Finite Group Databasebeta

← #10823174

Subgroups — #10823174

Order 768.

1570 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic
2C3×#2309C211yesmaximal, characteristic, complemented
3C22×#832C211yesmaximal, characteristic, complemented
4#82270C211yesmaximal, characteristic, complemented
5C2×#5002C211yesmaximal, characteristic, complemented
6C2×#4999C211yesmaximal, characteristic, complemented
7#71581C211yesmaximal, characteristic, complemented
8#69651C214yesmaximal, complemented
9#69650C214yesmaximal, complemented
10#23544—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×#214C2212yes—
16C2×#932—24no—
17C2×#932—24no—
18C2×#932—24no—
19C22×#213C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C3×#390—22no—
24C3×#390C2214yes—
25C3×#379—22no—
26C2×#832—24no—
27C2×#832C2214yes—
28#5002—24no—
29#4999—24no—
30#4987—22no—
31C2×#810C2212yescomplemented
32C2×#809C2212yescomplemented
33#4958—22no—
34#4939C2212yescomplemented
35#4929C2212yescomplemented
36C2×C4⋊(C3⋊C8)C2214yescomplemented
37#4529C2214yescomplemented
38C22×AGL(1,ℤ/8)—31no—
39C2×#424—31no—
40C2×#416—31no—
41C22×#102—31no—
42#2311—31no—
43#2309D311yescharacteristic, complemented
44#2055—34no—
45#2054—34no—
46#1736—31no—
47C25×C3—21no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4D414yes—
55C22×C3×D4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235D414yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6C2311yescharacteristic
68C3×#117—22no—
69C3×#117—22no—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C42—21no—
79C2×C3×C42C2312yes—
80C2×#214—24no—
81C2×#214—24no—
82C2×#214—24no—
83#932—232no—
84C2×#213—24no—
85C2×#213—24no—
86C2×#213—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2312yes—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208D414yes—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2312yes—
100#832—28no—
101C3⋊#95—24no—
102#810—24no—
103#809—24no—
104C4⋊(C3⋊C8)—28no—
105C22×#41—31no—
106C23×D4C2×D311yescharacteristic
107C2×AGL(1,ℤ/8)—64no—
108C2×AGL(1,ℤ/8)—64no—
109C2×AGL(1,ℤ/8)—64no—
110C22×SD16—32no—
111C22×D8—32no—
112C22×M16—31no—
113C23×C8—31no—
114C2×#117C2×D311yescharacteristic
115C2×#117C2×D311yescharacteristic
116C22×C4⋊C4—31no—
117#424—64no—
118C2×C4⋊C8—34no—
119#416—64no—
120C2×#103—32no—
121C2×#102—34no—
122C2×#102—32no—
123C2×#102—64no—
124#391—32no—
125#390—22no—
126#390—32no—
127#390C2×D314yes—
128#386—62no—
129#380—62no—
130#379—22no—
131#340—34no—
132C24×C3—22no—
133C24×C3—24no—
134C24×C3—42no—
135C24×C3—42no—
136C24×C3—82no—
137C24×C3—82no—
138C24×C3C2×D411yescharacteristic
139C23×D3—21no—
140C2×C3×D4—28no—
141C2×C3×D4—24no—
142C2×C3×D4—28no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4C2×D414yes—
153C22×C3×C4—22no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D414yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4#4112yes—
160C22×C3×C4#4112yes—
161C22×C3×C4#4112yes—
162C22×C3×C4C2411yescharacteristic
163C2×#67—24no—
164C2×#67—42no—
165C2×#67—42no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—42no—
169C22×Dic3—82no—
170#235—28no—
171#235—28no—
172#235—44no—
173#235—44no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12C2×D412yes—
180C2×C4×D3—24no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6C2×D412yes—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—84no—
195C3×C22⋊C4—84no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—42no—
199#214—216no—
200#213—216no—
201(C2×D3)⋊C4—84no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—42no—
205#208—28no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—44no—
212C2×C3⋊C8C2×D414yes—
213C25—21no—
214C2×#41—34no—
215C2×#41—62no—
216C2×#41—62no—
217C2×#41—62no—
218C2×#41—62no—
219C2×#41—62no—
220C22×Q8—31no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—31no—
228C22×D4—61no—
229C22×D4D1214yes—
230C22×D4C22×D312yes—
231C23×C4—61no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234AGL(1,ℤ/8)—632no—
235C2×SD16—64no—
236C2×SD16—64no—
237C2×SD16—64no—
238C2×D8—64no—
239C2×D8—64no—
240C2×D8—64no—
241C2×M16—34no—
242C2×M16—62no—
243C2×M16—62no—
244C22×C8—32no—
245C22×C8—32no—
246C22×C8—62no—
247C22×C8—62no—
248C22×C8—62no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—22no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4#6712yes—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4#6712yes—
264C2×C42—21no—
265C2×C42C22×D312yes—
266C4⋊C8—68no—
267#103—64no—
268#102—64no—
269#102—68no—
270#95—64no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—48no—
279C23×C3—84no—
280C23×C3—84no—
281C23×C3—88no—
282C23×C3—88no—
283C23×C3#12112yes—
284C23×C3#12112yes—
285C23×C3#12112yes—
286C23×C3C22×D411yescharacteristic
287C22×D3—22no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—42no—
291C22×D3—42no—
292C3×D4—24no—
293C3×D4—24no—
294C3×D4—42no—
295C3×D4—42no—
