Finite Group Databasebeta

← #10850463

Subgroups — #10850463

Order 768.

2090 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic, complemented
2C2×#5244C214yesmaximal, complemented
3#82653C212yesmaximal, complemented
4#82403C211yesmaximal, characteristic, complemented
5#71851C214yesmaximal, complemented
6#71686C212yesmaximal, complemented
7C4×#834C211yesmaximal, characteristic, complemented
8#25771—31nomaximal, Sylow
9C2×C3×H(5,𝔽2)C2212yes—
10C23×D12C2211yescharacteristic, complemented
11C3×#547—22no—
12C3×#546—24no—
13C3×D42—22no—
14C3×#537—21no—
15C3×#534—21no—
16C3×#534C2212yes—
17C3×#532—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#940—22no—
22C2×#937C2212yes—
23C2×#937C2218yescomplemented
24C2×#930—21no—
25#5244—216no—
26C2×#834C2212yes—
27C2×#834C2218yescomplemented
28#5212C2211yescharacteristic, complemented
29C4×#217C2212yescomplemented
30#4991—21no—
31#4988—22no—
32#4960—24no—
33#4960—21no—
34#4955C2212yescomplemented
35#4939—21no—
36#4526C2212yescomplemented
37#3496D311yescharacteristic, complemented
38C2×B(4,𝔽2)—34no—
39#2433—32no—
40#2408—31no—
41#2021—34no—
42#1964—32no—
43C4×#99—31no—
44C24×D3C2312yes—
45C3×H(5,𝔽2)—28no—
46C3×H(5,𝔽2)—44no—
47C2×C3×#41—22no—
48C2×C3×#41—24no—
49C22×C3×D4—22no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4D412yescomplemented
53C22×C3×D4C2314yes—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4D411yescharacteristic, complemented
56C23×C3×C4C2311yescharacteristic
57C22×#67—22no—
58C22×#67—28no—
59C22×D12—21no—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—44no—
64C3×C4⋊Q8—21no—
65C3×#127—21no—
66C3×#124—22no—
67C3×#123—24no—
68C3×#123—24no—
69C3×C22⋊Q8—24no—
70C3×#121—22no—
71C3×#121—22no—
72C3×#121—24no—
73C3×#121—22no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#120—24no—
80C3×#120—24no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D412yescomplemented
87C2×C3×C4⋊C4D411yescharacteristic, complemented
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4D411yescharacteristic, complemented
90C2×C3×C22⋊C4D414yescomplemented
91C2×C3×C22⋊C4C2311yescharacteristic
92C2×C3×C42D411yescharacteristic, complemented
93C2×#217C2312yes—
94C2×#217C2314yes—
95#940—48no—
96#937—28no—
97#937—216no—
98#937—48no—
99#930—44no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4—28no—
103C2×(C2×D3)⋊C4—22no—
104C2×C4⋊Dic3—21no—
105C2×C4×Dic3—22no—
106#834—216no—
107#834D418yescomplemented
108C3⋊#95—22no—
109C3⋊#95—21no—
110C2×H(5,𝔽2)C2×D312yes—
111C23×D4—31no—
112#547—22no—
113#546—24no—
114D42—22no—
115#537—21no—
116#534—21no—
117#534C2×D312yes—
118#532—21no—
119C2×#127—62no—
120C2×#121—21no—
121C2×#121—61no—
122C2×#121C2×D312yes—
123C2×#120—32no—
124C2×#120—38no—
125C2×C4×D4C2×D311yescharacteristic
126B(4,𝔽2)—616no—
127C2×#99—32no—
128C2×#99—38no—
129#394—61no—
130#392—61no—
131#392—64no—
132#390—62no—
133#390—61no—
134#380—31no—
135#379—32no—
136C4×C22⋊C4—32no—
137#342—32no—
138C24×C3—22no—
139C24×C3C2×D414yes—
140C24×C3C2×D411yescharacteristic
141C24×C3C2411yescharacteristic
142C23×D3—22no—
143C23×D3—24no—
144C23×D3—24no—
145C23×D3—24no—
146C23×D3—44no—
147C23×D3—82no—
148C3×#41—416no—
149C3×#41—44no—
150C3×#41—44no—
151C2×C3×Q8—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—28no—
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—28no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—44no—
165C2×C3×D4—416no—
166C2×C3×D4—44no—
167C2×C3×D4—44no—
168C2×C3×D4—82no—
169C2×C3×D4C2×D414yes—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C2×#67—416no—
185C2×#67—44no—
186C2×#67—416no—
187C2×#67—44no—
188C2×#67—88no—
189C2×#67—82no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C2×D12—24no—
