Finite Group Databasebeta

← #10799098

Subgroups — #10799098

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×#908C211yesmaximal, characteristic
2C2×C3×#405C211yesmaximal, characteristic, complemented
3C2×#5121C211yesmaximal, characteristic, complemented
4C2×#5113C211yesmaximal, characteristic, complemented
5C22×#821C211yesmaximal, characteristic, complemented
6#82270C211yesmaximal, characteristic, complemented
7#82269C211yesmaximal, characteristic, complemented
8#69826C212yesmaximal, complemented
9#69825C212yesmaximal, complemented
10#69776C212yesmaximal, complemented
11#69774C212yesmaximal, complemented
12#21235—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#908—24no—
21C2×#908—24no—
22C2×#908C418yescomplemented
23C22×D24C2212yes—
24C22×#204C2212yes—
25C3×#405—24no—
26C2×C3×#97C2212yescomplemented
27C2×C3×C4×C8C2212yescomplemented
28#5121—24no—
29#5113—24no—
30C2×#821—24no—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2211yescharacteristic, complemented
34C2×#821C2211yescharacteristic, complemented
35C2×#818C2212yescomplemented
36C2×#817C2212yescomplemented
37C2×#816C2211yescharacteristic, complemented
38C2×#816C2211yescharacteristic, complemented
39#4988—22no—
40#4987—22no—
41#4958—22no—
42#4955—22no—
43#4939C2212yes—
44#4939C2212yes—
45#4939C2212yescomplemented
46#4929C2212yescomplemented
47C22×AGL(1,ℤ/8)—31no—
48C2×#428—31no—
49C2×#416—31no—
50C22×#102—31no—
51C2×#405D311yescharacteristic, complemented
52#2311—31no—
53#2309—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C24×D3—21no—
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C23×Dic3—21no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235C2×C414yes—
68C22×D12—22no—
69C22×D12—22no—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—21no—
73C22×D12—22no—
74C22×D12C2×C414yes—
75C22×D12C2312yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C2×C3×M16C2×C414yes—
80C2×C3×M16D414yes—
81C2×C3×M16D414yes—
82C22×C3×C8—22no—
83C22×C3×C8—22no—
84C22×C3×C8D414yes—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2312yes—
88C3×#117—22no—
89C3×#117—22no—
90C2×C3×C4⋊C4D412yes—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42C2312yes—
93C2×#217—21no—
94C2×(C2×D3)⋊C4—24no—
95C2×(C2×D3)⋊C4—24no—
96C2×(C2×D3)⋊C4—24no—
97#926—22no—
98#926—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4×Dic3—21no—
106#908—232no—
107C2×D24—24no—
108C2×D24—24no—
109C2×D24C2×C418yescomplemented
110C2×#204—24no—
111C2×#204—24no—
112C2×#204C2×C418yescomplemented
113C3×#97—24no—
114C3×C4×C8—24no—
115C3⋊#95—24no—
116#821—24no—
117#821—24no—
118#821—22no—
119#821—22no—
120#818—24no—
121#817—24no—
122#816—22no—
123#816—22no—
124C22×#41—31no—
125C23×D4—31no—
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×M16C2×D311yescharacteristic
132C23×C8C2×D311yescharacteristic
133C2×#117—31no—
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4—31no—
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—64no—
140C2×#103—31no—
141C2×#103—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—64no—
147#405—24no—
148C2×#97C2×D312yescomplemented
149C2×C4×C8C2×D312yescomplemented
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×C3C2×D411yescharacteristic
159C23×D3—21no—
160C23×D3—22no—
161C23×D3—24no—
162C23×D3—42no—
163C23×D3—42no—
164C23×D3—82no—
165C23×D3—82no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—24no—
176C2×#67—42no—
177C2×#67—42no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—42no—
183C22×Dic3—82no—
184#235—28no—
185#235—28no—
186#235—44no—
187#235—44no—
188C2×D12—28no—
189C2×D12—28no—
190C2×D12—22no—
191C2×D12—21no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—22no—
197C2×D12—21no—
198C2×D12—24no—
199C2×D12—22no—
200C2×D12—21no—
201C2×D12—44no—
202C2×D12—42no—
203C2×D12—44no—
204C2×D12—42no—
