Finite Group Databasebeta

← #10823192

Subgroups — #10823192

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×#932C211yesmaximal, characteristic
2C2×C3×#390C211yesmaximal, characteristic, complemented
3C2×#5215C211yesmaximal, characteristic, complemented
4C2×#4966C211yesmaximal, characteristic, complemented
5#82070C211yesmaximal, characteristic, complemented
6C2×#4924C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#69651C214yesmaximal, complemented
9#69650C214yesmaximal, complemented
10#23546—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4—22no—
15C22×C3×C42C2211yescharacteristic
16C22×#214C2212yes—
17C2×#932—24no—
18C2×#932—24no—
19C2×#932—24no—
20C22×#213C2212yes—
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#390—21no—
25C3×#390—21no—
26C3×#390C2214yes—
27#5215—24no—
28C2×#832C2214yes—
29#5213—22no—
30C2×#810C2212yescomplemented
31#4966—24no—
32C2×#809C2212yescomplemented
33#4939C2212yescomplemented
34C4×(C2×D3)⋊C4—21no—
35C4×(C2×D3)⋊C4—21no—
36#4929C2212yescomplemented
37#4924—24no—
38C2×C4⋊(C3⋊C8)C2214yescomplemented
39#4529C2214yescomplemented
40C22×AGL(1,ℤ/8)—31no—
41C2×#423—31no—
42C2×#417—31no—
43C2×#418—31no—
44#2297—31no—
45C2×#390D311yescharacteristic, complemented
46#2055—34no—
47#2054—34no—
48#1736—31no—
49C25×C3—21no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4—21no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4D414yes—
57C22×C3×D4C2312yes—
58C23×C3×C4—21no—
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235D414yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6C2311yescharacteristic
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—48no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2312yes—
83C2×#217—21no—
84C2×#214—24no—
85C2×#214—24no—
86C2×#214—24no—
87#932—232no—
88C2×#213—24no—
89C2×#213—24no—
90C2×#213—24no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93#926—22no—
94#926—22no—
95C2×C4⋊Dic3C2312yes—
96C2×C4×Dic3—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—22no—
101C2×#208—22no—
102C2×#208D414yes—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#832—28no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108#810—24no—
109#809—24no—
110C4⋊(C3⋊C8)—28no—
111C22×#41—31no—
112C23×D4C2×D311yescharacteristic
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×M16—31no—
119C22×M16—31no—
120C2×#117—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C22⋊C4—22no—
123C22×C42C2×D311yescharacteristic
124#423—64no—
125C2×C4⋊C8—34no—
126#418—64no—
127#417—64no—
128C2×#103—32no—
129C2×#102—34no—
130C2×#102—32no—
131#391—32no—
132#390—21no—
133#390—21no—
134#390—32no—
135#390C2×D314yes—
136#386—62no—
137C4×C22⋊C4—61no—
138C4×C22⋊C4—61no—
139#340—34no—
140C24×C3—24no—
141C24×C3—22no—
142C24×C3—44no—
143C24×C3—44no—
144C24×C3—42no—
145C24×C3—42no—
146C24×C3C2×D411yescharacteristic
147C23×D3—21no—
148C2×C3×D4—28no—
149C2×C3×D4—24no—
150C2×C3×D4—28no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C2×C3×D4C2×D414yes—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—44no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C2×#67—24no—
178C2×#67—42no—
179C2×#67—42no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182#235—28no—
183#235—28no—
184#235—44no—
185#235—44no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
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×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C22⋊C4—48no—
205C3×C22⋊C4—48no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—22no—
210#217—42no—
211#214—216no—
212#213—216no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—44no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4×Dic3—42no—
218#208—28no—
219#208—44no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C2×D414yes—
226C25—21no—
227C2×#41—34no—
228C2×#41—62no—
229C2×#41—62no—
230C2×#41—62no—
231C2×#41—62no—
232C2×#41—62no—
233C22×Q8—31no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—31no—
241C22×D4—61no—
242C22×D4D1214yes—
243C22×D4C22×D312yes—
244C23×C4—21no—
245C23×C4—61no—
246C23×C4#6712yes—
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—34no—
256C2×M16—62no—
257C2×M16—62no—
258C2×M16—62no—
259C2×M16—62no—
260C2×M16—62no—
261C22×C8—32no—
262C22×C8—32no—
263#117—62no—
264#117—62no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4C22×D312yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—61no—
279C2×C42#6711yescharacteristic
280C2×C42#6711yescharacteristic
281C2×C42C22×D312yes—
282C4⋊C8—68no—
283#103—64no—
284#102—64no—
285#102—68no—
286#95—62no—
287#95—62no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—416no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—48no—
297C23×C3—84no—
298C23×C3—84no—
299C23×C3—88no—
300C23×C3#12112yes—
301C23×C3#12112yes—
302C23×C3#12112yes—
303C23×C3C22×D411yescharacteristic
304C22×D3—22no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—42no—
308C22×D3—42no—
309C3×D4—24no—
