Finite Group Databasebeta

← #10823172

Subgroups — #10823172

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
2C22×C3×#102C211yesmaximal, characteristic, complemented
3#84044C211yesmaximal, characteristic, complemented
4C2×#5113C211yesmaximal, characteristic, complemented
5C2×#5102C211yesmaximal, characteristic, complemented
6#82186C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#70217C214yesmaximal, complemented
9#70212C214yesmaximal, complemented
10#23544—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×#214C2212yes—
16C2×#932—24no—
17C2×#932—24no—
18C2×#932—24no—
19C22×#213C2212yes—
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C2×C3×#102—24no—
24C2×C3×#102C2214yes—
25#5247—22no—
26#5221—22no—
27#5221C2214yes—
28#5213—22no—
29#5113—24no—
30C2×#821C2212yescomplemented
31C2×#816C2212yescomplemented
32#5102—24no—
33C2×Dic3⋊C8C2214yescomplemented
34#4988C2212yescomplemented
35#4987—22no—
36#4973C2212yescomplemented
37#4529C2214yescomplemented
38C22×AGL(1,ℤ/8)—31no—
39C2×#424—31no—
40C2×#416—31no—
41C22×#102D311yescharacteristic, complemented
42#2311—31no—
43#2309—31no—
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×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C22×#67—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
68C22×C3×C8—22no—
69C22×C3×C8—22no—
70C22×C3×C8—22no—
71C22×C3×C8C2312yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2312yes—
76C2×#217—21no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217D412yes—
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—
88#926—22no—
89#926—22no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3D412yes—
93C2×C4⋊Dic3C2312yes—
94C2×C4×Dic3—21no—
95C2×C4×Dic3C2312yes—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208D414yes—
99C22×C3⋊C8C2312yes—
100C3×#102—28no—
101C3⋊#95—24no—
102#821—24no—
103#816—24no—
104Dic3⋊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×C8C2×D311yescharacteristic
114C2×#117—31no—
115C2×#117—31no—
116C22×C4⋊C4C2×D311yescharacteristic
117#424—64no—
118C2×C4⋊C8—34no—
119#416—64no—
120C2×#103—32no—
121C2×#102—24no—
122C2×#102—32no—
123C2×#102C2×D314yes—
124#391—32no—
125#390—34no—
126#390—32no—
127#390—62no—
128#386—62no—
129#380—62no—
130#379—62no—
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×C4—42no—
155C22×C3×C4—82no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4#4112yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×C3×C4C2411yescharacteristic
164C2×#67—24no—
165C2×#67—42no—
166C2×#67—42no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3C2×D414yes—
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—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—44no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196#217—24no—
197#217—42no—
198#217—84no—
199#217—84no—
200#214—216no—
201#213—216no—
202(C2×D3)⋊C4—84no—
203C4⋊Dic3—24no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C4×Dic3—42no—
209#208—28no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
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—21no—
226C22×D4—22no—
227C22×D4—31no—
228C22×D4—61no—
229C22×D4#6714yes—
230C22×D4C22×D312yes—
231C23×C4—21no—
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—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—32no—
248C22×C8C22×D312yes—
249#117—62no—
250#117—62no—
251#117—62no—
252#117—62no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C42—32no—
265C2×C42—61no—
266C4⋊C8—68no—
267#103—64no—
268#102—28no—
269#102—64no—
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—42no—
298C2×C3×C4—44no—
299C2×C3×C4—84no—
300C2×C3×C4#12014yes—
301C2×C3×C4#12112yes—
302C2×C3×C4#12112yes—
303C2×C3×C4#12112yes—
304C2×C3×C4#12314yes—
305C2×C3×C4#12314yes—
306C2×C3×C4#12414yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313#67—48no—
314C2×Dic3—24no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323D12—22no—
324D12—22no—
325D12—41no—
326D12—41no—
327C4×D3—48no—
328Dic6—22no—
329Dic6—22no—
330Dic6—41no—
331Dic6—41no—
332C3×C8—42no—
333C3⋊C8—24no—
334C24—22no—
335C24—24no—
336C24—42no—
337C24—42no—
338C24—61no—
339C24—82no—
340C24—82no—
341C24D3×D411yescharacteristic
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—42no—
363C2×D4—42no—
364C2×D4—44no—
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×#6714yes—
374C22×C4—22no—
375C22×C4—34no—
376C22×C4—42no—
377C22×C4—61no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—82no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4#23512yes—
384C22×C4#23512yes—
385C22×C4D3×D412yes—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D412yes—
388C22×C4#23712yes—
389C22×C4C2×#6712yes—
390C22×C4C23×D311yescharacteristic
391SD16—616no—
392D8—616no—
393M16—68no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—34no—
398C2×C8—44no—
399C2×C8—62no—
400C2×C8—62no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—64no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C22⋊C4—64no—
408C22⋊C4—122no—
409C22⋊C4—244no—
410C22⋊C4—244no—
411C22⋊C4—244no—
412C42—62no—
413C42—62no—
414C42—122no—
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—82no—
437C3×C4#39414yes—
438C3×C4C2×#12011yescharacteristic
439C3×C4C2×#12311yescharacteristic
440C3×C4C2×#12311yescharacteristic
441C3×C4C2×#12411yescharacteristic
442Dic3—42no—
443Dic3—44no—
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—122no—
460C23—122no—
461C23—121no—
462C23#88412yes—
463C23#88412yes—
464C23#93912yes—
465C23C2×D3×D411yescharacteristic
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—44no—
481C2×C4—42no—
482C2×C4—64no—
483C2×C4—62no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—84no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—128no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4#88314yes—
494C2×C4#88412yes—
495C2×C4#88412yes—
496C2×C4#88514yes—
497C2×C4#92914yes—
498C2×C4#93714yes—
499C2×C4#93912yes—
500C2×C4C2×#23511yescharacteristic
501C2×C4C2×#23511yescharacteristic
502C2×C4C2×D3×D411yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C2×C4C2×#23711yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—42no—
507C8—64no—
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—124no—
527C22—122no—
528C22—241no—
529C22—241no—
530C22—242no—
531C22#495914yes—
532C22C2×#88411yescharacteristic
533C22C2×#88411yescharacteristic
534C22#575712yes—
535C22#576212yes—
536C22C2×#93911yescharacteristic
537C4—82no—
538C4—124no—
539C4—122no—
540C4—242no—
541C4#495914yes—
542C4C2×#88311yescharacteristic
543C4C2×#88511yescharacteristic
544C4C2×#92911yescharacteristic
545C4C2×#93711yescharacteristic
546C3#2354411yescharacteristic, complemented, Sylow
547C2—24no—
548C2—84no—
549C2—242no—
550C2#7875212yes—
551C2#7876712yes—
552C2C2×#495911yescharacteristic
553C2#8382611yescharacteristic
554C2#8384811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenterC22×#102D3
hypocenterC3#23544
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C3×#102C2semidirect
3. #84044C2semidirect
4. C2×#5113C2semidirect
5. C2×#5102C2semidirect
6. #82186C2semidirect
7. #71558C2semidirect
8. #70217C2semidirect
9. #70212C2semidirect
30. C2×#821C22semidirect
31. C2×#816C22semidirect
33. C2×Dic3⋊C8C22semidirect
34. #4988C22semidirect
36. #4973C22semidirect
37. #4529C22semidirect
41. C22×#102D3semidirect
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
C22×C3×#1022
#840442
C2×#51132
C2×#51022
#821862
#715582
#702172
#702122
#235443