193C2×D12—28no—
194C2×D12—44no—
195C2×D12—42no—
196C2×D12—44no—
197C2×D12—48no—
198C2×D12—41no—
199C2×D12—42no—
200C2×D12—41no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C42—21no—
213C3×C42—21no—
214#217—24no—
215#217—24no—
216#217C2×D418yescomplemented
217(C2×D3)⋊C4—416no—
218(C2×D3)⋊C4—44no—
219(C2×D3)⋊C4—44no—
220(C2×D3)⋊C4—82no—
221C4⋊Dic3—42no—
222C4×Dic3—44no—
223C25—32no—
224H(5,𝔽2)—28no—
225H(5,𝔽2)—44no—
226C2×#41—22no—
227C2×#41—24no—
228C22×D4—22no—
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—62no—
232C22×D4—68no—
233C22×D4—61no—
234C22×D4—62no—
235C22×D4—62no—
236C22×D4—61no—
237C22×D4—124no—
238C22×D4#6712yescomplemented
239C22×D4C22×D314yes—
240C22×D4C22×D311yescharacteristic
241C23×C4#6711yescharacteristic, complemented
242C23×C4C22×D311yescharacteristic
243C4⋊Q8—21no—
244#127—21no—
245#127—128no—
246#124—22no—
247#123—24no—
248#123—24no—
249C22⋊Q8—24no—
250#121—22no—
251#121—22no—
252#121—24no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—124no—
260#120—24no—
261#120—24no—
262#120—616no—
263#120—68no—
264#120—128no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4#6711yescharacteristic, complemented
272C2×C4⋊C4#6712yescomplemented
273C2×C22⋊C4—22no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—32no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4#6711yescharacteristic, complemented
281C2×C22⋊C4#6714yescomplemented
282C2×C22⋊C4C22×D311yescharacteristic
283C2×C42—62no—
284C2×C42#6711yescharacteristic, complemented
285#99—38no—
286#99—616no—
287#95—62no—
288#95—61no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—24no—
292C23×C3—28no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—82no—
301C23×C3—82no—
302C23×C3#12712yescomplemented
303C23×C3C22×D412yes—
304C23×C3C22×D411yescharacteristic
305C23×C3C22×D412yes—
306C23×C3H(5,𝔽2)12yes—
307C22×D3—24no—
308C22×D3—28no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—44no—
312C22×D3—42no—
313C22×D3—48no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—48no—
318C22×D3—88no—
319C22×D3—88no—
320C3×Q8—44no—
321C3×D4—28no—
322C3×D4—44no—
323C3×D4—48no—
324C3×D4—44no—
325C3×D4—44no—
326C3×D4—48no—
327C3×D4—44no—
328C3×D4—82no—
329C3×D4—82no—
330C3×D4—82no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—24no—
338C2×C3×C4—28no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—48no—
342C2×C3×C4—44no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—81no—
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12012yes—
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4#12712yescomplemented
351#67—832no—
352#67—88no—
353C2×Dic3—28no—
354C2×Dic3—44no—
355C2×Dic3—44no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359D12—44no—
360D12—48no—
361D12—44no—
362C24—22no—
363C24—62no—
364C24—64no—
365C24—64no—
366C24—64no—
367C24—124no—
368C24—242no—
369C24C2×#6714yes—
370C24C2×#6711yescharacteristic
371C24C23×D311yescharacteristic
372#41—416no—
373#41—44no—
374#41—44no—
375C2×Q8—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—416no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—64no—
393C2×D4—68no—
394C2×D4—82no—
395C2×D4—124no—
396C2×D4—122no—
397C2×D4—124no—
398C2×D4—128no—
399C2×D4—1216no—
400C2×D4—124no—
401C2×D4—1216no—
402C2×D4—124no—
403C2×D4—121no—
404C2×D4—122no—
405C2×D4—121no—
406C2×D4—248no—
407C2×D4—242no—
408C2×D4D3×D414yes—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—64no—
416C22×C4—62no—
417C22×C4C2×#6714yes—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6712yes—
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6712yes—
424C22×C4C2×#6711yescharacteristic
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—122no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—24no—
