Finite Group Databasebeta

← #10801131

Subgroups — #10801131

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1007C211yesmaximal, characteristic, complemented
2#82455C211yesmaximal, characteristic, complemented
3C2×#4753C211yesmaximal, characteristic
4C4×#908C212yesmaximal, complemented
5C3×C4×#101C211yesmaximal, characteristic, complemented
6#70121C211yesmaximal, characteristic, complemented
7#70104C211yesmaximal, characteristic, complemented
8#70102C211yesmaximal, characteristic, complemented
9#69866C212yesmaximal, complemented
10#69005C212yesmaximal, complemented
11#69002C212yesmaximal, complemented
12#21475—31nomaximal, Sylow
13C2×#1007C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947C2211yescharacteristic
16#5812C2211yescharacteristic
17C2×#908C2212yes—
18#5574—22no—
19C4×#239—22no—
20#5572—22no—
21C4×D3×Q8—22no—
22#5504—22no—
23#5504C2212yes—
24C4×#235—22no—
25C4×#235C2212yes—
26C2×C4×D12C2211yescharacteristic
27C2×C3×#101C2211yescharacteristic
28C2×C3×#101C2211yescharacteristic
29C3×C4×M16C2212yes—
30#5229C2211yescharacteristic
31C2×#825C2212yes—
32C2×#825C2212yes—
33#5133C2212yes—
34#5131C2212yes—
35#5121C2212yes—
36#4989—22no—
37#4988C2211yescharacteristic
38#4975C2211yescharacteristic
39#4940—22no—
40#4932C2211yescharacteristic
41#4753C418yescomplemented
42#4711—22no—
43#4710—22no—
44C4×D24—22no—
45C4×#204—22no—
46C3×#343C2212yes—
47#4555C2212yes—
48#4555C2212yes—
49C4×2−1+4—31no—
50C2×#454—31no—
51C4×AGL(1,ℤ/8)—32no—
52#2326—31no—
53#1888—31no—
54#1883—32no—
55#1859—32no—
56#1857—31no—
57#1793—31no—
58#1759—32no—
59C4×#101D311yescharacteristic, complemented
60C22×C3×Q8C2311yescharacteristic
61#1007—22no—
62#1007—24no—
63#1007C2×C414yescomplemented
64C2×#239—22no—
65C2×D3×Q8—22no—
66C2×#235—22no—
67C2×#235C2312yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C42D412yes—
77C2×C3×C42C2311yescharacteristic
78#947—22no—
79#947C2×C414yescomplemented
80Q8×Dic3—22no—
81Q8×Dic3—22no—
82C4×#67—22no—
83C4×#67—24no—
84C4×#67—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88#926—22no—
89#926C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94#908—24no—
95#908C2×C418yescomplemented
96C2×D24—22no—
97C2×#204—22no—
98#893—24no—
99#893—22no—
100#892—24no—
101#892—22no—
102D3×C4⋊C4—24no—
103D3×C4⋊C4—22no—
104#887—24no—
105#887—22no—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12—22no—
109C4×D12—22no—
110C4×D12D412yes—
111C4×D12D412yes—
112#872—24no—
113#872—22no—
114C42×D3—24no—
115C42×D3—22no—
116C4×Dic6—22no—
117C4×Dic6—22no—
118C4×Dic6D412yes—
119C4×Dic6D412yes—
120C3×#101—22no—
121C3×#101C2×C414yescomplemented
122C3×#97—22no—
123C3×C4×C8—22no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—22no—
127#825—24no—
128#825C2×C418yescomplemented
129#821—22no—
130#821—22no—
131#821—22no—
132#818—22no—
133#817—22no—
134#816—22no—
135C2×2−1+4—31no—
136C2×AGL(1,ℤ/8)—32no—
137C2×#124—31no—
138#520—32no—
139#520—62no—
140#520—62no—
141#519—31no—
142#519—62no—
143C4×#41—32no—
144C4×#41—62no—
145C4×#41—62no—
146C2×C4×Q8—62no—
147C2×C4×Q8C2×D311yescharacteristic
148C2×C4×D4—31no—
149#454—38no—
150#442—62no—
151#440—62no—
152C4×SD16—62no—
153C4×D8—62no—
154#428—32no—
155C2×#104—32no—
156C2×#104—32no—
157#418—32no—
158#417—32no—
159C2×#101C2×D311yescharacteristic
160C2×#101C2×D311yescharacteristic
161C4×M16C2×D312yes—
162#391—31no—
163#390—31no—
164#388—62no—
165#385—62no—
166#384—31no—
167#376—31no—
168#343C2×D312yes—
169#339—32no—
170#339—32no—
171C23×D3—21no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8C22×C412yes—
175C2×C3×Q8#4112yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×#67—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185#239—28no—
