Finite Group Databasebeta

← #10876309

Subgroups — #10876309

Order 768.

1124 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4987C211yesmaximal, characteristic
2D3×#380C211yesmaximal, characteristic
3C3×#1823C211yesmaximal, characteristic, complemented
4#71553C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#69927C211yesmaximal, characteristic, complemented
7#69504C211yesmaximal, characteristic, complemented
8#68889C212yesmaximal, complemented
9#68874C212yesmaximal, complemented
10#68871C212yesmaximal, complemented
11#68830C212yesmaximal, complemented
12#27600—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4—21no—
16C22×C4⋊Dic3C2211yescharacteristic
17C2×D3×C4⋊C4C2211yescharacteristic
18C2×D3×C4⋊C4C2211yescharacteristic
19C2×D3×C22⋊C4—21no—
20C2×C3×C22⋊C8C2211yescharacteristic
21C3×#380C2211yescharacteristic
22#5212C2211yescharacteristic
23C2×C3⋊#95—21no—
24C2×#830C2211yescharacteristic
25C2×(C2×D3)⋊C8C2212yes—
26C2×(C2×D3)⋊C8C2212yes—
27C2×#818C2212yescomplemented
28#4987—21no—
29#4987—21no—
30#4987C2212yes—
31#4987C2212yes—
32#4987C2211yescharacteristic
33#4987C2211yescharacteristic
34C2×#807C2212yescomplemented
35#4942—21no—
36#4667—21no—
37D3×#95—21no—
38C3×#340C2212yescomplemented
39C3×#340C2212yescomplemented
40#4551C2212yescomplemented
41#4529C2212yescomplemented
42#4529C2212yescomplemented
43#4529C2212yescomplemented
44C2×#380—31no—
45C2×#380—31no—
46#2075—32no—
47#2075—32no—
48#1852—31no—
49#1852—31no—
50#1852—32no—
51#1823—32no—
52#1823—31no—
53#1823D311yescharacteristic, complemented
54#1779—31no—
55C24×D3D411yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—24no—
63C22×C4×D3—21no—
64C22×C4×D3—21no—
65C22×C4×D3—42no—
66C22×C4×D3D411yescharacteristic
67C22×C4×D3D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×C3×C8C2312yes—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×#217—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—44no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—42no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C22×C3⋊C8C2312yes—
92D3×C4⋊C4—24no—
93D3×C4⋊C4—42no—
94D3×C22⋊C4—44no—
95C3×C22⋊C8—22no—
96C3×#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102#830—22no—
103(C2×D3)⋊C8—24no—
104(C2×D3)⋊C8—42no—
105#818—42no—
106#807—42no—
107C24×C4—31no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C22⋊C4—61no—
113C22×C22⋊C4—61no—
114C2×#107—32no—
115C2×#107—32no—
116C2×C22⋊C8—31no—
117C2×C22⋊C8—32no—
118C2×C22⋊C8—32no—
119C2×C22⋊C8C2×D311yescharacteristic
120#380—31no—
121#380—31no—
122#380—31no—
123#380—32no—
124#380—32no—
125#380—61no—
126#380—61no—
127#380—61no—
128#380—61no—
129#380C2×D311yescharacteristic
130C2×#95—61no—
131C2×#95—61no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340—32no—
136#340C2×D312yescomplemented
137#340C2×D312yescomplemented
138C24×C3C2×D411yescharacteristic
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—41no—
144C23×D3—41no—
145C23×D3D814yes—
146C23×D3C2×D411yescharacteristic
147C23×D3C2×D411yescharacteristic
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—41no—
166C22×Dic3—42no—
167C22×Dic3—42no—
168C22×Dic3—42no—
169C22×Dic3C2×D411yescharacteristic
170C22×Dic3C2×D411yescharacteristic
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—48no—
183C2×C4×D3—48no—
184C2×C4×D3—41no—
185C2×C4×D3—41no—
186C2×C4×D3—42no—
187C2×C4×D3—42no—
188C2×C4×D3—42no—
189C2×C4×D3—42no—
190C2×C4×D3—42no—
191C2×C4×D3—42no—
192C2×C4×D3#4114yes—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—42no—
201C3×C22⋊C4—42no—
202#217—42no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—42no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—44no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—42no—
212Dic3⋊C4—42no—
213Dic3⋊C4—42no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C25—31no—
218C23×C4—21no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—64no—
225C23×C4—61no—
226C23×C4—61no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4—61no—
230C23×C4—122no—
231C23×C4C22×D311yescharacteristic
232C22×C8—32no—
233C22×C8C22×D312yes—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4—122no—
246C2×C4⋊C4C22×D311yescharacteristic
247C2×C4⋊C4C22×D311yescharacteristic
248C2×C4⋊C4C22×D312yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—61no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—124no—
256C2×C22⋊C4—124no—
257#107—122no—
258#107—122no—
259C22⋊C8—22no—
260C22⋊C8—64no—
261C22⋊C8—62no—
262C22⋊C8—122no—
263#95—21no—
264#95—61no—
265#95—61no—
266#95—62no—
267#95—62no—
268#95—61no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3#12312yes—
274C23×C3C2×D812yes—
275C23×C3#13712yes—
276C23×C3C22×D411yescharacteristic
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—21no—
284C22×D3—21no—
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—28no—
288C22×D3—28no—
