Finite Group Databasebeta

← #10804142

Subgroups — #10804142

Order 768.

1622 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5493C211yesmaximal, characteristic
2#73849C211yesmaximal, characteristic, complemented
3#73819C211yesmaximal, characteristic, complemented
4C3×#1819C211yesmaximal, characteristic, complemented
5#71542C211yesmaximal, characteristic, complemented
6#69498C211yesmaximal, characteristic, complemented
7#69498C211yesmaximal, characteristic, complemented
8#68970C214yesmaximal, complemented
9#68954C214yesmaximal, complemented
10#21749—31nomaximal, Sylow
11C23×#67C2211yescharacteristic
12C23×#67C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C2×#930C414yescomplemented
15#5493—24no—
16#5493—24no—
17C2×#886C412yescomplemented
18C2×#886C2212yes—
19C2×#882C412yescomplemented
20C2×#882C2212yes—
21C2×D3×C22⋊C4C2212yescomplemented
22C2×#877C2212yescomplemented
23C3×#379C2211yescharacteristic
24C3×#379C2211yescharacteristic
25#5247C2211yescharacteristic
26#5247C2211yescharacteristic
27#4957—24no—
28#4956—24no—
29#4955C2212yescomplemented
30C2×#806C2214yes—
31C2×#806C2214yes—
32#4943C2212yescomplemented
33C3×#342C2214yescomplemented
34#4526C2214yescomplemented
35C2×#534—31no—
36#2309—31no—
37#2308—31no—
38#1899—34no—
39#1898—34no—
40#1819—31no—
41#1819D311yescharacteristic, complemented
42#1781—31no—
43#1781—31no—
44C25×C3C2311yescharacteristic
45C24×D3C2×C411yescharacteristic, complemented
46C24×D3C2311yescharacteristic
47C23×C3×C4C2×C412yescomplemented
48C22×#67—22no—
49C22×#67—22no—
50C22×#67—44no—
51C22×#67—44no—
52C22×#67C2×C412yes—
53C22×#67C2×C412yes—
54C22×#67C2×C412yescomplemented
55C22×#67D414yes—
56C22×#67D414yes—
57C22×#67C2312yes—
58C23×Dic3C2×C411yescharacteristic, complemented
59C23×Dic3C2311yescharacteristic
60C22×D12C2×C414yes—
61C22×C4×D3—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4C2×C412yescomplemented
67C2×C3×C22⋊C4C2312yes—
68C2×C3×C22⋊C4C2312yes—
69C2×C3×C22⋊C4C2312yes—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217C2312yes—
73C2×#217C2312yes—
74#930—28no—
75#930—28no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4C2×C414yes—
79C2×(C2×D3)⋊C4C2×C414yescomplemented
80C2×C4⋊Dic3C2×C414yescomplemented
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—22no—
83#886—28no—
84#886—24no—
85#886—24no—
86#882—24no—
87#882—24no—
88#882—28no—
89D3×C22⋊C4—44no—
90#877—44no—
91C3⋊#95—24no—
92#806—44no—
93#806—44no—
94C23×D4—31no—
95C23×D4—31no—
96#534—64no—
97#534—64no—
98C2×#123—32no—
99C2×#123—32no—
100C2×#121—34no—
101C2×#120—32no—
102C2×#120—32no—
103C2×#117—32no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4C2×D311yescharacteristic
106C2×#99—34no—
107C2×#99—34no—
108#390—64no—
109#386—64no—
110#381—32no—
111#379—31no—
112#379—31no—
113#379—32no—
114#379C2×D311yescharacteristic
115#379C2×D311yescharacteristic
116#342—34no—
117#342C2×D314yescomplemented
118C24×C3—24no—
119C24×C3—82no—
120C24×C3C22×C411yescharacteristic
121C24×C3C22×C411yescharacteristic
122C24×C3C2×D412yes—
123C24×C3C2×D412yes—
124C24×C3C2411yescharacteristic
125C23×D3—24no—
126C23×D3—22no—
127C23×D3—42no—
128C23×D3—82no—
129C23×D3—82no—
130C23×D3C22×C412yes—
131C23×D3C22×C412yes—
132C23×D3C2×D412yes—
133C23×D3C2×D412yes—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—82no—
139C22×C3×C4C22×C414yes—
140C2×#67—24no—
141C2×#67—22no—
142C2×#67—22no—
