Finite Group Databasebeta

← #10828476

Subgroups — #10828476

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#4805C212yesmaximal
3D3×#389C211yesmaximal, characteristic
4C3×#2043C211yesmaximal, characteristic, complemented
5#71781C211yesmaximal, characteristic, complemented
6#70268C212yesmaximal
7#70238C212yesmaximal, complemented
8#70237C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic
11#68633C211yesmaximal, characteristic, complemented
12#24063—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16#5728—28no—
17C2×#897—21no—
18C2×#892—21no—
19C2×D3×C4⋊C4—21no—
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#890—21no—
23C2×#887—21no—
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C3×#389C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2212yes—
33#4984C2211yescharacteristic
34#4976—21no—
35#4972C2211yescharacteristic
36C2×#808C2212yes—
37#4928C2211yescharacteristic
38#4805—28no—
39D3×#95—21no—
40#4647—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#483—32no—
48C2×#389—31no—
49#2058—31no—
50#2058—31no—
51#2043—31no—
52#2043D311yescharacteristic, complemented
53#1872—32no—
54#1865—32no—
55#1865—32no—
56#1817—31no—
57C22×C3×Q8C2312yes—
58C22×C4×D3—21no—
59C22×C4×D3—21no—
60C22×C4×D3D411yescharacteristic
61C22×C4×D3D411yescharacteristic
62C22×C4×D3C2311yescharacteristic
63C22×Dic6—21no—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72#946—24no—
73#946D418yes—
74Q8×Dic3—24no—
75Q8×Dic3—24no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#897—28no—
94#892—28no—
95D3×C4⋊C4—22no—
96D3×C4⋊C4—22no—
97D3×C4⋊C4—24no—
98D3×C4⋊C4—44no—
99#890—28no—
100#887—28no—
101C42×D3—24no—
102#869—24no—
103C3×#103—24no—
104C3×#103—24no—
105C3×#95—21no—
106#835—24no—
107#835—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111(C2×D3)⋊C8—24no—
112#817—24no—
113#808—24no—
114#533—68no—
115C2×C4⋊Q8—31no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125—61no—
118C2×C22⋊Q8—32no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—32no—
121C2×C4×Q8—61no—
122C2×#117—61no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—61no—
126C22×C42—31no—
127#483—68no—
128C2×#106—32no—
129C2×#106—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103C2×D312yes—
133C2×#103C2×D312yes—
134C2×C22⋊C8—32no—
135#389—31no—
136#389—31no—
137#389—31no—
138#389—61no—
139#389—61no—
140#389C2×D311yescharacteristic
141C2×#95—61no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—31no—
146#336C2×D311yescharacteristic
147C23×D3C2×D411yescharacteristic
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8C2×D414yes—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3—22no—
171C2×C4×D3—24no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—41no—
175C2×C4×D3—42no—
176C2×C4×D3—42no—
177C2×C4×D3—42no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×C4×D3SD1614yes—
181C2×C4×D3C2×D414yes—
182C2×Dic6—24no—
183C2×Dic6—42no—
184C2×Dic6—42no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4#4112yes—
194C3×C42—22no—
195(C2×D3)⋊C4—28no—
196C4⋊Dic3—24no—
197C4⋊Dic3—28no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—42no—
202C4⋊Dic3C2×D414yes—
203C4⋊Dic3#4112yes—
204Dic3⋊C4—24no—
205Dic3⋊C4—28no—
206Dic3⋊C4—42no—
207Dic3⋊C4—42no—
208Dic3⋊C4—42no—
209C4×Dic3—28no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C22×Q8—61no—
217C22×Q8C22×D312yes—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—61no—
222C23×C4—61no—
223C22×C8—32no—
224C22×C8C22×D312yes—
225C4⋊Q8—24no—
226C4⋊Q8—64no—
227#125—68no—
228C22⋊Q8—38no—
229C22⋊Q8—68no—
230C22⋊Q8—64no—
231C4×Q8—68no—
232C4×Q8—64no—
233C4×Q8—64no—
234#117—68no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—124no—
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C22⋊C4—62no—
256C2×C42—31no—
257C2×C42—31no—
258C2×C42—31no—
259C2×C42—64no—
260C2×C42—62no—
261C2×C42C22×D311yescharacteristic
262#106—64no—
263#106—64no—
264#103—24no—
265#103—24no—
266#103—64no—
267#103—64no—
268C22⋊C8—64no—
269#95—21no—
270#95—61no—
271#95—61no—
272#95—61no—
273C23×C3C22×D411yescharacteristic
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—21no—
