Finite Group Databasebeta

← #10828493

Subgroups — #10828493

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×C3×#533C211yesmaximal, characteristic
2C2×#5045C212yesmaximal
3#82069C211yesmaximal, characteristic
4#71783C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#71781C211yesmaximal, characteristic, complemented
7#71781C211yesmaximal, characteristic, complemented
8#71693C212yesmaximal
9#71629C212yesmaximal, complemented
10#71628C212yesmaximal, complemented
11#71531C211yesmaximal, characteristic
12#24065—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2212yes—
29#5210—21no—
30#5210—21no—
31C2×#830C2212yes—
32#5045—28no—
33C2×#808C2212yes—
34C2×#807C2212yes—
35C4×Dic3⋊C4—21no—
36C4×Dic3⋊C4—21no—
37#4928C2211yescharacteristic
38#4928C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#533D311yescharacteristic
47C2×#485—32no—
48#2296—31no—
49#2058—31no—
50#2057—31no—
51#2043—31no—
52#2043—31no—
53#1873—32no—
54#1865—32no—
55#1863—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—24no—
64C3×#125—28no—
65C3×C22⋊Q8—28no—
66C3×C22⋊Q8—24no—
67C3×C22⋊Q8D418yes—
68C3×C4×Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×#117—28no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C42—24no—
80C2×C3×C42—22no—
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#217—21no—
87#926—22no—
88#926—22no—
89#926—24no—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2312yes—
102#835—24no—
103#835—24no—
104#835—24no—
105#835—24no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#830—24no—
111#808—24no—
112#807—24no—
113#533—28no—
114C2×C4⋊Q8C2×D311yescharacteristic
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#125—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2×D312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2×D312yes—
121C2×#117—21no—
122C2×#117—31no—
123C2×#117—31no—
124C22×C42C2×D311yescharacteristic
125#485—68no—
126C2×#107—32no—
127C2×#106—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×C22⋊C8—32no—
133#389—31no—
134#389—31no—
135#389—31no—
136#389—31no—
137#385—61no—
138#385—61no—
139C4×C4⋊C4—61no—
140C4×C4⋊C4—61no—
141#340—32no—
142#340—32no—
143#336—31no—
144#336—31no—
145#336—32no—
146C24×C3C2×D411yescharacteristic
147C2×C3×Q8—24no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×Q8—42no—
155C2×C3×Q8C2×D414yes—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—41no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—28no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4C2×D414yes—
185C3×C22⋊C4—28no—
186C3×C42—28no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191#217—24no—
192#217—42no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3#4112yes—
198C4⋊Dic3#4112yes—
199Dic3⋊C4—42no—
200Dic3⋊C4—42no—
201Dic3⋊C4—42no—
202Dic3⋊C4—42no—
203C4×Dic3—24no—
204C4×Dic3—42no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—21no—
212C22×Q8C22×D312yes—
213C23×C4#6711yescharacteristic
214C23×C4#6711yescharacteristic
215C23×C4C22×D311yescharacteristic
216C22×C8—32no—
217C22×C8—32no—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220#125—28no—
221C22⋊Q8—28no—
222C22⋊Q8—24no—
223C22⋊Q8#6718yes—
224C4×Q8—28no—
225C4×Q8—24no—
226C4×Q8—24no—
227#117—28no—
228#117—62no—
229#117—62no—
230#117—64no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4C22×D312yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—61no—
251C2×C42—61no—
252C2×C42#6711yescharacteristic
253C2×C42#6711yescharacteristic
254C2×C42C22×D311yescharacteristic
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#103—64no—
259#103—64no—
260#103—64no—
261#103—64no—
262C22⋊C8—64no—
263#95—61no—
264#95—61no—
265#95—61no—
266#95—61no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3#12012yes—
274C23×C3C22×D411yescharacteristic
275C3×Q8—24no—
276C3×Q8—24no—
277C3×Q8—48no—
278C3×Q8—42no—
279C3×Q8—42no—
280C2×C3×C4—28no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4#12012yes—
302C2×C3×C4#13512yes—
303C2×C3×C4#13712yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D412yes—
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C42−1+412yes—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C3⋊C8—42no—
322C3⋊C8—42no—
323C24C2×#6711yescharacteristic
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8C2×#6714yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6714yes—
354C22×C4C23×D311yescharacteristic
355C2×C8—62no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—62no—
360C2×C8—62no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—22no—
367C4⋊C4—32no—
368C4⋊C4—32no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—122no—
374C4⋊C4—122no—
375C4⋊C4—122no—
376C4⋊C4—122no—
377C4⋊C4C2×#6714yes—
378C22⋊C4—28no—
379C22⋊C4—64no—
380C22⋊C4—122no—
381C42—28no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—64no—
387C42—122no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3C2×#12011yescharacteristic
396C22×C3#52511yescharacteristic
397C22×C3#53611yescharacteristic
398C22×C3C2×#13511yescharacteristic
399C22×C3C2×#13711yescharacteristic
400C22×C3C2×#13711yescharacteristic
401C22×C3#57511yescharacteristic
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—44no—
407C3×C4—41no—
408C3×C4—44no—
409C3×C4#54712yes—
410C3×C4#54712yes—
411Dic3—42no—
412Dic3—42no—
413Dic3—81no—
414Dic3—81no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23#95312yes—
422C23C22×#6711yescharacteristic
423Q8—24no—
424Q8—24no—
425Q8—48no—
426Q8—42no—
427Q8—42no—
428C2×C4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4#94212yes—
462C2×C4#94212yes—
463C2×C4#95012yes—
464C2×C4#95312yes—
465C2×C4D3×#4111yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C2×C4#101012yes—
468C2×C4C22×#6712yes—
469C8—122no—
470C8—122no—
471C2×C3—42no—
472C2×C3#240611yescharacteristic
473C2×C3#300211yescharacteristic
474C2×C3#300411yescharacteristic
475C2×C3#310711yescharacteristic
476C2×C3#311611yescharacteristic
477C2×C3#323111yescharacteristic
478C2×C3#323711yescharacteristic
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22C2×#94211yescharacteristic
487C22C2×#94211yescharacteristic
488C22C2×#95011yescharacteristic
489C22#582211yescharacteristic
490C22#582511yescharacteristic
491C22#582811yescharacteristic
492C22C2×#95311yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—44no—
499C4—44no—
500C4—122no—
501C4—122no—
502C4—241no—
503C4—241no—
504C4#582412yes—
505C4#582412yes—
506C3#2406511yescharacteristic, complemented, Sylow
507C2—42no—
508C2#8332511yescharacteristic
509C2#8332711yescharacteristic
510C2#8335211yescharacteristic
511C2#8335411yescharacteristic
512C2#8405911yescharacteristic
513C2#8408011yescharacteristic
514C2#8409011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#24065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #71783C2semidirect
5. #71782C2semidirect
6. #71781C2semidirect
7. #71781C2semidirect
9. #71629C2semidirect
10. #71628C2semidirect
506. C3#24065semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50452
#820692
#717832
#717822
#717812
#716932
#716292
#716282
#715312
#240653