Finite Group Databasebeta

← #10810909

Subgroups — #10810909

Order 768.

1602 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
1C23×#204C211yesmaximal, characteristic
2C2×#4710C212yesmaximal, complemented
3C22⋊C4×D12C211yesmaximal, characteristic
4C22⋊C4×Dic6C211yesmaximal, characteristic, complemented
5C3×#1752C211yesmaximal, characteristic, complemented
6#71971C211yesmaximal, characteristic, complemented
7#69963C211yesmaximal, characteristic, complemented
8#69959C211yesmaximal, characteristic
9#69826C212yesmaximal, complemented
10#68686C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#22437—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C22×#204—28no—
17C22×#204C418yescomplemented
18C22×#204C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C22×#204C2211yescharacteristic
22C2×D3×C22⋊C4—21no—
23C2×C4×D12C2212yes—
24C2×C4×Dic6C2212yescomplemented
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×#97C2212yescomplemented
27C3×C4×C22⋊C4C2211yescharacteristic
28#5212C2211yescharacteristic
29#5212C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#818C2212yescomplemented
33C2×#816C2212yescomplemented
34C2×#816C2212yescomplemented
35#4955—21no—
36#4942—21no—
37Dic3×C22⋊C4—21no—
38#4939C2212yes—
39#4929C2212yescomplemented
40#4710—28no—
41#4665—22no—
42#4646—22no—
43C3×#336C2212yescomplemented
44#4551C2212yescomplemented
45C23×SD16—31no—
46C2×#440—32no—
47Q8×C22⋊C4—31no—
48D4×C22⋊C4—31no—
49#1973—32no—
50#1880—32no—
51#1877—32no—
52#1808—31no—
53#1787—31no—
54#1786—31no—
55#1752D311yescharacteristic, complemented
56C24×D3—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C22×D12—24no—
60C22×D12—21no—
61C22×D12—22no—
62C22×D12—42no—
63C22×D12C2×C414yes—
64C22×D12C2311yescharacteristic
65C22×D12C2311yescharacteristic
66C22×C4×D3—22no—
67C22×Dic6—24no—
68C22×Dic6—21no—
69C22×Dic6—22no—
70C22×Dic6—42no—
71C22×Dic6C2×C414yescomplemented
72C22×Dic6C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C22×C3×C8—24no—
75C22×C3×C8C2×C414yescomplemented
76C22×C3×C8C2312yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C42C2312yes—
82C2×#217—21no—
83C2×#217—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—22no—
91C2×#204—24no—
92C2×#204—24no—
93C2×#204—24no—
94C2×#204—232no—
95C2×#204C2×C414yes—
96C2×#204C2×C414yes—
97C2×#204C2×C414yescomplemented
98C2×#204C2×C414yescomplemented
99C2×#204D418yes—
100D3×C22⋊C4—44no—
101C4×D12—24no—
102C4×D12—42no—
103C4×Dic6—24no—
104C4×Dic6—42no—
105C3×C22⋊C8—22no—
106C3×#97—24no—
107C3×#95D412yes—
108C3⋊#95—21no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112#821—24no—
113#821—42no—
114#818—24no—
115#816—24no—
116#816—42no—
117C23×Q8—31no—
118C23×D4—31no—
119C22×SD16—38no—
120C22×SD16—31no—
121C22×SD16—31no—
122C22×SD16—31no—
123C22×SD16—31no—
124C22×SD16—68no—
125C23×C8C2×D311yescharacteristic
126C2×C4×Q8—32no—
127C2×C4×D4—32no—
128C22×C22⋊C4—61no—
129#440—68no—
130C2×#107—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8C2×D311yescharacteristic
136C2×#97C2×D312yescomplemented
137#391—32no—
138#391—62no—
139#390—32no—
140#386—62no—
141#380—31no—
142#380—31no—
143#380—61no—
144#379—61no—
145C4×C22⋊C4—61no—
146C4×C22⋊C4C2×D311yescharacteristic
147#340—32no—
148#336C2×D312yescomplemented
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C23×D3—21no—
152C23×D3—24no—
153C23×D3—44no—
154C23×D3—42no—
155C23×D3—42no—
156C23×D3—82no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C22×C412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4#4112yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—24no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—44no—
171C2×D12—21no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—28no—
176C2×D12—24no—
177C2×D12—21no—
178C2×D12—44no—
179C2×D12—48no—
180C2×D12—44no—
181C2×D12C22×C412yes—