289C22×D3—44no—
290C22×D3—44no—
291C22×D3—44no—
292C22×D3—44no—
293C22×D3—44no—
294C22×D3#12312yes—
295C22×D3C2×D812yes—
296C22×D3C2×D812yes—
297C22×D3#13712yes—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—84no—
312C2×C3×C4#12312yes—
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4112yes—
316C2×C3×C4H(5,𝔽2)12yes—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—44no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—84no—
333C2×Dic3#12312yes—
334C4×D3—24no—
335C4×D3—24no—
336C4×D3—44no—
337C4×D3—44no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—84no—
341C4×D3—84no—
342C3×C8—42no—
343C3⋊C8—42no—
344C24—31no—
345C24—31no—
346C24—34no—
347C24—62no—
348C24—62no—
349C24—62no—
350C24—62no—
351C24—121no—
352C24—121no—
353C24D3×D411yescharacteristic
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—31no—
360C22×C4—34no—
361C22×C4—31no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—128no—
384C22×C4—128no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—121no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4#23512yes—
396C22×C4D3×D411yescharacteristic
397C22×C4#23712yes—
398C22×C4C23×D311yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—122no—
415C4⋊C4—122no—
416C4⋊C4—122no—
417C22⋊C4—42no—
418C22⋊C4—122no—
419C22⋊C4—122no—
420C22⋊C4—124no—
421C22⋊C4—124no—
422C22⋊C4—122no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3#48014yes—
430C22×C3C2×#12311yescharacteristic
431C22×C3C2×#12311yescharacteristic
432C22×C3#53411yescharacteristic
433C22×C3C22×D811yescharacteristic
434C22×C3C2×#13711yescharacteristic
435C22×C3C2×#13711yescharacteristic
436C22×C3#57511yescharacteristic
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—416no—
447C2×D3—41no—
448C2×D3—41no—
449C2×D3—41no—
450C2×D3—41no—
451C2×D3#48014yes—
452C3×C4—22no—
453C3×C4—81no—
454C3×C4—82no—
455C3×C4—81no—
456C3×C4#55112yes—
457C3×C4#55112yes—
458Dic3—22no—
459Dic3—41no—
460Dic3—81no—
461Dic3—81no—
462Dic3—82no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—32no—
468C23—32no—
469C23—32no—
470C23—32no—
471C23—68no—
472C23—61no—
473C23—61no—
474C23—61no—
475C23—61no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23—68no—
483C23—124no—
484C23—124no—
485C23—124no—
486C23—124no—
487C23—124no—
488C23#88312yes—
489C23D3×D812yes—
490C23#91512yes—
491C23C2×D3×D411yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—32no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—84no—
514C2×C4—124no—
515C2×C4—124no—
516C2×C4—124no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—121no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—244no—
529C2×C4—244no—
530C2×C4—244no—
531C2×C4#88312yes—
532C2×C4C2×#23511yescharacteristic
533C2×C4C2×#23711yescharacteristic
534C2×C4D3×#4112yes—
535C2×C4#100912yes—
536C8—42no—
537C8—122no—
538C2×C3—24no—
539C2×C3#248311yescharacteristic
540C2×C3C2×#48011yescharacteristic
541C2×C3C2×#48011yescharacteristic
542C2×C3#309511yescharacteristic
543C2×C3#309511yescharacteristic
544C2×C3#318611yescharacteristic
545C2×C3#319511yescharacteristic
546D3—24no—
547D3—42no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—34no—
555C22—62no—
556C22—62no—
557C22—62no—
558C22—62no—
559C22—62no—
560C22—62no—
561C22—62no—
562C22—62no—
563C22—62no—
564C22—1216no—
565C22—121no—
566C22—121no—
567C22—121no—
568C22—121no—
569C22#477414yes—
570C22C2×#88311yescharacteristic
571C22D3×#12311yescharacteristic
572C22#565311yescharacteristic
573C22C2×D3×D811yescharacteristic
574C22C2×#91511yescharacteristic
575C22D3×#13711yescharacteristic
576C22#577711yescharacteristic
577C4—22no—
578C4—62no—
579C4—81no—
580C4—81no—
581C4—82no—
582C4—121no—
583C4—241no—
584C4—241no—
585C4—242no—
586C4#552712yes—
587C4#565812yes—
588C3#2760011yescharacteristic, complemented, Sylow
589C2—24no—
590C2—64no—
591C2—122no—
592C2#7413011yescharacteristic
593C2C2×#477411yescharacteristic
594C2#7896511yescharacteristic
595C2D3×#48011yescharacteristic
596C2#8129911yescharacteristic
597C2#8142811yescharacteristic
598C2#8144911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenter#1823D3
hypocenterC3#27600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. C3×#1823C2semidirect
4. #71553C2semidirect
6. #69927C2semidirect
7. #69504C2semidirect
8. #68889C2semidirect
9. #68874C2semidirect
10. #68871C2semidirect
11. #68830C2semidirect
27. C2×#818C22semidirect
34. C2×#807C22semidirect
38. C3×#340C22semidirect
39. C3×#340C22semidirect
40. #4551C22semidirect
41. #4529C22semidirect
42. #4529C22semidirect
43. #4529C22semidirect
53. #1823D3semidirect
136. #340C2×D3semidirect
137. #340C2×D3semidirect
588. C3#27600semidirect

Maximal subgroups

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

subgroupindex
C2×#49872
D3×#3802
C3×#18232
#715532
#699552
#699272
#695042
#688892
#688742
#688712
#688302
#276003