Finite Group Databasebeta

← #10848752

Subgroups — #10848752

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×C3×#489C211yesmaximal, characteristic, complemented
3#83008C211yesmaximal, characteristic
4C2×#5060C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70281C212yesmaximal, complemented
7#70273C212yesmaximal, complemented
8#70213C212yesmaximal, complemented
9#70205C212yesmaximal
10#69822C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25603—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×#946—21no—
16C2×#946C2211yescharacteristic
17C2×Q8×Dic3—21no—
18C22×#216C2212yes—
19C2×#938C2211yescharacteristic
20C22×#214C2212yes—
21#5728—24no—
22#5728—24no—
23C2×#897C2212yes—
24C2×#892—21no—
25C2×#890—21no—
26C2×#887C2212yes—
27C2×C42×D3C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C3×#489—24no—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#97C2211yescharacteristic
33#5229C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×(C2×D3)⋊C8C2212yes—
36C2×#816C2212yes—
37C2×#816C2212yes—
38#5060—24no—
39#4956—21no—
40#4956—21no—
41#4944—21no—
42#4944—21no—
43#4936C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#533—31no—
49C2×#497—31no—
50C2×#489D311yescharacteristic, complemented
51#2850—31no—
52#2001—32no—
53#2000—32no—
54#1993—31no—
55#1960—31no—
56#1880—32no—
57#1878—32no—
58#1749—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—21no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yescomplemented
70C3×#124—22no—
71C3×#124—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—21no—
78C2×#217—21no—
79C2×#217—21no—
80#946—28no—
81#946—42no—
82Q8×Dic3—28no—
83C2×#216—24no—
84C2×#216—24no—
85C2×#216—42no—
86#938—42no—
87C2×#214—24no—
88C2×#214—24no—
89C2×#214—42no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—21no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#897—24no—
105#897—24no—
106#892—28no—
107#890—28no—
108#887—24no—
109#887—24no—
110C42×D3—22no—
111C42×D3—22no—
112#869—22no—
113#869D414yescomplemented
114C3×#103—24no—
115C3×#102—24no—
116C3×#97—22no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121(C2×D3)⋊C8—24no—
122#816—42no—
123#816—42no—
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×#124C2×D311yescharacteristic
133C2×C22⋊Q8—31no—
134C2×C22⋊Q8—32no—
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—24no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103C2×D312yes—
146C2×#102C2×D312yes—
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#382—31no—
157#340—32no—
158#340—32no—
159#336—32no—
160C24×C3—21no—
161C23×D3C2×D411yescharacteristic
162C2×C3×Q8—24no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—42no—
168C2×C3×Q8—42no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—42no—
179C2×#67—42no—
180C2×#67—42no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—21no—
192C2×C4×D3—21no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—41no—
196C2×C4×D3—41no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6C2×D414yes—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4C2×D414yes—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213#217—42no—
214#217—42no—
215#217—42no—
216#216—48no—
217#214—48no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—42no—
220C4⋊Dic3—28no—
221C4⋊Dic3—24no—
222C4⋊Dic3—24no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225Dic3⋊C4—28no—
226Dic3⋊C4—42no—
227C4×Dic3—28no—
228C4×Dic3—22no—
229C4×Dic3—22no—
230C4×Dic3—21no—
231C4×Dic3—21no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8#4114yes—
235C22×Q8—21no—
236C22×Q8—32no—
237C22×Q8C22×D311yescharacteristic
238C22×D4—61no—
239C22×D4C22×D311yescharacteristic
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⋊Q8#6714yescomplemented
255#125—68no—
256#124—22no—
257#124—22no—
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—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C22×D312yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—62no—
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—24no—
291#103—122no—
292#103—122no—
293#102—24no—
294C22⋊C8—64no—
295#97—22no—
296#97—62no—
297#95—61no—
298#95—61no—
299#95—61no—
300#95—61no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3C22×D411yescharacteristic