182C2×D12C22×C412yes—
183C2×D12C2×D414yes—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—82no—
187C2×Dic6—21no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—28no—
192C2×Dic6—24no—
193C2×Dic6—21no—
194C2×Dic6—44no—
195C2×Dic6—48no—
196C2×Dic6—44no—
197C2×Dic6C22×C412yes—
198C2×Dic6C22×C412yes—
199C2×Dic6C2×D414yes—
200C2×C3×C8—22no—
201C2×C3×C8—28no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8C22×C412yes—
206C2×C3×C8C22×C412yes—
207C2×C3×C8C2×D414yes—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C42—22no—
211C3×C42—22no—
212#217—42no—
213#217—42no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—44no—
216(C2×D3)⋊C4—82no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221Dic3⋊C4—44no—
222C4×Dic3—44no—
223#204—28no—
224#204—28no—
225#204—28no—
226#204C22⋊C4116yescomplemented
227C25—61no—
228C22×Q8—34no—
229C22×Q8—31no—
230C22×Q8—31no—
231C22×Q8—64no—
232C22×Q8—61no—
233C22×Q8—62no—
234C22×Q8—122no—
235C22×D4—34no—
236C22×D4—31no—
237C22×D4—31no—
238C22×D4—64no—
239C22×D4—61no—
240C22×D4—62no—
241C22×D4—122no—
242C23×C4—61no—
243C23×C4—62no—
244C23×C4C22×D311yescharacteristic
245C2×SD16—34no—
246C2×SD16—34no—
247C2×SD16—34no—
248C2×SD16—34no—
249C2×SD16—38no—
250C2×SD16—632no—
251C2×SD16—64no—
252C2×SD16—64no—
253C2×SD16—64no—
254C22×C8—24no—
255C22×C8C4×D314yescomplemented
256C22×C8C22×D312yes—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C4×Q8—64no—
260C4×Q8—122no—
261C4×D4—64no—
262C4×D4—122no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—62no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—124no—
272C2×C22⋊C4D1211yescharacteristic
273C2×C22⋊C4D1211yescharacteristic
274C2×C42—62no—
275C2×C42C22×D312yes—
276#107—64no—
277#103—64no—
278#103—122no—
279#102—64no—
280#102—122no—
281C22⋊C8—22no—
282#97—24no—
283#95—62no—
284#95—61no—
285#95—61no—
286#95—62no—
287#95D1212yes—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3C4×D412yes—
291C23×C3AGL(1,ℤ/8)12yes—
292C23×C3#13712yes—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—22no—
297C22×D3—48no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—48no—
306C22×D3—84no—
307C22×D3—88no—
308C22×D3—88no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4C4×D412yes—
320C2×C3×C4C23×C411yescharacteristic
321C2×C3×C4C22×D412yes—
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4H(5,𝔽2)12yes—
324C2×Dic3—22no—
325C2×Dic3—24no—
326C2×Dic3—22no—
327C2×Dic3—48no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—22no—
336D12—22no—
337D12—22no—
338D12—24no—
339D12—48no—
340D12—42no—
341D12C2×C22⋊C414yes—
342C4×D3—88no—
343Dic6—22no—
344Dic6—24no—
345Dic6—22no—
346Dic6—22no—
347Dic6—48no—
348Dic6—42no—
349Dic6C2×C22⋊C414yescomplemented
350C3×C8—22no—
351C3×C8—42no—
352C3×C8C2×C22⋊C414yescomplemented
353C24—61no—
354C24—61no—
355C24—64no—
356C24—124no—
357C24—122no—
358C24—122no—
359C24—242no—
360C24C2×D1211yescharacteristic
361C2×Q8—32no—
362C2×Q8—32no—
363C2×Q8—34no—
364C2×Q8—68no—
365C2×Q8—64no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—124no—
372C2×Q8—124no—
373C2×Q8—128no—
374C2×D4—32no—
375C2×D4—32no—
376C2×D4—34no—
377C2×D4—68no—
378C2×D4—64no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—124no—
385C2×D4—128no—
386C2×D4—124no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—61no—
390C22×C4—64no—
391C22×C4—61no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—124no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4—242no—
398C22×C4C2×C4×D312yes—
399C22×C4C2×D1212yes—
400C22×C4C2×D1212yes—
401C22×C4C2×D1211yescharacteristic
402C22×C4#23512yes—
403C22×C4C23×D311yescharacteristic
404SD16—316no—
405SD16—68no—
406SD16—68no—
407SD16—68no—
408C2×C8—22no—
409C2×C8—28no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8C2×C4×D312yes—
414C2×C8C2×C4×D312yes—
