Finite Group Databasebeta

← #10847386

Subgroups — #10847386

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic, complemented
2C3×#2854C211yesmaximal, characteristic, complemented
3C2×#5038C212yesmaximal, complemented
4C2×#4524C211yesmaximal, characteristic, complemented
5#69658C211yesmaximal, characteristic, complemented
6#69657C212yesmaximal, complemented
7#69638C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69625C212yesmaximal, complemented
10#69604C211yesmaximal, characteristic, complemented
11#69515C212yesmaximal, complemented
12#25496—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×#123—21no—
15C2×C3×#121C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#213C2212yes—
18C2×#930—21no—
19C2×#930C2212yes—
20C23×C3⋊C8C2212yes—
21#5477—28no—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×D12C2212yes—
26C2×#869C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#388—21no—
32C3×#388—21no—
33C3×#386—21no—
34C3×#380C2211yescharacteristic
35C2×#832C2212yes—
36#5038—28no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#466—32no—
48#2854D311yescharacteristic, complemented
49#2057—31no—
50#2041—31no—
51#2023—31no—
52#2011—31no—
53#1893—32no—
54#1879—32no—
55#1862—32no—
56C2×#336—31no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—22no—
63C2×#235—28no—
64C2×#235—44no—
65C2×#235—44no—
66C22×D12—21no—
67C22×D12—21no—
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×Dic6—21no—
71C3×#123—44no—
72C3×#121—24no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—24no—
85C2×C3×C42D411yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#213—24no—
90C2×#213—24no—
91C2×#213—42no—
92#930—28no—
93#930—24no—
94#930D418yes—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2312yes—
99C22×C3⋊C8—24no—
100C22×C3⋊C8—24no—
101C22×C3⋊C8—42no—
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104#875—28no—
105#874—24no—
106C4×D12—28no—
107C4×D12—24no—
108C4×D12—24no—
109#869—24no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113#832—24no—
114#809—24no—
115#809—24no—
116#809—42no—
117#807—24no—
118C22×#41—61no—
119C22×D8—32no—
120C23×C8—32no—
121#532—68no—
122C2×C4⋊Q8—31no—
123C2×#127—31no—
124C2×#124—61no—
125C2×#123—21no—
126C2×#121—32no—
127C2×#121—61no—
128C2×#121C2×D311yescharacteristic
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C22×C42C2×D311yescharacteristic
132#466—68no—
133C2×#107—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138#390C2×D311yescharacteristic
139#390C2×D311yescharacteristic
140#389C2×D311yescharacteristic
141#389C2×D311yescharacteristic
142#388—21no—
143#388—21no—
144#386—21no—
145#380C2×D311yescharacteristic
146#340—32no—
147#340—32no—
148#336—31no—
149#336—32no—
150#336—31no—
151C24×C3—21no—
152C24×C3C2×D411yescharacteristic
153C23×D3—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—41no—
172C22×C3×C4D814yescomplemented
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C2×#67—28no—
181C2×#67—44no—
182C2×#67—44no—
183C22×Dic3—22no—
184#235—432no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—24no—
190C2×D12—24no—
191C2×D12—22no—
192C2×D12—42no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12—42no—
196C2×D12C2×D414yes—
197C2×C4×D3—28no—
198C2×C4×D3—44no—
199C2×C4×D3—44no—
200C2×Dic6—24no—
201C2×Dic6—42no—
202C2×Dic6—42no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42—22no—
219#213—48no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—28no—
222C4⋊Dic3—24no—
223C4⋊Dic3—22no—
224C4⋊Dic3C2×D414yes—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—48no—
231C2×C3⋊C8#4114yescomplemented
232C2×#41—68no—
233C2×#41—124no—
234C2×#41—124no—
235C22×Q8—61no—
236C22×D4—21no—
237C22×D4—32no—
238C22×D4—62no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4C22×D311yescharacteristic
242C23×C4—62no—
243C23×C4#6711yescharacteristic
244C23×C4#6711yescharacteristic
245C23×C4C22×D311yescharacteristic
246C2×D8—64no—
247C2×D8—64no—
248C2×D8—122no—
249C22×C8—32no—
250C22×C8—32no—
251C22×C8—64no—
252C22×C8—64no—
253C22×C8—122no—
254C4⋊Q8—64no—
255#127—64no—
256#124—68no—
257#123—44no—
258#121—24no—
259#121—38no—
260#121—68no—
261#121—64no—
262C4×D4—64no—
263C4×D4—64no—
264C4×D4—68no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C42—24no—
281C2×C42#6711yescharacteristic
282C2×C42#6711yescharacteristic
283C2×C42C22×D311yescharacteristic
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—122no—
290#95—21no—
291#95—21no—
292#95—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3#12012yes—
