Finite Group Databasebeta

← #10810902

Subgroups — #10810902

Order 768.

1602 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#214C211yesmaximal, characteristic
2C3×D4×C22⋊C4C211yesmaximal, characteristic
3C2×#5018C212yesmaximal, complemented
4C22⋊C4×Dic6C211yesmaximal, characteristic, complemented
5#72128C211yesmaximal, characteristic
6#71612C212yesmaximal, complemented
7#69660C212yesmaximal, complemented
8#69649C212yesmaximal, complemented
9#69525C211yesmaximal, characteristic, complemented
10#69512C211yesmaximal, characteristic, complemented
11#69447C211yesmaximal, characteristic, complemented
12#22437—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×C4×D4C2212yes—
16C22×C3×C22⋊C4—21no—
17C22×#214—28no—
18C22×#214C418yescomplemented
19C22×#214C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×C4×Dic6C2212yescomplemented
25C3×#390C2212yes—
26C3×#386—22no—
27C3×#380C2211yescharacteristic
28C3×#379—21no—
29C3×C4×C22⋊C4C2211yescharacteristic
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#5018—28no—
35C2×#810C2212yescomplemented
36C2×#810C2212yescomplemented
37C2×#807C2212yescomplemented
38#4942—21no—
39Dic3×C22⋊C4—21no—
40#4929C2212yescomplemented
41C2×#803C2212yescomplemented
42#4646—22no—
43#4529C2212yescomplemented
44#4524C2212yescomplemented
45C23×SD16—31no—
46C2×#440—32no—
47Q8×C22⋊C4—31no—
48D4×C22⋊C4D311yescharacteristic
49#1973—32no—
50#1880—32no—
51#1877—32no—
52#1808—31no—
53#1787—31no—
54#1786—31no—
55#1752—31no—
56C25×C3—21no—
57C22×C3×D4—24no—
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4—42no—
61C22×C3×D4C2×C414yes—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—22no—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C22×Dic6—24no—
68C22×Dic6—21no—
69C22×Dic6—22no—
70C22×Dic6—42no—
71C22×Dic6C2×C414yescomplemented
72C22×Dic6C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C3×C4×D4—24no—
75C3×C4×D4—42no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—44no—
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C22⋊C4D411yescharacteristic
84C2×C3×C42C2312yes—
85C2×#217—21no—
86C2×#214—24no—
87C2×#214—24no—
88C2×#214—232no—
89C2×#214—24no—
90C2×#214C2×C414yes—
91C2×#214C2×C414yes—
92C2×#214C2×C414yescomplemented
93C2×#214C2×C414yescomplemented
94C2×#214D418yes—
95C2×C4⋊Dic3C2312yes—
96C2×Dic3⋊C4—22no—
97C2×C4×Dic3—22no—
98C22×C3⋊C8—24no—
99C22×C3⋊C8C2×C414yescomplemented
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C4×Dic6—24no—
104C4×Dic6—42no—
105C3×#95—21no—
106C3×#95D412yes—
107#832—24no—
108#832—42no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112#830—22no—
113#810—24no—
114#810—42no—
115#807—24no—
116#803—24no—
117C23×Q8—31no—
118C23×D4C2×D311yescharacteristic
119C22×SD16—31no—
120C22×SD16—38no—
121C22×SD16—31no—
122C22×SD16—31no—
123C22×SD16—31no—
124C22×SD16—68no—
125C23×C8—31no—
126C2×C4×Q8—32no—
127C2×C4×D4C2×D312yes—
128C22×C22⋊C4—21no—
129#440—68no—
130C2×#107—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×#97—32no—
137#391—32no—
138#391—62no—
139#390C2×D312yes—
140#386—22no—
141#380—31no—
142#380—61no—
143#380C2×D311yescharacteristic
144#379—21no—
145C4×C22⋊C4—61no—
146C4×C22⋊C4C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149C24×C3—21no—
150C24×C3—21no—
151C24×C3—24no—
152C24×C3—44no—
153C24×C3—42no—
154C24×C3—42no—
155C24×C3—82no—
156C24×C3C2×D411yescharacteristic
157C2×C3×D4—28no—
158C2×C3×D4—24no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—44no—
165C2×C3×D4—48no—
166C2×C3×D4—44no—
167C2×C3×D4C22×C412yes—
168C2×C3×D4C22×C412yes—
169C2×C3×D4C2×D414yes—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—44no—
174C22×C3×C4—44no—
