Finite Group Databasebeta

← #10799092

Subgroups — #10799092

Order 768.

1678 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic
2C22×C3×#102C211yesmaximal, characteristic, complemented
3#84044C211yesmaximal, characteristic, complemented
4C2×#5113C211yesmaximal, characteristic, complemented
5C2×#5101C211yesmaximal, characteristic, complemented
6#82186C211yesmaximal, characteristic, complemented
7C2×#4964C211yesmaximal, characteristic, complemented
8#70192C212yesmaximal, complemented
9#70191C212yesmaximal, complemented
10#70190C212yesmaximal, complemented
11#70189C212yesmaximal, complemented
12#21235—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×#214C2212yes—
18C2×#932—24no—
19C2×#932—24no—
20C2×#932C418yescomplemented
21C22×#213C2212yes—
22C2×#926C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C2×C3×#102—24no—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28#5247—22no—
29#5221—22no—
30#5221C2212yes—
31#5221C2212yes—
32#5213—22no—
33#5113—24no—
34C2×#821C2211yescharacteristic, complemented
35C2×#821C2211yescharacteristic, complemented
36C2×#816C2211yescharacteristic, complemented
37C2×#816C2211yescharacteristic, complemented
38#5101—24no—
39C2×#815C2212yescomplemented
40C2×C8×Dic3C2212yescomplemented
41#4988C2212yescomplemented
42#4987—22no—
43#4973C2212yescomplemented
44#4964—24no—
45C2×#808C2212yescomplemented
46C2×#807C2212yescomplemented
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#416—31no—
50C22×#102D311yescharacteristic, complemented
51C2×#405—31no—
52#2311—31no—
53#2309—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C25×C3—21no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4C2×C414yes—
66C22×C3×D4C2312yes—
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235C2×C414yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8D414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2312yes—
88C2×#217—21no—
89C2×#217—24no—
90C2×#217—24no—
91C2×#217D412yes—
92C2×#214—24no—
93C2×#214—24no—
94C2×#214C2×C418yescomplemented
95#932—232no—
96C2×#213—24no—
97C2×#213—24no—
98C2×#213C2×C418yescomplemented
99C2×(C2×D3)⋊C4—24no—
100#926—22no—
101#926—22no—
102#926—22no—
103#926—22no—
104C2×C4⋊Dic3D412yes—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2312yes—
109C2×#208C2×C414yes—
110C2×#208D414yes—
111C2×#208D414yes—
112C22×C3⋊C8C2312yes—
113C3×#102—24no—
114C3×#102—24no—
115C3⋊#95—24no—
116#821—22no—
117#821—22no—
118#816—22no—
119#816—22no—
120#815—24no—
121C8×Dic3—24no—
122#808—24no—
123#807—24no—
124C22×#41—31no—
125C23×D4C2×D311yescharacteristic
126C2×AGL(1,ℤ/8)—38no—
127C2×AGL(1,ℤ/8)—64no—
128C2×AGL(1,ℤ/8)—64no—
129C22×SD16—32no—
130C22×D8—32no—
131C22×M16—31no—
132C23×C8C2×D311yescharacteristic
133C2×#117—31no—
134C2×#117—31no—
135C22×C4⋊C4C2×D311yescharacteristic
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—64no—
140C2×#103—31no—
141C2×#103—31no—
142C2×#102—24no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102C2×D312yes—
146C2×#102C2×D312yes—
147#405—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—32no—
151#390—32no—
152#390—32no—
153#390—32no—
154#390—62no—
155#386—62no—
156#380—62no—
157#379—62no—
158C24×C3—22no—
159C24×C3—24no—
160C24×C3—42no—
161C24×C3—42no—
162C24×C3—82no—
163C24×C3—82no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C2×C3×D4—28no—
167C2×C3×D4—28no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—42no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4—42no—
178C2×C3×D4C22×C414yes—
179C22×C3×C4—22no—
180C22×C3×C4—42no—
181C22×C3×C4—82no—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4112yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—24no—