278C22×D3#12012yes—
279C22×D3C2×SD1612yes—
280C22×D3#13712yes—
281C3×Q8—24no—
282C3×Q8—24no—
283C3×Q8—42no—
284C3×Q8—42no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4C2×SD1612yes—
299C2×C3×C4#13712yes—
300C2×C3×C4C22×D412yes—
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C42−1+412yes—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—24no—
310C2×Dic3—22no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3#12012yes—
321C2×Dic3C2×SD1612yes—
322C4×D3—28no—
323C4×D3—24no—
324C4×D3—24no—
325C4×D3—44no—
326C4×D3—48no—
327C4×D3—84no—
328Dic6—48no—
329C3×C8—42no—
330C3⋊C8—42no—
331C24—31no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—64no—
338C2×Q8—122no—
339C2×Q8—122no—
340C2×Q8D3×D414yes—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—34no—
347C22×C4—34no—
348C22×C4—31no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—64no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—121no—
361C22×C4—122no—
362C22×C4—122no—
363C22×C4—122no—
364C22×C4—122no—
365C22×C4—122no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8—62no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—34no—
380C4⋊C4—32no—
381C4⋊C4—42no—
382C4⋊C4—68no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—68no—
386C4⋊C4—62no—
387C4⋊C4—61no—
388C4⋊C4—62no—
389C4⋊C4—122no—
390C4⋊C4—122no—
391C4⋊C4—122no—
392C4⋊C4—122no—
393C4⋊C4#23712yes—
394C22⋊C4—68no—
395C42—22no—
396C42—62no—
397C42—62no—
398C42—68no—
399C42—62no—
400C22×C3C2×#12011yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#53611yescharacteristic
403C22×C3C22×SD1611yescharacteristic
404C22×C3C2×#13711yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C22×C3#57511yescharacteristic
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—22no—
410C2×D3—22no—
411C2×D3—22no—
412C2×D3—22no—
413C2×D3#44814yes—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4—81no—
419C3×C4#54712yes—
420C3×C4#54712yes—
421Dic3—22no—
422Dic3—22no—
423Dic3—42no—
424Dic3—44no—
425Dic3—41no—
426Dic3—81no—
427C23—32no—
428C23—32no—
429C23—32no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—61no—
434C23C2×D3×D411yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—128no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—68no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—124no—
468C2×C4—128no—
469C2×C4—121no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—244no—
477C2×C4D3×SD1612yes—
478C2×C4#91812yes—
479C2×C4C2×D3×D412yes—
480C2×C4C2×#23711yescharacteristic
481C2×C4D3×#4111yescharacteristic
482C2×C4#101012yes—
483C8—42no—
484C8—122no—
485C2×C3#240611yescharacteristic
486C2×C3C2×#44811yescharacteristic
487C2×C3#300711yescharacteristic
488C2×C3#310111yescharacteristic
489C2×C3#311311yescharacteristic
490C2×C3#322911yescharacteristic
491C2×C3#323611yescharacteristic
492D3—24no—
493C22—34no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—62no—
498C22—62no—
499C22—62no—
500C22D3×#12011yescharacteristic
501C22#560611yescharacteristic
502C22#567511yescharacteristic
503C22C2×D3×SD1611yescharacteristic
504C22C2×#91811yescharacteristic
505C22D3×#13711yescharacteristic
506C22#577711yescharacteristic
507C4—22no—
508C4—22no—
509C4—42no—
510C4—44no—
511C4—62no—
512C4—62no—
513C4—81no—
514C4—122no—
515C4—121no—
516C4—124no—
517C4—241no—
518C4#561112yes—
519C4#565512yes—
520C3#2406311yescharacteristic, complemented, Sylow
521C2—64no—
522C2#7710711yescharacteristic
523C2D3×#44811yescharacteristic
524C2#8018411yescharacteristic
525C2#8040811yescharacteristic
526C2#8045811yescharacteristic
527C2#8160711yescharacteristic
528C2#8174511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2043C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2043D3
hypocenterC3#24063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
4. C3×#2043C2semidirect
5. #71781C2semidirect
7. #70238C2semidirect
8. #70237C2semidirect
9. #69605C2semidirect
11. #68633C2semidirect
52. #2043D3semidirect
520. C3#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#48052
D3×#3892
C3×#20432
#717812
#702682
#702382
#702372
#696052
#686642
#686332
#240633