Finite Group Databasebeta

← #10830068

Subgroups — #10830068

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×#5718C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#72044C212yesmaximal, complemented
6#72037C212yesmaximal, complemented
7#71765C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70473C211yesmaximal, characteristic, complemented
10#70211C212yesmaximal, complemented
11#70169C212yesmaximal, complemented
12#24196—31nomaximal, Sylow
13C22×#237—21no—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#940C2212yes—
18C2×#939—22no—
19C2×#938—21no—
20C2×D4×Dic3C2212yes—
21C22×#213C2211yescharacteristic
22#5718—24no—
23#5718—24no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#459—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29#5221C2212yes—
30C2×#832C2212yes—
31C2×#832C2211yescharacteristic
32C2×#832C2212yes—
33#5210—22no—
34C2×#821C2212yes—
35#5014—24no—
36#4972C2212yes—
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#550—31no—
47C2×#459D311yescharacteristic, complemented
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—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C22×#67—22no—
63C22×#67—22no—
64C23×Dic3—22no—
65C2×#237—28no—
66C2×#237—48no—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×Dic6D411yescharacteristic
70C2×C3×D8—24no—
71C2×C3×D8—24no—
72C2×C3×D8—42no—
73C22×C3×C8C2312yes—
74C3×#127—22no—
75C3×#127D414yes—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C42C2311yescharacteristic
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—22no—
84#940—24no—
85#940—24no—
86#939—216no—
87#938—28no—
88D4×Dic3—24no—
89D4×Dic3—24no—
90C2×#213—22no—
91C2×#213—22no—
92C2×#213—22no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3D411yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4D412yes—
99C2×C4×Dic3C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102C4×D12—22no—
103C4×D12D414yes—
104C4×Dic6—22no—
105C4×Dic6—22no—
106C3×C4⋊C8—22no—
107C3×#102—24no—
108#832—22no—
109#832—24no—
110#832—42no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—22no—
114#821—42no—
115#809—22no—
116#808—22no—
117#803—22no—
118C22×#41—61no—
119C22×D8—31no—
120C22×D8C2×D312yes—
121#550—64no—
122#550—64no—
123C2×#127—32no—
124C2×#127C2×D311yescharacteristic
125C2×#121—61no—
126C2×#121—62no—
127C2×C4×Q8—31no—
128C2×C4×D4—32no—
129C2×C4×D4—31no—
130C2×C4×D4C2×D311yescharacteristic
131#459—24no—
132#442—64no—
133C2×#106—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#102—32no—
136C2×#102—31no—
137C2×#102—31no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102C2×D312yes—
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—21no—
152C24×C3—22no—
153C23×D3—21no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—44no—
167C2×C3×D4—82no—
168C2×C3×D4C2×D414yes—
169C2×C3×D4#4112yescomplemented
170C22×C3×C4—21no—
171C22×C3×C4—42no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—28no—
177C2×#67—28no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—82no—
181C2×#67—82no—
182C22×Dic3—28no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—44no—
186C22×Dic3—44no—
187C22×Dic3C2×D412yes—
188C22×Dic3C2×D412yes—
189#237—432no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12C2×D412yes—
196C2×C4×D3—24no—
197C2×C4×D3—44no—
198C2×Dic6—22no—
199C2×Dic6—24no—
200C3×D8—48no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C22⋊C4—42no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209#217—28no—
210#217—44no—
211#217—82no—
212#213—28no—
213(C2×D3)⋊C4—24no—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3C2×D412yes—
219Dic3⋊C4—24no—
220Dic3⋊C4—82no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8#4112yescomplemented
229C2×#41—68no—
230C2×#41—128no—
231C22×Q8—31no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—31no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4C22×D311yescharacteristic
238C22×D4C22×D312yes—
239C23×C4—21no—
240C23×C4—62no—
241C23×C4—61no—
242C2×D8—24no—
243C2×D8—24no—
244C2×D8—42no—
245C2×D8—62no—
246C2×D8—62no—
247C2×D8—62no—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8C22×D312yes—
251#127—22no—
252#127—64no—
253#127—64no—
254#127#6714yes—
255#121—68no—
256#121—616no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—34no—
262C4×D4—64no—
263C4×D4—64no—
264C4×D4—62no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C42—32no—
277C2×C42C22×D311yescharacteristic
278#106—62no—
279C4⋊C8—22no—
280#102—24no—
281#102—62no—