310C3×D4—24no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—48no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—48no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—84no—
323C2×C3×C4#12014yes—
324C2×C3×C4#12112yes—
325C2×C3×C4#12112yes—
326C2×C3×C4#12112yes—
327C2×C3×C4#12314yes—
328C2×C3×C4#12314yes—
329C2×C3×C4#12414yes—
330C2×C3×C4AGL(1,ℤ/8)14yescomplemented
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337#67—48no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344D12—22no—
345D12—22no—
346D12—41no—
347D12—41no—
348C4×D3—48no—
349Dic6—22no—
350Dic6—22no—
351Dic6—41no—
352Dic6—41no—
353C3⋊C8—24no—
354C3⋊C8—42no—
355C24—24no—
356C24—22no—
357C24—44no—
358C24—44no—
359C24—42no—
360C24—42no—
361C24—61no—
362C24C2×#6711yescharacteristic
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—28no—
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—32no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—64no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4C2×D1214yes—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—61no—
402C22×C4—64no—
403C22×C4—62no—
404C22×C4—122no—
405C22×C4—122no—
406C22×C4C2×D1212yes—
407C22×C4#23512yes—
408C22×C4D3×D412yes—
409C22×C4D3×D412yes—
410C22×C4#23712yes—
411C22×C4#23712yes—
412C22×C4C2×#6712yes—
413C22×C4C2×#6712yes—
414C22×C4C2×#6711yescharacteristic
415C22×C4C23×D311yescharacteristic
416SD16—616no—
417D8—616no—
418M16—68no—
419M16—124no—
420C2×C8—34no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8—62no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—42no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—124no—
434C22⋊C4—122no—
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—44no—
446C22×C3—48no—
447C22×C3—816no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—84no—
451C22×C3#39414yes—
452C22×C3C2×#12111yescharacteristic
453C22×C3C2×#12111yescharacteristic
454C22×C3C2×#12111yescharacteristic
455C22×C3C2×AGL(1,ℤ/8)12yes—
456C22×C3#57512yes—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—82no—
461C2×D3—82no—
462C3×C4—24no—
463C3×C4—42no—
464C3×C4—82no—
465C3×C4#39414yes—
466C3×C4C2×#12011yescharacteristic
467C3×C4C2×#12311yescharacteristic
468C3×C4C2×#12311yescharacteristic
469C3×C4C2×#12411yescharacteristic
470Dic3—42no—
471Dic3—82no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—416no—
476C23—48no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—62no—
482C23—84no—
483C23—88no—
484C23—84no—
485C23—121no—
486C23—122no—
487C23—121no—
488C23—122no—
489C23#93012yes—
490C23#93812yes—
491C23#93912yes—
492C23C22×#6711yescharacteristic
493Q8—62no—
494Q8—62no—
495Q8—121no—
496Q8—121no—
497D4—24no—
498D4—24no—
499D4—48no—
500D4—42no—
501D4—42no—
502D4—62no—
503D4—62no—
504D4—128no—
505D4—121no—
506D4—121no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—62no—
516C2×C4—84no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—128no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4#88214yes—
524C2×C4#88314yes—
525C2×C4#88514yes—
526C2×C4#88614yes—
527C2×C4#93012yes—
528C2×C4#93214yescomplemented
529C2×C4#93812yes—
530C2×C4#93912yes—
531C2×C4C22×D1211yescharacteristic
532C2×C4C2×#23511yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×#23711yescharacteristic
536C2×C4C2×#23711yescharacteristic
537C8—64no—
538C8—122no—
539C2×C3—24no—
540C2×C3—84no—
541C2×C3C2×#39411yescharacteristic
542C2×C3#303612yes—
543C2×C3#303811yescharacteristic
544C2×C3#304011yescharacteristic
545C2×C3#305512yes—
546D3—82no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—48no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—84no—
556C22—82no—
557C22—122no—
558C22—122no—
559C22—122no—
560C22—242no—
561C22—242no—
562C22#496114yes—
563C22C2×#93011yescharacteristic
564C22C2×#93212yes—
565C22C2×#93811yescharacteristic
566C22C2×#93911yescharacteristic
567C22#582212yes—
568C4—24no—
569C4—42no—
570C4—82no—
571C4—122no—
572C4—242no—
573C4#496114yes—
574C4C2×#88211yescharacteristic
575C4C2×#88311yescharacteristic
576C4C2×#88511yescharacteristic
577C4C2×#88611yescharacteristic
578C3#2354611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—84no—
581C2—242no—
582C2C2×#496111yescharacteristic
583C2#8230712yes—
584C2#8262112yes—
585C2#8294511yescharacteristic
586C2#8316511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#390C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenterC2×#390D3
hypocenterC3#23546
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×C3×#390C2semidirect
3. C2×#5215C2semidirect
4. C2×#4966C2semidirect
5. #82070C2semidirect
6. C2×#4924C2semidirect
7. #71558C2semidirect
8. #69651C2semidirect
9. #69650C2semidirect
30. C2×#810C22semidirect
32. C2×#809C22semidirect
33. #4939C22semidirect
36. #4929C22semidirect
38. C2×C4⋊(C3⋊C8)C22semidirect
39. #4529C22semidirect
45. C2×#390D3semidirect
330. C2×C3×C4AGL(1,ℤ/8)semidirect
528. C2×C4#932semidirect
578. C3#23546semidirect

Maximal subgroups

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

subgroupindex
C22×#9322
C2×C3×#3902
C2×#52152
C2×#49662
#820702
C2×#49242
#715582
#696512
#696502
#235463