Finite Group Databasebeta

← #10944096

Subgroups — #10944096

Order 768.

1546 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5798C211yesmaximal, characteristic
2C2×C3×#450C211yesmaximal, characteristic
3C2×#5218C211yesmaximal, characteristic
4C2×#5162C212yesmaximal, complemented
5#82786C211yesmaximal, characteristic, complemented
6#72047C212yesmaximal, complemented
7#71693C211yesmaximal, characteristic, complemented
8#70171C212yesmaximal, complemented
9#70154C212yesmaximal, complemented
10#69931C211yesmaximal, characteristic, complemented
11#69909C211yesmaximal, characteristic, complemented
12#32914—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C22×#217—22no—
17C2×#944C2211yescharacteristic, complemented
18#5798—28no—
19C2×D4×Dic3—22no—
20C2×D4×Dic3C2212yes—
21C2×D4×Dic3C2212yes—
22C22×#214C2212yes—
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C3×#450—24no—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29#5229C2211yescharacteristic, complemented
30C2×#835C2212yescomplemented
31#5218—24no—
32C2×#832C2212yes—
33C2×#830C2211yescharacteristic
34#5162—28no—
35C2×#816C2212yescomplemented
36C2×Dic3⋊C8C2212yescomplemented
37#4982—22no—
38#4977C2211yescharacteristic, complemented
39#4973C2211yescharacteristic, complemented
40#4952—22no—
41#4551C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#518—31no—
45C2×#461—32no—
46C2×#450—31no—
47C2×#450D311yescharacteristic
48#2621—31no—
49#1877—32no—
50#1877—32no—
51#1873—31no—
52#1873—31no—
53#1804—32no—
54#1798—31no—
55C25×C3—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—24no—
58C22×C3×D4—24no—
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4D414yes—
62C22×C3×D4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—22no—
65C23×Dic3—22no—
66C22×Dic6C2311yescharacteristic
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C22×C3×C8C2312yes—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×#217—216no—
77C2×#217—21no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—21no—
81C2×#217D412yes—
82#944—24no—
83D4×Dic3—216no—
84D4×Dic3—24no—
85D4×Dic3—24no—
86D4×Dic3—24no—
87D4×Dic3D418yes—
88C2×#214—24no—
89C2×#214—24no—
90C2×#214—24no—
91#926—24no—
92#926—24no—
93C2×C4⋊Dic3D412yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96#924—22no—
97#924—22no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—22no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C3×#102—24no—
105C3×C22⋊C8—22no—
106#835—24no—
107#832—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#830—22no—
112#816—24no—
113Dic3⋊C8—24no—
114C23×D4C2×D311yescharacteristic
115C22×SD16—32no—
116C22×SD16C2×D312yes—
117C2×C4⋊Q8—31no—
118C2×C22⋊Q8—31no—
119C2×C22⋊Q8C2×D311yescharacteristic
120#518—68no—
121C2×C4×D4—32no—
122C2×C4×D4—32no—
123C2×C4×D4—62no—
124C2×#117—31no—
125C2×#117—31no—
126C22×C22⋊C4—62no—
127#461—68no—
128#450—24no—
129#450—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#398—62no—
138#397—62no—
139#391—31no—
140#391—31no—
141#382—31no—
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3—24no—
146C24×C3—24no—
147C24×C3—22no—
148C24×C3—24no—
149C24×C3—82no—
150C24×C3—82no—
151C24×C3C2×D411yescharacteristic
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—216no—
159C2×C3×D4—216no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4C2×D414yes—
165C2×C3×D4C2×D414yes—
166C2×C3×D4#4114yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3—28no—
177C22×Dic3—28no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—82no—
181C22×Dic3—82no—
182C22×Dic3—82no—
183C22×Dic3—82no—
184C22×Dic3C2×D412yes—
185C22×Dic3C2×D412yes—
186C2×Dic6—22no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—22no—
190C3×SD16—48no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—42no—
197C3×C22⋊C4—42no—
198#217—24no—
199#217—28no—
200#217—24no—
201#217—28no—
202#217—42no—
203#214—48no—
204C4⋊Dic3—22no—
205C4⋊Dic3—24no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—42no—
210Dic3⋊C4—44no—
211C4×Dic3—24no—
212C4×Dic3—22no—
213C4×Dic3C2×D414yes—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C25—21no—
218C22×Q8—31no—
219C22×Q8C22×D311yescharacteristic
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4#6714yes—
225C22×D4C22×D312yes—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4C22×D311yescharacteristic
229C2×SD16—24no—
230C2×SD16—24no—
231C2×SD16—24no—
232C2×SD16—64no—
233C2×SD16—64no—
234C2×SD16—64no—
235C22×C8—32no—
236C22×C8C22×D312yes—
237C4⋊Q8—64no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—62no—
241C22⋊Q8—62no—
242C4×D4—38no—
243C4×D4—64no—
244C4×D4—64no—
245C4×D4—616no—
246C4×D4—64no—
247#117—64no—
