Finite Group Databasebeta

← #10826810

Subgroups — #10826810

Order 768.

1744 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2#87273C211yesmaximal, characteristic
3#84069C212yesmaximal, complemented
4C2×#5053C212yesmaximal, complemented
5#72139C211yesmaximal, characteristic, complemented
6#71706C214yesmaximal, complemented
7#71704C212yesmaximal, complemented
8#71595C212yesmaximal, complemented
9#23905—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic, complemented
12C2×#1004C2211yescharacteristic
13C3×#536—28no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×#124—21no—
16C2×C3×#123C2212yes—
17C2×C3×#123C2212yescomplemented
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×#117C2211yescharacteristic, complemented
21#5829—24no—
22#5824—24no—
23#5816—21no—
24#5804—21no—
25#5804C2212yes—
26#5798C2211yescharacteristic
27#5791—21no—
28C2×#929—21no—
29C2×#929C2214yes—
30C2×#929C2212yes—
31#5231—24no—
32#5230—24no—
33C2×#834C2212yes—
34C2×#834C2212yescomplemented
35#5211C2214yescomplemented
36#5211C2212yescomplemented
37#5053—28no—
38C2×#806C2214yes—
39#4527C2214yescomplemented
40#3485—31no—
41C2×#536D311yescharacteristic, complemented
42C2×#458—32no—
43#2614—32no—
44#2022—34no—
45#1899—32no—
46#1890—32no—
47#1782—31no—
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C2×C3×#41—42no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—24no—
54C22×C3×Q8—24no—
55C22×C3×Q8—42no—
56C22×C3×Q8D411yescharacteristic
57C22×C3×D4D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×#67C2312yes—
66#1004—24no—
67#1004—42no—
68C2×#237—22no—
69C2×D3×D4—22no—
70C2×#235—21no—
71C3×C4⋊Q8—28no—
72C3×#124—28no—
73C3×#123—24no—
74C3×#123—24no—
75C3×#123—24no—
76C3×#123D418yescomplemented
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—28no—
79C3×#117—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4C2312yes—
86C2×C3×C42—21no—
87#953—22no—
88C2×#217—21no—
89C2×#217C2314yes—
90C2×#217C2312yes—
91#947—22no—
92#946—24no—
93#944—22no—
94#939—24no—
95#939—24no—
96#937—22no—
97#937—22no—
98#935—22no—
99#935—22no—
100D4×Dic3—24no—
101#930—22no—
102#929—24no—
103#929—22no—
104#929—22no—
105#929—22no—
106#929—24no—
107#929—28no—
108#929—44no—
109C4×#67—24no—
110C2×(C2×D3)⋊C4—22no—
111C2×(C2×D3)⋊C4—24no—
112C2×(C2×D3)⋊C4D414yes—
113#926—21no—
114#924—22no—
115C2×Dic3⋊C4—22no—
116C2×Dic3⋊C4—24no—
117C2×Dic3⋊C4D414yes—
118#834—24no—
119#834—42no—
120C3⋊#95—24no—
121C3⋊#95—28no—
122#806—28no—
123C2×H(5,𝔽2)—31no—
124C22×#41C2×D311yescharacteristic
125C23×Q8C2×D311yescharacteristic, complemented
126#547—64no—
127#546—64no—
128#536—28no—
129#536—61no—
130#534—32no—
131#534—61no—
132C2×C4⋊Q8—21no—
133C2×#124—21no—
134#525—61no—
135C2×#123—32no—
136C2×#123—34no—
137C2×#123—61no—
138C2×#123C2×D312yes—
139C2×#123C2×D312yescomplemented
140C2×C22⋊Q8—21no—
141C2×C22⋊Q8—21no—
142#518—31no—
143C2×#117C2×D311yescharacteristic, complemented
144#458—68no—
145C2×#99—32no—
146C2×#99—34no—
147C2×#99—32no—
148#395—64no—
149#393—64no—
150#381—34no—
151#381—32no—
152#342—34no—
153C24×C3C2×D411yescharacteristic
154C24×C3C2×D411yescharacteristic
155C24×C3C2411yescharacteristic
156C23×D3—21no—
157C23×D3C2×D412yes—
158C3×#41—416no—
159C2×C3×Q8—24no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—41no—
164C2×C3×Q8—416no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×Q8—42no—
168C2×C3×Q8—42no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—41no—
177C2×C3×D4C2×D414yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—41no—
190C22×C3×C4—41no—
191C22×C3×C4C2×D414yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C2×#67—22no—
198C2×#67—24no—
199C2×#67—22no—
200C2×#67—22no—
201C2×#67—24no—
202C2×#67—24no—
203C2×#67—42no—
204C2×#67—42no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—24no—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D414yes—
210#237—48no—
211D3×D4—48no—
212#235—42no—
213#235—42no—
214C2×D12—21no—
215C2×C4×D3—22no—
216C2×Dic6—21no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4C2×D414yes—
224C3×C42—24no—
225#217—24no—
226#217—22no—
227#217—22no—
228#217—22no—
229#217—24no—
230#217—24no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—48no—
235C4⋊Dic3—22no—
236Dic3⋊C4—24no—
237Dic3⋊C4—24no—
238Dic3⋊C4—24no—
239Dic3⋊C4—48no—
240C4×Dic3—22no—
241H(5,𝔽2)—64no—
242H(5,𝔽2)—122no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—42no—
246C2×#41—62no—
247C2×#41—61no—
248C22×Q8—24no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—24no—
252C22×Q8—42no—
253C22×Q8#6711yescharacteristic
254C22×D4—32no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4#6711yescharacteristic
258C22×D4C22×D311yescharacteristic
259C22×D4C22×D311yescharacteristic
260C23×C4#6711yescharacteristic
261C23×C4#6711yescharacteristic
262C23×C4C22×D312yes—
263C23×C4C22×D311yescharacteristic
264C4⋊Q8—28no—
265C4⋊Q8—62no—
266#124—28no—
267#124—62no—
268#123—24no—
269#123—24no—
270#123—24no—
271#123—64no—
272#123—62no—
273#123—62no—
274#123—64no—
