Finite Group Databasebeta

← #10823443

Subgroups — #10823443

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5018C212yesmaximal, complemented
3C4×#924C211yesmaximal, characteristic, complemented
4#71794C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#71768C211yesmaximal, characteristic, complemented
7#71762C211yesmaximal, characteristic, complemented
8#71685C212yesmaximal, complemented
9#71679C212yesmaximal, complemented
10#71609C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#23576—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#532—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×#214C2212yes—
24C2×#924C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212C2211yescharacteristic
30#5210—21no—
31C4×#217—21no—
32C2×#830C2212yes—
33#5018—28no—
34C2×#810C2212yes—
35C2×#807C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37#4933—21no—
38C4×Dic3⋊C4—21no—
39#4929C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×#803C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#532D311yescharacteristic
47C2×#440—32no—
48C4×C22⋊Q8—31no—
49#2057—31no—
50#2041—31no—
51#2033—32no—
52#2024—31no—
53#2014—31no—
54#2000—32no—
55#1762—32no—
56#1749—31no—
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C3×C4⋊Q8—24no—
71C3×#127—24no—
72C3×#124—28no—
73C3×#121—28no—
74C3×#121—24no—
75C3×#121—24no—
76C3×C4×D4—28no—
77C3×C4×D4—24no—
78C3×C4×D4D418yes—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—24no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—24no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—22no—
97#924—22no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105C4×Dic6—24no—
106#832—24no—
107#832—24no—
108#832—24no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#830—24no—
115#810—24no—
116#807—24no—
117#803—24no—
118C22×#41—21no—
119C22×SD16—32no—
120#532—28no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#127C2×D311yescharacteristic
123C2×#124—21no—
124C2×C22⋊Q8—31no—
125C2×#121—21no—
126C2×#121C2×D312yes—
127C2×C4×Q8—31no—
128C2×C4×D4—21no—
129C2×C4×D4C2×D312yes—
130C22×C42C2×D311yescharacteristic
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#391—31no—
140#389—31no—
141#387—61no—
142#385—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C4⋊C4—61no—
146C4×C22⋊C4—61no—
147#340—32no—
148#340—32no—
149#336—31no—
150#336—31no—
151C24×C3—22no—
152C24×C3C2×D411yescharacteristic
153C3×#41—432no—
154C2×C3×Q8—24no—
155C2×C3×Q8—42no—
156C2×C3×Q8—42no—
157C2×C3×D4—24no—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—44no—
168C2×C3×D4—44no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—28no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—41no—
184C22×C3×C4—44no—
185C22×C3×C4—44no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—21no—
199C2×Dic6#4112yescomplemented
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4C2×D414yes—
203C3×C22⋊C4—28no—
204C3×C22⋊C4—28no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42C2×D414yes—
209#217—24no—
210#217—42no—
211#214—216no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3#4112yescomplemented
216Dic3⋊C4—24no—
217Dic3⋊C4—42no—
218Dic3⋊C4—42no—
219C4×Dic3—42no—
220C4×Dic3—42no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8#4114yescomplemented
227C2×#41—28no—
228C2×#41—44no—
229C2×#41—44no—
230C22×Q8—21no—
231C22×Q8—31no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C22×D312yes—
236C23×C4—22no—
237C23×C4#6711yescharacteristic
238C23×C4#6711yescharacteristic
239C23×C4C22×D311yescharacteristic
240C2×SD16—64no—
241C2×SD16—64no—
242C2×SD16—64no—
243C22×C8—32no—
244C22×C8—32no—
245C4⋊Q8—24no—
246#127—24no—
247#124—28no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250#121—28no—
251#121—24no—
252#121—24no—
253C4×Q8—64no—
254C4×D4—28no—
255C4×D4—24no—
256C4×D4#6718yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—61no—
269C2×C42—24no—
270C2×C42—61no—
271C2×C42—61no—
272C2×C42#6711yescharacteristic
273C2×C42#6711yescharacteristic
274C2×C42C22×D311yescharacteristic
275C2×C42C22×D311yescharacteristic
276#107—64no—
277#103—64no—
278#102—64no—
279#102—64no—
280#102—64no—