309C22×D3—21no—
310C22×D3—21no—
311C22×D3—21no—
312C22×D3—21no—
313C22×D3—21no—
314C22×D3—21no—
315C22×D3#13712yes—
316C3×Q8—48no—
317C3×Q8—42no—
318C3×Q8—42no—
319C3×D4—42no—
320C3×D4—42no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12012yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C2×#4112yes—
336C2×C3×C4H(5,𝔽2)12yes—
337#67—84no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—24no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3#13712yes—
356C4×D3—28no—
357C4×D3—44no—
358C4×D3—42no—
359C4×D3—42no—
360Dic6—24no—
361Dic6—24no—
362Dic6—42no—
363Dic6—42no—
364C3×C8—42no—
365C3⋊C8—24no—
366C24—21no—
367C24—31no—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—34no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—122no—
386C2×D4—122no—
387C2×D4—122no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4—61no—
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—22no—
414C2×C8—34no—
415C2×C8—62no—
416C2×C8—62no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—68no—
420C4⋊C4—68no—
421C4⋊C4—64no—
422C4⋊C4—64no—
423C4⋊C4—62no—
424C4⋊C4—62no—
425C4⋊C4—122no—
426C4⋊C4C2×#6714yes—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—68no—
430C22⋊C4—122no—
431C22⋊C4—122no—
432C22⋊C4—122no—
433C22⋊C4—122no—
434C42—68no—
435C42—62no—
436C42—62no—
437C42—61no—
438C42—61no—
439C42D3×D412yes—
440C42C2×#6712yes—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3C2×#12011yescharacteristic
449C22×C3#53211yescharacteristic
450C22×C3#53511yescharacteristic
451C22×C3C2×#13711yescharacteristic
452C22×C3C2×#13711yescharacteristic
453C22×C3#57511yescharacteristic
454C22×C3#57511yescharacteristic
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C2×D3—41no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—44no—
465C3×C4—81no—
466C3×C4#54612yes—
467C3×C4#55012yes—
468Dic3—22no—
469Dic3—44no—
470Dic3—41no—
471Dic3—41no—
472Dic3—81no—
473Dic3—81no—
474C23—32no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—61no—
483C23—61no—
484C23—61no—
485C23—61no—
486C23—61no—
487C23—61no—
488C23C2×D3×D411yescharacteristic
489Q8—48no—
490Q8—42no—
491Q8—42no—
492Q8—64no—
493Q8—64no—
494Q8—122no—
495Q8—122no—
496D4—42no—
497D4—42no—
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—68no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—64no—
520C2×C4—124no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4#93712yes—
532C2×C4#93712yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4D3×#4112yes—
535C2×C4#100912yes—
536C2×C4C22×#6711yescharacteristic
537C8—42no—
538C8—64no—
539C2×C3—161no—
540C2×C3#243111yescharacteristic
541C2×C3#300411yescharacteristic
542C2×C3#300511yescharacteristic
543C2×C3#317211yescharacteristic
544C2×C3#318111yescharacteristic
545C2×C3#321511yescharacteristic
546C2×C3#321611yescharacteristic
547D3—42no—
548C22—62no—
549C22—62no—
550C22—62no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22#566811yescharacteristic
563C22#567411yescharacteristic
564C22#577311yescharacteristic
565C22#577311yescharacteristic
566C22D3×#13711yescharacteristic
567C22D3×#13711yescharacteristic
568C22C2×#93711yescharacteristic
569C4—22no—
570C4—22no—
571C4—44no—
572C4—62no—
573C4—81no—
574C4—121no—
575C4—124no—
576C4—121no—
577C4—241no—
578C4—241no—
579C4#567612yes—
580C4#583412yes—
581C3#2560311yescharacteristic, complemented, Sylow
582C2—122no—
583C2—161no—
584C2#8166511yescharacteristic
585C2#8166911yescharacteristic
586C2#8168611yescharacteristic
587C2#8168711yescharacteristic
588C2#8295811yescharacteristic
589C2#8394511yescharacteristic
590C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenterC2×#489D3
hypocenterC3#25603
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
2. C2×C3×#489C2semidirect
4. C2×#5060C2semidirect
5. #71788C2semidirect
6. #70281C2semidirect
7. #70273C2semidirect
8. #70213C2semidirect
10. #69822C2semidirect
11. #69807C2semidirect
50. C2×#489D3semidirect
69. C3×C4⋊Q8D4semidirect
113. #869D4semidirect
254. C4⋊Q8#67semidirect
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×C3×#4892
#830082
C2×#50602
#717882
#702812
#702732
#702132
#702052
#698222
#698072
#256033