Finite Group Databasebeta

← #10822279

Subgroups — #10822279

Order 768.

2130 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C23×#204C211yesmaximal, characteristic, complemented
4C2×C3×#416C211yesmaximal, characteristic, complemented
5#82230C211yesmaximal, characteristic, complemented
6C2×#5001C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#70278C214yesmaximal, complemented
9#70210C214yesmaximal, complemented
10#23423—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5830—24no—
17#5830—24no—
18C2×#951—24no—
19C2×#951—24no—
20C2×#951—24no—
21C2×#947C2212yes—
22C22×#216C2212yescomplemented
23C2×#940C2212yes—
24C2×#937—22no—
25C22×#214C2212yescomplemented
26C2×#930—22no—
27C2×#926C2211yescharacteristic
28C22×#208C2211yescharacteristic
29C22×#204—24no—
30C22×#204—24no—
31C22×#204—24no—
32C22×#204C2214yes—
33C3×#416—24no—
34C2×C3×#103C2212yescomplemented
35C2×C3×#102C2212yescomplemented
36#5001—24no—
37#4973C2214yescomplemented
38C2×#809C2214yes—
39#4943—22no—
40#4929—22no—
41#4550C2214yescomplemented
42C22×#137—31no—
43C23×SD16—31no—
44C2×#535—31no—
45C2×#418—31no—
46C2×#416D311yescharacteristic, complemented
47#2310—31no—
48#2015—34no—
49#2014—34no—
50#1729—31no—
51C24×D3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C22×#67—24no—
64C23×Dic3—21no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12—21no—
70C22×D12—22no—
71C22×D12D414yes—
72C22×D12C2312yes—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6D414yescomplemented
80C22×Dic6C2312yes—
81C22×C3×C8—22no—
82C22×C3×C8—22no—
83C22×C3×C8D414yescomplemented
84C22×C3×C8C2312yes—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217D412yes—
91#951—232no—
92#947—24no—
93#947—24no—
94C2×#216—24no—
95C2×#216—24no—
96C2×#216—24no—
97#940—24no—
98#940—24no—
99#937—216no—
100C2×#214—24no—
101C2×#214—24no—
102C2×#214—24no—
103#930—216no—
104C2×(C2×D3)⋊C4—24no—
105#926—22no—
106#926—22no—
107C2×C4⋊Dic3D412yes—
108C2×C4×Dic3C2312yes—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yescomplemented
112C22×C3⋊C8C2312yes—
113C2×#204—232no—
114C2×#204—28no—
115C2×#204—28no—
116C2×#204—28no—
117C3×#103—24no—
118C3×#102—24no—
119C3⋊#95—28no—
120#809—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×D4—31no—
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—32no—
128C22×SD16—32no—
129C22×SD16—34no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×SD16—64no—
133C22×M16—31no—
134C23×C8C2×D311yescharacteristic
135#535—64no—
136#535—64no—
137C2×#127—32no—
138C2×#124—32no—
139C2×#121—62no—
140C2×#120—62no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#418—64no—
144#416—24no—
145C2×#103C2×D312yescomplemented
146C2×#102—34no—
147C2×#102C2×D312yescomplemented
148#391—34no—
149#391—62no—
150#381—62no—
151#336—34no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—42no—
157C23×D3—42no—
158C23×D3—84no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—21no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C2×#67—216no—
192C2×#67—88no—
193C22×Dic3—24no—
194C22×Dic3—22no—
195C22×Dic3—42no—
196C22×Dic3—42no—
197C22×Dic3C2×D414yes—
198C2×D12—28no—
199C2×D12—28no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—24no—
205C2×D12—22no—
206C2×D12—44no—
207C2×D12—42no—
208C2×D12—44no—
209C2×D12—42no—
210C2×D12C2×D414yes—
211C2×Dic6—28no—
212C2×Dic6—28no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—42no—
220C2×Dic6—44no—
221C2×Dic6—44no—
222C2×Dic6—42no—
223C2×Dic6C2×D414yes—
224C2×C3×C8—28no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8C2×D414yes—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C22⋊C4—42no—
231C3×C42—42no—
232#217—24no—
233#216—216no—
234#214—216no—
235(C2×D3)⋊C4—216no—
236C4⋊Dic3—24no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239#208—28no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×D414yes—
243#204—232no—
244C25—61no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41D1214yes—
251C22×Q8—34no—
252C22×Q8—32no—
253C22×Q8—62no—
254C22×Q8—61no—
255C22×Q8—62no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8C22×D311yescharacteristic
260C22×D4—21no—
261C22×D4—34no—
262C22×D4—32no—
263C22×D4—64no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—61no—
269C22×D4—62no—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274#137—632no—
275C2×Q16—64no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×SD16—68no—
279C2×SD16—68no—
280C2×SD16—632no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—68no—
285C2×M16—34no—
286C2×M16—62no—
287C2×M16—62no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—32no—
291C22×C8#6714yescomplemented
292C22×C8C22×D312yes—
293#127—64no—
294#127—64no—
295#124—64no—
296#124—64no—
297#121—616no—
298#120—616no—
299#117—22no—
300#117—22no—
301#117—62no—
302#117—62no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4C22×D312yes—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—32no—
307C2×C22⋊C4—64no—
308C2×C42—21no—
309C2×C42—32no—
310#103—24no—
311#102—24no—
312#102—68no—
313#95—68no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—44no—
319C23×C3#12012yes—
320C23×C3#12112yes—
321C23×C3#12712yes—
322C23×C3C22×D411yescharacteristic
323C22×D3—24no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—88no—
