Finite Group Databasebeta

← #10822708

Subgroups — #10822708

Order 768.

1610 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5001C211yesmaximal, characteristic, complemented
5C22×#809C211yesmaximal, characteristic, complemented
6C2×#4964C211yesmaximal, characteristic, complemented
7#71580C211yesmaximal, characteristic, complemented
8#69524C214yesmaximal, complemented
9#69523C214yesmaximal, complemented
10#23483—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15#5830—24no—
16#5830—24no—
17C2×#947—21no—
18C2×#940—21no—
19C2×#937—22no—
20C2×#930C2212yes—
21C2×#930C2212yes—
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#391—21no—
26C3×#390—21no—
27C3×#386—22no—
28C3×#380C2214yescomplemented
29#5234—24no—
30C2×#830C2214yescomplemented
31#5001—24no—
32C2×#809—24no—
33C2×#809C2214yes—
34C2×#809C2214yes—
35#4964—24no—
36C2×#808C2212yescomplemented
37C2×#807C2212yescomplemented
38#4529C2212yescomplemented
39#4529C2212yescomplemented
40C2×#535—31no—
41C2×#428—31no—
42C2×#418—31no—
43C22×#102—31no—
44C2×#408—31no—
45#2311D311yescharacteristic, complemented
46#2034—34no—
47#2033—34no—
48#1735—31no—
49C24×D3—21no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C22×C3×Q8D411yescharacteristic
57C22×C3×D4D412yes—
58C22×C3×D4D411yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C22×#67—22no—
63C22×#67—22no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—21no—
69C22×D12—22no—
70C22×D12D414yes—
71C22×D12C2312yes—
72C3×#117—22no—
73C3×#117—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C42—21no—
82C2×#217—21no—
83#947—28no—
84#940—28no—
85#937—216no—
86#930—24no—
87#930—24no—
88#930—24no—
89#930—24no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92#926—22no—
93#926D414yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4×Dic3—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208D414yescomplemented
100C22×C3⋊C8—22no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105C3×#95—24no—
106#830—28no—
107#809—28no—
108#809—28no—
109#808—24no—
110#807—24no—
111C22×#41C2×D311yescharacteristic
112C23×D4—31no—
113C22×M16—31no—
114C23×C8—31no—
115#535—64no—
116#535—64no—
117C2×#127—61no—
118C2×#124—61no—
119C2×#121—32no—
120C2×#121—32no—
121C2×#120—62no—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4C2×D311yescharacteristic
125#428—64no—
126C2×#107—32no—
127C2×#106—32no—
128#418—64no—
129C2×#102—34no—
130C2×#102—34no—
131C2×#102—64no—
132#408—64no—
133C2×C22⋊C8—34no—
134#391—21no—
135#390—21no—
136#386—22no—
137#380C2×D314yescomplemented
138#340—32no—
139#340—32no—
140C24×C3C2×D412yes—
141C24×C3C2×D411yescharacteristic
142C23×D3—22no—
143C23×D3—24no—
144C23×D3—42no—
145C23×D3—42no—
146C23×D3—82no—
147C23×D3—82no—
148C3×#41—44no—
149C3×#41—44no—
150C3×#41—44no—
151C3×#41—44no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—42no—
170C22×C3×C4—82no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—28no—
181C2×#67—28no—
182C2×#67—84no—
183C2×#67—84no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C2×D12—28no—
187C2×D12—28no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—24no—
193C2×D12—22no—
194C2×D12—44no—
195C2×D12—42no—
196C2×D12—42no—
197C2×D12—44no—
198C2×D12C2×D414yes—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C22⋊C4—42no—
205C3×C22⋊C4—84no—
206C3×C42—42no—
207#217—24no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—28no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3C2×D412yes—
213C4⋊Dic3C2×D412yes—
214C4×Dic3—24no—
215#208—28no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—44no—
222C2×C3⋊C8C2×D414yes—
223C25—61no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C22×Q8#6711yescharacteristic
231C22×D4—34no—
232C22×D4—32no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4—62no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—61no—
241C22×D4#6712yes—
242C22×D4#6711yescharacteristic
243C23×C4—21no—
244C23×C4C22×D312yes—
245C23×C4C22×D311yescharacteristic
246C2×M16—34no—
247C2×M16—62no—
248C2×M16—62no—
249C22×C8—32no—
250C22×C8—32no—
251C22×C8—62no—
252C22×C8—62no—
253C22×C8—62no—
254#127—68no—
255#124—68no—
256#121—64no—
257#121—64no—
258#121—64no—
259#121—64no—
260#120—616no—
261#117—22no—
262#117—22no—
263#117—34no—
264#117—62no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4C22×D312yes—
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—21no—
278C2×C42—61no—
279#107—64no—
280#106—64no—
281#102—68no—
282#102—68no—
283C22⋊C8—68no—
284#95—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3#12012yes—
293C23×C3#12112yes—
294C23×C3#12312yes—
295C23×C3C22×D411yescharacteristic
296C22×D3—24no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3—48no—
