×D12—22no—
74C22×D12—22no—
75C22×D12—22no—
76C22×D12—22no—
77C22×D12—21no—
78C22×D12—22no—
79C22×D12D414yes—
80C22×D12C2312yes—
81C3×#137—232no—
82C2×C3×Q16—24no—
83C2×C3×Q16—24no—
84C2×C3×Q16—24no—
85C2×C3×SD16—24no—
86C2×C3×SD16—24no—
87C2×C3×SD16—24no—
88C2×C3×M16—22no—
89C2×C3×M16—22no—
90C2×C3×M16D414yes—
91C22×C3×C8C2312yes—
92C2×#217—21no—
93C2×#217—24no—
94#949—232no—
95#947—216no—
96#947—28no—
97#947—28no—
98C2×#215—24no—
99C2×#215—24no—
100C2×#215—24no—
101C2×#213—24no—
102C2×#213—24no—
103C2×#213—24no—
104C2×(C2×D3)⋊C4—216no—
105C2×(C2×D3)⋊C4—28no—
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#835—28no—
119C3⋊#95—24no—
120#821—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8C2×D311yescharacteristic
123C23×D4—31no—
124C2×#137—24no—
125C2×#137—24no—
126C2×#137—24no—
127C2×AGL(1,ℤ/8)—64no—
128C2×AGL(1,ℤ/8)—64no—
129C2×AGL(1,ℤ/8)—64no—
130C22×Q16C2×D312yescomplemented
131C22×SD16—32no—
132C22×SD16C2×D312yescomplemented
133C22×D8—32no—
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—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—24no—
155C23×D3—22no—
156C23×D3—48no—
157C23×D3—42no—
158C23×D3—42no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—28no—
167C2×C3×Q8—28no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—44no—
177C2×C3×Q8—44no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D414yes—
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D412yes—
191C22×C3×C4—24no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—24no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—24no—
208C22×Dic3—22no—
209C22×Dic3—44no—
210C22×Dic3C2×D414yes—
211C2×D12—28no—
212C2×D12—28no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—24no—
218C2×D12—22no—
219C2×D12—44no—
220C2×D12—44no—
221C2×D12—42no—
222C2×D12—42no—
223C2×D12C2×D414yes—
224C3×Q16—216no—
225C3×SD16—216no—
226C3×M16—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8C2×D414yes—
230#217—48no—
231#217—42no—
232#215—216no—
233#213—216no—
234(C2×D3)⋊C4—232no—
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—61no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41#6714yes—
252C22×Q8—22no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8#6714yes—
259C22×Q8C22×D312yes—
260C22×Q8C22×D311yescharacteristic
261C22×D4—21no—
262C22×D4—34no—
263C22×D4—32no—
264C22×D4—62no—
265C22×D4—61no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4—62no—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—32no—
274C23×C4C22×D311yescharacteristic
275#137—232no—
276AGL(1,ℤ/8)—632no—
277C2×Q16—24no—
278C2×Q16—24no—
279C2×Q16—24no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
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—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—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—44no—
320C23×C3#12012yes—
321C23×C3#12112yes—
322C23×C3#12712yes—
323C23×C3C22×D411yescharacteristic
324C22×D3—24no—
325C22×D3—416no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—48no—
329C22×D3—48no—
330C22×D3—88no—
331C22×D3—88no—
332C3×Q8—22no—
333C3×Q8—24no—
334C3×Q8—22no—
335C3×Q8—24no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×Q8—41no—
339C3×Q8—41no—
340C3×Q8—48no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—48no—
344C3×D4—41no—
345C3×D4—41no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—48no—
353C2×C3×C4—48no—
354C2×C3×C4#12014yes—
355C2×C3×C4#12012yes—
356C2×C3×C4#12014yes—
357C2×C3×C4#12114yes—
358C2×C3×C4#12114yes—
359C2×C3×C4#12112yes—
360C2×C3×C4#12712yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—48no—