205C2×D12C22×C414yes—
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—
216C3×M16C22⋊C4116yescomplemented
217C2×C3×C8—28no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8C22×C414yes—
223C2×C3×C8C2×D412yes—
224C2×C3×C8C2×D412yes—
225C2×C3×C8C2×D414yes—
226C2×C3×C8C2×D414yes—
227C3×C4⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C42—22no—
230C3×C42—22no—
231#217—42no—
232(C2×D3)⋊C4—84no—
233(C2×D3)⋊C4—84no—
234(C2×D3)⋊C4—84no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—22no—
239C4⋊Dic3—21no—
240C4⋊Dic3—22no—
241C4⋊Dic3—42no—
242C4×Dic3—42no—
243D24—216no—
244#204—216no—
245C25—61no—
246C2×#41—34no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C22×Q8—31no—
253C22×D4—34no—
254C22×D4—31no—
255C22×D4—32no—
256C22×D4—61no—
257C22×D4—62no—
258C22×D4—62no—
259C22×D4—61no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4—62no—
263C23×C4—61no—
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×M16C4×D314yes—
274C2×M16D1214yes—
275C2×M16#6714yes—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8D1214yes—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D312yes—
282#117—22no—
283#117—22no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4D1212yes—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4D1212yes—
298C2×C42—61no—
299C2×C42C22×D312yes—
300#107—64no—
301#106—64no—
302#103—62no—
303#103—62no—
304#102—64no—
305#102—64no—
306#102—62no—
307#102—62no—
308#97—24no—
309C4×C8—24no—
310#95—64no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3C4×D412yes—
314C23×C3#12112yes—
315C23×C3#12712yes—
316C23×C3C22×D411yescharacteristic
317C22×D3—22no—
318C22×D3—24no—
319C22×D3—41no—
320C22×D3—42no—
321C22×D3—41no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—44no—
325C22×D3—44no—
326C22×D3—42no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—84no—
330C22×D3—84no—
331C22×D3—88no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4C2×C22⋊C414yes—
336C2×C3×C4C4×D414yes—
337C2×C3×C4C4×D412yes—
338C2×C3×C4#12114yes—
339C2×C3×C4#12112yes—
340C2×C3×C4#12414yes—
341C2×C3×C4#12712yes—
342C2×C3×C4C23×C411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348#67—48no—
349C2×Dic3—22no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—44no—
357C2×Dic3—42no—
358C2×Dic3—84no—
359D12—24no—
360D12—22no—
361D12—24no—
362D12—22no—
363D12—42no—
364D12—41no—
365D12—42no—
366D12—41no—
367D12—48no—
368C4×D3—48no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3×C8—22no—
374C3×C8—22no—
375C3×C8C2×C22⋊C418yes—
376C24—61no—
377C24—62no—
378C24—64no—
379C24—122no—
380C24—122no—
381C24—242no—
382C24—242no—
383C24C2×D1211yescharacteristic
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—32no—
395C2×D4—34no—
396C2×D4—64no—
397C2×D4—68no—
398C2×D4—64no—
399C2×D4—68no—
400C2×D4—62no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—62no—
404C2×D4—62no—
405C2×D4—61no—
406C2×D4—62no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—62no—
410C2×D4—122no—
411C2×D4—122no—
412C2×D4—122no—
413C2×D4—124no—
414C2×D4—122no—
415C2×D4—124no—
416C22×C4—61no—
417C22×C4—64no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—242no—
425C22×C4C2×C4×D312yes—
426C22×C4C2×D1214yes—
427C22×C4C2×D1211yescharacteristic
428C22×C4C2×D1212yes—
429C22×C4C2×D1212yes—
430C22×C4#23512yes—
431C22×C4#23512yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—616no—
435D8—616no—
436M16(C2×D3)⋊C4116yescomplemented
437C2×C8—28no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8C2×C4×D314yes—
443C2×C8C2×D1212yes—
444C2×C8C2×D1212yes—
445C2×C8C2×D1214yes—
446C2×C8C2×#6714yes—
447C4⋊C4—24no—
448C4⋊C4—62no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—62no—
453C4⋊C4—61no—
454C4⋊C4—122no—
455C22⋊C4—24no—
456C22⋊C4—122no—
457C22⋊C4—244no—
458C22⋊C4—244no—
459C22⋊C4—244no—
460C42—22no—
461C42—22no—
