Finite Group Databasebeta

← #10828492

Subgroups — #10828492

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#4815C212yesmaximal
3#75963C211yesmaximal, characteristic
4C3×#2043C211yesmaximal, characteristic, complemented
5#71781C211yesmaximal, characteristic, complemented
6#70269C212yesmaximal
7#70238C212yesmaximal, complemented
8#70235C212yesmaximal, complemented
9#69604C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic
11#68633C211yesmaximal, characteristic, complemented
12#24065—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16#5728—28no—
17C2×#897—21no—
18C2×#892—21no—
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#890—21no—
22C2×#887—21no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C3×#389C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#817C2212yes—
32#4989—21no—
33#4984C2211yescharacteristic
34Dic3×C4⋊C4—21no—
35#4972C2211yescharacteristic
36C2×#807C2212yes—
37#4928C2211yescharacteristic
38#4815—28no—
39#4668—21no—
40#4649—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#485—32no—
48#2296—31no—
49#2058—31no—
50#2057—31no—
51#2043—31no—
52#2043D311yescharacteristic, complemented
53#1873—32no—
54#1865—32no—
55#1863—32no—
56#1817—31no—
57C22×C3×Q8C2312yes—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×Dic6—21no—
62C22×C3×C8C2312yes—
63C3×C4⋊Q8—24no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70#946—24no—
71#946D418yes—
72Q8×Dic3—24no—
73Q8×Dic3—24no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—21no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#897—28no—
94#892—28no—
95#892—22no—
96#892—22no—
97#892—24no—
98#890—28no—
99#887—28no—
100C42×D3—24no—
101#869—24no—
102C3×#103—24no—
103C3×#103—24no—
104C3×#95—21no—
105#835—24no—
106#835—24no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110(C2×D3)⋊C8—24no—
111#817—24no—
112#807—24no—
113#533—68no—
114C2×C4⋊Q8—31no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#125—61no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—61no—
119C2×C4×Q8—32no—
120C2×C4×Q8—61no—
121C2×#117—31no—
122C2×#117—31no—
123C2×#117—61no—
124C22×C42—31no—
125#485—68no—
126C2×#107—32no—
127C2×#106—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103C2×D312yes—
131C2×#103C2×D312yes—
132C2×C22⋊C8—32no—
133#389—31no—
134#389—31no—
135#389—31no—
136#389C2×D311yescharacteristic
137#385—61no—
138#385—61no—
139C4×C4⋊C4—61no—
140C4×C4⋊C4—61no—
141#340—32no—
142#340—32no—
143#336—32no—
144#336—31no—
145#336C2×D311yescharacteristic
146C23×D3C2×D411yescharacteristic
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D414yes—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—21no—
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—41no—
174C2×C4×D3C2×D414yes—
175C2×Dic6—24no—
176C2×Dic6—42no—
177C2×Dic6—42no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4#4112yes—
187C3×C42—22no—
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—28no—
190(C2×D3)⋊C4—42no—
191C4⋊Dic3—24no—
192C4⋊Dic3—28no—
193C4⋊Dic3—22no—
194C4⋊Dic3—21no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3C2×D414yes—
198C4⋊Dic3#4112yes—
199Dic3⋊C4—28no—
200Dic3⋊C4—42no—
201Dic3⋊C4—42no—
202C4×Dic3—28no—
203C4×Dic3—24no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C4×Dic3—42no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—61no—
212C22×Q8C22×D312yes—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C22×C8—32no—
217C22×C8C22×D312yes—
218C4⋊Q8—24no—
219C4⋊Q8—64no—
220#125—68no—
221C22⋊Q8—38no—
222C22⋊Q8—68no—
223C22⋊Q8—64no—
224C4×Q8—68no—
225C4×Q8—64no—
226C4×Q8—64no—
227#117—62no—
228#117—68no—
229#117—62no—
230#117—64no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C22⋊C4—61no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—61no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—31no—
251C2×C42—64no—
252C2×C42—61no—
253C2×C42—62no—
254C2×C42—61no—
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#103—24no—
259#103—24no—
260#103—64no—
261#103—64no—
262C22⋊C8—64no—
263#95—21no—