143C2×#67—24no—
144C2×#67—24no—
145C2×#67—44no—
146C2×#67—44no—
147C2×#67—42no—
148C2×#67—42no—
149C2×#67—44no—
150C2×#67—44no—
151C2×#67—44no—
152C2×#67—44no—
153C2×#67—48no—
154C2×#67—48no—
155C2×#67C22⋊C418yes—
156C2×#67C22⋊C418yes—
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—42no—
162C22×Dic3C22×C412yes—
163C22×Dic3C22×C412yes—
164C22×Dic3C2×D412yes—
165C22×Dic3C2×D412yes—
166C2×D12—28no—
167C2×D12—84no—
168C2×C4×D3—44no—
169C2×C4×D3—84no—
170C3×C22⋊C4—24no—
171C3×C22⋊C4—22no—
172C3×C22⋊C4—22no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—82no—
176C3×C22⋊C4—82no—
177#217—24no—
178#217—24no—
179#217—82no—
180#217—82no—
181(C2×D3)⋊C4—28no—
182(C2×D3)⋊C4—28no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—84no—
185C4⋊Dic3—28no—
186Dic3⋊C4—44no—
187C4×Dic3—44no—
188C25—31no—
189C25—31no—
190C25C22×D311yescharacteristic
191C22×D4—32no—
192C22×D4—32no—
193C22×D4—34no—
194C22×D4—34no—
195C22×D4—32no—
196C22×D4—32no—
197C22×D4—34no—
198C22×D4—62no—
199C22×D4—62no—
200C22×D4—124no—
201C22×D4—124no—
202C23×C4—31no—
203C23×C4—31no—
204C23×C4—62no—
205C23×C4C4×D312yescomplemented
206#123—64no—
207#123—64no—
208#123—68no—
209#121—68no—
210#121—68no—
211#120—68no—
212#120—64no—
213#120—64no—
214#117—124no—
215C2×C4⋊C4—34no—
216C2×C4⋊C4—62no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—34no—
222C2×C22⋊C4—34no—
223C2×C22⋊C4—32no—
224C2×C22⋊C4—32no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—124no—
230C2×C22⋊C4C4×D312yescomplemented
231C2×C22⋊C4C22×D312yes—
232C2×C22⋊C4C22×D312yes—
233C2×C22⋊C4C22×D312yes—
234C2×C42—62no—
235#99—124no—
236#99—124no—
237#95—64no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—88no—
250C23×C3C2×C22⋊C412yes—
251C23×C3C2×C22⋊C412yes—
252C23×C3C23×C411yescharacteristic
253C23×C3C22×D411yescharacteristic
254C23×C3C22×D411yescharacteristic
255C23×C3H(5,𝔽2)14yes—
256C22×D3—22no—
257C22×D3—22no—
258C22×D3—24no—
259C22×D3—24no—
260C22×D3—22no—
261C22×D3—22no—
262C22×D3—22no—
263C22×D3—44no—
264C22×D3—44no—
265C22×D3—44no—
266C22×D3—44no—
267C22×D3—44no—
268C22×D3—82no—
269C22×D3—82no—
270C22×D3—88no—
271C22×D3—88no—
272C22×D3C2×C22⋊C414yes—
273C2×C3×C4—24no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—82no—
280C2×C3×C4—82no—
281#67—48no—
282#67—48no—
283#67—48no—
284#67—44no—
285#67—44no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—24no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—44no—
294C2×Dic3—44no—
295C2×Dic3—44no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3C2×C22⋊C414yes—
299D12—88no—
300C4×D3—88no—
301C24—24no—
302C24—32no—
303C24—32no—
304C24—32no—
305C24—32no—
306C24—64no—
307C24—62no—
308C24—82no—
309C24—122no—
310C24—242no—
311C24—242no—
312C24C2×C4×D311yescharacteristic
313C24C2×C4×D311yescharacteristic
314C24D3×D412yes—
315C24D3×D412yes—
316C24C23×D311yescharacteristic
317C2×D4—38no—
318C2×D4—38no—
319C2×D4—68no—
320C2×D4—64no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—64no—
324C2×D4—64no—
325C2×D4—124no—
326C2×D4—124no—
327C2×D4—124no—
328C2×D4—124no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—122no—
332C2×D4—122no—
333C2×D4—128no—
334C2×D4—128no—
335C2×D4—244no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—32no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—64no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—82no—
349C22×C4—124no—
350C22×C4—122no—