275#123—62no—
276#123—62no—
277#123—62no—
278#123—68no—
279#123—124no—
280#123#6718yescomplemented
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—62no—
284C22⋊Q8—64no—
285#121—62no—
286#121—64no—
287#121—64no—
288#120—62no—
289#120—62no—
290#120—62no—
291C4×D4—64no—
292C4×D4—64no—
293#117—24no—
294#117—61no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—34no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4—64no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—34no—
305C2×C22⋊C4—34no—
306C2×C22⋊C4—62no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—64no—
309C2×C22⋊C4C22×D312yes—
310C2×C42—21no—
311#99—64no—
312#99—68no—
313#99—122no—
314#95—68no—
315#95—64no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3#12012yes—
325C23×C3C22×D412yes—
326C23×C3C22×D411yescharacteristic
327C23×C3C22×D412yes—
328C23×C3H(5,𝔽2)12yes—
329C22×D3—22no—
330C22×D3—24no—
331C22×D3—42no—
332C22×D3—44no—
333C22×D3—82no—
334C3×Q8—44no—
335C3×Q8—48no—
336C3×Q8—48no—
337C3×D4—28no—
338C3×D4—44no—
339C3×D4—44no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—28no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—28no—
353C2×C3×C4—44no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4#12012yes—
360C2×C3×C4#12012yes—
361C2×C3×C4#12012yes—
362#67—44no—
363#67—42no—
364#67—42no—
365#67—44no—
366#67—84no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—42no—
371C2×Dic3—44no—
372C2×Dic3—48no—
373C2×Dic3—44no—
374C2×Dic3—82no—
375D12—42no—
376C4×D3—44no—
377Dic6—42no—
378C24—32no—
379C24—61no—
380C24C2×#6711yescharacteristic
381C24C2×#6711yescharacteristic
382C24C23×D311yescharacteristic
383#41—416no—
384#41—122no—
385#41—128no—
386#41—122no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—41no—
392C2×Q8—416no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—61no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—41no—
406C2×D4—61no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—62no—
412C2×D4—64no—
413C2×D4—128no—
414C2×D4—122no—
415C2×D4—122no—
416C2×D4C2×#6714yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—64no—
436C22×C4D3×D414yes—
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6714yes—
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—64no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—128no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—22no—
453C22⋊C4—64no—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—62no—
457C22⋊C4—62no—
458C22⋊C4—64no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—128no—
463C22⋊C4C2×#6714yes—
464C42—24no—
465C42—62no—
466C22×C3—24no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×#12011yescharacteristic
480C22×C3#53411yescharacteristic
481C22×C3D4214yes—
482C2×D3—42no—
483C2×D3—44no—
484C2×D3—84no—
485C2×D3—162no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—44no—
490C3×C4—41no—
491C3×C4—44no—
492Dic3—42no—
493Dic3—44no—
494Dic3—84no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—62no—
504C23—64no—
505C23—122no—
506C23—124no—
507C23—242no—
508C23#95312yes—
509C23C2×D3×D412yes—
510C23#100412yes—
511C23C22×#6711yescharacteristic
512C23C22×#6712yes—
513Q8—44no—
514Q8—48no—
515Q8—48no—
516Q8—122no—
517D4—28no—
518D4—44no—
519D4—44no—
520D4—122no—
521D4—124no—
522D4—124no—
523D4—122no—
524D4—122no—
525D4—244no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—124no—
549C2×C4—122no—
550C2×C4—124no—
551C2×C4—128no—
552C2×C4—124no—
553C2×C4—242no—
554C2×C4#93712yes—
555C2×C4#93712yes—
556C2×C4#95312yes—
557C2×C3—24no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—42no—
561C2×C3—81no—
562C2×C3#240311yescharacteristic
563C2×C3#302812yes—
564D3—82no—
565C22—22no—
566C22—24no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—122no—
578C22—124no—
579C22—244no—
580C22—482no—
581C22C2×#93711yescharacteristic
582C22D4×#6714yes—
583C22#580411yescharacteristic
584C22C2×#95311yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—44no—
589C4—41no—
590C4—44no—
591C4—122no—
592C4—124no—
593C4—244no—
594C3#2390511yescharacteristic, complemented, Sylow
595C2—21no—
596C2—24no—
597C2—21no—
598C2—42no—
599C2—81no—
600C2—242no—
601C2#8341712yes—
602C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#536C2
socleC22×C3C2×#120
prime radical#87273C2
hypercenterC2×#536D3
hypocenterC3#23905
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. #84069C2semidirect
4. C2×#5053C2semidirect
5. #72139C2semidirect
6. #71706C2semidirect
7. #71704C2semidirect
8. #71595C2semidirect
11. C23×C3×Q8C22semidirect
17. C2×C3×#123C22semidirect
20. C2×C3×#117C22semidirect
34. C2×#834C22semidirect
35. #5211C22semidirect
36. #5211C22semidirect
39. #4527C22semidirect
41. C2×#536D3semidirect
76. C3×#123D4semidirect
125. C23×Q8C2×D3semidirect
139. C2×#123C2×D3semidirect
143. C2×#117C2×D3semidirect
280. #123#67semidirect
594. C3#23905semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
#872732
#840692
C2×#50532
#721392
#717062
#717042
#715952
#239053