305C23×C3C2×D812yes—
306C23×C3AGL(1,ℤ/8)12yes—
307C23×C3C22×D411yescharacteristic
308C22×D3—24no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—84no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—28no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—44no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4#12012yes—
343C2×C3×C4C2×D812yes—
344C2×C3×C4C2×D812yes—
345C2×C3×C4#13512yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350#67—416no—
351C2×Dic3—24no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354D12—24no—
355D12—24no—
356D12—48no—
357D12—48no—
358D12—42no—
359D12—42no—
360C4×D3—416no—
361Dic6—48no—
362C3⋊C8—24no—
363C3⋊C8—42no—
364C24—21no—
365C24—62no—
366C24C2×#6711yescharacteristic
367#41—1232no—
368C2×Q8—64no—
369C2×Q8—122no—
370C2×Q8—122no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—34no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—64no—
380C2×D4—64no—
381C2×D4—68no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—122no—
388C2×D4—122no—
389C2×D4—124no—
390C2×D4—124no—
391C2×D4—122no—
392C2×D4—122no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—68no—
405C22×C4—62no—
406C22×C4—124no—
407C22×C4—124no—
408C22×C4#21314yescomplemented
409C22×C4D3×D414yes—
410C22×C4C2×#6711yescharacteristic
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415C22×C4C23×D311yescharacteristic
416D8—128no—
417C2×C8—34no—
418C2×C8—62no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—128no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—34no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—64no—
432C4⋊C4—62no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—68no—
440C22⋊C4—68no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3#44714yes—
459C22×C3C2×#12011yescharacteristic
460C22×C3#53211yescharacteristic
461C22×C3#53711yescharacteristic
462C22×C3C22×D811yescharacteristic
463C22×C3C2×#13511yescharacteristic
464C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
465C22×C3#57511yescharacteristic
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4#54612yes—
482C3×C4#54712yes—
483Dic3—44no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—64no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23—122no—
502C23C2×#21312yes—
503C23#93712yes—
504C23#94912yes—
505C23C22×#6711yescharacteristic
506Q8—128no—
507D4—42no—
508D4—42no—
509D4—64no—
510D4—64no—
511D4—84no—
512D4—128no—
513D4—128no—
514D4—1216no—
515D4—122no—
516D4—122no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—64no—
542C2×C4—1216no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4C2×#21312yes—
546C2×C4C2×#21312yes—
547C2×C4#93712yes—
548C2×C4#95012yes—
549C2×C4C2×D3×D412yes—
550C2×C4D3×#4112yes—
551C2×C4#100911yescharacteristic
552C2×C4#101011yescharacteristic
553C8—64no—
554C8—122no—
555C2×C3—24no—
556C2×C3—161no—
557C2×C3#243011yescharacteristic
558C2×C3C2×#44711yescharacteristic
559C2×C3#300211yescharacteristic
560C2×C3#316911yescharacteristic
561C2×C3#318011yescharacteristic
562C2×C3#324511yescharacteristic
563C2×C3#324711yescharacteristic
564D3—84no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—124no—
579C22—124no—
580C22—124no—
581C22—242no—
582C22—242no—
583C22—242no—
584C22—242no—
585C22#503614yes—
586C22C22×#21311yescharacteristic
587C22C2×#93711yescharacteristic
588C22C2×#94911yescharacteristic
589C22C2×#95011yescharacteristic
590C22#582211yescharacteristic
591C22#583211yescharacteristic
592C22#583311yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—124no—
602C4#565412yes—
603C4#565512yes—
604C3#2549611yescharacteristic, complemented, Sylow
605C2—24no—
606C2—161no—
607C2—244no—
608C2C2×#503611yescharacteristic
609C2#8284811yescharacteristic
610C2#8292611yescharacteristic
611C2#8313711yescharacteristic
612C2#8314811yescharacteristic
613C2#8329111yescharacteristic
614C2#8331011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2854C2
socleC23×C3C22×D4
hypercenter#2854D3
hypocenterC3#25496
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C3×#2854C2semidirect
3. C2×#5038C2semidirect
4. C2×#4524C2semidirect
5. #69658C2semidirect
6. #69657C2semidirect
7. #69638C2semidirect
8. #69627C2semidirect
9. #69625C2semidirect
10. #69604C2semidirect
11. #69515C2semidirect
48. #2854D3semidirect
172. C22×C3×C4D8semidirect
231. C2×C3⋊C8#41semidirect
408. C22×C4#213semidirect
604. C3#25496semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C3×#28542
C2×#50382
C2×#45242
#696582
#696572
#696382
#696272
#696252
#696042
#695152
#254963