Finite Group Databasebeta

← #10823193

Subgroups — #10823193

Order 768.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#908C211yesmaximal, characteristic
2C2×C3×#423C211yesmaximal, characteristic, complemented
3C2×#5133C211yesmaximal, characteristic, complemented
4C2×#5131C211yesmaximal, characteristic, complemented
5#82070C211yesmaximal, characteristic, complemented
6C2×#4939C211yesmaximal, characteristic, complemented
7#71985C211yesmaximal, characteristic, complemented
8#70482C214yesmaximal, complemented
9#70481C214yesmaximal, complemented
10#23546—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×(C2×D3)⋊C4—22no—
17C2×#926C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#908—24no—
20C2×#908—24no—
21C2×#908—24no—
22C22×D24C2212yes—
23C22×#204C2212yes—
24C3×#423—24no—
25C2×C3×C4⋊C8C2214yescomplemented
26#5213—22no—
27#5133—24no—
28#5131—24no—
29C2×#821C2214yes—
30C2×#821C2212yescomplemented
31C2×#816C2212yescomplemented
32#4939—21no—
33#4939—21no—
34#4939C2214yes—
35#4939C2212yescomplemented
36C4×(C2×D3)⋊C4—21no—
37C4×(C2×D3)⋊C4—21no—
38#4929C2212yescomplemented
39#4551C2214yescomplemented
40C22×AGL(1,ℤ/8)—31no—
41C2×#423D311yescharacteristic, complemented
42C2×#417—31no—
43C2×#418—31no—
44#2297—31no—
45C2×#390—31no—
46#2055—34no—
47#2054—34no—
48#1736—31no—
49C24×D3—21no—
50C23×C3×C4D412yes—
51C23×C3×C4C2311yescharacteristic
52C22×#67—21no—
53C23×Dic3—21no—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235D414yes—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—22no—
66C22×D12D414yes—
67C22×D12C2312yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6C2311yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16D414yes—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C2×C3×C42—22no—
80C2×C3×C42—22no—
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42C2312yes—
84C2×#217—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—48no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2312yes—
97C2×C4×Dic3—21no—
98#908—232no—
99C2×D24—24no—
100C2×D24—24no—
101C2×D24—24no—
102C2×#204—24no—
103C2×#204—24no—
104C2×#204—24no—
105C3×C4⋊C8—28no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108#821—28no—
109#821—24no—
110#816—24no—
111C22×#41—31no—
112C23×D4—31no—
113C2×AGL(1,ℤ/8)—64no—
114C2×AGL(1,ℤ/8)—64no—
115C2×AGL(1,ℤ/8)—64no—
116C22×SD16—32no—
117C22×D8—32no—
118C22×M16C2×D311yescharacteristic
119C22×M16C2×D311yescharacteristic
120C2×#117—31no—
121C22×C4⋊C4—31no—
122C22×C22⋊C4—62no—
123C22×C42C2×D311yescharacteristic
124#423—24no—
125C2×C4⋊C8C2×D314yescomplemented
126#418—64no—
127#417—64no—
128C2×#103—32no—
129C2×#102—32no—
130C2×#102—34no—
131#391—32no—
132#390—32no—
133#390—34no—
134#390—61no—
135#390—61no—
136#386—62no—
137C4×C22⋊C4—61no—
138C4×C22⋊C4—61no—
139#340—34no—
140C24×C3C2×D411yescharacteristic
141C23×D3—24no—
142C23×D3—21no—
143C23×D3—22no—
144C23×D3—44no—
145C23×D3—44no—
146C23×D3—42no—
147C23×D3—42no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C2×#67—24no—
162C2×#67—42no—
163C2×#67—42no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—44no—
168C22×Dic3—42no—
169#235—28no—
170#235—28no—
171#235—44no—
172#235—44no—
173C2×D12—28no—
174C2×D12—28no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—24no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—44no—
189C2×D12—44no—
190C2×D12C2×D414yes—
191C2×D12C2×D412yes—
192C2×C4×D3—24no—
193C2×C4×D3—42no—
194C2×C4×D3—42no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6C2×D412yes—
201C3×M16—28no—