296C3×D4—48no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4#12014yes—
304C2×C3×C4#12112yes—
305C2×C3×C4#12112yes—
306C2×C3×C4#12112yes—
307C2×C3×C4#12314yes—
308C2×C3×C4#12314yes—
309C2×C3×C4#12414yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316#67—48no—
317C2×Dic3—22no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—44no—
322C2×Dic3—84no—
323D12—22no—
324D12—22no—
325D12—41no—
326D12—41no—
327C4×D3—48no—
328Dic6—22no—
329Dic6—22no—
330Dic6—41no—
331Dic6—41no—
332C3⋊C8—24no—
333C3⋊C8—42no—
334C24—22no—
335C24—24no—
336C24—42no—
337C24—42no—
338C24—61no—
339C24—82no—
340C24—82no—
341C24C2×#6711yescharacteristic
342#41—68no—
343#41—68no—
344#41—124no—
345#41—124no—
346C2×Q8—32no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—32no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—64no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—122no—
372C2×D4—122no—
373C2×D4C2×D1214yes—
374C22×C4—22no—
375C22×C4—61no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—122no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4—242no—
382C22×C4C2×D1212yes—
383C22×C4#23512yes—
384C22×C4D3×D412yes—
385C22×C4D3×D412yes—
386C22×C4#23712yes—
387C22×C4#23712yes—
388C22×C4C2×#6714yes—
389C22×C4C2×#6711yescharacteristic
390C22×C4C23×D311yescharacteristic
391SD16—616no—
392D8—616no—
393M16—68no—
394C2×C8—34no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C2×C8—124no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C22⋊C4—24no—
408C22⋊C4—42no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—244no—
412C42—22no—
413C42—22no—
414C42—42no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—816no—
423C22×C3—84no—
424C22×C3—84no—
425C22×C3#39414yes—
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12111yescharacteristic
428C22×C3C2×#12111yescharacteristic
429C22×C3#57612yes—
430C22×C3#57712yes—
431C2×D3—42no—
432C2×D3—44no—
433C2×D3—81no—
434C2×D3—81no—
435C2×D3—82no—
436C3×C4—44no—
437C3×C4—82no—
438C3×C4#39414yes—
439C3×C4C2×#12011yescharacteristic
440C3×C4C2×#12311yescharacteristic
441C3×C4C2×#12311yescharacteristic
442C3×C4C2×#12411yescharacteristic
443Dic3—42no—
444Dic3—82no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—48no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—62no—
454C23—84no—
455C23—84no—
456C23—88no—
457C23—88no—
458C23—121no—
459C23—121no—
460C23—122no—
461C23—122no—
462C23#93012yes—
463C23#93812yes—
464C23#93912yes—
465C23C22×#6711yescharacteristic
466Q8—62no—
467Q8—62no—
468Q8—121no—
469Q8—121no—
470D4—24no—
471D4—24no—
472D4—48no—
473D4—42no—
474D4—42no—
475D4—62no—
476D4—62no—
477D4—128no—
478D4—121no—
479D4—121no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—62no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—128no—
490C2×C4—122no—
491C2×C4—124no—
492C2×C4—244no—
493C2×C4#88214yes—
494C2×C4#88314yes—
495C2×C4#88514yes—
496C2×C4#88614yes—
497C2×C4#93012yes—
498C2×C4#93812yes—
499C2×C4#93912yes—
500C2×C4C22×D1211yescharacteristic
501C2×C4C2×#23511yescharacteristic
502C2×C4C2×D3×D411yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C2×C4C2×#23711yescharacteristic
505C2×C4C2×#23711yescharacteristic
506C8—64no—
507C8—122no—
508C2×C3—24no—
509C2×C3—84no—
510C2×C3C2×#39411yescharacteristic
511C2×C3#304111yescharacteristic
512C2×C3#304211yescharacteristic
513C2×C3#304512yes—
514C2×C3#306212yes—
515D3—82no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—816no—
524C22—84no—
525C22—84no—
526C22—122no—
527C22—124no—
528C22—241no—
529C22—241no—
530C22—242no—
531C22#496114yes—
532C22C2×#93011yescharacteristic
533C22C2×#93811yescharacteristic
534C22C2×#93911yescharacteristic
535C22#583512yes—
536C22#583612yes—
537C4—44no—
538C4—82no—
539C4—122no—
540C4—242no—
541C4#496114yes—
542C4C2×#88211yescharacteristic
543C4C2×#88311yescharacteristic
544C4C2×#88511yescharacteristic
545C4C2×#88611yescharacteristic
546C3#2354411yescharacteristic, complemented, Sylow
547C2—24no—
548C2—84no—
549C2—242no—
550C2C2×#496111yescharacteristic
551C2#8254011yescharacteristic
552C2#8254411yescharacteristic
553C2#8337912yes—
554C2#8338512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2309C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenter#2309D3
hypocenterC3#23544
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C3×#2309C2semidirect
3. C22×#832C2semidirect
4. #82270C2semidirect
5. C2×#5002C2semidirect
6. C2×#4999C2semidirect
7. #71581C2semidirect
8. #69651C2semidirect
9. #69650C2semidirect
31. C2×#810C22semidirect
32. C2×#809C22semidirect
34. #4939C22semidirect
35. #4929C22semidirect
36. C2×C4⋊(C3⋊C8)C22semidirect
37. #4529C22semidirect
43. #2309D3semidirect
546. C3#23544semidirect

Maximal subgroups

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

subgroupindex
C22×#9322
C3×#23092
C22×#8322
#822702
C2×#50022
C2×#49992
#715812
#696512
#696502
#235443