302C22×D3—84no—
303C22×D3—84no—
304C22×D3—88no—
305C22×D3—88no—
306C3×Q8—44no—
307C3×D4—48no—
308C3×D4—44no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—84no—
322C2×C3×C4#12012yes—
323C2×C3×C4#12114yes—
324C2×C3×C4#12114yes—
325C2×C3×C4#12112yes—
326C2×C3×C4#12314yes—
327C2×C3×C4#12312yes—
328C2×C3×C4#12414yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335#67—88no—
336#67—88no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341D12—24no—
342D12—24no—
343D12—42no—
344D12—42no—
345D12—48no—
346C3⋊C8—24no—
347C3⋊C8—42no—
348C24—62no—
349C24—64no—
350C24—122no—
351C24—122no—
352C24—242no—
353C24—242no—
354C24C2×#6712yes—
355C24C2×#6711yescharacteristic
356#41—44no—
357#41—44no—
358#41—44no—
359#41—44no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—34no—
370C2×D4—68no—
371C2×D4—68no—
372C2×D4—68no—
373C2×D4—68no—
374C2×D4—64no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—124no—
381C2×D4—124no—
382C2×D4—122no—
383C2×D4—122no—
384C2×D4—244no—
385C2×D4—244no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—42no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—82no—
398C22×C4#23512yes—
399C22×C4#23512yes—
400C22×C4D3×D412yes—
401C22×C4D3×D412yes—
402C22×C4D3×D412yes—
403C22×C4#23712yes—
404C22×C4C2×#6712yes—
405C22×C4C2×#6711yescharacteristic
406C22×C4C23×D311yescharacteristic
407M16—68no—
408C2×C8—34no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—124no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—32no—
420C4⋊C4—32no—
421C4⋊C4—42no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C22⋊C4—42no—
425C22⋊C4—68no—
426C22⋊C4—68no—
427C22⋊C4—64no—
428C22⋊C4—84no—
429C42—42no—
430C42—64no—
431C22×C3—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3#39414yes—
441C22×C3C2×#12011yescharacteristic
442C22×C3C2×#12111yescharacteristic
443C22×C3C2×#12311yescharacteristic
444C22×C3#57612yes—
445C22×C3#57712yes—
446C2×D3—44no—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—816no—
450C2×D3—84no—
451C2×D3—84no—
452C3×C4—42no—
453C3×C4—82no—
454C3×C4—82no—
455C3×C4#39414yes—
456C3×C4C2×#12111yescharacteristic
457C3×C4C2×#12111yescharacteristic
458C3×C4C2×#12311yescharacteristic
459C3×C4C2×#12411yescharacteristic
460Dic3—42no—
461Dic3—42no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—64no—
470C23—128no—
471C23—122no—
472C23—122no—
473C23—124no—
474C23—124no—
475C23—244no—
476C23—244no—
477C23—248no—
478C23—248no—
479C23#92912yes—
480C23#93712yes—
481C23#93912yes—
482C23C22×#6711yescharacteristic
483Q8—44no—
484D4—48no—
485D4—44no—
486D4—64no—
487D4—64no—
488D4—128no—
489D4—122no—
490D4—122no—
491D4—248no—
492D4—248no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—84no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4#88314yes—
511C2×C4#88414yes—
512C2×C4#88414yes—
513C2×C4#88514yes—
514C2×C4#92912yes—
515C2×C4#93712yes—
516C2×C4#93912yes—
517C2×C4C2×#23511yescharacteristic
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×#23711yescharacteristic
523C8—64no—
524C8—122no—
525C2×C3—24no—
526C2×C3—82no—
527C2×C3C2×#39411yescharacteristic
528C2×C3#300811yescharacteristic
529C2×C3#300911yescharacteristic
530C2×C3#306212yes—
531C2×C3#309612yes—
532D3—84no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—124no—
543C22—124no—
544C22—124no—
545C22—2416no—
546C22—244no—
547C22—244no—
548C22#495914yes—
549C22C2×#92911yescharacteristic
550C22C2×#93711yescharacteristic
551C22C2×#93911yescharacteristic
552C22#583512yes—
553C22#583712yes—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—122no—
558C4—122no—
559C4#495914yes—
560C4C2×#88311yescharacteristic
561C4C2×#88411yescharacteristic
562C4C2×#88411yescharacteristic
563C4C2×#88511yescharacteristic
564C3#2348311yescharacteristic, complemented, Sylow
565C2—24no—
566C2—82no—
567C2—244no—
568C2C2×#495911yescharacteristic
569C2#8256612yes—
570C2#8338311yescharacteristic
571C2#8340411yescharacteristic
572C2#8342212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
prime radicalC2×#5830C2
hypercenter#2311D3
hypocenterC3#23483
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C3×#2311C2semidirect
3. C2×#5234C2semidirect
4. C2×#5001C2semidirect
5. C22×#809C2semidirect
6. C2×#4964C2semidirect
7. #71580C2semidirect
8. #69524C2semidirect
9. #69523C2semidirect
28. C3×#380C22semidirect
30. C2×#830C22semidirect
36. C2×#808C22semidirect
37. C2×#807C22semidirect
38. #4529C22semidirect
39. #4529C22semidirect
45. #2311D3semidirect
99. C2×#208D4semidirect
137. #380C2×D3semidirect
564. C3#23483semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C3×#23112
C2×#52342
C2×#50012
C22×#8092
C2×#49642
#715802
#695242
#695232
#234833