186#239—28no—
187D3×Q8—28no—
188D3×Q8—28no—
189#235—28no—
190#235—28no—
191#235—22no—
192#235—22no—
193#235C22×C414yes—
194#235#4114yes—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12C22×C412yes—
200C2×D12C2×D412yes—
201C2×C4×D3—24no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—24no—
205C2×C4×D3—22no—
206C2×C4×D3—42no—
207C2×C4×D3—82no—
208C2×Dic6—22no—
209C2×Dic6C2×D412yes—
210C3×M16—22no—
211C3×M16—22no—
212C3×M16—22no—
213C3×M16C22×C414yes—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222#217—22no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—28no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—41no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—82no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3C2×D412yes—
234C4⋊Dic3C2×D412yes—
235Dic3⋊C4—24no—
236Dic3⋊C4—22no—
237Dic3⋊C4—22no—
238Dic3⋊C4—82no—
239C4×Dic3—24no—
240C4×Dic3—24no—
241C4×Dic3—22no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—21no—
245C4×Dic3—22no—
246C4×Dic3C22×C412yes—
247D24—28no—
248#204—28no—
2492−1+4—34no—
2502−1+4—64no—
2512−1+4—62no—
252C2×#41—32no—
253C2×#41—62no—
254C2×#41—62no—
255C22×Q8—62no—
256C22×Q8C22×D311yescharacteristic
257C22×D4—31no—
258C23×C4—61no—
259AGL(1,ℤ/8)—38no—
260AGL(1,ℤ/8)—64no—
261C2×SD16—62no—
262C2×D8—62no—
263C2×M16C22×D312yes—
264C2×M16C22×D312yes—
265#124—34no—
266#124—62no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—32no—
270C4×Q8—32no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8—62no—
274C4×Q8—64no—
275C4×Q8—62no—
276C4×Q8—62no—
277C4×D4—32no—
278C4×D4—32no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—64no—
283C4×D4—62no—
284C4×D4—62no—
285C4×D4—64no—
286C4×D4—62no—
287C4×D4—62no—
288#117—32no—
289#117—64no—
290#117—62no—
291#117—64no—
292#117—62no—
293#117—62no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4—64no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4D1212yes—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C22⋊C4—62no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—62no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—64no—
306C2×C42—62no—
307C2×C42D1212yes—
308C2×C42C22×D311yescharacteristic
309#107—62no—
310#106—62no—
311#104—38no—
312#104—64no—
313#103—62no—
314#102—62no—
315#102—62no—
316#102—62no—
317#101—22no—
318#101C4×D314yescomplemented
319#97—22no—
320C4×C8—22no—
321#95—61no—
322#95—61no—
323#95—62no—
324C23×C3C22×D411yescharacteristic
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C3×Q8—24no—
332C3×Q8—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4C4×D414yes—
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12012yes—
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4C23×C411yescharacteristic
344C2×C3×C4C22×D412yes—
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4112yes—
347#67—24no—
348#67—24no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360D12—28no—
361D12—24no—
362D12—22no—
363D12—42no—
364D12C4×D414yes—
365C4×D3—216no—
366C4×D3—24no—
367C4×D3—24no—
368C4×D3—44no—
369C4×D3—82no—
370C4×D3—84no—
371C4×D3—84no—
372Dic6—28no—
373Dic6—22no—
374Dic6C4×D414yes—
375C3×C8—24no—
376C3×C8—42no—
377C24—61no—
378#41—34no—
379#41—34no—
380#41—68no—
381#41—68no—
382#41—68no—
383#41—68no—
384#41—62no—
385#41—62no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—32no—
389C2×Q8—62no—
390C2×Q8—68no—
391C2×Q8—68no—
392C2×Q8C2×C4×D312yes—
393C2×Q8#23512yes—
394C2×D4—32no—
395C2×D4—32no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—61no—
399C2×D4—62no—
400C2×D4—62no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—122no—
