Finite Group Databasebeta

← #10826949

Subgroups — #10826949

Order 768.

1908 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C22⋊Q8C212yesmaximal
2C2×#5186C212yesmaximal, complemented
3C2×#4809C212yesmaximal, complemented
4C3×#2001C211yesmaximal, characteristic, complemented
5#71700C212yesmaximal, complemented
6#70268C212yesmaximal, complemented
7#69964C211yesmaximal, characteristic, complemented
8#69955C211yesmaximal, characteristic
9#68878C212yesmaximal, complemented
10#23921—31nomaximal, Sylow
11C22×D3×Q8C2212yes—
12C23×C4×D3C2211yescharacteristic
13C22×C3×Q16C2211yescharacteristic, complemented
14C2×C3×C22⋊Q8C2212yes—
15C2×#946C2212yes—
16C22×#216C2212yescomplemented
17C2×#924C2212yes—
18C22×Dic12C2211yescharacteristic, complemented
19D3×C22⋊Q8—28no—
20D3×C22⋊Q8—28no—
21C2×#897C2212yes—
22C2×#896—22no—
23C2×D3×C4⋊C4—22no—
24C2×D3×C4⋊C4C2212yes—
25C2×D3×C22⋊C4—22no—
26C2×#878—22no—
27C2×C3×#103C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#835C2212yescomplemented
30C2×#830C2211yescharacteristic
31#5186—28no—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yescomplemented
35C2×#810C2212yescomplemented
36#4809—28no—
37C3×#340C2211yescharacteristic, complemented
38#4551C2211yescharacteristic, complemented
39#4529C2212yescomplemented
40C22×C22⋊Q8—32no—
41C2×#451—32no—
42C2×#451—32no—
43#2001—31no—
44#2001—32no—
45#2001D311yescharacteristic, complemented
46#1872—32no—
47#1872—32no—
48#1779—31no—
49C24×D3D411yescharacteristic
50C22×C3×Q8C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3D411yescharacteristic
53C2×D3×Q8—24no—
54C2×D3×Q8—24no—
55C2×D3×Q8—24no—
56C2×D3×Q8—24no—
57C2×D3×Q8—42no—
58C2×D3×Q8—42no—
59C22×C4×D3—22no—
60C22×C4×D3—22no—
61C22×C4×D3—24no—
62C22×C4×D3—24no—
63C22×C4×D3—41no—
64C22×C4×D3—41no—
65C22×C4×D3D412yes—
66C22×C4×D3C2312yes—
67C22×Dic6—22no—
68C22×Dic6C2312yes—
69C2×C3×Q16—24no—
70C2×C3×Q16—41no—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×#217—22no—
79#946—24no—
80#946D418yes—
81C2×#216—24no—
82C2×#216—24no—
83C2×#216—42no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3C2312yes—
88#924—24no—
89#924—24no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C22×C3⋊C8C2312yes—
94C2×Dic12—24no—
95C2×Dic12—41no—
96#897—24no—
97#897D418yes—
98#896—216no—
99D3×C4⋊C4—216no—
100D3×C4⋊C4—24no—
101D3×C4⋊C4—24no—
102D3×C22⋊C4—216no—
103#878—216no—
104C3×#103—24no—
105C3×C22⋊C8—41no—
106#835—24no—
107#830—41no—
108(C2×D3)⋊C8—24no—
109(C2×D3)⋊C8—24no—
110#816—24no—
111#810—24no—
112C23×Q8—32no—
113C24×C4—31no—
114C22×Q16—32no—
115C22×Q16—31no—
116C22×Q16C2×D311yescharacteristic, complemented
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×C22⋊Q8—32no—
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—62no—
122C2×C22⋊Q8—68no—
123C2×C22⋊Q8—68no—
124C2×C22⋊Q8C2×D312yes—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—62no—
127C22×C22⋊C4—62no—
128#451—68no—
129#451—68no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103C2×D312yescomplemented
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—31no—
137C2×C22⋊C8C2×D311yescharacteristic
138#340—31no—
139#340—32no—
140#340C2×D311yescharacteristic, complemented
141C24×C3C2×D411yescharacteristic
142C23×D3—24no—
143C23×D3—24no—
144C23×D3—41no—
145C23×D3—41no—
146C23×D3Q1614yescomplemented
147C23×D3C2×D412yes—
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—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—24no—
