Finite Group Databasebeta

← #10876144

Subgroups — #10876144

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#390C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic, complemented
3C3×#2071C211yesmaximal, characteristic, complemented
4#70198C212yesmaximal
5#70156C212yesmaximal, complemented
6#69656C211yesmaximal, characteristic, complemented
7#69585C211yesmaximal, characteristic, complemented
8#68875C212yesmaximal, complemented
9#68792C212yesmaximal, complemented
10#68682C211yesmaximal, characteristic, complemented
11#68664C211yesmaximal, characteristic, complemented
12#27585—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C2×#892—21no—
15C2×D3×C4⋊C4C2211yescharacteristic
16C2×D3×C22⋊C4—21no—
17C2×D3×C22⋊C4—21no—
18C2×C42×D3C2211yescharacteristic
19C2×C3×#102C2212yes—
20C3×#390C2211yescharacteristic
21C3×#384C2211yescharacteristic, complemented
22#5221C2211yescharacteristic
23C2×#832C2212yes—
24C2×#821C2212yes—
25C2×(C2×D3)⋊C8C2212yes—
26C2×#817C2212yescomplemented
27#4988C2211yescharacteristic
28#4987—21no—
29#4987C2212yes—
30#4977—21no—
31Dic3×C4⋊C4—21no—
32Dic3×C4⋊C4C2212yescomplemented
33C2×#809C2212yes—
34C2×#808C2212yescomplemented
35#4957—21no—
36#4957—21no—
37#4939C2211yescharacteristic
38#4935C2211yescharacteristic, complemented
39C3×#340C2212yescomplemented
40C3×#336C2211yescharacteristic, complemented
41#4550C2212yescomplemented
42#4529C2212yescomplemented
43#4524C2211yescharacteristic, complemented
44C2×#390—31no—
45#2285—31no—
46#2094—31no—
47#2094—31no—
48#2071—31no—
49#2071D311yescharacteristic, complemented
50#1926—32no—
51#1926—32no—
52#1853—32no—
53#1817—31no—
54#1786—32no—
55C24×D3—21no—
56C22×C3×D4C2311yescharacteristic
57C22×#67—21no—
58C2×D3×D4—22no—
59C2×D3×D4—22no—
60C2×D3×D4—24no—
61C2×D3×D4—42no—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×C3×C8C2312yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—21no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92#892—44no—
93D3×C4⋊C4—42no—
94D3×C22⋊C4—44no—
95D3×C22⋊C4—44no—
96C42×D3—24no—
97C3×#102—24no—
98#832—42no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#821—42no—
102(C2×D3)⋊C8—24no—
103#817—42no—
104#809—24no—
105#808—42no—
106C23×D4—31no—
107C2×#117—61no—
108C22×C4⋊C4—31no—
109C22×C22⋊C4—61no—
110C22×C22⋊C4—61no—
111C22×C42—31no—
112C2×#106—32no—
113C2×#106—32no—
114C2×#102—32no—
115C2×#102—32no—
116C2×#102—32no—
117C2×#102C2×D312yes—
118C2×C22⋊C8—32no—
119#390—31no—
120#390—31no—
121#390—31no—
122#390—61no—
123#390—61no—
124#390C2×D311yescharacteristic
125#384—31no—
126#384C2×D311yescharacteristic, complemented
127#382—61no—
128#380—32no—
129#380—61no—
130C4×C4⋊C4—32no—
131C4×C4⋊C4—61no—
132#340—32no—
133#340C2×D312yescomplemented
134#336—32no—
135#336—31no—
136#336C2×D311yescharacteristic, complemented
137C24×C3—21no—
138C23×D3—24no—
139C23×D3—21no—
140C23×D3—42no—
141C23×D3—42no—
142C23×D3—42no—
143C23×D3—42no—
144C23×D3—42no—
145C23×D3—42no—
146C23×D3C2×D411yescharacteristic
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4#4112yes—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C2×#67—24no—
157C2×#67—42no—
158C2×#67—42no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164D3×D4—48no—
165D3×D4—48no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—21no—
169C2×D12#4112yes—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—41no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×C4×D3—42no—
181C2×C4×D3SD1614yes—
182C2×C4×D3#4114yes—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C42—22no—
194#217—42no—
195#217—42no—
196(C2×D3)⋊C4—42no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—42no—
199(C2×D3)⋊C4—42no—
200(C2×D3)⋊C4—42no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—42no—
205C4⋊Dic3—42no—
206Dic3⋊C4—42no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C4×Dic3—44no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C25—61no—
215C22×D4—31no—
216C22×D4—64no—
217C22×D4—61no—
218C22×D4—62no—
219C22×D4—62no—
220C22×D4—122no—
221C22×D4C22×D311yescharacteristic
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—61no—
226C22×C8—32no—
227C22×C8C22×D312yes—
228#117—124no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—122no—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—61no—
242C2×C22⋊C4—61no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—124no—
249C2×C22⋊C4—124no—
250C2×C42—31no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—64no—
254C2×C42—62no—
255C2×C42C22×D311yescharacteristic
256#106—122no—
257#106—122no—
258#102—24no—
259#102—64no—
260#102—122no—
261#102—122no—
262C22⋊C8—64no—
263#95—62no—
264#95—62no—
265C23×C3—22no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3C22×D411yescharacteristic
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—22no—
278C22×D3—48no—
279C22×D3—48no—
280C22×D3—48no—
281C22×D3—41no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—84no—
288C22×D3—84no—
289C22×D3—84no—
290C22×D3—84no—
291C22×D3#12312yes—
292C22×D3C2×SD1612yes—
293C22×D3AGL(1,ℤ/8)12yes—