191C2×#67—42no—
192C2×#67—42no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D414yes—
197#235—28no—
198#235—28no—
199#235—44no—
200#235—44no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C22×C412yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C22×C412yes—
216C2×C3×C8—28no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D412yes—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—42no—
226#217—24no—
227#217—42no—
228#217—84no—
229#217—84no—
230#214—216no—
231#213—216no—
232(C2×D3)⋊C4—84no—
233C4⋊Dic3—24no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4×Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—42no—
241#208C22⋊C4116yescomplemented
242C2×C3⋊C8C22×C414yes—
243C2×C3⋊C8C2×D414yes—
244C2×C3⋊C8C2×D414yes—
245C25—21no—
246C2×#41—34no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C22×Q8—31no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—31no—
260C22×D4—61no—
261C22×D4C4×D314yes—
262C22×D4C22×D312yes—
263C23×C4—21no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266AGL(1,ℤ/8)—632no—
267C2×SD16—38no—
268C2×SD16—64no—
269C2×SD16—64no—
270C2×D8—38no—
271C2×D8—64no—
272C2×D8—64no—
273C2×M16—34no—
274C2×M16—34no—
275C2×M16—34no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—32no—
279C22×C8#6714yes—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282#117—62no—
283#117—62no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4C22×D312yes—
293C2×C22⋊C4—32no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—64no—
298C2×C42—32no—
299C2×C42—61no—
300#107—64no—
301#106—64no—
302#103—62no—
303#103—62no—
304#102—24no—
305#102—24no—
306#102—62no—
307#102—62no—
308#97—64no—
309C4×C8—64no—
310#95—64no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—48no—
319C23×C3—88no—
320C23×C3—84no—
321C23×C3—84no—
322C23×C3—88no—
323C23×C3C4×D412yes—
324C23×C3#12112yes—
325C23×C3#12712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—42no—
331C22×D3—42no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—44no—
339C2×C3×C4—84no—
340C2×C3×C4C2×C22⋊C414yes—
341C2×C3×C4C4×D412yes—
342C2×C3×C4C4×D414yes—
343C2×C3×C4#12112yes—
344C2×C3×C4#12114yes—
345C2×C3×C4#12414yes—
346C2×C3×C4#12712yes—
347C2×C3×C4C23×C411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353#67—48no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364D12—22no—
365D12—22no—
366D12—41no—
367D12—41no—
368C4×D3—48no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3×C8—22no—
374C3×C8—22no—
375C3⋊C8C2×C22⋊C418yes—
376C24—22no—
377C24—24no—
378C24—42no—
379C24—42no—
380C24—61no—
381C24—82no—
382C24—82no—
383C24D3×D411yescharacteristic
384#41—68no—
385#41—68no—
386#41—124no—
387#41—124no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—32no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—64no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4C2×C4×D314yes—
416C22×C4—22no—
417C22×C4—34no—
418C22×C4—42no—
419C22×C4—61no—
420C22×C4—64no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—82no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4C2×C4×D312yes—
427C22×C4#23512yes—
428C22×C4D3×D412yes—
429C22×C4D3×D412yes—
430C22×C4D3×D411yescharacteristic
431C22×C4#23712yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—616no—
435D8—616no—
436M16—316no—
437C2×C8—28no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—34no—
443C2×C8—34no—
444C2×C8—34no—
445C2×C8C2×#6712yes—
446C2×C8C2×#6712yes—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C22⋊C4—64no—
456C22⋊C4—122no—
457C22⋊C4—244no—
458C22⋊C4—244no—
459C22⋊C4—244no—
460C42—62no—