165C22×Dic3—41no—
166C22×Dic3C2×D412yes—
167D3×Q8—48no—
168D3×Q8—48no—
169D3×Q8—48no—
170D3×Q8—48no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—28no—
176C2×C4×D3—28no—
177C2×C4×D3—24no—
178C2×C4×D3—24no—
179C2×C4×D3—44no—
180C2×C4×D3—44no—
181C2×C4×D3—48no—
182C2×C4×D3—42no—
183C2×C4×D3—44no—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—44no—
187C2×C4×D3C2×D414yes—
188C2×C4×D3C2×D414yes—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—28no—
194C2×Dic6—22no—
195C2×Dic6—44no—
196C2×Dic6—44no—
197C2×Dic6C2×D414yes—
198C3×Q16—44no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D414yes—
206C3×C22⋊C4—28no—
207#217—28no—
208#216—48no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—28no—
211C4⋊Dic3—22no—
212C4⋊Dic3—28no—
213C4⋊Dic3C2×D414yes—
214Dic3⋊C4—28no—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220Dic12—44no—
221C25—31no—
222C22×Q8—32no—
223C22×Q8—62no—
224C22×Q8—64no—
225C22×Q8—64no—
226C22×Q8—64no—
227C22×Q8—64no—
228C22×Q8—122no—
229C22×Q8—122no—
230C22×Q8C22×D312yes—
231C23×C4—32no—
232C23×C4—31no—
233C23×C4—32no—
234C23×C4—64no—
235C23×C4—64no—
236C23×C4—62no—
237C23×C4—62no—
238C23×C4—121no—
239C23×C4—121no—
240C23×C4C22×D311yescharacteristic
241C2×Q16—24no—
242C2×Q16—41no—
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—64no—
246C2×Q16—121no—
247C2×Q16—122no—
248C22×C8—32no—
249C22×C8C22×D311yescharacteristic
250C22×C8C22×D311yescharacteristic
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—38no—
254C22⋊Q8—38no—
255C22⋊Q8—616no—
256C22⋊Q8—64no—
257C22⋊Q8—616no—
258C22⋊Q8—64no—
259C22⋊Q8—64no—
260C22⋊Q8—64no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—616no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—616no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276#103—24no—
277#103—64no—
278#103—64no—
279#103—64no—
280C22⋊C8—41no—
281C22⋊C8—64no—
282C22⋊C8—64no—
283C22⋊C8—121no—
284C23×C3—22no—
285C23×C3—21no—
286C23×C3—22no—
287C23×C3—21no—
288C23×C3C2×Q1612yes—
289C23×C3C22×D411yescharacteristic
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—216no—
295C22×D3—44no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3—48no—
299C22×D3#12014yes—
300C22×D3C2×Q1614yes—
301C22×D3#13714yes—
302C3×Q8—24no—
303C3×Q8—24no—
304C3×Q8—42no—
305C3×Q8—42no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—21no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4C22×D412yes—
318C2×C3×C4C22×D412yes—
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—44no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3#12014yes—
331C4×D3—24no—
332C4×D3—24no—
333C4×D3—24no—
334C4×D3—24no—
335C4×D3—48no—
336C4×D3—416no—
337C4×D3—416no—
338Dic6—24no—
339Dic6—24no—
340Dic6—416no—
341Dic6—42no—
342Dic6—42no—
343C3×C8—41no—
344C3×C8—41no—
345C3⋊C8—42no—
346C24—32no—
347C24—34no—
348C24—64no—
349C24—64no—
350C24—121no—
351C24—121no—
352C24D3×D411yescharacteristic
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—34no—
359C2×Q8—68no—
360C2×Q8—62no—
361C2×Q8—62no—
362C2×Q8—62no—
363C2×Q8—62no—
364C2×Q8—62no—
365C2×Q8—124no—
366C2×Q8—124no—
367C2×Q8—128no—
368C2×Q8—128no—
369C2×Q8—128no—
370C2×Q8—128no—
371C2×Q8D3×D414yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—32no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—64no—
382C22×C4—64no—
383C22×C4—68no—
384C22×C4—68no—
385C22×C4—64no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—122no—
393C22×C4—124no—
