Finite Group Databasebeta

← #10804089

Subgroups — #10804089

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic
2C2×#4726C211yesmaximal, characteristic, complemented
3C2×D3×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×#1817C211yesmaximal, characteristic, complemented
5#71531C211yesmaximal, characteristic, complemented
6#69955C211yesmaximal, characteristic, complemented
7#69955C211yesmaximal, characteristic, complemented
8#68669C214yesmaximal, complemented
9#68668C214yesmaximal, complemented
10#21740—31nomaximal, Sylow
11C23×D12C411yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C23×C4×D3C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C411yescharacteristic
16C22×(C2×D3)⋊C4C411yescharacteristic
17C22×C4⋊Dic3C411yescharacteristic
18C22×ΓL(1,𝔽25)C2212yescomplemented
19C22×D3×C8C2212yescomplemented
20C2×C4×D12—24no—
21C2×C4×D12—24no—
22C2×C4×D12C412yes—
23C2×C4×D12C412yes—
24C2×C4×D12C2212yes—
25C2×C4×D12C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×(C2×D3)⋊C8C2212yescomplemented
31C2×(C2×D3)⋊C8C2212yescomplemented
32C2×(C2×D3)⋊C8C2214yescomplemented
33C2×(C2×D3)⋊C8C2214yescomplemented
34#4726—24no—
35D3×C22⋊C8—24no—
36C3×#336C2214yescomplemented
37#4524C2214yescomplemented
38C22×C4×D4—31no—
39C2×#407—31no—
40C22×C22⋊C8—31no—
41#1870—34no—
42#1869—34no—
43#1817—31no—
44#1817D311yescharacteristic, complemented
45#1779—31no—
46#1779—31no—
47C24×D3C2×C411yescharacteristic
48C24×D3C2×C411yescharacteristic
49C23×C3×C4C2×C411yescharacteristic
50C23×C3×C4C2×C411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2×C411yescharacteristic
53C23×Dic3C2×C411yescharacteristic
54C22×D12—24no—
55C22×D12—28no—
56C22×D12C2×C412yes—
57C22×D12C2×C412yes—
58C22×C4×D3—22no—
59C22×C4×D3—22no—
60C22×C4×D3—44no—
61C22×C4×D3—44no—
62C22×C4×D3C2×C412yes—
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3C2×C411yescharacteristic
65C22×C4×D3D414yes—
66C22×C4×D3D414yes—
67C22×C4×D3C2312yes—
68C22×C4×D3C2311yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42C2×C412yes—
75C2×C3×C42C2312yes—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4C2×C412yes—
79C2×(C2×D3)⋊C4C2×C412yes—
80C2×(C2×D3)⋊C4C2×C412yes—
81C2×(C2×D3)⋊C4C2×C412yes—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3C2×C412yes—
84C2×C4⋊Dic3C2×C412yes—
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2312yes—
87C2×ΓL(1,𝔽25)—24no—
88C2×ΓL(1,𝔽25)—44no—
89C2×D3×C8—24no—
90C2×D3×C8—44no—
91C4×D12—28no—
92C4×D12—28no—
93C4×D12—24no—
94C4×D12—24no—
95C4×D12—24no—
96C4×D12—24no—
97C3×C22⋊C8—22no—
98C3×C22⋊C8—22no—
99#830—22no—
100#830—22no—
101(C2×D3)⋊C8—24no—
102(C2×D3)⋊C8—24no—
103(C2×D3)⋊C8—44no—
104(C2×D3)⋊C8—44no—
105C23×D4—31no—
106C24×C4—31no—
107C24×C4—31no—
108C22×M16—32no—
109C23×C8—32no—
110C2×C4×D4—32no—
111C2×C4×D4—32no—
112C2×C4×D4—32no—
113C2×C4×D4—32no—
114C2×C4×D4—64no—
115C2×C4×D4—64no—
116C22×C4⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—31no—
119C22×C42C2×D311yescharacteristic
120#407—64no—
121C2×C22⋊C8—31no—
122C2×C22⋊C8—31no—
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8—34no—
126C2×C22⋊C8—34no—
127C2×C22⋊C8—64no—
128C2×C22⋊C8C2×D311yescharacteristic
129C2×C22⋊C8C2×D311yescharacteristic
130#336—34no—
131#336C2×D314yescomplemented
132C24×C3C22×C411yescharacteristic
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—24no—
137C23×D3—24no—
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—42no—
141C23×D3—42no—
142C23×D3C22⋊C414yes—
143C23×D3C22×C411yescharacteristic
144C23×D3C22×C412yes—
145C23×D3C22×C411yescharacteristic
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C22×C411yescharacteristic
154C22×C3×C4C22×C412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—42no—
160C22×Dic3—42no—
161C22×Dic3C22⋊C414yes—