437C22⋊C4—38no—
438C22⋊C4—64no—
439C22⋊C4—64no—
440C22⋊C4—1216no—
441C22⋊C4—124no—
442C22⋊C4—124no—
443C22⋊C4—242no—
444C42—21no—
445C42—21no—
446C42—124no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—81no—
461C22×C3—88no—
462C22×C3—164no—
463C22×C3C2×#12011yescharacteristic
464C22×C3C2×#12711yescharacteristic
465C22×C3#53511yescharacteristic
466C22×C3D4214yes—
467C2×D3—416no—
468C2×D3—44no—
469C2×D3—44no—
470C2×D3—44no—
471C2×D3—88no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3—82no—
475C2×D3—82no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—44no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4B(4,𝔽2)12yescomplemented
483Dic3—48no—
484Dic3—82no—
485C23—21no—
486C23—21no—
487C23—24no—
488C23—28no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—64no—
497C23—68no—
498C23—82no—
499C23—82no—
500C23—122no—
501C23—122no—
502C23—124no—
503C23—122no—
504C23—128no—
505C23—128no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—248no—
510C23—248no—
511C23#94012yescomplemented
512C23C2×D3×D412yes—
513C23#100412yes—
514C23C22×#6712yes—
515C23C22×#6711yescharacteristic
516Q8—44no—
517D4—28no—
518D4—44no—
519D4—48no—
520D4—48no—
521D4—44no—
522D4—44no—
523D4—44no—
524D4—82no—
525D4—82no—
526D4—82no—
527D4—124no—
528D4—128no—
529D4—124no—
530D4—2432no—
531D4—248no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—68no—
548C2×C4—81no—
549C2×C4—124no—
550C2×C4—124no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4#94012yescomplemented
555C2×C4#95311yescharacteristic
556C2×C4#95312yes—
557C2×C4#95311yescharacteristic
558C2×C3—24no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—42no—
562C2×C3—84no—
563C2×C3#243311yescharacteristic
564C2×C3C2×B(4,𝔽2)11yescharacteristic
565C2×C3#302711yescharacteristic
566D3—84no—
567C22—21no—
568C22—22no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—81no—
581C22—88no—
582C22—1216no—
583C22—124no—
584C22—124no—
585C22—124no—
586C22—164no—
587C22—248no—
588C22—242no—
589C22—242no—
590C22—242no—
591C22—242no—
592C22D4×#6714yes—
593C22C2×#94011yescharacteristic
594C22#580511yescharacteristic
595C22C2×#95311yescharacteristic
596C4—22no—
597C4—21no—
598C4—22no—
599C4—44no—
600C4—41no—
601C4—41no—
602C4—128no—
603C4—242no—
604C4#524412yescomplemented
605C3#2577111yescharacteristic, complemented, Sylow
606C2—24no—
607C2—21no—
608C2—21no—
609C2—42no—
610C2—84no—
611C2—244no—
612C2#8404911yescharacteristic
613C2C2×#524411yescharacteristic
614C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#25771
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C3×#3496C2semidirect
2. C2×#5244C2semidirect
3. #82653C2semidirect
4. #82403C2semidirect
5. #71851C2semidirect
6. #71686C2semidirect
7. C4×#834C2semidirect
10. C23×D12C22semidirect
23. C2×#937C22semidirect
27. C2×#834C22semidirect
28. #5212C22semidirect
29. C4×#217C22semidirect
34. #4955C22semidirect
36. #4526C22semidirect
37. #3496D3semidirect
52. C22×C3×D4D4semidirect
55. C23×C3×C4D4semidirect
86. C2×C3×C4⋊C4D4semidirect
87. C2×C3×C4⋊C4D4semidirect
89. C2×C3×C22⋊C4D4semidirect
90. C2×C3×C22⋊C4D4semidirect
92. C2×C3×C42D4semidirect
107. #834D4semidirect
216. #217C2×D4semidirect
238. C22×D4#67semidirect
241. C23×C4#67semidirect
271. C2×C4⋊C4#67semidirect
272. C2×C4⋊C4#67semidirect
280. C2×C22⋊C4#67semidirect
281. C2×C22⋊C4#67semidirect
284. C2×C42#67semidirect
302. C23×C3#127semidirect
350. C2×C3×C4#127semidirect
482. C3×C4B(4,𝔽2)semidirect
511. C23#940semidirect
554. C2×C4#940semidirect
604. C4#5244semidirect
605. C3#25771semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
C2×#52442
#826532
#824032
#718512
#716862
C4×#8342
#257713