Finite Group Databasebeta

← #10830066

Subgroups — #10830066

Order 768.

1428 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic, complemented
3#82501C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71767C212yesmaximal, complemented
6#71765C211yesmaximal, characteristic, complemented
7#71762C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71613C212yesmaximal, complemented
10#71607C212yesmaximal, complemented
11#71605C211yesmaximal, characteristic, complemented
12#24196—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#550—24no—
15C3×#550—24no—
16C2×C3×#127C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121—22no—
19C2×C3×#121—21no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C22×#213C2212yes—
25C2×C4×D12C2211yescharacteristic
26#5231—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5072—24no—
33#5014—24no—
34#4993—21no—
35#4989—22no—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550D311yescharacteristic, complemented
47C2×#459—31no—
48C2×#442—31no—
49#2580—31no—
50#2041—32no—
51#2011—32no—
52#1985—31no—
53#1961—31no—
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—22no—
67C23×C3×C4—21no—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C3×#127—24no—
71C3×#127—24no—
72C3×#127—22no—
73C3×#127D414yes—
74C3×#121—28no—
75C3×#121—216no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4D414yes—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C2×#213—22no—
90C2×#213—22no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—22no—
94C2×#213—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2312yes—
100C2×Dic3⋊C4—22no—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2311yescharacteristic
104C4×D12—22no—
105C4×D12—22no—
106#832—22no—
107#832—24no—
108#832—24no—
109#832—42no—
110#832—42no—
111C3⋊#95—22no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#809—22no—
115#808—22no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—21no—
119C22×D8—31no—
120C22×D8—32no—
121#550—24no—
122#550—24no—
123C2×#127C2×D312yes—
124C2×#127C2×D311yescharacteristic
125C2×#121—22no—
126C2×#121—21no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×C4×D4C2×D312yes—
130C2×C4×D4C2×D311yescharacteristic
131#459—64no—
132#442—64no—
133C2×#106—31no—
134C2×C4⋊C8—31no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#97—31no—
142#399—61no—
143#393—61no—
144#390—32no—
145#389—32no—
146#385—62no—
147#382—31no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—22no—
152C24×C3—21no—
153C23×D3—21no—
154C3×#41—432no—
155C2×C3×Q8—22no—
156C2×C3×Q8—24no—
157C2×C3×D4—28no—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—82no—
175C2×C3×D4—82no—
176C2×C3×D4—82no—
177C2×C3×D4C2×D414yes—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—28no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—22no—
191C22×Dic3—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12#4112yescomplemented
198C2×C4×D3—42no—
199C2×C4×D3—82no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C2×D412yes—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—24no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42C2×D412yes—
209C3×C42C2×D412yes—
210#213—28no—
211#213—48no—
212(C2×D3)⋊C4—42no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—82no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219Dic3⋊C4—82no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8#4112yescomplemented
229C2×#41—28no—
230C2×#41—48no—
231C22×Q8#6711yescharacteristic
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—31no—
237C22×D4C22×D312yes—
238C22×D4C22×D311yescharacteristic
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—61no—
242C2×D8—62no—
243C2×D8—62no—
244C2×D8—62no—
245C2×D8—64no—
246C2×D8—64no—
247C2×D8—122no—
248C22×C8—31no—
249C22×C8—32no—
250C22×C8—31no—
251#127—24no—
252#127—24no—
253#127—22no—
254#127#6714yes—
255#121—28no—
256#121—216no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—62no—
263C4×D4—62no—
264C4×D4#6714yes—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4#6712yes—
269C2×C4⋊C4#6711yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C42C22×D312yes—
277C2×C42C22×D311yescharacteristic
278#106—62no—
279C4⋊C8—62no—
280#102—64no—
281#102—62no—
282#102—64no—
283#102—62no—
284#102—122no—
285#102—122no—
286#97—62no—
287#95—61no—
288#95—61no—
289#95—62no—
290C23×C3—24no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C22×D3—22no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C3×Q8—28no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—24no—
