Finite Group Databasebeta

← #10944004

Subgroups — #10944004

Order 768.

1868 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic, complemented
3C2×#5083C212yesmaximal, complemented
4C2×#5046C212yesmaximal
5#71692C212yesmaximal, complemented
6#71530C211yesmaximal, characteristic, complemented
7#69687C214yesmaximal, complemented
8#69584C212yesmaximal, complemented
9#32905—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C23×C4×D3—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic, complemented
13C2×C3×C22⋊Q8C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×#215C2214yes—
16C22×(C2×D3)⋊C4—21no—
17C22×C4⋊Dic3C2211yescharacteristic
18C2×C4×D12—21no—
19C2×C4×D12—21no—
20C2×C4×D12—28no—
21C2×C4×D12C2212yes—
22C2×C4×D12C2212yes—
23C3×#391C2212yescomplemented
24C3×#389—21no—
25C3×#389—21no—
26C3×#384C2211yescharacteristic, complemented
27C3×#380—21no—
28C3×#380—21no—
29C2×#835C2214yescomplemented
30C2×#830C2212yes—
31#5083—28no—
32#5046—28no—
33C2×#809C2214yes—
34C2×C4⋊(C3⋊C8)C2212yescomplemented
35#4529C2214yescomplemented
36#4524C2212yescomplemented
37C22×C4×D4—31no—
38C2×#461—32no—
39C2×#450—32no—
40#2857D311yescharacteristic, complemented
41#1877—34no—
42#1872—32no—
43#1802—32no—
44#1797—31no—
45C24×D3—21no—
46C22×C3×Q8C2312yes—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C22×D12—21no—
52C22×D12—22no—
53C22×D12—24no—
54C22×D12—24no—
55C22×D12D414yes—
56C22×D12C2312yes—
57C22×C4×D3—28no—
58C22×C4×D3—22no—
59C22×C4×D3—22no—
60C22×C4×D3—44no—
61C22×C4×D3—44no—
62C3×C4⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—22no—
72C2×C3×C42—24no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#215—28no—
78C2×#215—28no—
79C2×#215—28no—
80C2×(C2×D3)⋊C4—28no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3C2312yes—
86C22×C3⋊C8C2314yes—
87C4×D12—28no—
88C4×D12—28no—
89C4×D12—24no—
90C4×D12—24no—
91C4×D12—24no—
92C4×D12D418yes—
93C3×#95—22no—
94C3×#95—22no—
95#835—28no—
96#830—24no—
97#809—28no—
98C4⋊(C3⋊C8)—24no—
99C23×D4—31no—
100C24×C4—61no—
101C22×SD16—34no—
102C2×C4⋊Q8C2×D311yescharacteristic, complemented
103C2×C22⋊Q8C2×D312yes—
104C2×C4×D4—32no—
105C2×C4×D4—32no—
106C2×C4×D4—68no—
107C2×C4×D4—61no—
108C2×C4×D4—61no—
109C22×C4⋊C4—31no—
110C22×C22⋊C4—61no—
111C22×C42C2×D311yescharacteristic
112#461—68no—
113#450—68no—
114C2×C4⋊C8—32no—
115C2×#103—34no—
116C2×#102—34no—
117C2×C22⋊C8—32no—
118#391C2×D312yescomplemented
119#389—21no—
120#389—21no—
121#384C2×D311yescharacteristic, complemented
122#380—21no—
123#380—21no—
124#340—34no—
125#336—32no—
126C24×C3C2×D411yescharacteristic
127C23×D3—24no—
128C23×D3—24no—
129C23×D3—24no—
130C23×D3—22no—
131C23×D3—42no—
132C23×D3—42no—
133C23×D3—42no—
134C23×D3—42no—
135C2×C3×Q8—24no—
136C2×C3×Q8—22no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—41no—
148C22×C3×C4SD1614yescomplemented
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4112yes—
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—42no—
160C22×Dic3—42no—
161C2×D12—24no—
162C2×D12—24no—
163C2×D12—216no—
164C2×D12—24no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—216no—
169C2×D12C2×D414yes—
170C2×D12C2×D414yes—
171C2×D12#4114yes—
172C2×C4×D3—28no—
173C2×C4×D3—28no—
174C2×C4×D3—44no—
175C2×C4×D3—432no—
176C2×C4×D3—44no—
