Finite Group Databasebeta

← #10804113

Subgroups — #10804113

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5524C211yesmaximal, characteristic
2C2×#4727C212yesmaximal, complemented
3C3×#1818C211yesmaximal, characteristic, complemented
4#71584C211yesmaximal, characteristic, complemented
5#69955C212yesmaximal, complemented
6#68669C212yesmaximal, complemented
7#68669C212yesmaximal, complemented
8#68668C212yesmaximal, complemented
9#68668C212yesmaximal, complemented
10#21744—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2212yescomplemented
15C22×D3×C8C2212yescomplemented
16#5524—28no—
17C2×#897C412yes—
18C2×#895C412yes—
19C2×#886C412yes—
20C2×#882C412yes—
21C2×C4×D12C2212yes—
22C2×C4×D12C2212yes—
23C2×C3×C22⋊C8C2212yes—
24C2×#830C2212yes—
25C2×(C2×D3)⋊C8C2212yes—
26C2×(C2×D3)⋊C8C2212yes—
27C2×(C2×D3)⋊C8C2212yes—
28C2×(C2×D3)⋊C8C2212yes—
29C2×(C2×D3)⋊C8C2212yescomplemented
30C2×(C2×D3)⋊C8C2212yescomplemented
31#4727—28no—
32C3×#336C2211yescharacteristic, complemented
33C3×#336C2212yescomplemented
34C3×#336C2211yescharacteristic, complemented
35#4524C2211yescharacteristic, complemented
36#4524C2212yescomplemented
37#4524C2211yescharacteristic, complemented
38C2×#525—31no—
39C2×#408—32no—
40#1870—32no—
41#1870—32no—
42#1869—32no—
43#1869—32no—
44#1818—31no—
45#1818D311yescharacteristic, complemented
46#1779—32no—
47C24×D3C2×C411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×#67C2×C412yes—
50C23×Dic3C2×C411yescharacteristic
51C2×#235—41no—
52C2×#235—42no—
53C2×#235—41no—
54C2×#235D418yes—
55C22×D12C2×C412yes—
56C22×D12C2×C412yes—
57C22×D12C2×C411yescharacteristic
58C22×C4×D3—24no—
59C22×C4×D3—42no—
60C22×C4×D3—41no—
61C22×C4×D3—41no—
62C22×C4×D3C2×C412yes—
63C22×C4×D3C2312yes—
64C22×C4×D3C2312yes—
65C22×Dic6C2×C411yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×#117—24no—
69C2×C3×C4⋊C4C2×C412yes—
70C2×C3×C22⋊C4C2×C412yes—
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×(C2×D3)⋊C4C2×C412yes—
74C2×(C2×D3)⋊C4C2×C412yes—
75C2×(C2×D3)⋊C4C2×C412yes—
76C2×(C2×D3)⋊C4C2×C412yes—
77C2×C4⋊Dic3C2×C412yes—
78C2×C4⋊Dic3C2×C412yes—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81C2×ΓL(1,𝔽25)—24no—
82C2×ΓL(1,𝔽25)—42no—
83C2×ΓL(1,𝔽25)—42no—
84C2×D3×C8—24no—
85C2×D3×C8—42no—
86C2×D3×C8—42no—
87#897—28no—
88#895—28no—
89#886—28no—
90#882—28no—
91C4×D12—24no—
92C4×D12—24no—
93C4×D12—24no—
94C4×D12—24no—
95C3×C22⋊C8—24no—
96#830—24no—
97(C2×D3)⋊C8—24no—
98(C2×D3)⋊C8—24no—
99(C2×D3)⋊C8—42no—
100(C2×D3)⋊C8—42no—
101(C2×D3)⋊C8—42no—
102(C2×D3)⋊C8—42no—
103C22×#41—31no—
104C24×C4—31no—
105C22×M16—32no—
106C23×C8—32no—
107#525—68no—
108C2×#123—32no—
109C2×C22⋊Q8—32no—
110C2×#121—32no—
111C2×#120—32no—
112C2×C4×D4—32no—
113C2×C4×D4—32no—
114C2×#117C2×D311yescharacteristic
115#408—68no—
116C2×C22⋊C8—32no—
117C2×C22⋊C8—32no—
118C2×C22⋊C8—32no—
119C2×C22⋊C8—32no—
120C2×C22⋊C8—32no—
121C2×C22⋊C8—32no—
122C2×C22⋊C8—32no—
123C2×C22⋊C8C2×D312yes—
124#336—31no—
125#336—32no—
126#336—31no—
127#336C2×D311yescharacteristic, complemented
128#336C2×D312yescomplemented
129#336C2×D311yescharacteristic, complemented
130C24×C3C22×C411yescharacteristic
131C23×D3—22no—
132C23×D3—24no—
133C23×D3—41no—
134C23×D3—41no—
135C23×D3—42no—
136C23×D3C22×C412yes—
137C23×D3C22×C412yes—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4C22×C412yes—
142C22×C3×C4C22×C412yes—
