Finite Group Databasebeta

← #10829940

Subgroups — #10829940

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic, complemented
4C2×C4×#215C211yesmaximal, characteristic, complemented
5#71764C211yesmaximal, characteristic, complemented
6#71728C212yesmaximal, complemented
7#71722C212yesmaximal, complemented
8#71634C211yesmaximal, characteristic, complemented
9#71626C211yesmaximal, characteristic, complemented
10#71613C212yesmaximal, complemented
11#71607C212yesmaximal, complemented
12#24186—31nomaximal, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C2×#946—21no—
24C22×#215C2211yescharacteristic
25C22×#213C2212yes—
26C2×C4×D12C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5072—24no—
33C4×#215—24no—
34#4994—22no—
35#4987—21no—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4939C2212yes—
39#4935C2211yescharacteristic
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559D311yescharacteristic, complemented
47C2×#459—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1957—31no—
54#1879—32no—
55#1864—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C3×#127—22no—
66C3×#127D414yes—
67C3×#124—28no—
68C3×#124—24no—
69C3×#124—24no—
70C3×#123—216no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C3×C4×D4—28no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88#946—44no—
89C2×#215—22no—
90C2×#215—22no—
91C2×#215—22no—
92C2×#213—24no—
93C2×#213—24no—
94C2×#213—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2312yes—
102C2×Dic3⋊C4—22no—
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2311yescharacteristic
106C4×D12—22no—
107C4×D12—22no—
108#835—22no—
109#835—42no—
110#832—24no—
111#832—24no—
112#832—42no—
113C3⋊#95—22no—
114#809—22no—
115#808—22no—
116C4⋊(C3⋊C8)—22no—
117C4×C3⋊C8—22no—
118C22×SD16—31no—
119C22×D8—32no—
120#559—24no—
121#559—24no—
122C2×#127C2×D311yescharacteristic
123C2×#124—21no—
124C2×#124C2×D312yes—
125C2×#123—22no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—31no—
131C2×C4×D4C2×D312yes—
132#459—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102—31no—
141C2×#102—32no—
142C2×C4×C8—31no—
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—22no—
152C23×D3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D412yes—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—81no—
172C2×C3×D4—81no—
173C2×C3×D4C2×D414yes—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—28no—
177C22×C3×C4—82no—
178C22×C3×C4—82no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—21no—
185C22×Dic3—22no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12#4112yes—
192C2×C4×D3—42no—
193C2×C4×D3—42no—
194C2×C4×D3—42no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—28no—
202C3×C22⋊C4—28no—
203C3×C42—24no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208#215—28no—
209#213—48no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—42no—
212(C2×D3)⋊C4—82no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—42no—
218C4⋊Dic3—81no—
219Dic3⋊C4—44no—
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#4112yes—
229C22×Q8#6711yescharacteristic
230C22×Q8C22×D311yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—31no—
234C22×D4C22×D312yes—
235C23×C4—22no—
236C23×C4—61no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×D8—64no—
241C2×D8—64no—
242C2×D8—122no—
243C22×C8—31no—
244C22×C8—32no—
245C22×C8—31no—
246#127—22no—
247#127#6714yes—
248#124—28no—
249#124—24no—
250#124—24no—
251#123—216no—
252C22⋊Q8—124no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8#6714yes—
257C4×D4—28no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—62no—
261C4×D4—62no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4#6711yescharacteristic
269C2×C4⋊C4#6712yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—21no—
278C2×C42C22×D312yes—
279C2×C42C22×D311yescharacteristic
280#106—62no—
281C4⋊C8—62no—
282#103—62no—
283#103—122no—
284#102—64no—
285#102—64no—
286#102—62no—
287#102—122no—
288C4×C8—62no—
289#95—62no—
290C23×C3—24no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3C22×D411yescharacteristic
298C22×D3—22no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×Q8—44no—
310C3×D4—24no—
311C3×D4—24no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—48no—
315C3×D4—84no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—88no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12012yes—
