Finite Group Databasebeta

← #10900940

Subgroups — #10900940

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2C2×#4805C212yesmaximal
3#76663C211yesmaximal, characteristic
4C3×#2024C211yesmaximal, characteristic, complemented
5#71783C211yesmaximal, characteristic, complemented
6#70231C212yesmaximal, complemented
7#69668C211yesmaximal, characteristic, complemented
8#68871C212yesmaximal
9#68827C212yesmaximal, complemented
10#68664C211yesmaximal, characteristic
11#68633C211yesmaximal, characteristic, complemented
12#29641—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946C2211yescharacteristic
16C2×#944—21no—
17D3×C4⋊Q8—28no—
18C2×D3×C4⋊C4—21no—
19C2×D3×C4⋊C4C2212yes—
20C2×#888—21no—
21C2×#888C2212yes—
22C2×C42×D3C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C3×#103C2212yes—
25C3×#391C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2212yes—
28C2×(C2×D3)⋊C8C2212yes—
29C2×#817C2212yes—
30#4987C2211yescharacteristic
31#4984C2211yescharacteristic
32C2×#808C2212yes—
33#4928C2211yescharacteristic
34#4805—28no—
35#4668—21no—
36#4664—21no—
37#4654—21no—
38#4652—21no—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C22×C4⋊Q8—31no—
47C2×#483—32no—
48#2855—31no—
49#2058—32no—
50#2058—31no—
51#2058—31no—
52#2024—32no—
53#2024—31no—
54#2024D311yescharacteristic, complemented
55#1823—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C2×D3×Q8—28no—
60C2×D3×Q8—44no—
61C2×D3×Q8—44no—
62C22×C4×D3—22no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6—21no—
67C22×Dic6—22no—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—24no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74#946—24no—
75#944—28no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91D3×C4⋊C4—28no—
92D3×C4⋊C4—24no—
93D3×C4⋊C4D418yes—
94#888—28no—
95#888—24no—
96#888—24no—
97C42×D3—24no—
98#869—24no—
99C3×#103—24no—
100C3×#95—21no—
101C3×#95—21no—
102#835—24no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108(C2×D3)⋊C8—24no—
109#817—24no—
110#808—24no—
111C23×Q8—61no—
112C2×C4⋊Q8—32no—
113C2×C4⋊Q8—31no—
114C2×C4⋊Q8—61no—
115C2×C4⋊Q8—61no—
116C2×C4⋊Q8—68no—
117C2×C4⋊Q8C2×D311yescharacteristic
118C2×C22⋊Q8—31no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—61no—
121C22×C42—31no—
122#483—68no—
123C2×#106—32no—
124C2×#106—32no—
125C2×#103—32no—
126C2×#103C2×D312yes—
127C2×C22⋊C8—32no—
128#391—31no—
129#391C2×D311yescharacteristic
130#389—31no—
131#389—31no—
132#387—61no—
133#387—61no—
134#385—61no—
135#381—61no—
136#380—31no—
137#340—32no—
138#340—32no—
139#340—32no—
140#340C2×D312yes—
141#336—32no—
142#336—31no—
143#336C2×D311yescharacteristic
144C23×D3C2×D411yescharacteristic
145C2×C3×Q8—24no—
146C2×C3×Q8—22no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—21no—
150C2×C3×Q8—42no—
151C2×C3×Q8—42no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164D3×Q8—432no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—21no—
170C2×C4×D3—22no—
171C2×C4×D3—21no—
172C2×C4×D3—22no—
173C2×C4×D3—28no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×C4×D3SD1614yes—
177C2×C4×D3C2×D414yes—
178C2×Dic6—24no—
179C2×Dic6—28no—
180C2×Dic6—44no—
181C2×Dic6—42no—
182C2×Dic6—42no—
183C2×Dic6—44no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4C2×D414yes—
191C3×C42—22no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—24no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—42no—
200C4⋊Dic3C2×D414yes—
201Dic3⋊C4—28no—
202Dic3⋊C4—28no—
203Dic3⋊C4—42no—
204Dic3⋊C4—42no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—21no—
212C22×Q8—62no—
213C22×Q8—68no—
214C22×Q8—61no—
215C22×Q8—124no—
216C22×Q8—124no—
217C22×Q8C22×D311yescharacteristic
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—62no—
222C22×C8—32no—
223C22×C8C22×D312yes—
224C4⋊Q8—24no—
225C4⋊Q8—64no—
226C4⋊Q8—64no—
227C4⋊Q8—68no—
228C4⋊Q8—68no—
229C4⋊Q8—64no—
230C22⋊Q8—64no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—68no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C4⋊C4C22×D312yes—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—31no—
251C2×C42—64no—
252C2×C42C22×D311yescharacteristic
253#106—64no—
254#106—64no—
255#103—24no—
256#103—64no—
257C22⋊C8—64no—
258#95—21no—
259#95—21no—
260#95—61no—
261#95—61no—
262#95—61no—
263#95—61no—
264#95—61no—
265C23×C3C22×D411yescharacteristic
266C22×D3—21no—
267C22×D3—21no—