461C42—62no—
462C42—122no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—48no—
468C22×C3—44no—
469C22×C3—816no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—84no—
473C22×C3#39014yes—
474C22×C3C2×C4×D411yescharacteristic
475C22×C3C2×#12111yescharacteristic
476C22×C3C2×#12711yescharacteristic
477C22×C3#57612yes—
478C22×C3#57712yes—
479C2×D3—44no—
480C2×D3—42no—
481C2×D3—81no—
482C2×D3—81no—
483C2×D3—82no—
484C3×C4—82no—
485C3×C4#39014yes—
486C3×C4C22×C22⋊C411yescharacteristic
487C3×C4C2×C4×D411yescharacteristic
488C3×C4C2×#12111yescharacteristic
489C3×C4C2×#12411yescharacteristic
490Dic3—42no—
491Dic3—44no—
492Dic3—81no—
493Dic3—81no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—48no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—62no—
503C23—88no—
504C23—84no—
505C23—84no—
506C23—88no—
507C23—121no—
508C23—122no—
509C23—122no—
510C23—121no—
511C23#67⋊C412yes—
512C23#88412yes—
513C23#94012yes—
514C23C2×D3×D411yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—121no—
518Q8—121no—
519D4—24no—
520D4—24no—
521D4—48no—
522D4—42no—
523D4—42no—
524D4—62no—
525D4—62no—
526D4—128no—
527D4—121no—
528D4—121no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—84no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—128no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4D3×C22⋊C414yes—
544C2×C4#67⋊C412yes—
545C2×C4#88412yes—
546C2×C4#88514yes—
547C2×C4C4×#6714yes—
548C2×C4#93914yes—
549C2×C4#94012yes—
550C2×C4C22×C4×D311yescharacteristic
551C2×C4C2×#23511yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—22no—
557C8—22no—
558C8—38no—
559C2×C3—24no—
560C2×C3—84no—
561C2×C3C2×#39011yescharacteristic
562C2×C3#294612yes—
563C2×C3#306212yes—
564C2×C3#315411yescharacteristic
565C2×C3#315511yescharacteristic
566D3—82no—
567C22—22no—
568C22—24no—
569C22—22no—
570C22—24no—
571C22—48no—
572C22—44no—
573C22—816no—
574C22—82no—
575C22—82no—
576C22—84no—
577C22—124no—
578C22—122no—
579C22—241no—
580C22—241no—
581C22—242no—
582C22#495714yes—
583C22C2×#67⋊C411yescharacteristic
584C22C2×#88411yescharacteristic
585C22#575712yes—
586C22#576212yes—
587C22C2×#94011yescharacteristic
588C4—82no—
589C4—124no—
590C4—122no—
591C4—241no—
592C4—241no—
593C4#495714yes—
594C4C2×D3×C22⋊C411yescharacteristic
595C4C2×#88511yescharacteristic
596C4C2×C4×#6711yescharacteristic
597C4C2×#93911yescharacteristic
598C3#2123511yescharacteristic, complemented, Sylow
599C2—24no—
600C2—84no—
601C2—242no—
602C2#7872612yes—
603C2#7875212yes—
604C2C2×#495711yescharacteristic
605C2#8383011yescharacteristic
606C2#8385411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC23×C3C22×D4
prime radicalC22×#932C2
cosocleC22×C3×C4C24
hypercenterC22×#102D3
hypocenterC3#21235
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C3×#102C2semidirect
3. #84044C2semidirect
4. C2×#5113C2semidirect
5. C2×#5101C2semidirect
6. #82186C2semidirect
7. C2×#4964C2semidirect
8. #70192C2semidirect
9. #70191C2semidirect
10. #70190C2semidirect
11. #70189C2semidirect
20. C2×#932C4semidirect
34. C2×#821C22semidirect
35. C2×#821C22semidirect
36. C2×#816C22semidirect
37. C2×#816C22semidirect
39. C2×#815C22semidirect
40. C2×C8×Dic3C22semidirect
41. #4988C22semidirect
43. #4973C22semidirect
45. C2×#808C22semidirect
46. C2×#807C22semidirect
50. C22×#102D3semidirect
94. C2×#214C2×C4semidirect
98. C2×#213C2×C4semidirect
241. #208C22⋊C4semidirect
598. C3#21235semidirect

Maximal subgroups

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

subgroupindex
C22×#9322
C22×C3×#1022
#840442
C2×#51132
C2×#51012
#821862
C2×#49642
#701922
#701912
#701902
#701892
#212353