Finite Group Databasebeta

← #10801459

Subgroups — #10801459

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C42×Q8C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3C2×#5026C212yesmaximal, complemented
4C4×#874C211yesmaximal, characteristic
5#71633C212yesmaximal, complemented
6#71626C211yesmaximal, characteristic, complemented
7#69487C211yesmaximal, characteristic
8#69474C212yesmaximal, complemented
9#69466C211yesmaximal, characteristic, complemented
10#69462C212yesmaximal, complemented
11#69391C211yesmaximal, characteristic, complemented
12#21505—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C22×#215C2212yes—
17C2×#874C2211yescharacteristic
18C2×C4×D12—21no—
19C2×C4×D12—21no—
20C2×C4×D12C2212yes—
21C3×C4×C4⋊C4—21no—
22C3×C4×C4⋊C4—21no—
23C3×C4×C4⋊C4—21no—
24C3×C4×C4⋊C4C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C43—21no—
28C3×C43C2211yescharacteristic
29C2×#835C2212yes—
30#5027C418yescomplemented
31#5026—28no—
32C2×#809C2212yes—
33C2×#809C2212yes—
34C2×#809C2212yes—
35C2×#807C2212yes—
36#4939—21no—
37#4939—21no—
38#4939C2212yes—
39#4935C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2212yes—
46C2×#460—31no—
47C2×#440—32no—
48C4×#127—31no—
49C42×Q8D311yescharacteristic, complemented
50#1982—31no—
51#1947—31no—
52#1925—32no—
53#1880—32no—
54#1803—31no—
55#1767—31no—
56#1762—32no—
57C22×C3×Q8C2311yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8C2×C414yescomplemented
67C3×C4×Q8D414yescomplemented
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×C3×C42—21no—
79C2×C3×C42—21no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#215—24no—
88C2×#215—24no—
89C2×#215C2×C418yescomplemented
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2312yes—
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2312yes—
98#874—22no—
99#874C2×C414yes—
100C4×D12—24no—
101C4×D12—24no—
102C4×D12—44no—
103C4×D12—44no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107#835—24no—
108#809—24no—
109#809—24no—
110#809C2×C418yescomplemented
111#807—24no—
112C4⋊(C3⋊C8)—22no—
113C4⋊(C3⋊C8)C2×C414yescomplemented
114#803—24no—
115C22×SD16—32no—
116C2×#127—31no—
117C2×C4×Q8C2×D311yescharacteristic
118C2×C4×Q8C2×D311yescharacteristic
119C2×C4×Q8C2×D311yescharacteristic
120C2×C4×D4—32no—
121C2×C4×D4—61no—
122C2×C4×D4—61no—
123#460—38no—
124#440—68no—
125C2×#107—32no—
126C2×C4⋊C8—31no—
127C2×C4⋊C8—31no—
128C2×#103—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#97—32no—
133#390—32no—
134#390—61no—
135#390—61no—
136#384—31no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2×D311yescharacteristic
141C4×C4⋊C4C2×D311yescharacteristic
142C4×C4⋊C4C2×D311yescharacteristic
143C43—21no—
144C43C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148C23×D3—22no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8C22×C412yes—
153C2×C3×Q8C2×D412yes—
154C2×C3×Q8#4112yes—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—24no—
172C2×D12—24no—
173C2×D12—22no—
174C2×D12—42no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—42no—
178C2×D12C22×C414yes—
179C2×C4×D3—44no—
180C2×C4×D3—44no—
181C2×C4×D3—44no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4C22×C412yes—
189C3×C4⋊C4C2×D412yes—
190C3×C4⋊C4#4112yes—
191C3×C42—24no—
192C3×C42—24no—
193C3×C42—24no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—41no—
200C3×C42—42no—
201C3×C42—42no—
202C3×C42—42no—
203C3×C42C22×C412yes—
204C3×C42C2×D412yes—
205C3×C42#4112yes—
206C3×C42#4112yes—
207#215—216no—
208(C2×D3)⋊C4—44no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—44no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4⋊Dic3—42no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8C22×C414yes—
221C22×Q8C22×D311yescharacteristic
222C22×D4—32no—
223C22×D4—61no—
224C22×D4—61no—
225C23×C4—62no—
226C2×SD16—38no—
227C2×SD16—64no—
228C2×SD16—64no—
229C22×C8—32no—
230C22×C8—32no—
231#127—34no—
232#127—62no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8C4×D314yescomplemented
238C4×Q8D1214yescomplemented
239C4×D4—64no—
240C4×D4—64no—
241C4×D4—124no—
242C4×D4—124no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42#6711yescharacteristic
262C2×C42#6711yescharacteristic
263C2×C42#6711yescharacteristic
264C2×C42#6711yescharacteristic
265C2×C42C22×D311yescharacteristic
266C2×C42C22×D311yescharacteristic
267C2×C42C22×D311yescharacteristic
268#107—64no—
269C4⋊C8—34no—
270C4⋊C8—62no—
271#103—64no—
272#102—38no—
273#102—64no—
274#102—64no—
275#97—64no—
276#95—21no—
277#95—21no—
278#95—21no—
279C23×C3C22×D411yescharacteristic
280C22×D3—24no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8C4×D414yescomplemented
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4C4×D412yes—
