Finite Group Databasebeta

← #10848751

Subgroups — #10848751

Order 768.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3C2×#4715C211yesmaximal, characteristic, complemented
4#74256C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6#69807C211yesmaximal, characteristic, complemented
7#69552C211yesmaximal, characteristic, complemented
8#68909C212yesmaximal, complemented
9#68908C212yesmaximal, complemented
10#68888C212yesmaximal, complemented
11#68887C212yesmaximal
12#25603—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—21no—
15C2×Q8×Dic3—21no—
16C22×Dic12C2212yes—
17C22×#204C2212yes—
18#5728—24no—
19#5728—24no—
20C2×#897C2212yes—
21C2×#897C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×#892—21no—
24C2×#890—21no—
25C2×#887C2212yes—
26C2×#875C2211yescharacteristic
27C2×C42×D3C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C3×#382C2211yescharacteristic
31C2×(C2×D3)⋊C8C2212yes—
32#5089—24no—
33#4988C2211yescharacteristic
34#4973C2211yescharacteristic
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×#803C2211yescharacteristic
40#4715—24no—
41#4670—21no—
42#4670—21no—
43#4665—21no—
44#4665—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47#4550C2212yes—
48C2×#533—31no—
49C2×#497—31no—
50C2×#489—31no—
51#2850—31no—
52#2001—32no—
53#2000—32no—
54#1993D311yescharacteristic, complemented
55#1960—31no—
56#1880—32no—
57#1878—32no—
58#1749—31no—
59C22×C3×Q8—21no—
60C22×D12—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3C2311yescharacteristic
65C22×Dic6C2311yescharacteristic
66C22×Dic6C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yescomplemented
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42C2311yescharacteristic
75#946—28no—
76Q8×Dic3—28no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C2×Dic12—24no—
95C2×Dic12—24no—
96C2×Dic12—42no—
97C2×#204—24no—
98C2×#204—24no—
99C2×#204—42no—
100#897—24no—
101#897—24no—
102#897—42no—
103#895—42no—
104#892—28no—
105#890—28no—
106#887—24no—
107#887—24no—
108#875—22no—
109#875—22no—
110C42×D3—22no—
111C42×D3—22no—
112#869—22no—
113#869D414yescomplemented
114C3×#97—22no—
115C3×#95—21no—
116C3×#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119(C2×D3)⋊C8—24no—
120#810—24no—
121#810—42no—
122#810—42no—
123#809—24no—
124#803—22no—
125C22×Q16—32no—
126C22×SD16—32no—
127#533—64no—
128#533—64no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8C2×D311yescharacteristic
131C2×#125—61no—
132C2×#124—31no—
133C2×C22⋊Q8—32no—
134C2×C22⋊Q8—31no—
135C2×C22⋊Q8—61no—
136C2×#121—31no—
137C2×C4×Q8—32no—
138C2×C4×Q8—61no—
139C2×#117—61no—
140C22×C42—31no—
141#497—64no—
142#489—64no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#102—32no—
147C2×C22⋊C8—32no—
148C2×#97—31no—
149C2×#97C2×D311yescharacteristic
150#391—31no—
151#390—31no—
152#388—61no—
153#388—61no—
154#386—61no—
155#386—61no—
156#382C2×D311yescharacteristic
157#340C2×D312yes—
158#340C2×D312yes—
159#336—32no—
160C23×D3—21no—
161C23×D3C2×D411yescharacteristic
162C2×C3×Q8—24no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12—42no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—41no—
192C2×C4×D3—41no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6C2×D414yes—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8#4114yes—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4C2×D414yes—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—42no—
215(C2×D3)⋊C4—42no—
216(C2×D3)⋊C4—42no—
217(C2×D3)⋊C4—42no—
218(C2×D3)⋊C4—42no—
219(C2×D3)⋊C4—42no—
220C4⋊Dic3—28no—
221C4⋊Dic3—24no—
222C4⋊Dic3—24no—
223C4⋊Dic3—42no—
224Dic3⋊C4—28no—
225C4×Dic3—28no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233Dic12—48no—
234#204—48no—
235C22×Q8—21no—
236C22×Q8—32no—
237C22×Q8—31no—
238C22×D4—31no—
239C22×D4—61no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—31no—
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8—32no—
250C22×C8C22×D312yes—
251C4⋊Q8—22no—
252C4⋊Q8—34no—
253C4⋊Q8—62no—
254C4⋊Q8D1214yescomplemented
255#125—68no—
256#124—62no—
257#124—62no—
258C22⋊Q8—64no—
259C22⋊Q8—68no—
260C22⋊Q8—64no—
261C22⋊Q8—122no—
262#121—122no—
263C4×Q8—64no—
264C4×Q8—68no—
265C4×Q8—64no—
266#117—68no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D312yes—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—62no—
287C2×C42—62no—
288C2×C42—62no—
289C2×C42C22×D311yescharacteristic
290#103—64no—
291#103—122no—
292#103—122no—
293#102—64no—
294C22⋊C8—64no—
295#97—22no—
296#97—62no—
297#95—21no—