264#95—61no—
265#95—61no—
266#95—61no—
267C23×C3C22×D411yescharacteristic
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—21no—
274C22×D3#12012yes—
275C3×Q8—24no—
276C3×Q8—24no—
277C3×Q8—42no—
278C3×Q8—42no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4#13512yes—
293C2×C3×C4#13712yes—
294C2×C3×C4C22×D412yes—
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C42−1+412yes—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3#12012yes—
315C2×Dic3#13712yes—
316C4×D3—28no—
317C4×D3—24no—
318C4×D3—24no—
319C4×D3—44no—
320Dic6—48no—
321C3×C8—42no—
322C3⋊C8—42no—
323C24—31no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—64no—
330C2×Q8—122no—
331C2×Q8—122no—
332C2×Q8D3×D414yes—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—34no—
339C22×C4—31no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—121no—
352C22×C4D3×D411yescharacteristic
353C22×C4D3×D411yescharacteristic
354C22×C4C23×D311yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—62no—
359C2×C8—62no—
360C2×C8—62no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—34no—
366C4⋊C4—32no—
367C4⋊C4—42no—
368C4⋊C4—68no—
369C4⋊C4—64no—
370C4⋊C4—68no—
371C4⋊C4—62no—
372C4⋊C4—61no—
373C4⋊C4—62no—
374C4⋊C4—122no—
375C4⋊C4—122no—
376C4⋊C4—122no—
377C4⋊C4#23712yes—
378C22⋊C4—68no—
379C22⋊C4—64no—
380C22⋊C4—122no—
381C42—22no—
382C42—62no—
383C42—62no—
384C42—68no—
385C42—64no—
386C42—62no—
387C42—122no—
388C22×C3C2×#12011yescharacteristic
389C22×C3#52511yescharacteristic
390C22×C3#53611yescharacteristic
391C22×C3C2×#13511yescharacteristic
392C22×C3C2×#13711yescharacteristic
393C22×C3C2×#13711yescharacteristic
394C22×C3#57511yescharacteristic
395C2×D3—22no—
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—41no—
399C2×D3—41no—
400C2×D3—41no—
401C2×D3—41no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—42no—
405C3×C4—44no—
406C3×C4—81no—
407C3×C4#54712yes—
408C3×C4#54712yes—
409Dic3—22no—
410Dic3—22no—
411Dic3—42no—
412Dic3—44no—
413Dic3—41no—
414Dic3—81no—
415C23—32no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23C2×D3×D411yescharacteristic
423Q8—24no—
424Q8—24no—
425Q8—42no—
426Q8—42no—
427Q8—128no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—68no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—64no—
449C2×C4—64no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—124no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4#91612yes—
464C2×C4#91812yes—
465C2×C4C2×D3×D412yes—
466C2×C4C2×#23711yescharacteristic
467C2×C4D3×#4111yescharacteristic
468C2×C4#101012yes—
469C8—42no—
470C8—122no—
471C2×C3#240611yescharacteristic
472C2×C3#300211yescharacteristic
473C2×C3#300411yescharacteristic
474C2×C3#310711yescharacteristic
475C2×C3#311611yescharacteristic
476C2×C3#323111yescharacteristic
477C2×C3#323711yescharacteristic
478D3—42no—
479C22—62no—
480C22—62no—
481C22—62no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22—121no—
486C22D3×#12011yescharacteristic
487C22#560611yescharacteristic
488C22#567511yescharacteristic
489C22C2×#91611yescharacteristic
490C22C2×#91811yescharacteristic
491C22D3×#13711yescharacteristic
492C22#577711yescharacteristic
493C4—22no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—62no—
498C4—62no—
499C4—81no—
500C4—122no—
501C4—121no—
502C4—124no—
503C4—241no—
504C4#561112yes—
505C4#565512yes—
506C3#2406511yescharacteristic, complemented, Sylow
507C2—122no—
508C2#7710711yescharacteristic
509C2#8003811yescharacteristic
510C2#8018311yescharacteristic
511C2#8040911yescharacteristic
512C2#8045911yescharacteristic
513C2#8160911yescharacteristic
514C2#8174611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2043C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2043D3
hypocenterC3#24065
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. #70235C2semidirect
9. #69604C2semidirect
11. #68633C2semidirect
52. #2043D3semidirect
506. C3#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#48152
#759632
C3×#20432
#717812
#702692
#702382
#702352
#696042
#686642
#686332
#240653