177C2×C4×D3—44no—
178C2×C4×D3—44no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—44no—
185C3×C4⋊C4—42no—
186C3×C22⋊C4—44no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D414yes—
191#215—416no—
192(C2×D3)⋊C4—28no—
193(C2×D3)⋊C4—28no—
194C4⋊Dic3—24no—
195C4⋊Dic3—22no—
196C4⋊Dic3C2×D414yes—
197C2×C3⋊C8—24no—
198C2×C3⋊C8—24no—
199C2×C3⋊C8—24no—
200C25—61no—
201C22×Q8C22×D312yes—
202C22×D4—34no—
203C22×D4—32no—
204C22×D4—64no—
205C22×D4—64no—
206C22×D4—62no—
207C22×D4—61no—
208C23×C4—68no—
209C23×C4—61no—
210C23×C4—62no—
211C23×C4—62no—
212C23×C4—124no—
213C23×C4—124no—
214C23×C4#6711yescharacteristic
215C23×C4#6711yescharacteristic
216C23×C4C22×D311yescharacteristic
217C2×SD16—68no—
218C2×SD16—68no—
219C2×SD16—68no—
220C22×C8—34no—
221C4⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C4×D4—38no—
225C4×D4—64no—
226C4×D4—64no—
227C4×D4—68no—
228C4×D4—68no—
229C4×D4—64no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—68no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C42—24no—
244C2×C42#6711yescharacteristic
245C2×C42#6711yescharacteristic
246C2×C42C22×D311yescharacteristic
247C2×C42C22×D311yescharacteristic
248C4⋊C8—64no—
249#103—68no—
250#102—68no—
251C22⋊C8—64no—
252#95—22no—
253#95—22no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3#12112yes—
259C23×C3C2×SD1612yes—
260C23×C3#13712yes—
261C23×C3C22×D411yescharacteristic
262C22×D3—216no—
263C22×D3—24no—
264C22×D3—24no—
265C22×D3—24no—
266C22×D3—48no—
267C22×D3—48no—
268C22×D3—48no—
269C22×D3—48no—
270C22×D3—48no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—48no—
276C3×Q8—44no—
277C3×Q8—44no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—28no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4#12112yes—
300C2×C3×C4C2×SD1612yes—
301C2×C3×C4C2×SD1612yes—
302C2×C3×C4AGL(1,ℤ/8)12yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D412yes—
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4112yes—
307C2×Dic3—24no—
308C2×Dic3—48no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311D12—216no—
312D12—24no—
313D12—24no—
314D12—24no—
315D12#12118yes—
316C4×D3—416no—
317C4×D3—416no—
318C3⋊C8—44no—
319C24—64no—
320C24—64no—
321C24—64no—
322C24—62no—
323C24—122no—
324C24—122no—
325C24—122no—
326C24—122no—
327C24C2×#6711yescharacteristic
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×D4—34no—
333C2×D4—34no—
334C2×D4—34no—
335C2×D4—616no—
336C2×D4—64no—
337C2×D4—64no—
338C2×D4—616no—
339C2×D4—64no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—68no—
353C22×C4—64no—
354C22×C4—62no—
355C22×C4—68no—
356C22×C4—124no—
357C22×C4—1232no—
358C22×C4—122no—
359C22×C4—122no—
360C22×C4—124no—
361C22×C4—124no—
362C22×C4—124no—
363C22×C4#21514yescomplemented
364C22×C4D3×D412yes—
365C22×C4#23712yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C23×D311yescharacteristic
372SD16—1216no—
373C2×C8—64no—
374C2×C8—64no—
375C2×C8—64no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—34no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—64no—
385C4⋊C4—62no—
386C22⋊C4—44no—
387C22⋊C4—68no—
388C22⋊C4—68no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42D3×D414yes—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3#46114yescomplemented
400C22×C3C2×#12011yescharacteristic