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D414yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—42no—
147C2×#67—42no—
148C2×#67C22⋊C418yes—
149C22×Dic3—22no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3C22×C412yes—
153C22×Dic3C22×C412yes—
154#235—48no—
155#235—48no—
156C2×D12—24no—
157C2×D12—24no—
158C2×D12—24no—
159C2×D12—24no—
160C2×D12—42no—
161C2×D12—42no—
162C2×D12—41no—
163C2×D12—41no—
164C2×D12C22⋊C414yes—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—24no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—42no—
177C2×C4×D3—42no—
178C2×C4×D3—48no—
179C2×C4×D3—48no—
180C2×C4×D3C2×D414yes—
181C2×C4×D3C2×D414yes—
182C2×Dic6—41no—
183C2×Dic6—41no—
184C2×Dic6C22⋊C414yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C42—22no—
194C3×C42—22no—
195(C2×D3)⋊C4—24no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4—24no—
199C4⋊Dic3—24no—
200C4⋊Dic3—24no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207ΓL(1,𝔽25)—48no—
208D3×C8—48no—
209C25—31no—
210C2×#41—38no—
211C2×#41—121no—
212C2×#41—122no—
213C2×#41—121no—
214C22×Q8—31no—
215C22×D4—32no—
216C22×D4—31no—
217C22×D4—32no—
218C22×D4—32no—
219C23×C4—32no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—32no—
223C23×C4—64no—
224C23×C4—122no—
225C23×C4—121no—
226C23×C4—121no—
227C23×C4C22×D311yescharacteristic
228C2×M16—64no—
229C2×M16—122no—
230C2×M16—122no—
231C22×C8—32no—
232C22×C8—32no—
233C22×C8—64no—
234C22×C8—122no—
235C22×C8—122no—
236C22×C8C22×D312yes—
237C22×C8C22×D312yes—
238#123—68no—
239C22⋊Q8—68no—
240#121—68no—
241#120—68no—
242C4×D4—64no—
243C4×D4—64no—
244C4×D4—64no—
245C4×D4—64no—
246#117—24no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4C4×D312yes—
250C2×C22⋊C4—32no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—32no—
254C2×C22⋊C4C4×D312yes—
255C2×C42C22×D311yescharacteristic
256C2×C42C22×D311yescharacteristic
257C22⋊C8—24no—
258C22⋊C8—64no—
259C22⋊C8—64no—
260C22⋊C8—64no—
261C22⋊C8—122no—
262C22⋊C8—122no—
263C22⋊C8—122no—
264C22⋊C8—122no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3C2×C22⋊C412yes—
270C23×C3C23×C411yescharacteristic
271C22×D3—22no—
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—44no—
284C22×D3—48no—
285C22×D3—48no—
286C22×D3—44no—
287C22×D3—44no—
288C22×D3C2×C22⋊C414yes—
289C22×D3#13114yes—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4C2×C22⋊C412yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304C2×C3×C4H(5,𝔽2)12yes—
305#67—48no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—44no—
312C2×Dic3C2×C22⋊C414yes—
313D12—44no—
314D12—48no—
315D12—48no—
316C4×D3—24no—
317C4×D3—24no—
318C4×D3—24no—
319C4×D3—24no—
320C4×D3—48no—
321C4×D3—42no—
322C4×D3—42no—
323C4×D3—42no—
324C4×D3—42no—
325Dic6—44no—
326C3×C8—42no—
327C3×C8—42no—
328C3⋊C8—42no—
329C3⋊C8—42no—
330C24—32no—
331C24—32no—
332C24—64no—
333C24—62no—
334C24—121no—
335C24—121no—
336C24—122no—
337C24C2×C4×D311yescharacteristic
338#41—128no—
339#41—128no—
340C2×Q8—34no—
341C2×Q8—121no—
342C2×Q8—121no—
343C2×D4—38no—
344C2×D4—34no—
345C2×D4—64no—
346C2×D4—64no—
347C2×D4—64no—
348C2×D4—64no—
349C2×D4—121no—
350C2×D4—121no—
351C2×D4—122no—
352C2×D4—122no—
353C2×D4—122no—
354C2×D4—122no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—32no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—32no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—64no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—122no—