281C22⋊C8—64no—
282#97—64no—
283#95—61no—
284#95—61no—
285#95—61no—
286#95—61no—
287#95—61no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—24no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3AGL(1,ℤ/8)12yes—
302C23×C3C22×D411yescharacteristic
303C3×Q8—48no—
304C3×D4—24no—
305C3×D4—24no—
306C3×D4—48no—
307C3×D4—48no—
308C3×D4—416no—
309C3×D4—42no—
310C3×D4—42no—
311C2×C3×C4—28no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—44no—
325C2×C3×C4—416no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4#12012yes—
331C2×C3×C4#12012yes—
332C2×C3×C4AGL(1,ℤ/8)12yes—
333C2×C3×C4#13712yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C4C2×#4111yescharacteristic
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350Dic6—24no—
351Dic6—42no—
352C3⋊C8—24no—
353C3⋊C8—42no—
354C24—22no—
355C24C2×#6711yescharacteristic
356#41—432no—
357C2×Q8—24no—
358C2×Q8—32no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—61no—
362C2×Q8—62no—
363C2×Q8—62no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4C2×#6714yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401C22×C4C2×#6711yescharacteristic
402C22×C4C23×D311yescharacteristic
403SD16—616no—
404C2×C8—34no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C2×C8—62no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—32no—
413C4⋊C4—64no—
414C4⋊C4—61no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—122no—
418C4⋊C4—122no—
419C4⋊C4C2×#6714yes—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—64no—
423C22⋊C4—122no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—122no—
428C42—122no—
429C42C2×#6714yes—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#52511yescharacteristic
446C22×C3#53211yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
448C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
449C22×C3C2×#13711yescharacteristic
450C22×C3#57511yescharacteristic
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—41no—
456C3×C4—44no—
457C3×C4#52512yes—
458C3×C4#53212yes—
459Dic3—42no—
460Dic3—42no—
461Dic3—81no—
462Dic3—81no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23#93212yes—
477C23C22×#6711yescharacteristic
478Q8—48no—
479Q8—64no—
480Q8—122no—
481D4—24no—
482D4—24no—
483D4—48no—
484D4—48no—
485D4—416no—
486D4—42no—
487D4—42no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—416no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4#93212yes—
520C2×C4#95112yes—
521C2×C4#95312yes—
522C2×C4#95312yes—
523C2×C4D3×#4111yescharacteristic
524C2×C4D3×#4112yes—
525C2×C4D3×#4111yescharacteristic
526C2×C4C22×#6712yes—
527C8—64no—
528C8—122no—
529C2×C3—42no—
530C2×C3—84no—
531C2×C3#238811yescharacteristic
532C2×C3#300311yescharacteristic
533C2×C3#300711yescharacteristic
534C2×C3#311211yescharacteristic
535C2×C3#311511yescharacteristic
536C2×C3#318011yescharacteristic
537C2×C3#318211yescharacteristic
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22C2×#93211yescharacteristic
553C22C2×#93211yescharacteristic
554C22#582211yescharacteristic
555C22C2×#95111yescharacteristic
556C22#582811yescharacteristic
557C22#583211yescharacteristic
558C22C2×#95311yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—44no—
564C4—41no—
565C4—122no—
566C4—122no—
567C4—241no—
568C4—241no—
569C4#582812yes—
570C4#583212yes—
571C3#2357611yescharacteristic, complemented, Sylow
572C2—42no—
573C2—84no—
574C2#8329311yescharacteristic
575C2#8329411yescharacteristic
576C2#8330111yescharacteristic
577C2#8331611yescharacteristic
578C2#8403811yescharacteristic
579C2#8408411yescharacteristic
580C2#8409511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#23576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5018C2semidirect
3. C4×#924C2semidirect
4. #71794C2semidirect
5. #71782C2semidirect
6. #71768C2semidirect
7. #71762C2semidirect
8. #71685C2semidirect
9. #71679C2semidirect
10. #71609C2semidirect
11. #71529C2semidirect
199. C2×Dic6#41semidirect
215. C4⋊Dic3#41semidirect
226. C2×C3⋊C8#41semidirect
571. C3#23576semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50182
C4×#9242
#717942
#717822
#717682
#717622
#716852
#716792
#716092
#715292
#235763