394C22×C4—124no—
395C22×C4—128no—
396C22×C4—124no—
397C22×C4—124no—
398C22×C4—124no—
399C22×C4—124no—
400C22×C4—121no—
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403Q16—44no—
404Q16—124no—
405Q16—128no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—34no—
416C4⋊C4—68no—
417C4⋊C4—68no—
418C4⋊C4—68no—
419C4⋊C4—68no—
420C4⋊C4—62no—
421C4⋊C4D3×D414yes—
422C22⋊C4—28no—
423C22⋊C4—68no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22×C3—22no—
427C22×C3—24no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3C2×#12012yes—
432C22×C3#53411yescharacteristic
433C22×C3C22×Q1611yescharacteristic
434C22×C3C2×#13712yes—
435C22×C3#57511yescharacteristic
436C2×D3—24no—
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—24no—
441C2×D3—24no—
442C2×D3—416no—
443C2×D3#45118yescomplemented
444C3×C4—44no—
445C3×C4—41no—
446C3×C4—44no—
447C3×C4D4211yescharacteristic
448C3×C4D4211yescharacteristic
449C3×C4D4212yes—
450Dic3—24no—
451Dic3—44no—
452Dic3—44no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—21no—
457C23—34no—
458C23—34no—
459C23—34no—
460C23—616no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—62no—
465C23—124no—
466C23—124no—
467C23—124no—
468C23—128no—
469C23D3×Q1612yes—
470C23C2×D3×D411yescharacteristic
471Q8—24no—
472Q8—24no—
473Q8—42no—
474Q8—42no—
475Q8—64no—
476Q8—64no—
477Q8—1216no—
478Q8—122no—
479Q8—122no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—34no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—128no—
501C2×C4—1216no—
502C2×C4—1216no—
503C2×C4—124no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4C2×D3×D412yes—
509C2×C4C2×D3×D412yes—
510C2×C4#100411yescharacteristic
511C2×C4#100911yescharacteristic
512C8—41no—
513C8—41no—
514C8—122no—
515C2×C3—42no—
516C2×C3#240311yescharacteristic
517C2×C3C2×#45112yes—
518C2×C3#300712yes—
519C2×C3#318911yescharacteristic
520C2×C3#319511yescharacteristic
521D3—28no—
522C22—24no—
523C22—22no—
524C22—38no—
525C22—41no—
526C22—42no—
527C22—41no—
528C22—64no—
529C22—64no—
530C22—64no—
531C22—64no—
532C22—64no—
533C22—64no—
534C22—1216no—
535C22D3×#12012yes—
536C22#560511yescharacteristic
537C22C2×D3×Q1611yescharacteristic
538C22#576811yescharacteristic
539C22D3×#13712yes—
540C4—44no—
541C4—41no—
542C4—44no—
543C4—64no—
544C4—124no—
545C4—124no—
546C4#559011yescharacteristic
547C4#560912yes—
548C4#571711yescharacteristic
549C3#2392111yescharacteristic, complemented, Sylow
550C2—42no—
551C2—68no—
552C2#7675311yescharacteristic
553C2D3×#45112yes—
554C2#8022812yes—
555C2#8076311yescharacteristic
556C2#8090811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenter#2001D3
hypocenterC3#23921
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
2. C2×#5186C2semidirect
3. C2×#4809C2semidirect
4. C3×#2001C2semidirect
5. #71700C2semidirect
6. #70268C2semidirect
7. #69964C2semidirect
9. #68878C2semidirect
13. C22×C3×Q16C22semidirect
16. C22×#216C22semidirect
18. C22×Dic12C22semidirect
27. C2×C3×#103C22semidirect
29. C2×#835C22semidirect
34. C2×#816C22semidirect
35. C2×#810C22semidirect
37. C3×#340C22semidirect
38. #4551C22semidirect
39. #4529C22semidirect
45. #2001D3semidirect
116. C22×Q16C2×D3semidirect
133. C2×#103C2×D3semidirect
140. #340C2×D3semidirect
146. C23×D3Q16semidirect
443. C2×D3#451semidirect
549. C3#23921semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊Q82
C2×#51862
C2×#48092
C3×#20012
#717002
#702682
#699642
#699552
#688782
#239213