410C22×C4—242no—
411C22×C4C2×D1212yes—
412C22×C4C2×D1212yes—
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C23×D311yescharacteristic
416SD16—68no—
417D8—68no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16C2×C4×D314yes—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—32no—
427C4⋊C4—32no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—242no—
435C22⋊C4—64no—
436C22⋊C4—62no—
437C22⋊C4—68no—
438C22⋊C4—62no—
439C22⋊C4—62no—
440C22⋊C4—121no—
441C22⋊C4—121no—
442C22⋊C4—242no—
443C42—22no—
444C42—22no—
445C42—32no—
446C42—64no—
447C42—64no—
448C42—62no—
449C42—62no—
450C42—61no—
451C42—62no—
452C42—62no—
453C42D3×D412yes—
454C42D3×D412yes—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3C2×C4×D411yescharacteristic
458C22×C3C2×#12011yescharacteristic
459C22×C3#52511yescharacteristic
460C2×D3—24no—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—44no—
464C2×D3—42no—
465C2×D3—82no—
466C2×D3—81no—
467C2×D3—81no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—22no—
471C3×C4—24no—
472C3×C4—42no—
473C3×C4C2×C4×D412yes—
474C3×C4#52512yes—
475Dic3—24no—
476Dic3—42no—
477Dic3—42no—
478Dic3—42no—
479Dic3—81no—
480Dic3—81no—
481C23—62no—
482C23—62no—
483C23—122no—
484C23—122no—
485C23—121no—
486C23—121no—
487C23C22×D1211yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—34no—
491Q8—68no—
492Q8—62no—
493D4—34no—
494D4—68no—
495D4—64no—
496D4—64no—
497D4—64no—
498D4—62no—
499D4—122no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—616no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—124no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—122no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—242no—
522C2×C4—244no—
523C2×C4—244no—
524C2×C4C4×D1214yes—
525C2×C4#88211yescharacteristic
526C2×C4#88211yescharacteristic
527C2×C4#88212yes—
528C2×C4C22×C4×D311yescharacteristic
529C2×C4C2×#23511yescharacteristic
530C2×C4C2×D3×D412yes—
531C2×C4D3×#4112yes—
532C8—24no—
533C8—42no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3C4×#12011yescharacteristic
537C2×C3#302011yescharacteristic
538C2×C3#302111yescharacteristic
539D3—44no—
540D3—82no—
541C22—21no—
542C22—21no—
543C22—64no—
544C22—124no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22—242no—
549C22—241no—
550C22—241no—
551C22C2×C4×D1211yescharacteristic
552C22C2×#88211yescharacteristic
553C22#552411yescharacteristic
554C4—24no—
555C4—21no—
556C4—21no—
557C4—22no—
558C4—42no—
559C4—64no—
560C4—122no—
561C4—122no—
562C4—122no—
563C4—241no—
564C4—241no—
565C4C4×D3×D412yes—
566C4#554712yes—
567C3#2147511yescharacteristic, complemented, Sylow
568C2—21no—
569C2—21no—
570C2—124no—
571C2—242no—
572C2C4×#88211yescharacteristic
573C2#7870911yescharacteristic
574C2#7871711yescharacteristic
subgroupG/H
centerC22C2×#882
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#101C2
socleC22×C3C2×#120
prime radicalC2×#4753C2
cosocleC22×C3×C4C24
hypercenterC4×#101D3
hypocenterC3#21475
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
1. C4×#1007C2semidirect
2. #82455C2semidirect
4. C4×#908C2semidirect
5. C3×C4×#101C2semidirect
6. #70121C2semidirect
7. #70104C2semidirect
8. #70102C2semidirect
9. #69866C2semidirect
10. #69005C2semidirect
11. #69002C2semidirect
41. #4753C4semidirect
59. C4×#101D3semidirect
63. #1007C2×C4semidirect
79. #947C2×C4semidirect
95. #908C2×C4semidirect
121. C3×#101C2×C4semidirect
128. #825C2×C4semidirect
318. #101C4×D3semidirect
567. C3#21475semidirect

Maximal subgroups

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

subgroupindex
C4×#10072
#824552
C2×#47532
C4×#9082
C3×C4×#1012
#701212
#701042
#701022
#698662
#690052
#690022
#214753