202C3×M16—44no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8C2×D414yes—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—22no—
213#217—42no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—48no—
216(C2×D3)⋊C4—44no—
217(C2×D3)⋊C4—48no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—42no—
223C4×Dic3—42no—
224D24—216no—
225#204—216no—
226C25—61no—
227C2×#41—34no—
228C2×#41—62no—
229C2×#41—62no—
230C2×#41—62no—
231C2×#41—62no—
232C2×#41—62no—
233C22×Q8—31no—
234C22×D4—34no—
235C22×D4—31no—
236C22×D4—32no—
237C22×D4—61no—
238C22×D4—61no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4—62no—
244C23×C4—61no—
245C23×C4—61no—
246C23×C4D1212yes—
247C23×C4C22×D311yescharacteristic
248AGL(1,ℤ/8)—632no—
249C2×SD16—64no—
250C2×SD16—64no—
251C2×SD16—64no—
252C2×D8—64no—
253C2×D8—64no—
254C2×D8—64no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16#6714yes—
261C22×C8C22×D312yes—
262C22×C8C22×D312yes—
263#117—62no—
264#117—62no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—128no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—61no—
279C2×C42D1211yescharacteristic
280C2×C42D1211yescharacteristic
281C2×C42C22×D312yes—
282C4⋊C8—28no—
283#103—64no—
284#102—64no—
285#102—68no—
286#95—62no—
287#95—62no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3#12112yes—
291C23×C3#12112yes—
292C23×C3#12112yes—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—416no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—44no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—44no—
305C22×D3—48no—
306C22×D3—84no—
307C22×D3—84no—
308C22×D3—88no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—48no—
315C2×C3×C4#12014yes—
316C2×C3×C4#12112yes—
317C2×C3×C4#12112yes—
318C2×C3×C4#12112yes—
319C2×C3×C4#12314yes—
320C2×C3×C4#12314yes—
321C2×C3×C4#12414yes—
322C2×C3×C4AGL(1,ℤ/8)14yescomplemented
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329#67—48no—
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—44no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—84no—
339D12—24no—
340D12—22no—
341D12—24no—
342D12—22no—
343D12—42no—
344D12—41no—
345D12—42no—
346D12—41no—
347D12—48no—
348C4×D3—48no—
349Dic6—22no—
350Dic6—22no—
351Dic6—41no—
352Dic6—41no—
353C3×C8—24no—
354C3×C8—42no—
355C24—64no—
356C24—61no—
357C24—62no—
358C24—124no—
359C24—124no—
360C24—122no—
361C24—122no—
362C24C2×D1211yescharacteristic
363#41—68no—
364#41—68no—
365#41—124no—
366#41—124no—
367C2×Q8—32no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—61no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×D4—32no—
374C2×D4—34no—
375C2×D4—64no—
376C2×D4—68no—
377C2×D4—64no—
378C2×D4—68no—
379C2×D4—62no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—61no—
385C2×D4—62no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—62no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—124no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—61no—
399C22×C4—64no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4—124no—
405C22×C4—122no—
406C22×C4C2×D1212yes—
407C22×C4C2×D1212yes—
408C22×C4C2×D1211yescharacteristic
409C22×C4#23512yes—
410C22×C4D3×D412yes—
411C22×C4D3×D412yes—
412C22×C4#23912yes—
413C22×C4#23912yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416SD16—616no—
417D8—616no—
418M16—28no—
419M16—44no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8C2×#6714yes—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—122no—
431C22⋊C4—128no—
432C22⋊C4—124no—
433C22⋊C4—122no—
434C22⋊C4—128no—
435C22⋊C4—124no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—122no—