268C22×D3—21no—
269C22×D3—21no—
270C22×D3#12012yes—
271C22×D3C2×SD1612yes—
272C22×D3AGL(1,ℤ/8)12yes—
273C3×Q8—48no—
274C3×Q8—42no—
275C3×Q8—42no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—24no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4C2×SD1612yes—
290C2×C3×C4#13712yes—
291C2×C3×C4C22×D412yes—
292C2×C3×C42−1+411yescharacteristic
293C2×C3×C42−1+412yes—
294C2×C3×C42−1+411yescharacteristic
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—24no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3#12012yes—
311C2×Dic3C2×SD1612yes—
312C2×Dic3#13712yes—
313C4×D3—28no—
314C4×D3—28no—
315C4×D3—24no—
316C4×D3—24no—
317C4×D3—416no—
318Dic6—416no—
319Dic6—48no—
320C3×C8—42no—
321C3⋊C8—42no—
322C24—31no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—21no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—64no—
331C2×Q8—68no—
332C2×Q8—122no—
333C2×Q8—124no—
334C2×Q8—124no—
335C2×Q8—1232no—
336C2×Q8—122no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—34no—
344C22×C4—34no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—68no—
350C22×C4—62no—
351C22×C4—61no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—124no—
358C22×C4—124no—
359C22×C4D3×D411yescharacteristic
360C22×C4D3×D411yescharacteristic
361C22×C4C23×D311yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—34no—
372C4⋊C4—68no—
373C4⋊C4—68no—
374C4⋊C4—64no—
375C4⋊C4—62no—
376C4⋊C4—62no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—122no—
380C4⋊C4—122no—
381C4⋊C4—122no—
382C4⋊C4D3×D414yes—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C42—22no—
386C42—62no—
387C42—62no—
388C42—62no—
389C22×C3C2×#12011yescharacteristic
390C22×C3#53611yescharacteristic
391C22×C3#53611yescharacteristic
392C22×C3C22×SD1611yescharacteristic
393C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C3C2×#13711yescharacteristic
395C22×C3C2×#13711yescharacteristic
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3#45014yes—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—44no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4#53612yes—
409C3×C4#53612yes—
410Dic3—22no—
411Dic3—22no—
412Dic3—22no—
413Dic3—44no—
414Dic3—81no—
415Dic3—81no—
416C23—32no—
417C23—32no—
418C23—32no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23C2×D3×D411yescharacteristic
424Q8—48no—
425Q8—42no—
426Q8—42no—
427Q8—128no—
428Q8—1216no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—68no—
446C2×C4—68no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—64no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—1216no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4D3×SD1612yes—
466C2×C4#91812yes—
467C2×C4C2×D3×D412yes—
468C2×C4#100711yescharacteristic
469C2×C4#101011yescharacteristic
470C2×C4#101012yes—
471C8—42no—
472C8—122no—
473C2×C3#261411yescharacteristic
474C2×C3C2×#45011yescharacteristic
475C2×C3#300411yescharacteristic
476C2×C3#322911yescharacteristic
477C2×C3#322911yescharacteristic
478C2×C3#323811yescharacteristic
479C2×C3#323811yescharacteristic
480D3—24no—
481C22—34no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—62no—
486C22—62no—
487C22—62no—
488C22D3×#12011yescharacteristic
489C22#562811yescharacteristic
490C22#567511yescharacteristic
491C22C2×D3×SD1611yescharacteristic
492C22C2×#91811yescharacteristic
493C22D3×AGL(1,ℤ/8)11yescharacteristic
494C22D3×#13711yescharacteristic
495C4—22no—
496C4—22no—
497C4—44no—
498C4—62no—
499C4—62no—
500C4—62no—
501C4—81no—
502C4—81no—
503C4—124no—
504C4—241no—
505C4—241no—
506C4#562812yes—
507C4#567512yes—
508C3#2964111yescharacteristic, complemented, Sylow
509C2—64no—
510C2#7687011yescharacteristic
511C2D3×#45011yescharacteristic
512C2#8018311yescharacteristic
513C2#8101511yescharacteristic
514C2#8111511yescharacteristic
515C2#8160711yescharacteristic
516C2#8169011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2024C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2024D3
hypocenterC3#29641
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
4. C3×#2024C2semidirect
5. #71783C2semidirect
6. #70231C2semidirect
7. #69668C2semidirect
9. #68827C2semidirect
11. #68633C2semidirect
54. #2024D3semidirect
508. C3#29641semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#48052
#766632
C3×#20242
#717832
#702312
#696682
#688712
#688272
#686642
#686332
#296413