248#117—64no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—616no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—61no—
263C2×C42—31no—
264C2×C42—31no—
265C2×C42—61no—
266C4⋊C8—64no—
267#103—64no—
268#103—64no—
269#102—24no—
270#102—64no—
271C22⋊C8—22no—
272C22⋊C8—62no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3—216no—
277C23×C3—21no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—21no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—84no—
289C23×C3—84no—
290C23×C3—88no—
291C23×C3—88no—
292C23×C3#12112yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—42no—
295C3×Q8—42no—
296C3×D4—216no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—24no—
300C3×D4#12118yes—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4#12112yes—
311C2×C3×C4C22×D412yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4112yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—88no—
329C2×Dic3—88no—
330C2×Dic3#12014yes—
331Dic6—42no—
332Dic6—42no—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—82no—
340C24—82no—
341C24D3×D411yescharacteristic
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—62no—
347C2×Q8—61no—
348C2×Q8—62no—
349C2×Q8—61no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—216no—
353C2×D4—216no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4D3×D414yes—
359C2×D4#23714yes—
360C2×D4C2×#6714yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—32no—
366C22×C4—32no—
367C22×C4—68no—
368C22×C4—62no—
369C22×C4—68no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—242no—
373C22×C4—242no—
374C22×C4—242no—
375C22×C4—242no—
376C22×C4D3×D411yescharacteristic
377C22×C4#23712yes—
378C22×C4C2×#6712yes—
379C22×C4C23×D311yescharacteristic
380SD16—48no—
381SD16—128no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—34no—
391C4⋊C4—42no—
392C4⋊C4—64no—
393C4⋊C4—62no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—124no—
397C4⋊C4—122no—
398C22⋊C4—42no—
399C22⋊C4—64no—
400C22⋊C4—68no—
401C22⋊C4—64no—
402C22⋊C4—68no—
403C22⋊C4—122no—
404C42—34no—
405C42—64no—
406C42—62no—
407C22×C3—22no—
408C22×C3—28no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—816no—
420C22×C3C2×#12011yescharacteristic
421C22×C3C2×#12111yescharacteristic
422C22×C3#52511yescharacteristic
423C22×C3#57712yes—
424C22×C3#57712yes—
425C3×C4—22no—
426C3×C4—81no—
427C3×C4—81no—
428C3×C4C2×#12112yes—
429C3×C4#52512yes—
430Dic3—24no—
431Dic3—42no—
432Dic3—44no—
433Dic3—81no—
434Dic3—81no—
435C23—216no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—21no—
440C23—24no—
441C23—21no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—84no—
448C23—84no—
449C23—88no—
450C23—88no—
451C23#93912yes—
452C23C2×D3×D411yescharacteristic
453Q8—42no—
454Q8—42no—
455Q8—122no—
456Q8—122no—
457D4—216no—
458D4—24no—
459D4—24no—
460D4—24no—
461D4#93918yes—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—34no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—248no—
486C2×C4—248no—
487C2×C4#93912yes—
488C2×C4C2×D3×D412yes—
489C2×C4C2×#23712yes—
490C2×C4C2×#23711yescharacteristic
491C2×C4C22×#6711yescharacteristic
492C8—42no—
493C8—122no—
494C2×C3—42no—
495C2×C3—48no—
496C2×C3#284711yescharacteristic
497C2×C3#300912yes—
498C2×C3#304211yescharacteristic
499C2×C3#304211yescharacteristic
500C2×C3#312312yes—
501C22—22no—
502C22—22no—
503C22—28no—
504C22—22no—
505C22—42no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—42no—
511C22—44no—
512C22—44no—
513C22—816no—
514C22#558611yescharacteristic
515C22D3×#12011yescharacteristic
516C22#576212yes—
517C22#577512yes—
518C22C2×#93911yescharacteristic
519C4—22no—
520C4—64no—
521C4—81no—
522C4—81no—
523C4—122no—
524C4—124no—
525C4—241no—
526C4—241no—
527C4#558612yes—
528C4C2×#93912yes—
529C3#3291411yescharacteristic, complemented, Sylow
530C2—42no—
531C2—48no—
532C2#8013712yes—
533C2#8014812yes—
534C2#8260111yescharacteristic
535C2#8384811yescharacteristic
536C2#8393511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#450C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#450D3
hypocenterC3#32914
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
4. C2×#5162C2semidirect
5. #82786C2semidirect
6. #72047C2semidirect
7. #71693C2semidirect
8. #70171C2semidirect
9. #70154C2semidirect
10. #69931C2semidirect
11. #69909C2semidirect
17. C2×#944C22semidirect
29. #5229C22semidirect
30. C2×#835C22semidirect
35. C2×#816C22semidirect
36. C2×Dic3⋊C8C22semidirect
38. #4977C22semidirect
39. #4973C22semidirect
41. #4551C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
529. C3#32914semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
C2×C3×#4502
C2×#52182
C2×#51622
#827862
#720472
#716932
#701712
#701542
#699312
#699092
#329143