162C22×Dic3C22×C412yes—
163C22×Dic3C22×C411yescharacteristic
164C22×Dic3C22×C411yescharacteristic
165C2×D12—232no—
166C2×D12—24no—
167C2×D12—24no—
168C2×D12—24no—
169C2×D12—24no—
170C2×D12—24no—
171C2×D12M1618yescomplemented
172C2×C4×D3—24no—
173C2×C4×D3—24no—
174C2×C4×D3—24no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—24no—
178C2×C4×D3—24no—
179C2×C4×D3—48no—
180C2×C4×D3—48no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×C4×D3—48no—
186C2×C4×D3—48no—
187C2×C4×D3C22⋊C414yes—
188C2×C4×D3C22⋊C414yes—
189C2×C4×D3C2×D412yes—
190C2×C4×D3C2×D412yes—
191C2×C4×D3C2×D412yes—
192C2×C4×D3C2×D412yes—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—24no—
196C2×C3×C8—24no—
197C3×C42—24no—
198C3×C42—22no—
199C3×C42—22no—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—24no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—24no—
209C2×C3⋊C8—24no—
210ΓL(1,𝔽25)—48no—
211D3×C8—48no—
212C25—31no—
213C25—31no—
214C22×D4—32no—
215C22×D4—32no—
216C22×D4—68no—
217C22×D4—64no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—34no—
224C23×C4—34no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—62no—
229C23×C4—62no—
230C23×C4—124no—
231C23×C4—124no—
232C23×C4C4×D311yescharacteristic
233C23×C4C4×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C2×M16—64no—
236C2×M16—124no—
237C22×C8—32no—
238C22×C8—32no—
239C22×C8—64no—
240C22×C8—124no—
241C22×C8C22×D312yes—
242C22×C8C22×D312yes—
243C4×D4—68no—
244C4×D4—68no—
245C4×D4—64no—
246C4×D4—64no—
247C4×D4—64no—
248C4×D4—64no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—64no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—32no—
254C2×C22⋊C4—32no—
255C2×C22⋊C4—32no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C4×D312yes—
261C2×C42C22×D312yes—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—64no—
265C22⋊C8—64no—
266C22⋊C8—62no—
267C22⋊C8—62no—
268C22⋊C8—124no—
269C22⋊C8—124no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3C2×C22⋊C412yes—
273C23×C3#13112yes—
274C23×C3#13112yes—
275C23×C3C23×C411yescharacteristic
276C22×D3—22no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—28no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—21no—
287C22×D3—22no—
288C22×D3—21no—
289C22×D3—22no—
290C22×D3—28no—
291C22×D3—28no—
292C22×D3—44no—
293C22×D3—48no—
294C22×D3—48no—
295C22×D3—48no—
296C22×D3—48no—
297C22×D3—44no—
298C22×D3—44no—
299C22×D3C2×C22⋊C412yes—
300C22×D3C2×C22⋊C412yes—
301C22×D3C2×M1614yes—
302C22×D3C2×M1614yes—
303C2×C3×C4—28no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—44no—
313C2×C3×C4C2×C22⋊C412yes—
314C2×C3×C4C2×M1614yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4H(5,𝔽2)14yes—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—44no—
324C2×Dic3—44no—
325C2×Dic3C2×C22⋊C412yes—
326C2×Dic3C2×C22⋊C412yes—
327D12—216no—
328D12—216no—
329C4×D3—24no—
330C4×D3—24no—
331C4×D3—24no—
332C4×D3—24no—
333C4×D3—44no—
334C4×D3—44no—
335C4×D3—48no—
336C4×D3—44no—
337C4×D3—44no—
338C3×C8—42no—
339C3×C8—42no—
340C3⋊C8—42no—
341C3⋊C8—42no—
342C24—31no—
343C24—31no—
344C24—32no—
345C24—34no—
346C24—64no—
347C24—64no—
348C24—62no—
349C24—62no—
350C24—62no—
351C24—62no—
352C24—62no—
353C24—122no—
354C24—122no—
355C24C2×C4×D311yescharacteristic
356C2×D4—38no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—632no—
362C2×D4—64no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—34no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—32no—
379C22×C4—64no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—64no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—128no—
388C22×C4—128no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4—128no—
394C22×C4—128no—
395C22×C4—122no—
396C22×C4—122no—
397C22×C4C2×C4×D312yes—
