Finite Group Databasebeta

← #10831706

Subgroups — #10831706

Order 768.

1296 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic
3#82388C211yesmaximal, characteristic, complemented
4C2×#5008C211yesmaximal, characteristic, complemented
5#72037C212yesmaximal
6#71605C211yesmaximal, characteristic, complemented
7#70565C212yesmaximal, complemented
8#70169C212yesmaximal, complemented
9#70163C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24338—31nomaximal, Sylow
13C23×Dic6—21no—
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×D4×Dic3C2212yes—
18C2×#924—21no—
19D4×Dic6—24no—
20D4×Dic6—24no—
21C2×#888C2212yes—
22C2×#878—22no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#442—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5221C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33#5210—22no—
34#5008—24no—
35#4972C2212yes—
36C2×#808C2212yes—
37#4949—21no—
38#4947—21no—
39#4936C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×D4×Q8—31no—
47C2×#476—31no—
48C2×#442D311yescharacteristic
49#2580—31no—
50#2058—32no—
51#2023—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×Dic3—22no—
62C22×Dic6—28no—
63C22×Dic6—21no—
64C22×Dic6—22no—
65C22×Dic6—48no—
66C22×Dic6D411yescharacteristic
67C2×C3×D8—22no—
68C2×C3×D8—22no—
69C2×C3×D8Q814yescomplemented
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42C2311yescharacteristic
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—22no—
84D4×Dic3—24no—
85D4×Dic3—24no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2312yes—
90#924—28no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4D412yes—
94C2×C4×Dic3C2312yes—
95C22×C3⋊C8C2312yes—
96#888—24no—
97#888—24no—
98#878—216no—
99#869—22no—
100#869D414yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×#106—22no—
104C3×#102—22no—
105C3×#102—22no—
106C3×#97—22no—
107#832—24no—
108#832—24no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112#808—24no—
113C4⋊(C3⋊C8)—22no—
114C23×Q8—61no—
115C22×D8C2×D311yescharacteristic
116D4×Q8—64no—
117D4×Q8—64no—
118C2×C4⋊Q8—32no—
119C2×C4⋊Q8—31no—
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—61no—
122C2×C4×Q8—31no—
123C2×C4×D4—32no—
124C2×C4×D4C2×D311yescharacteristic
125C2×C4×D4C2×D311yescharacteristic
126#476—64no—
127#442—24no—
128C2×#106—32no—
129C2×#106C2×D311yescharacteristic
130C2×C4⋊C8—31no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102C2×D311yescharacteristic
134C2×#102C2×D311yescharacteristic
135C2×#97C2×D311yescharacteristic
136#399—61no—
137#393—61no—
138#390—32no—
139#389—32no—
140#385—62no—
141#382—31no—
142#340—32no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—32no—
147C24×C3—21no—
148C24×C3—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4C2×D412yes—
158C2×C3×D4C2×Q812yes—
159C2×C3×D4C2×Q812yescomplemented
160C2×C3×D4#4112yes—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—44no—
165C22×C3×C4—42no—
166C22×C3×C4—82no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—28no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—44no—
174C22×Dic3—44no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×Dic6—22no—
178C2×Dic6—24no—
179C2×Dic6—28no—
180C2×Dic6—24no—
181C2×Dic6—432no—
182C2×Dic6—44no—
183C2×Dic6—44no—
184C2×Dic6—44no—
185C3×D8—28no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8C2×Q812yescomplemented
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—42no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#217—28no—
201#217—44no—
202#217—82no—
203C4⋊Dic3—22no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—24no—
211Dic3⋊C4—82no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C22×Q8—31no—
218C22×Q8—62no—
219C22×Q8—68no—
220C22×Q8—61no—
221C22×Q8—128no—
222C22×D4C22×D311yescharacteristic
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—62no—
227C2×D8—22no—
228C2×D8—22no—
229C2×D8Dic614yescomplemented
230C22×C8—32no—
231C22×C8C22×D311yescharacteristic
232C22×C8C22×D311yescharacteristic
233C4⋊Q8—34no—
234C4⋊Q8—64no—
235C4⋊Q8—64no—
236C4⋊Q8—62no—
237C22⋊Q8—616no—
238C22⋊Q8—68no—
239C4×Q8—62no—
240C4×Q8—62no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—64no—
245C4×D4—64no—
246C4×D4#6714yes—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42—32no—
262C2×C42C22×D311yescharacteristic
263#106—22no—