175C22×C3×C4—82no—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4#4112yes—
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—21no—
183C22×Dic3—24no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—44no—
187C2×Dic6—21no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—28no—
192C2×Dic6—24no—
193C2×Dic6—21no—
194C2×Dic6—44no—
195C2×Dic6—48no—
196C2×Dic6—44no—
197C2×Dic6C22×C412yes—
198C2×Dic6C22×C412yes—
199C2×Dic6C2×D414yes—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—82no—
208C3×C42—22no—
209C3×C42—22no—
210#217—42no—
211#214—28no—
212#214—28no—
213#214—28no—
214#214C22⋊C4116yescomplemented
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217Dic3⋊C4—44no—
218C4×Dic3—44no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—28no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8C22×C412yes—
225C2×C3⋊C8C22×C412yes—
226C2×C3⋊C8C2×D414yes—
227C25—21no—
228C22×Q8—34no—
229C22×Q8—31no—
230C22×Q8—31no—
231C22×Q8—64no—
232C22×Q8—62no—
233C22×Q8—61no—
234C22×Q8—122no—
235C22×D4—24no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—42no—
239C22×D4C4×D314yes—
240C22×D4C22×D311yescharacteristic
241C22×D4C22×D311yescharacteristic
242C23×C4—22no—
243C23×C4—61no—
244C23×C4C22×D311yescharacteristic
245C2×SD16—34no—
246C2×SD16—38no—
247C2×SD16—34no—
248C2×SD16—34no—
249C2×SD16—34no—
250C2×SD16—64no—
251C2×SD16—632no—
252C2×SD16—64no—
253C2×SD16—64no—
254C22×C8—31no—
255C22×C8—34no—
256C22×C8—31no—
257C22×C8—32no—
258C22×C8—64no—
259C4×Q8—64no—
260C4×Q8—122no—
261C4×D4—24no—
262C4×D4—42no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4C22×D312yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4#6711yescharacteristic
273C2×C22⋊C4#6711yescharacteristic
274C2×C42—62no—
275C2×C42C22×D312yes—
276#107—64no—
277#103—64no—
278#103—122no—
279#102—64no—
280#102—122no—
281C22⋊C8—62no—
282#97—64no—
283#95—21no—
284#95—62no—
285#95—61no—
286#95—62no—
287#95#6712yes—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—48no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—48no—
302C23×C3—84no—
303C23×C3—88no—
304C23×C3—88no—
305C23×C3C4×D412yes—
306C23×C3AGL(1,ℤ/8)12yes—
307C23×C3#13712yes—
308C23×C3C22×D411yescharacteristic
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—24no—
313C3×D4—48no—
314C3×D4—42no—
315C3×D4C2×C22⋊C414yes—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—88no—
330C2×C3×C4C4×D412yes—
331C2×C3×C4C23×C411yescharacteristic
332C2×C3×C4C22×D412yes—
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4H(5,𝔽2)12yes—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—48no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343Dic6—22no—
344Dic6—24no—
345Dic6—22no—
346Dic6—22no—
347Dic6—48no—
348Dic6—42no—
349Dic6C2×C22⋊C414yescomplemented
350C3⋊C8—22no—
351C3⋊C8—42no—
352C3⋊C8C2×C22⋊C414yescomplemented
353C24—21no—
354C24—21no—
355C24—24no—
356C24—44no—
357C24—42no—
358C24—42no—
359C24—82no—
360C24C2×#6711yescharacteristic
361C2×Q8—32no—
362C2×Q8—32no—
363C2×Q8—34no—
364C2×Q8—68no—
365C2×Q8—64no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—124no—
372C2×Q8—124no—
373C2×Q8—128no—
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4C2×C4×D312yes—
385C2×D4C2×C4×D312yes—
386C2×D4D3×D414yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—61no—
393C22×C4—64no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4—82no—
397C22×C4—124no—
398C22×C4C2×C4×D312yes—
399C22×C4#23512yes—
400C22×C4C2×#6712yes—
401C22×C4C2×#6712yes—
402C22×C4C2×#6711yescharacteristic
403C22×C4C23×D311yescharacteristic
404SD16—316no—
405SD16—68no—
406SD16—68no—
407SD16—68no—
408C2×C8—32no—
409C2×C8—34no—
410C2×C8—32no—