294C3×D4—24no—
295C3×D4—42no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4C2×SD1612yes—
307C2×C3×C4AGL(1,ℤ/8)12yes—
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4112yes—
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4H(5,𝔽2)12yes—
312#67—48no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3#12312yes—
327C2×Dic3C2×SD1612yes—
328D12—24no—
329D12—42no—
330C4×D3—28no—
331C4×D3—24no—
332C4×D3—24no—
333C4×D3—44no—
334C4×D3—84no—
335C3×C8—42no—
336C3⋊C8—42no—
337C24—21no—
338C24—31no—
339C24—64no—
340C24—61no—
341C24—122no—
342C24—122no—
343C24—122no—
344C24—122no—
345C24—122no—
346C24—122no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—32no—
351C2×D4—64no—
352C2×D4—61no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—122no—
356C2×D4—128no—
357C2×D4—128no—
358C2×D4—122no—
359C2×D4#23712yes—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—34no—
365C22×C4—34no—
366C22×C4—31no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—121no—
377C22×C4—122no—
378C22×C4—122no—
379C22×C4—122no—
380C22×C4D3×D411yescharacteristic
381C22×C4D3×D411yescharacteristic
382C22×C4C23×D311yescharacteristic
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—122no—
398C4⋊C4—122no—
399C4⋊C4—122no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—122no—
403C22⋊C4—122no—
404C22⋊C4—122no—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C22⋊C4—122no—
409C42—22no—
410C42—62no—
411C42—62no—
412C42—62no—
413C42—124no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3C2×#12311yescharacteristic
422C22×C3#52511yescharacteristic
423C22×C3#53511yescharacteristic
424C22×C3C22×SD1611yescharacteristic
425C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
426C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
427C22×C3#57511yescharacteristic
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—816no—
438C2×D3—81no—
439C2×D3—81no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3#48114yes—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—82no—
446C3×C4—81no—
447C3×C4#55112yes—
448C3×C4#55112yes—
449Dic3—22no—
450Dic3—22no—
451Dic3—41no—
452Dic3—81no—
453Dic3—82no—
454C23—22no—
455C23—32no—
456C23—32no—
457C23—32no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—62no—
469C23—128no—
470C23—128no—
471C23—128no—
472C23—121no—
473C23—121no—
474C23—121no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—244no—
479C23—244no—
480C23—244no—
481C23—244no—
482C23C2×D3×D411yescharacteristic
483D4—24no—
484D4—42no—
485D4—64no—
486D4—128no—
487D4—122no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—68no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—124no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—244no—
518C2×C4#91112yes—
519C2×C4D3×SD1612yes—
520C2×C4C2×#23711yescharacteristic
521C2×C4D3×#4112yes—
522C2×C4D3×#4111yescharacteristic
523C2×C4#100912yes—
524C8—42no—
525C8—122no—
526C2×C3—82no—
527C2×C3#248211yescharacteristic
528C2×C3C2×#48111yescharacteristic
529C2×C3#309411yescharacteristic
530C2×C3#310111yescharacteristic
531C2×C3#310911yescharacteristic
532C2×C3#320811yescharacteristic
533C2×C3#321311yescharacteristic
534D3—24no—
535D3—82no—
536C22—34no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—62no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—62no—
545C22—62no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—2416no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22#560611yescharacteristic
559C22D3×#12311yescharacteristic
560C22#567411yescharacteristic
561C22C2×#91111yescharacteristic
562C22C2×D3×SD1611yescharacteristic
563C22D3×AGL(1,ℤ/8)11yescharacteristic
564C22#577711yescharacteristic
565C4—22no—
566C4—22no—
567C4—62no—
568C4—62no—
569C4—81no—
570C4—82no—
571C4—121no—
572C4—242no—
573C4—241no—
574C4#565812yes—
575C4#570512yes—
576C3#2758511yescharacteristic, complemented, Sylow
577C2—64no—
578C2—82no—
579C2—242no—
580C2#7767611yescharacteristic
581C2#8031411yescharacteristic
582C2#8040811yescharacteristic
583C2#8128311yescharacteristic
584C2D3×#48111yescharacteristic
585C2#8156811yescharacteristic
586C2#8168411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenter#2071D3
hypocenterC3#27585
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #74318C2semidirect
3. C3×#2071C2semidirect
5. #70156C2semidirect
6. #69656C2semidirect
7. #69585C2semidirect
8. #68875C2semidirect
9. #68792C2semidirect
10. #68682C2semidirect
11. #68664C2semidirect
21. C3×#384C22semidirect
26. C2×#817C22semidirect
32. Dic3×C4⋊C4C22semidirect
34. C2×#808C22semidirect
38. #4935C22semidirect
39. C3×#340C22semidirect
40. C3×#336C22semidirect
41. #4550C22semidirect
42. #4529C22semidirect
43. #4524C22semidirect
49. #2071D3semidirect
126. #384C2×D3semidirect
133. #340C2×D3semidirect
136. #336C2×D3semidirect
576. C3#27585semidirect

Maximal subgroups

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

subgroupindex
D3×#3902
#743182
C3×#20712
#701982
#701562
#696562
#695852
#688752
#687922
#686822
#686642
#275853