333C2×C3×C4#13512yes—
334C2×C3×C4AGL(1,ℤ/8)12yes—
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—41no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347D12—22no—
348D12—22no—
349D12—41no—
350D12—41no—
351C4×D3—84no—
352C3⋊C8—22no—
353C3⋊C8—41no—
354C3⋊C8—42no—
355C24—22no—
356C24—61no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×#6712yes—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—32no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—61no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4C2×#6714yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—28no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4C2×#6712yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—68no—
400D8—128no—
401C2×C8—32no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—62no—
405C2×C8—61no—
406C2×C8—62no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—62no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—124no—
419C4⋊C4—122no—
420C4⋊C4—241no—
421C4⋊C4C2×#6712yes—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—124no—
426C22⋊C4—122no—
427C22⋊C4—242no—
428C42—24no—
429C42—22no—
430C42—22no—
431C42C2×#6712yes—
432C42C2×#6712yes—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3C2×#12011yescharacteristic
441C22×C3#54811yescharacteristic
442C22×C3#55011yescharacteristic
443C22×C3C2×#13511yescharacteristic
444C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
445C22×C3#57611yescharacteristic
446C22×C3#57711yescharacteristic
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—81no—
451C2×D3—81no—
452C2×D3—81no—
453C2×D3—81no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4#44914yes—
460C3×C4#52511yescharacteristic
461C3×C4#53711yescharacteristic
462C3×C4#54611yescharacteristic
463C3×C4#54711yescharacteristic
464Dic3—82no—
465Dic3—81no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—62no—
474C23—121no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—121no—
479C23—121no—
480C23C22×#6711yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—44no—
484Q8—41no—
485Q8—41no—
486D4—24no—
487D4—24no—
488D4—48no—
489D4—42no—
490D4—42no—
491D4—62no—
492D4—62no—
493D4—84no—
494D4—121no—
495D4—121no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—88no—
511C2×C4—122no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—121no—
517C2×C4—244no—
518C2×C4#94912yes—
519C2×C4#95012yes—
520C2×C4#95312yes—
521C2×C4#95312yes—
522C2×C4D3×#4111yescharacteristic
523C2×C4D3×#4111yescharacteristic
524C2×C4#100911yescharacteristic
525C2×C4#101011yescharacteristic
526C2×C4C22×#6711yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—62no—
529C8—122no—
530C8—121no—
531C2×C3—84no—
532C2×C3#240711yescharacteristic
533C2×C3C2×#44911yescharacteristic
534C2×C3#301211yescharacteristic
535C2×C3#336111yescharacteristic
536C2×C3#337311yescharacteristic
537C2×C3#339411yescharacteristic
538C2×C3#339911yescharacteristic
539D3—82no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—122no—
548C22—122no—
549C22—122no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22—241no—
554C22C2×#94911yescharacteristic
555C22C2×#95011yescharacteristic
556C22#583111yescharacteristic
557C22#583411yescharacteristic
558C22#583511yescharacteristic
559C22#583611yescharacteristic
560C22C2×#95311yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—241no—
567C4—242no—
568C4#523814yes—
569C4#582411yescharacteristic
570C4#582811yescharacteristic
571C4#582911yescharacteristic
572C4#583311yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—84no—
575C2—242no—
576C2#8371011yescharacteristic
577C2#8371111yescharacteristic
578C2#8371311yescharacteristic
579C2#8371511yescharacteristic
580C2C2×#523811yescharacteristic
581C2#8409711yescharacteristic
582C2#8410211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenterC2×#559D3
hypocenterC3#24186
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. C2×C3×#559C2semidirect
3. #82500C2semidirect
4. C2×C4×#215C2semidirect
5. #71764C2semidirect
6. #71728C2semidirect
7. #71722C2semidirect
8. #71634C2semidirect
9. #71626C2semidirect
10. #71613C2semidirect
11. #71607C2semidirect
46. C2×#559D3semidirect
573. C3#24186semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50722
#825002
C2×C4×#2152
#717642
#717282
#717222
#716342
#716262
#716132
#716072
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