411C2×C8—68no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—62no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—62no—
419C4⋊C4—62no—
420C4⋊C4—124no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—82no—
427C22⋊C4—122no—
428C42—22no—
429C42—22no—
430C42—124no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—816no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3#44014yescomplemented
447C22×C3C2×C4×D411yescharacteristic
448C22×C3D4212yes—
449C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
450C22×C3C2×#13711yescharacteristic
451C22×C3#57712yes—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—82no—
456C3×C4C22×C22⋊C411yescharacteristic
457C3×C4#51811yescharacteristic
458C3×C4#54612yes—
459Dic3—24no—
460Dic3—42no—
461Dic3—82no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—22no—
467C23—48no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—48no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—84no—
477C23—88no—
478C23—88no—
479C23C4×#6712yes—
480C23#93212yes—
481C23#95112yes—
482C23C22×#6711yescharacteristic
483Q8—34no—
484Q8—62no—
485Q8—64no—
486Q8—62no—
487Q8—62no—
488Q8—128no—
489Q8—122no—
490D4—22no—
491D4—22no—
492D4—22no—
493D4—24no—
494D4—48no—
495D4—42no—
496D4D3×C22⋊C414yes—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—88no—
514C2×C4—128no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4C4×#6712yes—
520C2×C4C22×C4×D311yescharacteristic
521C2×C4C2×#23511yescharacteristic
522C2×C4C2×D3×D412yes—
523C2×C4#100412yes—
524C8—34no—
525C8—62no—
526C8—122no—
527C2×C3—24no—
528C2×C3—44no—
529C2×C3—82no—
530C2×C3D4×C22⋊C411yescharacteristic
531C2×C3C2×#44011yescharacteristic
532C2×C3#294911yescharacteristic
533C2×C3#326512yes—
534C2×C3#326612yes—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—44no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—816no—
548C22—82no—
549C22—82no—
550C22#501814yescomplemented
551C22C2×C4×#6711yescharacteristic
552C22C2×#93211yescharacteristic
553C22D4×#6712yes—
554C22C2×#95111yescharacteristic
555C22#583612yes—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—64no—
560C4—82no—
561C4—122no—
562C4—242no—
563C4C2×D3×C22⋊C411yescharacteristic
564C4#548611yescharacteristic
565C4#549012yes—
566C3#2243711yescharacteristic, complemented, Sylow
567C2—24no—
568C2—44no—
569C2—82no—
570C2C22⋊C4×#6711yescharacteristic
571C2C2×#501811yescharacteristic
572C2#8254311yescharacteristic
573C2#8349112yes—
574C2#8350612yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×D4×C22⋊C4C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
cosocleC22×C3×C4C24
hypercenterD4×C22⋊C4D3
hypocenterC3#22437
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ D4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#5018C2semidirect
4. C22⋊C4×Dic6C2semidirect
6. #71612C2semidirect
7. #69660C2semidirect
8. #69649C2semidirect
9. #69525C2semidirect
10. #69512C2semidirect
11. #69447C2semidirect
18. C22×#214C4semidirect
24. C2×C4×Dic6C22semidirect
35. C2×#810C22semidirect
36. C2×#810C22semidirect
37. C2×#807C22semidirect
40. #4929C22semidirect
41. C2×#803C22semidirect
43. #4529C22semidirect
44. #4524C22semidirect
71. C22×Dic6C2×C4semidirect
92. C2×#214C2×C4semidirect
93. C2×#214C2×C4semidirect
99. C22×C3⋊C8C2×C4semidirect
214. #214C22⋊C4semidirect
349. Dic6C2×C22⋊C4semidirect
352. C3⋊C8C2×C22⋊C4semidirect
446. C22×C3#440semidirect
550. C22#5018semidirect
566. C3#22437semidirect

Maximal subgroups

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

subgroupindex
C23×#2142
C3×D4×C22⋊C42
C2×#50182
C22⋊C4×Dic62
#721282
#716122
#696602
#696492
#695252
#695122
#694472
#224373