398C22×C4C2×C4×D311yescharacteristic
399C22×C4C2×C4×D312yes—
400C22×C4D3×D412yes—
401C22×C4D3×D412yes—
402C22×C4C23×D311yescharacteristic
403M16—128no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—64no—
411C2×C8—64no—
412C2×C8—128no—
413C4⋊C4—64no—
414C4⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C42—24no—
420C42—22no—
421C42—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3#40814yes—
427C22×C3C22×C22⋊C411yescharacteristic
428C22×C3#51812yes—
429C22×C3C22×M1612yes—
430C22×C3C2×#13111yescharacteristic
431C22×C3C2×#13111yescharacteristic
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—24no—
435C2×D3—24no—
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—24no—
439C2×D3—22no—
440C2×D3—24no—
441C2×D3—24no—
442C2×D3—416no—
443C2×D3—416no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3#40714yes—
451C3×C4—22no—
452C3×C4—24no—
453C3×C4—42no—
454C3×C4#53412yes—
455C3×C4#53412yes—
456Dic3—22no—
457Dic3—22no—
458Dic3—42no—
459C23—22no—
460C23—22no—
461C23—32no—
462C23—34no—
463C23—32no—
464C23—34no—
465C23—68no—
466C23—68no—
467C23—61no—
468C23—61no—
469C23—62no—
470C23—61no—
471C23—62no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—61no—
476C23—62no—
477C23—68no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—124no—
482C23—128no—
483C23—128no—
484C23—128no—
485C23—128no—
486C23—124no—
487C23—124no—
488C23D3×C22⋊C412yes—
489C23#90112yes—
490C23#90712yes—
491C23C22×C4×D311yescharacteristic
492D4—616no—
493D4—616no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—44no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—124no—
516C2×C4—124no—
517C2×C4—128no—
518C2×C4—124no—
519C2×C4—124no—
520C2×C4—124no—
521C2×C4—124no—
522C2×C4D3×C22⋊C412yes—
523C2×C4D3×M1614yes—
524C2×C4C2×D3×D411yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4#100914yes—
527C8—42no—
528C8—42no—
529C8—122no—
530C8—122no—
531C2×C3—24no—
532C2×C3#230811yescharacteristic
533C2×C3C2×#40711yescharacteristic
534C2×C3C2×#40811yescharacteristic
535C2×C3#297412yes—
536C2×C3#297412yes—
537D3—24no—
538D3—42no—
539D3—44no—
540C22—22no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—34no—
545C22—62no—
546C22—62no—
547C22—64no—
548C22—64no—
549C22—62no—
550C22—64no—
551C22—62no—
552C22—62no—
553C22—64no—
554C22—64no—
555C22—1216no—
556C22—1216no—
557C22—122no—
558C22—122no—
559C22—122no—
560C22—122no—
561C22—122no—
562C22—122no—
563C22#472714yes—
564C22C2×D3×C22⋊C411yescharacteristic
565C22#552112yes—
566C22C2×#90111yescharacteristic
567C22C2×D3×M1612yes—
568C22C2×#90711yescharacteristic
569C4—24no—
570C4—22no—
571C4—42no—
572C4—62no—
573C4—62no—
574C4—122no—
575C4#565312yes—
576C4#565312yes—
577C3#2174011yescharacteristic, complemented, Sylow
578C2—24no—
579C2—64no—
580C2—122no—
581C2—124no—
582C2#7411911yescharacteristic
583C2C2×#472711yescharacteristic
584C2D3×#40711yescharacteristic
585C2#7928712yes—
586C2#7933612yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1817C2
prime radicalC23×D12C4
cosocleC22×C3×C4C24
hypercenter#1817D3
hypocenterC3#21740
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#4726C2semidirect
3. C2×D3×C22⋊C8C2semidirect
4. C3×#1817C2semidirect
5. #71531C2semidirect
6. #69955C2semidirect
7. #69955C2semidirect
8. #68669C2semidirect
9. #68668C2semidirect
18. C22×ΓL(1,𝔽25)C22semidirect
19. C22×D3×C8C22semidirect
30. C2×(C2×D3)⋊C8C22semidirect
31. C2×(C2×D3)⋊C8C22semidirect
32. C2×(C2×D3)⋊C8C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
36. C3×#336C22semidirect
37. #4524C22semidirect
44. #1817D3semidirect
131. #336C2×D3semidirect
171. C2×D12M16semidirect
577. C3#21740semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C2×#47262
C2×D3×C22⋊C82
C3×#18172
#715312
#699552
#686692
#686682
#217403