351C22×C4—244no—
352C22×C4C2×C4×D314yes—
353C4⋊C4—68no—
354C4⋊C4—124no—
355C22⋊C4—24no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—68no—
361C22⋊C4—68no—
362C22⋊C4—64no—
363C22⋊C4—64no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—124no—
367C22⋊C4—244no—
368C22⋊C4—242no—
369C22⋊C4—242no—
370C42—124no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3—84no—
384C22×C3—84no—
385C22×C3C22×C22⋊C411yescharacteristic
386C22×C3#51812yes—
387C22×C3#53412yes—
388C22×C3#53412yes—
389C2×D3—24no—
390C2×D3—28no—
391C2×D3—44no—
392C2×D3—44no—
393C2×D3—42no—
394C2×D3—44no—
395C2×D3—84no—
396C2×D3—84no—
397C2×D3—88no—
398C2×D3—82no—
399C2×D3—82no—
400C3×C4—44no—
401C3×C4—82no—
402C3×C4—82no—
403Dic3—24no—
404Dic3—42no—
405Dic3—82no—
406Dic3—82no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—34no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—62no—
420C23—62no—
421C23—64no—
422C23—64no—
423C23—62no—
424C23—62no—
425C23—62no—
426C23—88no—
427C23—124no—
428C23—124no—
429C23—124no—
430C23—124no—
431C23—124no—
432C23—242no—
433C23—242no—
434C23—248no—
435C23—248no—
436C23D3×C22⋊C412yes—
437C23D3×C22⋊C412yes—
438C23C22×C4×D311yescharacteristic
439C23C2×D3×D411yescharacteristic
440C23C2×D3×D411yescharacteristic
441C23#100914yes—
442D4—128no—
443D4—128no—
444D4—128no—
445D4—124no—
446D4—124no—
447D4—248no—
448C2×C4—24no—
449C2×C4—34no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—124no—
465C2×C4—124no—
466C2×C4—124no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—248no—
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3#230811yescharacteristic
476C2×C3#288312yes—
477D3—44no—
478D3—82no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—68no—
492C22—64no—
493C22—84no—
494C22—84no—
495C22—124no—
496C22—124no—
497C22—124no—
498C22—122no—
499C22—244no—
500C22—244no—
501C22—242no—
502C22—242no—
503C22—248no—
504C22C2×D3×C22⋊C411yescharacteristic
505C22#552112yes—
506C22#565312yes—
507C22#565312yes—
508C4—44no—
509C4—64no—
510C4—82no—
511C4—82no—
512C4—122no—
513C4—242no—
514C4—242no—
515C3#2174911yescharacteristic, complemented, Sylow
516C2—22no—
517C2—22no—
518C2—22no—
519C2—42no—
520C2—42no—
521C2—124no—
522C2—242no—
523C2#7411911yescharacteristic
524C2#7868812yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1819C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5493C2
cosocleC24×C3C24
hypercenter#1819D3
hypocenterC3#21749
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #73849C2semidirect
3. #73819C2semidirect
4. C3×#1819C2semidirect
5. #71542C2semidirect
6. #69498C2semidirect
7. #69498C2semidirect
8. #68970C2semidirect
9. #68954C2semidirect
14. C2×#930C4semidirect
17. C2×#886C4semidirect
19. C2×#882C4semidirect
21. C2×D3×C22⋊C4C22semidirect
22. C2×#877C22semidirect
29. #4955C22semidirect
32. #4943C22semidirect
33. C3×#342C22semidirect
34. #4526C22semidirect
41. #1819D3semidirect
45. C24×D3C2×C4semidirect
47. C23×C3×C4C2×C4semidirect
54. C22×#67C2×C4semidirect
58. C23×Dic3C2×C4semidirect
66. C2×C3×C22⋊C4C2×C4semidirect
79. C2×(C2×D3)⋊C4C2×C4semidirect
80. C2×C4⋊Dic3C2×C4semidirect
117. #342C2×D3semidirect
205. C23×C4C4×D3semidirect
230. C2×C22⋊C4C4×D3semidirect
515. C3#21749semidirect

Maximal subgroups

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

subgroupindex
C2×#54932
#738492
#738192
C3×#18192
#715422
#694982
#689702
#689542
#217493