298#95—21no—
299#95—61no—
300#95—61no—
301C23×C3C22×D411yescharacteristic
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3#13712yes—
316C3×Q8—48no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4#12012yes—
331C2×C3×C4#12012yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4112yes—
335C2×C3×C4H(5,𝔽2)12yes—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—24no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3#13712yes—
351D12—42no—
352D12—42no—
353D12—84no—
354C4×D3—28no—
355C4×D3—44no—
356C4×D3—42no—
357C4×D3—42no—
358Dic6—24no—
359Dic6—24no—
360Dic6—42no—
361Dic6—42no—
362Dic6—42no—
363Dic6—42no—
364C3×C8—24no—
365C3⋊C8—42no—
366C24—31no—
367C24—61no—
368C2×Q8—24no—
369C2×Q8—34no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—62no—
373C2×Q8—61no—
374C2×Q8—61no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—122no—
386C2×D4—122no—
387C2×D4—122no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—121no—
405C22×C4—121no—
406C22×C4D3×D411yescharacteristic
407C22×C4D3×D411yescharacteristic
408C22×C4C23×D311yescharacteristic
409Q16—128no—
410SD16—128no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—62no—
416C2×C8#23914yes—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—68no—
422C4⋊C4—68no—
423C4⋊C4—64no—
424C4⋊C4—64no—
425C4⋊C4—122no—
426C4⋊C4C2×D1214yes—
427C22⋊C4—68no—
428C22⋊C4—122no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C22⋊C4—122no—
432C22⋊C4—122no—
433C22⋊C4—122no—
434C42—68no—
435C42—62no—
436C42—62no—
437C42—61no—
438C42—61no—
439C42C2×D1212yes—
440C42D3×D412yes—
441C22×C3C2×#12011yescharacteristic
442C22×C3#53211yescharacteristic
443C22×C3#53511yescharacteristic
444C22×C3C2×#13711yescharacteristic
445C22×C3C2×#13711yescharacteristic
446C22×C3#57511yescharacteristic
447C22×C3#57511yescharacteristic
448C2×D3—22no—
449C2×D3—22no—
450C2×D3—22no—
451C2×D3—41no—
452C2×D3—41no—
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—81no—
456C2×D3—81no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—44no—
465C3×C4—81no—
466C3×C4—81no—
467C3×C4#54612yes—
468C3×C4#55012yes—
469Dic3—22no—
470Dic3—44no—
471Dic3—41no—
472Dic3—41no—
473Dic3—81no—
474C23—32no—
475C23—61no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23—121no—
488C23C2×D3×D411yescharacteristic
489Q8—48no—
490Q8—64no—
491Q8—64no—
492Q8—122no—
493Q8—122no—
494Q8—122no—
495Q8—122no—
496D4—122no—
497D4—122no—
498D4—244no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—32no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—68no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—64no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—124no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4#88212yes—
532C2×C4#88212yes—
533C2×C4C22×D1211yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×#23912yes—
536C2×C4#100912yes—
537C8—24no—
538C8—122no—
539C2×C3#243111yescharacteristic
540C2×C3#300411yescharacteristic
541C2×C3#300511yescharacteristic
542C2×C3#317211yescharacteristic
543C2×C3#318111yescharacteristic
544C2×C3#321511yescharacteristic
545C2×C3#321611yescharacteristic
546D3—42no—
547D3—161no—
548C22—62no—
549C22—62no—
550C22—62no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22—241no—
562C22C2×#88211yescharacteristic
563C22#567411yescharacteristic
564C22#572511yescharacteristic
565C22#577311yescharacteristic
566C22#577311yescharacteristic
567C22D3×#13711yescharacteristic
568C22D3×#13711yescharacteristic
569C4—22no—
570C4—22no—
571C4—44no—
572C4—62no—
573C4—81no—
574C4—81no—
575C4—121no—
576C4—121no—
577C4—124no—
578C4—241no—
579C4#557712yes—
580C4#563012yes—
581C3#2560311yescharacteristic, complemented, Sylow
582C2—122no—
583C2—481no—
584C2#7662611yescharacteristic
585C2#7892711yescharacteristic
586C2#7895811yescharacteristic
587C2#8168611yescharacteristic
588C2#8168711yescharacteristic
589C2#8195911yescharacteristic
590C2#8196311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1993D3
hypocenterC3#25603
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
2. C2×#5089C2semidirect
3. C2×#4715C2semidirect
5. C3×#1993C2semidirect
6. #69807C2semidirect
7. #69552C2semidirect
8. #68909C2semidirect
9. #68908C2semidirect
10. #68888C2semidirect
54. #1993D3semidirect
69. C3×C4⋊Q8D4semidirect
113. #869D4semidirect
254. C4⋊Q8D12semidirect
581. C3#25603semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#50892
C2×#47152
#742562
C3×#19932
#698072
#695522
#689092
#689082
#688882
#688872
#256033