Finite Group Databasebeta

← #10801356

Subgroups — #10801356

Order 768.

1272 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic
3C2×#4756C212yesmaximal, complemented
4#74319C211yesmaximal, characteristic, complemented
5C3×#1802C211yesmaximal, characteristic, complemented
6#70457C211yesmaximal, characteristic, complemented
7Dic3×C4⋊C8C211yesmaximal, characteristic, complemented
8#70172C212yesmaximal, complemented
9#69567C211yesmaximal, characteristic, complemented
10#68791C212yesmaximal, complemented
11#68780C212yesmaximal, complemented
12C4×#459—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×D4×Dic3C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C22×#213C2212yes—
17C22×C4×Dic3—21no—
18C2×#893—21no—
19C2×#893—21no—
20C2×#893C2212yes—
21C2×#874C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2212yes—
24C3×#384C2211yescharacteristic
25C4×#217—21no—
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×C8×Dic3C2212yes—
29#5015C418yescomplemented
30#4988—21no—
31#4988—21no—
32#4988C2212yes—
33Dic3×C4⋊C4C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35C2×#809C2212yes—
36C2×#807C2212yes—
37Dic3×C22⋊C4—21no—
38Dic3×C22⋊C4—21no—
39C4×C4⋊Dic3C2211yescharacteristic
40C42×Dic3C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4756—28no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#459—31no—
47C2×C4×D8—32no—
48C4×#127—31no—
49C42×D4—31no—
50#1982—31no—
51#1947—31no—
52#1925—32no—
53#1879—32no—
54#1802D311yescharacteristic, complemented
55C4×C4⋊C8—31no—
56C4×#102—32no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×Dic3—21no—
60C23×Dic3—21no—
61C22×D12—21no—
62C22×D12—21no—
63C22×D12C2312yes—
64C22×C4×D3—22no—
65C22×C3×C8C2312yes—
66C3×C4×D4—22no—
67C3×C4×D4C2×C414yescomplemented
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—21no—
75C2×#217—21no—
76D4×Dic3—22no—
77D4×Dic3—22no—
78D4×Dic3—22no—
79D4×Dic3D414yescomplemented
80C2×#213—24no—
81C2×#213—24no—
82C2×#213C2×C418yescomplemented
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4×Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—44no—
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99#893—24no—
100#893—24no—
101#893—44no—
102#893—44no—
103#874—22no—
104#874C2×C414yescomplemented
105C3×C4⋊C8—22no—
106C3×#102—24no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—24no—
111#821—24no—
112C8×Dic3—24no—
113#809C2×C418yescomplemented
114#807—24no—
115C4⋊(C3⋊C8)C2×C414yescomplemented
116C22×D8—32no—
117C2×#127—31no—
118C2×C4×D4—31no—
119C2×C4×D4—31no—
120C2×C4×D4—32no—
121C2×C4×D4—61no—
122C2×C4×D4—61no—
123C2×C4×D4C2×D311yescharacteristic
124C22×C42—61no—
125#459—38no—
126C4×D8—68no—
127C2×#107—32no—
128C2×C4⋊C8—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102C2×D312yes—
134C2×C4×C8—32no—
135#390—32no—
136#390—61no—
137#390—61no—
138#384C2×D311yescharacteristic
139C4×C4⋊C4—31no—
140C4×C4⋊C4—31no—
141C4×C4⋊C4—31no—
142C4×C22⋊C4—61no—
143C4×C22⋊C4—61no—
144C4×C22⋊C4—61no—
145C43—31no—
146#340—32no—
147#340C2×D312yes—
148#336—32no—
149C24×C3—21no—
150C23×D3—22no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4C22×C412yes—
155C2×C3×D4C2×D412yes—
156C2×C3×D4#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—24no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—24no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C22×Dic3—42no—
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—24no—
182C2×D12—24no—
183C2×D12—22no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12C22×C414yes—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3—44no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4C22×C412yes—
201C3×C22⋊C4—24no—
202C3×C42C22×C412yes—
203C3×C42#4112yes—
204#217—24no—
205#217—24no—
206#213—216no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—44no—
209(C2×D3)⋊C4—44no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3C2×D412yes—
213C4⋊Dic3#4112yes—
214C4×Dic3—22no—
215C4×Dic3—21no—
216C4×Dic3—21no—
217C4×Dic3—22no—
218C4×Dic3—41no—
219C4×Dic3—42no—
220C4×Dic3—42no—
221C4×Dic3—42no—
222C4×Dic3D814yescomplemented
223C4×Dic3C2×D412yes—
224C4×Dic3#4112yes—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8C22×C414yes—
228C22×D4—32no—
229C22×D4—61no—
230C22×D4—61no—
231C22×D4C22×D311yescharacteristic
232C23×C4—21no—
233C23×C4—62no—
234C23×C4—61no—
235C23×C4—61no—
236C2×D8—38no—
237C2×D8—64no—
238C2×D8—64no—
239C22×C8—32no—
240C22×C8C22×D312yes—
241#127—34no—
242#127—62no—
243C4×D4—22no—
244C4×D4—34no—
245C4×D4—62no—
246C4×D4—64no—
247C4×D4—64no—
248C4×D4—62no—
249C4×D4—62no—
250C4×D4—124no—
251C4×D4—124no—
252C4×D4C4×D314yescomplemented
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4C22×D312yes—
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—61no—
272C2×C42—61no—
273C2×C42—61no—
274C2×C42—124no—
275C2×C42C22×D311yescharacteristic
276#107—64no—
277C4⋊C8—22no—
278C4⋊C8—34no—
279#102—24no—
280#102—38no—
281#102—64no—
282#102—64no—
283C4×C8—64no—
284#95—61no—
285#95—61no—
286#95—61no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3C22×D411yescharacteristic
295C22×D3—24no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4C4×D414yescomplemented
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—48no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4C4×D412yes—