375C22×C4—122no—
376C22×C4—128no—
377C22×C4—128no—
378C22×C4—121no—
379C22×C4—121no—
380C22×C4C2×C4×D312yes—
381C22×C4C2×C4×D312yes—
382C22×C4C2×C4×D311yescharacteristic
383C22×C4D3×D414yes—
384C22×C4C23×D311yescharacteristic
385M16—128no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—128no—
399C4⋊C4—24no—
400C4⋊C4—64no—
401C4⋊C4—64no—
402C22⋊C4—24no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C42—22no—
408C42—22no—
409C22×C3—22no—
410C22×C3—24no—
411C22×C3—42no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3C22×C22⋊C411yescharacteristic
415C22×C3#51811yescharacteristic
416C22×C3#51811yescharacteristic
417C22×C3C2×#13112yes—
418C22×C3#56812yes—
419C2×D3—24no—
420C2×D3—24no—
421C2×D3—24no—
422C2×D3—24no—
423C2×D3—24no—
424C2×D3—24no—
425C2×D3—416no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C3×C4—22no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4#53412yes—
438C3×C4#53412yes—
439Dic3—24no—
440Dic3—42no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—34no—
446C23—34no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23—128no—
460C23—124no—
461C23—124no—
462C23—128no—
463C23—124no—
464C23D3×C22⋊C412yes—
465C23C22×C4×D311yescharacteristic
466Q8—124no—
467D4—124no—
468D4—128no—
469D4—128no—
470D4—128no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—34no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—128no—
492C2×C4—124no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4D3×C22⋊C412yes—
498C2×C4C2×D3×D412yes—
499C2×C4#100911yescharacteristic
500C2×C4#100911yescharacteristic
501C2×C4#100912yes—
502C8—42no—
503C8—42no—
504C8—122no—
505C8—122no—
506C2×C3—42no—
507C2×C3#230811yescharacteristic
508C2×C3#288012yes—
509C2×C3#297612yes—
510C2×C3#297711yescharacteristic
511C2×C3#297811yescharacteristic
512D3—44no—
513D3—44no—
514C22—24no—
515C22—22no—
516C22—42no—
517C22—41no—
518C22—41no—
519C22—64no—
520C22—64no—
521C22—64no—
522C22—64no—
523C22—64no—
524C22—64no—
525C22—1216no—
526C22—122no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22C2×D3×C22⋊C411yescharacteristic
535C22#552111yescharacteristic
536C22#552111yescharacteristic
537C22D3×#13112yes—
538C22#575012yes—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—64no—
543C4—122no—
544C4#565312yes—
545C4#565312yes—
546C3#2174411yescharacteristic, complemented, Sylow
547C2—42no—
548C2—124no—
549C2—124no—
550C2#7411911yescharacteristic
551C2#7866312yes—
552C2#7937612yes—
553C2#7937711yescharacteristic
554C2#7937811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1818C2
prime radicalC2×#5524C2
cosocleC22×C3×C4C24
hypercenter#1818D3
hypocenterC3#21744
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. C2×#4727C2semidirect
3. C3×#1818C2semidirect
4. #71584C2semidirect
5. #69955C2semidirect
6. #68669C2semidirect
7. #68669C2semidirect
8. #68668C2semidirect
9. #68668C2semidirect
14. C22×ΓL(1,𝔽25)C22semidirect
15. C22×D3×C8C22semidirect
29. C2×(C2×D3)⋊C8C22semidirect
30. C2×(C2×D3)⋊C8C22semidirect
32. C3×#336C22semidirect
33. C3×#336C22semidirect
34. C3×#336C22semidirect
35. #4524C22semidirect
36. #4524C22semidirect
37. #4524C22semidirect
45. #1818D3semidirect
127. #336C2×D3semidirect
128. #336C2×D3semidirect
129. #336C2×D3semidirect
546. C3#21744semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
C2×#47272
C3×#18182
#715842
#699552
#686692
#686682
#217443