264#106—64no—
265C4⋊C8—62no—
266#102—22no—
267#102—22no—
268#102—64no—
269#102—64no—
270#97—22no—
271#95—62no—
272#95—61no—
273#95—61no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—41no—
291C3×D4—41no—
292C3×D4C22⋊Q814yes—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—48no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—84no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×Q811yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C42−1+411yescharacteristic
308C2×C3×C42−1+411yescharacteristic
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—416no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3AGL(1,ℤ/8)14yes—
322Dic6—28no—
323Dic6—416no—
324Dic6—48no—
325C3×C8—22no—
326C3×C8—41no—
327C3⋊C8—42no—
328C24—21no—
329C24—21no—
330C2×Q8—64no—
331C2×Q8—68no—
332C2×Q8—62no—
333C2×Q8—64no—
334C2×Q8—124no—
335C2×Q8—124no—
336C2×Q8—124no—
337C2×Q8—1232no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×Dic612yes—
347C2×D4C2×Dic612yescomplemented
348C2×D4#23512yes—
349C2×D4C2×#6712yes—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—68no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—82no—
361C22×C4—124no—
362C22×C4—124no—
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C23×D311yescharacteristic
366D8—28no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8—62no—
375C2×C8C2×Dic612yescomplemented
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—34no—
380C4⋊C4—68no—
381C4⋊C4—64no—
382C4⋊C4—62no—
383C4⋊C4—64no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C4⋊C4—62no—
387C4⋊C4—242no—
388C4⋊C4C2×#6712yes—
389C22⋊C4—24no—
390C22⋊C4—42no—
391C22⋊C4—68no—
392C22⋊C4—124no—
393C22⋊C4—242no—
394C42—62no—
395C42—62no—
396C42D3×D412yes—
397C42C2×#6712yes—
398C22×C3—24no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3D4×Q811yescharacteristic
411C22×C3C2×AGL(1,ℤ/8)12yes—
412C22×C3#57712yes—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—81no—
417C3×C4C2×C22⋊Q811yescharacteristic
418C3×C4#53611yescharacteristic
419C3×C4#54811yescharacteristic
420C3×C4D4×Q811yescharacteristic
421Dic3—24no—
422Dic3—44no—
423Dic3—42no—
424Dic3—82no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23C2×D3×D411yescharacteristic
436Q8—68no—
437Q8—128no—
438Q8—1216no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446D4#92414yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—34no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—64no—
461C2×C4—84no—
462C2×C4—1216no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4#93712yes—
469C2×C4#93712yes—
470C2×C4C22×Dic611yescharacteristic
471C2×C4C2×#23511yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4#101011yescharacteristic
474C2×C4#101011yescharacteristic
475C2×C4C22×#6711yescharacteristic
476C8—22no—
477C8—41no—
478C8—122no—
479C2×C3—44no—
480C2×C3—82no—
481C2×C3#240911yescharacteristic
482C2×C3#300311yescharacteristic
483C2×C3#301011yescharacteristic
484C2×C3#337312yes—
485C2×C3#338412yes—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—44no—
493C22—81no—
494C22—81no—
495C22—82no—
496C22D4×Dic611yescharacteristic
497C22#555211yescharacteristic
498C22D3×AGL(1,ℤ/8)12yes—
499C22#577512yes—
500C22C2×#93711yescharacteristic
501C4—22no—
502C4—22no—
503C4—42no—
504C4—64no—
505C4—81no—
506C4—124no—
507C4—122no—
508C4—242no—
509C4D4×Dic611yescharacteristic
510C4#555211yescharacteristic
511C4C2×#92411yescharacteristic
512C4#582511yescharacteristic
513C3#2433811yescharacteristic, complemented, Sylow
514C2—44no—
515C2—82no—
516C2#7985312yes—
517C2#7985512yes—
518C2#8234911yescharacteristic
519C2#8394411yescharacteristic
520C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#442D3
hypocenterC3#24338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic6C2semidirect
3. #82388C2semidirect
4. C2×#5008C2semidirect
6. #71605C2semidirect
7. #70565C2semidirect
8. #70169C2semidirect
9. #70163C2semidirect
10. #69838C2semidirect
11. #69828C2semidirect
69. C2×C3×D8Q8semidirect
159. C2×C3×D4C2×Q8semidirect
191. C2×C3×C8C2×Q8semidirect
229. C2×D8Dic6semidirect
347. C2×D4C2×Dic6semidirect
375. C2×C8C2×Dic6semidirect
513. C3#24338semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic62
C2×C3×#4422
#823882
C2×#50082
#720372
#716052
#705652
#701692
#701632
#698382
#698282
#243383