315C2×C3×C4C4×D412yes—
316C2×C3×C4#12112yes—
317C2×C3×C4#12112yes—
318C2×C3×C4AGL(1,ℤ/8)12yes—
319C2×C3×C4AGL(1,ℤ/8)12yes—
320C2×C3×C4#13712yes—
321C2×C3×C4C23×C411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330D12—24no—
331D12—24no—
332D12—48no—
333D12—42no—
334D12—42no—
335D12—48no—
336C4×D3—88no—
337C3⋊C8—24no—
338C3⋊C8—42no—
339C24—62no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8C2×C4×D312yes—
344C2×Q8C2×D1212yes—
345C2×Q8#23512yes—
346C2×D4—34no—
347C2×D4—64no—
348C2×D4—64no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—122no—
355C2×D4—122no—
356C2×D4—122no—
357C2×D4—122no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—62no—
363C22×C4—124no—
364C22×C4—124no—
365C22×C4—124no—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C23×D311yescharacteristic
373SD16—616no—
374C2×C8—34no—
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—122no—
389C4⋊C4—122no—
390C4⋊C4C2×C4×D312yes—
391C4⋊C4C2×D1212yes—
392C4⋊C4#23512yes—
393C22⋊C4—124no—
394C22⋊C4—124no—
395C22⋊C4—124no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—21no—
403C42—21no—
404C42—41no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42C2×C4×D312yes—
409C42C2×D1212yes—
410C42#23512yes—
411C42#23512yes—
412C22×C3C2×C4×D411yescharacteristic
413C22×C3C2×#12111yescharacteristic
414C22×C3#52511yescharacteristic
415C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C3C2×#13711yescharacteristic
418C22×C3#57511yescharacteristic
419C2×D3—44no—
420C2×D3—44no—
421C2×D3—44no—
422C2×D3—82no—
423C2×D3—82no—
424C2×D3—82no—
425C2×D3—82no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—24no—
431C3×C4—41no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4#44014yescomplemented
436C3×C4C2×C4×D411yescharacteristic
437C3×C4C4×#4111yescharacteristic
438C3×C4#52911yescharacteristic
439C3×C4#53211yescharacteristic
440Dic3—82no—
441C23—64no—
442C23—122no—
443C23—122no—
444C23—122no—
445C23—122no—
446C23—122no—
447C23—122no—
448C23C22×#6711yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—22no—
452Q8C4×D1214yescomplemented
453D4—64no—
454D4—64no—
455D4—128no—
456D4—122no—
457D4—122no—
458D4—128no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—248no—
486C2×C4C4×#6712yes—
487C2×C4C4×#6712yes—
488C2×C4#93012yes—
489C2×C4#93012yes—
490C2×C4#94212yes—
491C2×C4#94912yes—
492C2×C4#94912yes—
493C2×C4C22×C4×D311yescharacteristic
494C2×C4C22×D1211yescharacteristic
495C2×C4C2×#23511yescharacteristic
496C2×C4C2×#23511yescharacteristic
497C2×C4C2×#23511yescharacteristic
498C2×C4C2×#23511yescharacteristic
499C8—64no—
500C8—122no—
501C2×C3C4×#12111yescharacteristic
502C2×C3C2×#44011yescharacteristic
503C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
504C2×C3#303711yescharacteristic
505C2×C3#303811yescharacteristic
506C2×C3#311211yescharacteristic
507C2×C3#311511yescharacteristic
508D3—84no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—242no—
513C22—242no—
514C22—242no—
515C22—242no—
516C22C2×C4×#6711yescharacteristic
517C22C2×#93011yescharacteristic
518C22#579111yescharacteristic
519C22C2×#94211yescharacteristic
520C22C2×#94911yescharacteristic
521C22C2×#94911yescharacteristic
522C22#582211yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—24no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—242no—
533C4#502614yescomplemented
534C4C2×C4×D1211yescharacteristic
535C4C4×#23511yescharacteristic
536C4#547711yescharacteristic
537C4#547911yescharacteristic
538C3#2150511yescharacteristic, complemented, Sylow
539C2—244no—
540C2C4×#93011yescharacteristic
541C2C2×#502611yescharacteristic
542C2#8244111yescharacteristic
543C2#8244411yescharacteristic
544C2C4×#94911yescharacteristic
545C2#8252611yescharacteristic
546C2#8253111yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C42×Q8C2
socleC23×C3C22×D4
prime radicalC2×#874C22
cosocleC22×C3×C4C24
hypercenterC42×Q8D3
hypocenterC3#21505
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C42×Q8C2semidirect
3. C2×#5026C2semidirect
5. #71633C2semidirect
6. #71626C2semidirect
8. #69474C2semidirect
9. #69466C2semidirect
10. #69462C2semidirect
11. #69391C2semidirect
30. #5027C4semidirect
49. C42×Q8D3semidirect
66. C3×C4×Q8C2×C4semidirect
67. C3×C4×Q8D4semidirect
89. C2×#215C2×C4semidirect
110. #809C2×C4semidirect
113. C4⋊(C3⋊C8)C2×C4semidirect
237. C4×Q8C4×D3semidirect
238. C4×Q8D12semidirect
290. C3×Q8C4×D4semidirect
435. C3×C4#440semidirect
452. Q8C4×D12semidirect
533. C4#5026semidirect
538. C3#21505semidirect

Maximal subgroups

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

subgroupindex
C3×C42×Q82
C2×#50272
C2×#50262
C4×#8742
#716332
#716262
#694872
#694742
#694662
#694622
#693912
#215053