371C2×Dic3—42no—
372C2×Dic3—44no—
373D12—24no—
374D12—24no—
375D12—42no—
376D12—42no—
377D12—48no—
378C3×C8—24no—
379C3⋊C8—24no—
380C24—21no—
381C24—64no—
382C24—62no—
383C24—128no—
384C24—122no—
385C24—122no—
386C24D3×D411yescharacteristic
387#41—28no—
388#41—28no—
389#41—44no—
390#41—44no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8C2×#6714yes—
409C2×Q8C2×#6712yes—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—34no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—68no—
420C2×D4—64no—
421C2×D4—68no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—122no—
428C2×D4—124no—
429C2×D4—122no—
430C2×D4—124no—
431C2×D4C2×#6712yes—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—34no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—64no—
442C22×C4—62no—
443C22×C4—124no—
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—216no—
453SD16—216no—
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—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—82no—
479C22×C3—82no—
480C22×C3#39214yes—
481C22×C3C2×#12011yescharacteristic
482C22×C3C2×#12111yescharacteristic
483C22×C3C2×#12711yescharacteristic
484C22×C3#57514yes—
485C2×D3—44no—
486C2×D3—48no—
487C2×D3—816no—
488C2×D3—84no—
489C2×D3—84no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4#39214yes—
493C3×C4C2×#12011yescharacteristic
494C3×C4C2×#12011yescharacteristic
495C3×C4C2×#12111yescharacteristic
496C3×C4C2×#12111yescharacteristic
497Dic3—44no—
498Dic3—82no—
499C23—24no—
500C23—22no—
501C23—44no—
502C23—41no—
503C23—41no—
504C23—64no—
505C23—1216no—
506C23—128no—
507C23—128no—
508C23—122no—
509C23—122no—
510C23—248no—
511C23—248no—
512C23#93712yes—
513C23#93912yes—
514C23#94012yes—
515C23C2×D3×D411yescharacteristic
516Q8—22no—
517Q8—24no—
518Q8—24no—
519Q8—22no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—22no—
527D4—48no—
528D4—41no—
529D4—41no—
530D4—64no—
531D4—64no—
532D4—128no—
533D4—122no—
534D4—122no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—128no—
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×C3C2×#39211yescharacteristic
567C2×C3#300512yes—
568C2×C3#304012yes—
569C2×C3#315312yes—
570D3—84no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—82no—
577C22—82no—
578C22—128no—
579C22—124no—
580C22—2416no—
581C22—244no—
582C22—244no—
583C22#522314yes—
584C22#577714yes—
585C22C2×#93711yescharacteristic
586C22C2×#93911yescharacteristic
587C22C2×#94011yescharacteristic
588C4—42no—
589C4—44no—
590C4—124no—
591C4—242no—
592C4#522314yes—
593C4C2×#93711yescharacteristic
594C4C2×#93811yescharacteristic
595C4C2×#93911yescharacteristic
596C4C2×#95311yescharacteristic
597C3#2343111yescharacteristic, complemented, Sylow
598C2—24no—
599C2—82no—
600C2—244no—
601C2#8396812yes—
602C2#8397312yes—
603C2#8398012yes—
604C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#137C2
socleC23×C3C22×D4
prime radicalC22×#949C2
hypercenterC22×#137D3
hypocenterC3#23431
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C22×#947C2semidirect
4. #84086C2semidirect
5. C2×#5225C2semidirect
6. C2×#5131C2semidirect
7. #72072C2semidirect
8. #72055C2semidirect
9. #71979C2semidirect
17. C22×C3×Q16C22semidirect
18. C22×C3×SD16C22semidirect
33. #5229C22semidirect
35. C2×#835C22semidirect
36. #5221C22semidirect
41. #4550C22semidirect
42. C22×#137D3semidirect
130. C22×Q16C2×D3semidirect
132. C22×SD16C2×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×C3×#1372
C22×#9492
C22×#9472
#840862
C2×#52252
C2×#51312
#720722
#720552
#719792
#234313