316C2×C3×C4C4×D412yes—
317C2×C3×C4C2×D812yes—
318C2×C3×C4#13512yes—
319C2×C3×C4C23×C411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—22no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—48no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—48no—
340C2×Dic3#12112yes—
341C2×Dic3#12112yes—
342C2×Dic3C2×D812yes—
343C2×Dic3C2×D812yes—
344C2×Dic3AGL(1,ℤ/8)12yes—
345D12—24no—
346D12—24no—
347D12—48no—
348D12—42no—
349D12—42no—
350D12—48no—
351C4×D3—88no—
352C3×C8—42no—
353C3⋊C8—24no—
354C24—21no—
355C24—62no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—34no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4C2×C4×D312yes—
372C2×D4D3×D412yes—
373C2×D4#23912yes—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—64no—
384C22×C4—64no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—124no—
388C22×C4—124no—
389C22×C4—124no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4D3×D411yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4C23×D311yescharacteristic
397D8—616no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—34no—
402C2×C8—62no—
403C2×C8—62no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4C2×C4×D312yes—
414C22⋊C4—24no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—124no—
418C22⋊C4—124no—
419C22⋊C4—124no—
420C42—34no—
421C42—32no—
422C42—32no—
423C42—62no—
424C42—62no—
425C42—61no—
426C42—61no—
427C42—121no—
428C42—122no—
429C42—122no—
430C42—122no—
431C42C2×C4×D312yes—
432C42#23712yes—
433C22×C3—24no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3C2×C4×D411yescharacteristic
441C22×C3C2×#12111yescharacteristic
442C22×C3#52511yescharacteristic
443C22×C3C22×D811yescharacteristic
444C22×C3C2×#13511yescharacteristic
445C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
446C22×C3#57511yescharacteristic
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—82no—
451C2×D3—82no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—82no—
457C3×C4C4×D814yescomplemented
458C3×C4C2×C4×D411yescharacteristic
459C3×C4C4×#4111yescharacteristic
460C3×C4#52911yescharacteristic
461C3×C4#53211yescharacteristic
462Dic3—22no—
463Dic3—22no—
464Dic3—42no—
465Dic3—41no—
466Dic3—42no—
467Dic3#45914yescomplemented
468C23—22no—
469C23—22no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—64no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23C2×D3×D411yescharacteristic
483D4—22no—
484D4—22no—
485D4—22no—
486D4—64no—
487D4—64no—
488D4—128no—
489D4—128no—
490D4—122no—
491D4—122no—
492D4#89314yescomplemented
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—32no—
497C2×C4—32no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—128no—
517C2×C4—128no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—248no—
523C2×C4#67⋊C412yes—
524C2×C4#67⋊C412yes—
525C2×C4D3×D812yes—
526C2×C4#91612yes—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×#23711yescharacteristic
530C2×C4C2×#23911yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C8—42no—
534C8—64no—
535C2×C3—44no—
536C2×C3C4×#12111yescharacteristic
537C2×C3C2×C4×D811yescharacteristic
538C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
539C2×C3C2×#45911yescharacteristic
540C2×C3#303511yescharacteristic
541C2×C3#310011yescharacteristic
542C2×C3#310611yescharacteristic
543D3—84no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—124no—
552C22—124no—
553C22—124no—
554C22—242no—
555C22—242no—
556C22—242no—
557C22—242no—
558C22C2×#67⋊C411yescharacteristic
559C22D3×#12111yescharacteristic
560C22#562711yescharacteristic
561C22C2×D3×D811yescharacteristic
562C22C2×#91611yescharacteristic
563C22D3×AGL(1,ℤ/8)11yescharacteristic
564C22#577711yescharacteristic
565C4—22no—
566C4—34no—
567C4—42no—
568C4—62no—
569C4—62no—
570C4—82no—
571C4—122no—
572C4—121no—
573C4—122no—
574C4#475614yescomplemented
575C4C2×#89311yescharacteristic
576C4#552211yescharacteristic
577C4#559611yescharacteristic
578C4#562411yescharacteristic
579C3C4×#45911yescharacteristic, complemented, Sylow
580C2—44no—
581C2—244no—
582C2#7386511yescharacteristic
583C2C2×#475611yescharacteristic
584C2#7886611yescharacteristic
585C2D3×#45911yescharacteristic
586C2#8042711yescharacteristic
587C2#8094811yescharacteristic
588C2#8104511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
cosocleC22×C3×C4C24
hypercenter#1802D3
hypocenterC3C4×#459
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. C4×D4×Dic3C2semidirect
3. C2×#4756C2semidirect
4. #74319C2semidirect
5. C3×#1802C2semidirect
6. #70457C2semidirect
7. Dic3×C4⋊C8C2semidirect
8. #70172C2semidirect
9. #69567C2semidirect
10. #68791C2semidirect
11. #68780C2semidirect
29. #5015C4semidirect
54. #1802D3semidirect
67. C3×C4×D4C2×C4semidirect
79. D4×Dic3D4semidirect
82. C2×#213C2×C4semidirect
104. #874C2×C4semidirect
113. #809C2×C4semidirect
115. C4⋊(C3⋊C8)C2×C4semidirect
222. C4×Dic3D8semidirect
252. C4×D4C4×D3semidirect
305. C3×D4C4×D4semidirect
457. C3×C4C4×D8semidirect
467. Dic3#459semidirect
492. D4#893semidirect
574. C4#4756semidirect
579. C3C4×#459semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Dic32
C2×#50152
C2×#47562
#743192
C3×#18022
#704572
Dic3×C4⋊C82
#701722
#695672
#687912
#687802
C4×#4593