441C22×C3—24no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3#39414yes—
446C22×C3C2×#12111yescharacteristic
447C22×C3C2×#12111yescharacteristic
448C22×C3C2×#12111yescharacteristic
449C22×C3C2×AGL(1,ℤ/8)12yes—
450C22×C3#57512yes—
451C2×D3—42no—
452C2×D3—44no—
453C2×D3—48no—
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—816no—
457C2×D3—82no—
458C2×D3—82no—
459C2×D3—82no—
460C2×D3—82no—
461C2×D3—84no—
462C3×C4—24no—
463C3×C4—42no—
464C3×C4#39414yes—
465C3×C4C2×#12011yescharacteristic
466C3×C4C2×#12311yescharacteristic
467C3×C4C2×#12311yescharacteristic
468C3×C4C2×#12411yescharacteristic
469Dic3—42no—
470Dic3—82no—
471Dic3—82no—
472C23—22no—
473C23—22no—
474C23—62no—
475C23—64no—
476C23—1216no—
477C23—124no—
478C23—122no—
479C23—122no—
480C23—124no—
481C23—128no—
482C23—121no—
483C23—122no—
484C23—121no—
485C23—122no—
486C23—248no—
487C23—244no—
488C23—244no—
489C23#89512yes—
490C23#89512yes—
491C23#93012yes—
492C23C22×D1211yescharacteristic
493Q8—62no—
494Q8—62no—
495Q8—121no—
496Q8—121no—
497D4—64no—
498D4—62no—
499D4—62no—
500D4—64no—
501D4—128no—
502D4—128no—
503D4—122no—
504D4—121no—
505D4—122no—
506D4—121no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—62no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—128no—
517C2×C4—124no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—244no—
523C2×C4#89414yes—
524C2×C4#89414yes—
525C2×C4#89512yes—
526C2×C4#89512yes—
527C2×C4#90814yescomplemented
528C2×C4#93012yes—
529C2×C4#93714yes—
530C2×C4#94714yes—
531C2×C4C2×#23511yescharacteristic
532C2×C4C2×D3×D411yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×#23911yescharacteristic
535C2×C4C2×#23911yescharacteristic
536C2×C4C22×#6711yescharacteristic
537C8—24no—
538C8—42no—
539C2×C3—24no—
540C2×C3C2×#39411yescharacteristic
541C2×C3#303612yes—
542C2×C3#303811yescharacteristic
543C2×C3#304011yescharacteristic
544C2×C3#305512yes—
545D3—82no—
546D3—84no—
547C22—24no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—122no—
552C22—124no—
553C22—128no—
554C22—122no—
555C22—122no—
556C22—2416no—
557C22—244no—
558C22—242no—
559C22—242no—
560C22—242no—
561C22—242no—
562C22#499114yes—
563C22C2×#89511yescharacteristic
564C22C2×#89511yescharacteristic
565C22C2×#90812yes—
566C22#574812yes—
567C22C2×#93011yescharacteristic
568C4—24no—
569C4—42no—
570C4—122no—
571C4—242no—
572C4—242no—
573C4#499114yes—
574C4C2×#89411yescharacteristic
575C4C2×#89411yescharacteristic
576C4C2×#93711yescharacteristic
577C4C2×#94711yescharacteristic
578C3#2354611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—244no—
581C2—242no—
582C2#7905512yes—
583C2#7905611yescharacteristic
584C2#7905811yescharacteristic
585C2C2×#499111yescharacteristic
586C2#8377212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#423C2
socleC23×C3C22×D4
prime radicalC22×#908C2
hypercenterC2×#423D3
hypocenterC3#23546
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. C2×C3×#423C2semidirect
3. C2×#5133C2semidirect
4. C2×#5131C2semidirect
5. #82070C2semidirect
6. C2×#4939C2semidirect
7. #71985C2semidirect
8. #70482C2semidirect
9. #70481C2semidirect
25. C2×C3×C4⋊C8C22semidirect
30. C2×#821C22semidirect
31. C2×#816C22semidirect
35. #4939C22semidirect
38. #4929C22semidirect
39. #4551C22semidirect
41. C2×#423D3semidirect
125. C2×C4⋊C8C2×D3semidirect
322. C2×C3×C4AGL(1,ℤ/8)semidirect
527. C2×C4#908semidirect
578. C3#23546semidirect

Maximal subgroups

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

subgroupindex
C22×#9082
C2×C3×#4232
C2×#51332
C2×#51312
#820702
C2×#49392
#719852
#704822
#704812
#235463