312C3×D4—24no—
313C3×D4—48no—
314C3×D4—416no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—84no—
320C3×D4—88no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—48no—
330C2×C3×C4—416no—
331C2×C3×C4—42no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12012yes—
334C2×C3×C4AGL(1,ℤ/8)14yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C42−1+411yescharacteristic
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346D12—22no—
347D12—22no—
348D12—41no—
349D12—41no—
350C4×D3—84no—
351C3⋊C8—22no—
352C3⋊C8—41no—
353C3⋊C8—42no—
354C24—22no—
355C24—21no—
356C24—61no—
357#41—432no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—32no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×#6714yes—
387C2×D4C2×#6712yes—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—122no—
397C22×C4—242no—
398C22×C4C2×#6712yes—
399C22×C4C2×#6712yes—
400C22×C4C2×#6711yescharacteristic
401C22×C4C2×#6711yescharacteristic
402C22×C4C23×D311yescharacteristic
403D8—68no—
404D8—128no—
405C2×C8—32no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—62no—
409C2×C8—61no—
410C2×C8—62no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—62no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—242no—
422C4⋊C4C2×#6712yes—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—124no—
426C22⋊C4—122no—
427C22⋊C4—242no—
428C42—22no—
429C42—22no—
430C42C2×#6712yes—
431C42C2×#6712yes—
432C22×C3—42no—
433C22×C3—44no—
434C22×C3—42no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—42no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#54811yescharacteristic
446C22×C3#55011yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)12yes—
448C22×C3#57612yes—
449C2×D3—44no—
450C2×D3—42no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C3×C4—22no—
455C3×C4—24no—
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4#52511yescharacteristic
460C3×C4#53711yescharacteristic
461C3×C4#54611yescharacteristic
462C3×C4#54711yescharacteristic
463Dic3—82no—
464Dic3—81no—
465C23—24no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—62no—
478C23—122no—
479C23—121no—
480C23—122no—
481C23—121no—
482C23C22×#6711yescharacteristic
483Q8—28no—
484D4—24no—
485D4—22no—
486D4—24no—
487D4—22no—
488D4—416no—
489D4—48no—
490D4—41no—
491D4—42no—
492D4—41no—
493D4—42no—
494D4—62no—
495D4—62no—
496D4—84no—
497D4—88no—
498D4—121no—
499D4—121no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—416no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—121no—
516C2×C4—244no—
517C2×C4#93214yes—
518C2×C4#95312yes—
519C2×C4#95312yes—
520C2×C4D3×#4111yescharacteristic
521C2×C4D3×#4111yescharacteristic
522C2×C4#100911yescharacteristic
523C2×C4#101011yescharacteristic
524C2×C4C22×#6711yescharacteristic
525C2×C4C22×#6711yescharacteristic
526C8—62no—
527C8—122no—
528C8—121no—
529C2×C3—82no—
530C2×C3—84no—
531C2×C3#240711yescharacteristic
532C2×C3#300311yescharacteristic
533C2×C3#300811yescharacteristic
534C2×C3#336612yes—
535C2×C3#339912yes—
536D3—82no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—44no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—124no—
550C22—122no—
551C22—241no—
552C22—242no—
553C22—241no—
554C22C2×#93212yes—
555C22#583111yescharacteristic
556C22#583411yescharacteristic
557C22#583512yes—
558C22C2×#95311yescharacteristic
559C4—24no—
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—242no—
565C4—241no—
566C4#582411yescharacteristic
567C4#582811yescharacteristic
568C4#582911yescharacteristic
569C4#583311yescharacteristic
570C3#2419611yescharacteristic, complemented, Sylow
571C2—84no—
572C2—82no—
573C2—242no—
574C2#8358112yes—
575C2#8358912yes—
576C2#8403811yescharacteristic
577C2#8409711yescharacteristic
578C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#550C2
socleC23×C3C22×D4
hypercenterC2×#550D3
hypocenterC3#24196
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×C3×#550C2semidirect
2. C2×#5072C2semidirect
3. #82501C2semidirect
4. C2×#5014C2semidirect
5. #71767C2semidirect
6. #71765C2semidirect
7. #71762C2semidirect
8. #71635C2semidirect
9. #71613C2semidirect
10. #71607C2semidirect
11. #71605C2semidirect
46. C2×#550D3semidirect
197. C2×D12#41semidirect
228. C2×C3⋊C8#41semidirect
570. C3#24196semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×#50722
#825012
C2×#50142
#717672
#717652
#717622
#716352
#716132
#716072
#716052
#241963