Finite Group Databasebeta

← #10944137

Subgroups — #10944137

Order 768.

1372 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic, complemented
2C2×#5228C211yesmaximal, characteristic, complemented
3C2×#5227C211yesmaximal, characteristic
4C2×#5028C212yesmaximal, complemented
5#82287C211yesmaximal, characteristic
6#71633C212yesmaximal, complemented
7#71632C212yesmaximal, complemented
8#71626C212yesmaximal, complemented
9#71586C211yesmaximal, characteristic
10#71585C211yesmaximal, characteristic, complemented
11#71583C211yesmaximal, characteristic, complemented
12#32918—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C3×#519—28no—
15C2×C3×C4×Q8—22no—
16C2×C3×C4×Q8C2212yes—
17C2×C3×C4×Q8C2212yes—
18C2×C3×#117—22no—
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×#216C2212yes—
22C22×#215C2212yes—
23C2×#930C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×#875C2211yescharacteristic
26#5228—24no—
27#5227—24no—
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32#5028—28no—
33#4994—22no—
34C2×#810C2212yes—
35C2×#809C2212yes—
36#4961—22no—
37#4939C2211yescharacteristic
38#4935C2211yescharacteristic
39#4929C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C2×#519D311yescharacteristic, complemented
45C2×#464—32no—
46C2×#451—31no—
47C2×#448—31no—
48#2620—31no—
49#1880—32no—
50#1878—32no—
51#1875—31no—
52#1873—31no—
53#1803—32no—
54#1798—31no—
55C22×C3×Q8—24no—
56C22×C3×Q8—24no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8D414yes—
60C22×C3×Q8C2312yes—
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×D12C2311yescharacteristic
65C22×Dic6C2311yescharacteristic
66C3×C4×Q8—216no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×C4×Q8D418yescomplemented
71C3×#117—216no—
72C3×#117—24no—
73C3×#117—24no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4D412yes—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—21no—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#216—24no—
87C2×#216—24no—
88C2×#216—24no—
89C2×#215—24no—
90C2×#215—24no—
91C2×#215—24no—
92#930—22no—
93#930—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100#924—22no—
101#924—22no—
102C2×Dic3⋊C4—21no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#875—24no—
106#835—24no—
107#835—24no—
108C3⋊#95—22no—
109#830—22no—
110#830—22no—
111#810—24no—
112#809—24no—
113C4⋊(C3⋊C8)—24no—
114C23×Q8C2×D311yescharacteristic
115C22×Q16—32no—
116C22×SD16—32no—
117C2×#124—31no—
118C2×C22⋊Q8—31no—
119C2×#121—31no—
120#519—28no—
121C2×C4×Q8—22no—
122C2×C4×Q8C2×D312yes—
123C2×C4×Q8C2×D312yes—
124C2×#117—22no—
125C2×#117C2×D311yescharacteristic
126C2×#117C2×D311yescharacteristic
127#464—68no—
128#451—64no—
129#448—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8—31no—
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3C2×D411yescharacteristic
146C23×D3—21no—
147C2×C3×Q8—24no—
148C2×C3×Q8—216no—
149C2×C3×Q8—24no—
150C2×C3×Q8—216no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8C2×D414yes—
156C2×C3×Q8C2×D414yes—
157C2×C3×Q8#4114yes—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4#4112yes—
170C22×C3×C4C2411yescharacteristic
171C2×#67—42no—
172C2×#67—82no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—22no—
180C2×Dic6—22no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—22no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4C2×D414yes—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—24no—
191C3×C42—28no—
192C3×C42—24no—
193C3×C42—28no—
194C3×C42—22no—
195C3×C42C2×D414yes—
196#217—42no—
197#216—48no—
198#215—48no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—42no—
201(C2×D3)⋊C4—82no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—82no—
207Dic3⋊C4—42no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—22no—
217C22×Q8—21no—
218C22×Q8—31no—
219C22×Q8D1214yes—
220C22×Q8C22×D312yes—
221C22×D4—31no—
222C22×D4—61no—
223C23×C4—21no—
224C23×C4C22×D311yescharacteristic
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—64no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C22×C8—32no—
232C22×C8—32no—
233#124—64no—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236#121—62no—
237#121—62no—