330C22×D3—88no—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—48no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—48no—
344C2×C3×C4—42no—
345C2×C3×C4—44no—
346C2×C3×C4#12014yes—
347C2×C3×C4#12012yes—
348C2×C3×C4#12014yes—
349C2×C3×C4#12114yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12114yes—
352C2×C3×C4#12712yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359#67—816no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—28no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—48no—
366D12—24no—
367D12—24no—
368D12—42no—
369D12—42no—
370D12—48no—
371Dic6—24no—
372Dic6—24no—
373Dic6—42no—
374Dic6—42no—
375Dic6—48no—
376C3×C8—24no—
377C3⋊C8—24no—
378C24—21no—
379C24—62no—
380C24—64no—
381C24—122no—
382C24—122no—
383C24—244no—
384C24D3×D411yescharacteristic
385#41—28no—
386#41—28no—
387#41—44no—
388#41—44no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—34no—
395C2×Q8—64no—
396C2×Q8—62no—
397C2×Q8—68no—
398C2×Q8—68no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—122no—
404C2×Q8—124no—
405C2×Q8—124no—
406C2×Q8—122no—
407C2×Q8C2×D1212yes—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—34no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—616no—
418C2×D4—64no—
419C2×D4—68no—
420C2×D4—68no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—124no—
427C2×D4—122no—
428C2×D4—124no—
429C2×D4—122no—
430C2×D4—248no—
431C2×D4C2×D1212yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—34no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4C2×D1212yes—
443C22×C4#23512yes—
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D412yes—
446C22×C4D3×D412yes—
447C22×C4D3×D412yes—
448C22×C4C2×#6712yes—
449C22×C4C23×D311yescharacteristic
450Q16—616no—
451SD16—632no—
452SD16—616no—
453M16—68no—
454C2×C8—28no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—34no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8C2×#6714yes—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—64no—
464C22⋊C4—42no—
465C22⋊C4—616no—
466C22⋊C4—64no—
467C42—42no—
468C42—62no—
469C42—62no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3#39214yes—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×#12111yescharacteristic
480C22×C3C2×#12711yescharacteristic
481C22×C3#57714yes—
482C2×D3—44no—
483C2×D3—48no—
484C2×D3—816no—
485C2×D3—84no—
486C2×D3—84no—
487C3×C4—42no—
488C3×C4—82no—
489C3×C4#39214yes—
490C3×C4C2×#12011yescharacteristic
491C3×C4C2×#12011yescharacteristic
492C3×C4C2×#12111yescharacteristic
493C3×C4C2×#12111yescharacteristic
494Dic3—44no—
495Dic3—44no—
496C23—24no—
497C23—22no—
498C23—41no—
499C23—44no—
500C23—41no—
501C23—64no—
502C23—128no—
503C23—122no—
504C23—128no—
505C23—122no—
506C23—248no—
507C23—248no—
508C23—248no—
509C23#88212yes—
510C23#88412yes—
511C23#94012yes—
512C23C2×D3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517Q8—64no—
518Q8—64no—
519Q8—128no—
520Q8—122no—
521Q8—122no—
522D4—22no—
523D4—22no—
524D4—48no—
525D4—41no—
526D4—41no—
527D4—64no—
528D4—64no—
529D4—128no—
530D4—122no—
531D4—122no—
532D4—2416no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—48no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—68no—
542C2×C4—128no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4#88212yes—
546C2×C4#88214yes—
547C2×C4#88412yes—
548C2×C4#88414yes—
549C2×C4#93014yes—
550C2×C4#93714yes—
551C2×C4#94012yes—
552C2×C4C22×D1211yescharacteristic
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×D3×D411yescharacteristic
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C22×#6711yescharacteristic
558C8—24no—
559C8—64no—
560C2×C3—24no—
561C2×C3—82no—
562C2×C3C2×#39211yescharacteristic
563C2×C3#300912yes—
564C2×C3#304312yes—
565C2×C3#315412yes—
566D3—84no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—42no—
571C22—44no—
572C22—82no—
573C22—82no—
574C22—124no—
575C22—128no—
576C22—2416no—
577C22—244no—
578C22—244no—
579C22#496014yes—
580C22C2×#88211yescharacteristic
581C22C2×#88411yescharacteristic
582C22#577014yes—
583C22C2×#94011yescharacteristic
584C4—42no—
585C4—82no—
586C4—124no—
587C4—124no—
588C4#496014yes—
589C4C2×#88211yescharacteristic
590C4C2×#88411yescharacteristic
591C4C2×#93011yescharacteristic
592C4C2×#93711yescharacteristic
593C3#2342311yescharacteristic, complemented, Sylow
594C2—24no—
595C2—82no—
596C2—244no—
597C2#7883512yes—
598C2#7886412yes—
599C2C2×#496011yescharacteristic
600C2#8390512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#416C2
socleC23×C3C22×D4
prime radicalC2×#5830C2
hypercenterC2×#416D3
hypocenterC3#23423
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C23×#204C2semidirect
4. C2×C3×#416C2semidirect
5. #82230C2semidirect
6. C2×#5001C2semidirect
7. #71965C2semidirect
8. #70278C2semidirect
9. #70210C2semidirect
22. C22×#216C22semidirect
25. C22×#214C22semidirect
34. C2×C3×#103C22semidirect
35. C2×C3×#102C22semidirect
37. #4973C22semidirect
41. #4550C22semidirect
46. C2×#416D3semidirect
79. C22×Dic6D4semidirect
83. C22×C3×C8D4semidirect
111. C2×#208D4semidirect
145. C2×#103C2×D3semidirect
147. C2×#102C2×D3semidirect
291. C22×C8#67semidirect
593. C3#23423semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C22×#9512
C23×#2042
C2×C3×#4162
#822302
C2×#50012
#719652
#702782
#702102
#234233