462C42—122no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3#39014yes—
468C22×C3C2×C4×D411yescharacteristic
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×#12711yescharacteristic
471C22×C3#57612yes—
472C22×C3#57712yes—
473C2×D3—44no—
474C2×D3—48no—
475C2×D3—42no—
476C2×D3—44no—
477C2×D3—816no—
478C2×D3—81no—
479C2×D3—82no—
480C2×D3—81no—
481C2×D3—82no—
482C2×D3—82no—
483C2×D3—84no—
484C3×C4—44no—
485C3×C4#39014yes—
486C3×C4C22×C22⋊C411yescharacteristic
487C3×C4C2×C4×D411yescharacteristic
488C3×C4C2×#12111yescharacteristic
489C3×C4C2×#12411yescharacteristic
490Dic3—42no—
491Dic3—81no—
492Dic3—81no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—62no—
497C23—64no—
498C23—128no—
499C23—121no—
500C23—122no—
501C23—124no—
502C23—122no—
503C23—122no—
504C23—124no—
505C23—121no—
506C23—122no—
507C23—248no—
508C23—248no—
509C23—244no—
510C23—244no—
511C23C4×D1212yes—
512C23#87412yes—
513C23#93012yes—
514C23C22×D1211yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—121no—
518Q8—121no—
519D4—64no—
520D4—62no—
521D4—62no—
522D4—64no—
523D4—128no—
524D4—128no—
525D4—122no—
526D4—121no—
527D4—122no—
528D4—121no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—62no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—128no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—124no—
542C2×C4—244no—
543C2×C4C4×D1212yes—
544C2×C4C4×D1214yes—
545C2×C4#87412yes—
546C2×C4#87514yes—
547C2×C4C2×(C2×D3)⋊C414yes—
548C2×C4#93012yes—
549C2×C4#93014yes—
550C2×C4C22×C4×D311yescharacteristic
551C2×C4C22×D1211yescharacteristic
552C2×C4C22×D1211yescharacteristic
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23511yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—22no—
557C8—22no—
558C8C2×(C2×D3)⋊C418yes—
559C2×C3—24no—
560C2×C3C2×#39011yescharacteristic
561C2×C3#294612yes—
562C2×C3#306212yes—
563C2×C3#315411yescharacteristic
564C2×C3#315511yescharacteristic
565D3—82no—
566D3—84no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—122no—
572C22—124no—
573C22—128no—
574C22—124no—
575C22—2416no—
576C22—244no—
577C22—242no—
578C22—242no—
579C22—241no—
580C22—242no—
581C22—241no—
582C22#493914yes—
583C22C2×C4×D1211yescharacteristic
584C22C2×#87411yescharacteristic
585C22#575112yes—
586C22#575212yes—
587C22C2×#93011yescharacteristic
588C4—44no—
589C4—122no—
590C4—241no—
591C4—242no—
592C4—241no—
593C4#493914yes—
594C4C2×C4×D1211yescharacteristic
595C4C2×#87511yescharacteristic
596C4C22×(C2×D3)⋊C411yescharacteristic
597C4C2×#93011yescharacteristic
598C3#2123511yescharacteristic, complemented, Sylow
599C2—24no—
600C2—244no—
601C2—242no—
602C2#7856012yes—
603C2#7856411yescharacteristic
604C2#7856511yescharacteristic
605C2C2×#493911yescharacteristic
606C2#8380212yes—
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#405C2
socleC23×C3C22×D4
prime radicalC22×#908C2
cosocleC22×C3×C4C24
hypercenterC2×#405D3
hypocenterC3#21235
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. C2×C3×#405C2semidirect
3. C2×#5121C2semidirect
4. C2×#5113C2semidirect
5. C22×#821C2semidirect
6. #82270C2semidirect
7. #82269C2semidirect
8. #69826C2semidirect
9. #69825C2semidirect
10. #69776C2semidirect
11. #69774C2semidirect
22. C2×#908C4semidirect
26. C2×C3×#97C22semidirect
27. C2×C3×C4×C8C22semidirect
33. C2×#821C22semidirect
34. C2×#821C22semidirect
35. C2×#818C22semidirect
36. C2×#817C22semidirect
37. C2×#816C22semidirect
38. C2×#816C22semidirect
45. #4939C22semidirect
46. #4929C22semidirect
51. C2×#405D3semidirect
109. C2×D24C2×C4semidirect
112. C2×#204C2×C4semidirect
148. C2×#97C2×D3semidirect
149. C2×C4×C8C2×D3semidirect
216. C3×M16C22⋊C4semidirect
436. M16(C2×D3)⋊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×#9082
C2×C3×#4052
C2×#51212
C2×#51132
C22×#8212
#822702
#822692
#698262
#698252
#697762
#697742
#212353