238C4×Q8—216no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8#6718yescomplemented
243#117—216no—
244#117—24no—
245#117—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4#6712yes—
253C2×C4⋊C4C22×D312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4#6712yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42C22×D311yescharacteristic
264C2×C42C22×D311yescharacteristic
265C4⋊C8—64no—
266#103—64no—
267#103—64no—
268#103—64no—
269#102—64no—
270C22⋊C8—62no—
271C22⋊C8—62no—
272#95—62no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3#12112yes—
278C23×C3C22×D411yescharacteristic
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—216no—
285C3×Q8—24no—
286C3×Q8—24no—
287C3×Q8—24no—
288C3×Q8#12118yes—
289C2×C3×C4—216no—
290C2×C3×C4—24no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4#12014yes—
306C2×C3×C4#12112yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D412yes—
309C2×C3×C4C2×#4112yes—
310C2×C3×C4C2×#4111yescharacteristic
311#67—84no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319D12—42no—
320D12—42no—
321Dic6—42no—
322Dic6—42no—
323C3⋊C8—42no—
324C3⋊C8—42no—
325C24—61no—
326C24C2×#6711yescharacteristic
327C2×Q8—24no—
328C2×Q8—216no—
329C2×Q8—24no—
330C2×Q8—216no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8C2×D1214yes—
340C2×Q8#23514yes—
341C2×Q8C2×#6714yes—
342C2×D4—61no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—122no—
347C2×D4—242no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4C2×D1212yes—
359C22×C4#23512yes—
360C22×C4C2×#6712yes—
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6712yes—
363C22×C4C23×D311yescharacteristic
364Q16—128no—
365SD16—128no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—122no—
381C4⋊C4—242no—
382C4⋊C4C2×#6714yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C22⋊C4—124no—
388C22⋊C4—242no—
389C42—28no—
390C42—24no—
391C42—28no—
392C42—22no—
393C42C2×#6714yes—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3C2×#12011yescharacteristic
400C22×C3C2×#12111yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#57712yes—
403C22×C3#57812yes—
404C2×D3—81no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—28no—
410C3×C4—22no—
411C3×C4—24no—
412C3×C4—42no—
413C3×C4—44no—
414C3×C4C2×#12112yes—
415C3×C4#52512yes—
416Dic3—81no—
417Dic3—81no—
418Dic3—81no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—121no—
424C23—121no—
425C23—122no—
426C23—122no—
427C23—121no—
428C23#93012yes—
429C23C22×#6711yescharacteristic
430Q8—216no—
431Q8—24no—
432Q8—24no—
433Q8—24no—
434Q8—122no—
435Q8—122no—
436Q8#93018yes—
437D4—122no—
438D4—122no—
439D4—244no—
440C2×C4—216no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4#93012yes—
464C2×C4#95314yes—
465C2×C4C22×D1211yescharacteristic
466C2×C4C2×#23512yes—
467C2×C4C2×#23511yescharacteristic
468C2×C4C22×#6712yes—
469C8—122no—
470C8—122no—
471C2×C3—42no—
472C2×C3#284711yescharacteristic
473C2×C3#301312yes—
474C2×C3#304311yescharacteristic
475C2×C3#304411yescharacteristic
476C2×C3#312112yes—
477D3—161no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—241no—
484C22—241no—
485C22—242no—
486C22—241no—
487C22—242no—
488C22C2×#93011yescharacteristic
489C22#579111yescharacteristic
490C22#583712yes—
491C22#583812yes—
492C22C2×#95311yescharacteristic
493C4—28no—
494C4—22no—
495C4—24no—
496C4—42no—
497C4—44no—
498C4—241no—
499C4—241no—
500C4—241no—
501C4C2×#93012yes—
502C4#579112yes—
503C3#3291811yescharacteristic, complemented, Sylow
504C2—42no—
505C2—481no—
506C2#8255111yescharacteristic
507C2#8255212yes—
508C2#8255411yescharacteristic
509C2#8403211yescharacteristic
510C2#8411412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#519C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#519D3
hypocenterC3#32918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
1. C2×C3×#519C2semidirect
2. C2×#5228C2semidirect
4. C2×#5028C2semidirect
6. #71633C2semidirect
7. #71632C2semidirect
8. #71626C2semidirect
10. #71585C2semidirect
11. #71583C2semidirect
44. C2×#519D3semidirect
70. C3×C4×Q8D4semidirect
242. C4×Q8#67semidirect
503. C3#32918semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×#52282
C2×#52272
C2×#50282
#822872
#716332
#716322
#716262
#715862
#715852
#715832
#329183