401C22×C3C2×#12111yescharacteristic
402C22×C3#52511yescharacteristic
403C22×C3C22×SD1611yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C22×C3#57511yescharacteristic
407C2×D3—28no—
408C2×D3—432no—
409C2×D3—44no—
410C2×D3—44no—
411C2×D3—44no—
412C2×D3—44no—
413C2×D3—44no—
414C2×D3—44no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—82no—
420C3×C4—82no—
421C3×C4#45014yescomplemented
422C3×C4C2×#12112yes—
423C3×C4#52512yes—
424Dic3—44no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—616no—
430C23—64no—
431C23—64no—
432C23—64no—
433C23—128no—
434C23—128no—
435C23—128no—
436C23—128no—
437C23—128no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23—128no—
443C23#93812yes—
444C23C2×#21512yes—
445C23#94212yes—
446C23C22×#6711yescharacteristic
447Q8—44no—
448Q8—44no—
449D4—38no—
450D4—616no—
451D4—64no—
452D4—64no—
453D4—64no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—64no—
474C2×C4—1216no—
475C2×C4—128no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—1216no—
479C2×C4#93712yes—
480C2×C4#93712yes—
481C2×C4#93812yes—
482C2×C4C2×#21512yes—
483C2×C4C2×#21512yes—
484C2×C4#94912yes—
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×D3×D412yes—
487C2×C4C2×#23711yescharacteristic
488C2×C4D3×#4112yes—
489C8—124no—
490C2×C3—24no—
491C2×C3#284711yescharacteristic
492C2×C3C2×#45011yescharacteristic
493C2×C3#300711yescharacteristic
494C2×C3C2×#46111yescharacteristic
495C2×C3#303611yescharacteristic
496C2×C3#310111yescharacteristic
497C2×C3#311011yescharacteristic
498D3—48no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—68no—
506C22—1232no—
507C22—124no—
508C22—124no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—124no—
513C22#508314yescomplemented
514C22C2×#93711yescharacteristic
515C22C2×#93811yescharacteristic
516C22C22×#21511yescharacteristic
517C22C2×#94211yescharacteristic
518C22C2×#94911yescharacteristic
519C22#582211yescharacteristic
520C22#582811yescharacteristic
521C4—22no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—82no—
526C4—82no—
527C4—124no—
528C4#504614yescomplemented
529C4D3×#12112yes—
530C4#560612yes—
531C3#3290511yescharacteristic, complemented, Sylow
532C2—24no—
533C2—128no—
534C2#8272311yescharacteristic
535C2C2×#504611yescharacteristic
536C2#8273511yescharacteristic
537C2C2×#508311yescharacteristic
538C2#8329711yescharacteristic
539C2#8335111yescharacteristic
540C2#8335311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#2857D3
hypocenterC3#32905
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C3×#2857C2semidirect
3. C2×#5083C2semidirect
5. #71692C2semidirect
6. #71530C2semidirect
7. #69687C2semidirect
8. #69584C2semidirect
12. C2×C3×C4⋊Q8C22semidirect
23. C3×#391C22semidirect
26. C3×#384C22semidirect
29. C2×#835C22semidirect
34. C2×C4⋊(C3⋊C8)C22semidirect
35. #4529C22semidirect
36. #4524C22semidirect
40. #2857D3semidirect
102. C2×C4⋊Q8C2×D3semidirect
118. #391C2×D3semidirect
121. #384C2×D3semidirect
148. C22×C3×C4SD16semidirect
363. C22×C4#215semidirect
399. C22×C3#461semidirect
421. C3×C4#450semidirect
513. C22#5083semidirect
528. C4#5046semidirect
531. C3#32905semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C3×#28572
C2×#50832
C2×#50462
#716922
#715302
#696872
#695842
#329053