282#102—62no—
283#102—64no—
284#102—122no—
285#102—122no—
286#97—62no—
287#95—62no—
288#95—61no—
289#95—61no—
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×D4—22no—
309C3×D4—22no—
310C3×D4—24no—
311C3×D4—24no—
312C3×D4—48no—
313C3×D4—42no—
314C3×D4—41no—
315C3×D4—42no—
316C3×D4—41no—
317C3×D4—84no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—84no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12012yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331#67—416no—
332#67—88no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—416no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3AGL(1,ℤ/8)14yes—
345D12—22no—
346D12—22no—
347D12—41no—
348D12—41no—
349C4×D3—48no—
350Dic6—28no—
351C3×C8—42no—
352C3⋊C8—22no—
353C3⋊C8—41no—
354C24—21no—
355C24—22no—
356C24—61no—
357#41—1232no—
358C2×Q8—64no—
359C2×Q8—62no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—32no—
373C2×D4—44no—
374C2×D4—68no—
375C2×D4—68no—
376C2×D4—61no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—82no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—242no—
385C2×D4—242no—
386C2×D4#23712yescomplemented
387C2×D4C2×#6714yes—
388C22×C4—21no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—42no—
392C22×C4—68no—
393C22×C4—61no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—82no—
397C22×C4—124no—
398C22×C4—124no—
399C22×C4—124no—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403D8—48no—
404D8—68no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—32no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—32no—
417C4⋊C4—64no—
418C4⋊C4—62no—
419C4⋊C4—61no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—242no—
423C22⋊C4—42no—
424C22⋊C4—68no—
425C22⋊C4—64no—
426C22⋊C4—124no—
427C22⋊C4—242no—
428C42—62no—
429C42—62no—
430C42D3×D412yes—
431C42C2×#6712yes—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#54811yescharacteristic
446C22×C3#55011yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)12yes—
448C22×C3#57612yes—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—81no—
457C3×C4#52511yescharacteristic
458C3×C4#53711yescharacteristic
459C3×C4#54611yescharacteristic
460C3×C4#54711yescharacteristic
461Dic3—24no—
462Dic3—42no—
463Dic3—42no—
464Dic3—82no—
465C23—24no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—62no—
478C23—122no—
479C23—122no—
480C23—121no—
481C23—121no—
482C23C2×D3×D411yescharacteristic
483Q8—68no—
484D4—22no—
485D4—24no—
486D4—24no—
487D4—22no—
488D4—48no—
489D4—41no—
490D4—42no—
491D4—41no—
492D4—42no—
493D4—62no—
494D4—62no—
495D4—84no—
496D4—1216no—
497D4—121no—
498D4—121no—
499D4—248no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—34no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—84no—
512C2×C4—128no—
513C2×C4—1216no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4#93712yes—
519C2×C4#93712yes—
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×#23711yescharacteristic
522C2×C4#100411yescharacteristic
523C2×C4D3×#4111yescharacteristic
524C2×C4#101011yescharacteristic
525C2×C4C22×#6711yescharacteristic
526C8—42no—
527C8—62no—
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—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—81no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—242no—
553C22—242no—
554C22#559811yescharacteristic
555C22#559911yescharacteristic
556C22#575712yes—
557C22D3×AGL(1,ℤ/8)12yes—
558C22C2×#93711yescharacteristic
559C4—22no—
560C4—22no—
561C4—64no—
562C4—81no—
563C4—122no—
564C4—122no—
565C4—242no—
566C4#560611yescharacteristic
567C4#560811yescharacteristic
568C4#580611yescharacteristic
569C4#582411yescharacteristic
570C3#2419611yescharacteristic, complemented, Sylow
571C2—84no—
572C2—82no—
573C2—242no—
574C2#8032112yes—
575C2#8032812yes—
576C2#8270111yescharacteristic
577C2#8383311yescharacteristic
578C2#8394411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#24196
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#5718C2semidirect
2. C2×C3×#459C2semidirect
3. #82388C2semidirect
4. C2×#5014C2semidirect
5. #72044C2semidirect
6. #72037C2semidirect
7. #71765C2semidirect
8. #70480C2semidirect
9. #70473C2semidirect
10. #70211C2semidirect
11. #70169C2semidirect
47. C2×#459D3semidirect
169. C2×C3×D4#41semidirect
228. C2×C3⋊C8#41semidirect
386. C2×D4#237semidirect
570. C3#24196semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×C3×#4592
#823882
C2×#50142
#720442
#720372
#717652
#704802
#704732
#702112
#701692
#241963