415C2×C8D3×D414yes—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—62no—
420C4⋊C4—124no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—124no—
424C22⋊C4—124no—
425C22⋊C4—122no—
426C22⋊C4—122no—
427C22⋊C4—242no—
428C42—22no—
429C42—22no—
430C42—124no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3#44014yescomplemented
436C22×C3C2×C4×D411yescharacteristic
437C22×C3D4212yes—
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3C2×#13711yescharacteristic
440C22×C3#57712yes—
441C2×D3—24no—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—816no—
450C2×D3—82no—
451C2×D3—82no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4C22×C22⋊C411yescharacteristic
456C3×C4#51811yescharacteristic
457C3×C4#54612yes—
458Dic3—24no—
459Dic3—42no—
460Dic3—82no—
461Dic3—82no—
462C23—22no—
463C23—22no—
464C23—62no—
465C23—64no—
466C23—62no—
467C23—128no—
468C23—121no—
469C23—121no—
470C23—122no—
471C23—128no—
472C23—121no—
473C23—121no—
474C23—122no—
475C23—122no—
476C23—244no—
477C23—248no—
478C23—248no—
479C23C4×D1212yes—
480C23#90812yes—
481C23#90912yes—
482C23C22×D1211yescharacteristic
483Q8—34no—
484Q8—62no—
485Q8—64no—
486Q8—62no—
487Q8—62no—
488Q8—128no—
489Q8—122no—
490D4—34no—
491D4—62no—
492D4—62no—
493D4—62no—
494D4—64no—
495D4—128no—
496D4—122no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—128no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—248no—
519C2×C4C4×D1212yes—
520C2×C4C22×C4×D311yescharacteristic
521C2×C4C2×#23511yescharacteristic
522C2×C4C2×D3×D412yes—
523C2×C4#100912yes—
524C8—22no—
525C8—42no—
526C8D3×C22⋊C414yescomplemented
527C2×C3—24no—
528C2×C3D4×C22⋊C411yescharacteristic
529C2×C3C2×#44011yescharacteristic
530C2×C3#294911yescharacteristic
531C2×C3#326512yes—
532C2×C3#326612yes—
533D3—44no—
534D3—82no—
535C22—22no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—64no—
540C22—124no—
541C22—124no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22—2416no—
548C22—242no—
549C22—242no—
550C22#471014yescomplemented
551C22C2×C4×D1211yescharacteristic
552C22D4×D1212yes—
553C22C2×#90811yescharacteristic
554C22C2×#90911yescharacteristic
555C22#575112yes—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—64no—
560C4—122no—
561C4—242no—
562C4—242no—
563C4C2×D3×C22⋊C411yescharacteristic
564C4#552111yescharacteristic
565C4#555012yes—
566C3#2243711yescharacteristic, complemented, Sylow
567C2—24no—
568C2—124no—
569C2—242no—
570C2C22⋊C4×D1211yescharacteristic
571C2C2×#471011yescharacteristic
572C2#7857211yescharacteristic
573C2#7949412yes—
574C2#7949512yes—
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1752C2
socleC23×C3C22×D4
prime radicalC23×D12C22
cosocleC22×C3×C4C24
hypercenter#1752D3
hypocenterC3#22437
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
2. C2×#4710C2semidirect
4. C22⋊C4×Dic6C2semidirect
5. C3×#1752C2semidirect
6. #71971C2semidirect
7. #69963C2semidirect
9. #69826C2semidirect
10. #68686C2semidirect
11. #68672C2semidirect
17. C22×#204C4semidirect
24. C2×C4×Dic6C22semidirect
26. C2×C3×#97C22semidirect
32. C2×#818C22semidirect
33. C2×#816C22semidirect
34. C2×#816C22semidirect
39. #4929C22semidirect
43. C3×#336C22semidirect
44. #4551C22semidirect
55. #1752D3semidirect
71. C22×Dic6C2×C4semidirect
75. C22×C3×C8C2×C4semidirect
97. C2×#204C2×C4semidirect
98. C2×#204C2×C4semidirect
136. C2×#97C2×D3semidirect
148. #336C2×D3semidirect
226. #204C22⋊C4semidirect
255. C22×C8C4×D3semidirect
349. Dic6C2×C22⋊C4semidirect
352. C3×C8C2×C22⋊C4semidirect
435. C22×C3#440semidirect
526. C8D3×C22⋊C4semidirect
550. C22#4710semidirect
566. C3#22437semidirect

Maximal subgroups

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

subgroupindex
C23×#2042
C2×#47102
C22⋊C4×D122
C22⋊C4×Dic62
C3×#17522
#719712
#699632
#699592
#698262
#686862
#686722
#224373