Finite Group Databasebeta

← #10822704

Subgroups — #10822704

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×C3×#535C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5215C211yesmaximal, characteristic, complemented
5C22×#832C211yesmaximal, characteristic, complemented
6C2×#4995C211yesmaximal, characteristic, complemented
7#71680C214yesmaximal, complemented
8#71679C214yesmaximal, complemented
9#71580C211yesmaximal, characteristic, complemented
10#23483—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C3×#535—24no—
14C3×#535—24no—
15C2×C3×#127—21no—
16C2×C3×#124—21no—
17C2×C3×#121C2212yes—
18C2×C3×#121C2212yes—
19C2×C3×#120—22no—
20C2×C3×#117C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5234—24no—
26#5229—21no—
27#5221—21no—
28#5220—22no—
29#5215—24no—
30C2×#832—24no—
31C2×#832C2214yes—
32C2×#832C2214yes—
33#5212C2214yescomplemented
34C2×#830C2214yescomplemented
35#4995—24no—
36C2×#808C2212yescomplemented
37C2×#807C2212yescomplemented
38#4529C2212yescomplemented
39#4529C2212yescomplemented
40C2×#535D311yescharacteristic
41C2×#428—31no—
42C2×#418—31no—
43C22×#102—31no—
44C2×#408—31no—
45#2311—31no—
46#2034—34no—
47#2033—34no—
48#1735—31no—
49C25×C3—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×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4D412yes—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4D414yes—
68C22×C3×D4C2312yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—21no—
72C3×#127—28no—
73C3×#124—28no—
74C3×#121—24no—
75C3×#121—24no—
76C3×#121—24no—
77C3×#121—24no—
78C3×#120—216no—
79C3×#117—22no—
80C3×#117D414yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—21no—
87C2×#217—21no—
88C2×#217—24no—
89#926—22no—
90#926—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2312yes—
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—
105#832—28no—
106#832—28no—
107C3⋊#95—24no—
108#830—28no—
109#808—24no—
110#807—24no—
111C22×#41C2×D311yescharacteristic
112C23×D4C2×D311yescharacteristic
113C22×M16—31no—
114C23×C8—31no—
115#535—24no—
116#535—24no—
117C2×#127—21no—
118C2×#124—21no—
119C2×#121C2×D312yes—
120C2×#121C2×D312yes—
121C2×#120—22no—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4—31no—
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—61no—
135#390—61no—
136#386—62no—
137#380—34no—
138#340—32no—
139#340—32no—
140C24×C3—22no—
141C24×C3—24no—
142C24×C3—42no—
143C24×C3—42no—
144C24×C3—82no—
145C24×C3—82no—
146C24×C3C2×D412yes—
147C24×C3C2×D411yescharacteristic
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—28no—
159C2×C3×D4—28no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—28no—
163C2×C3×D4—22no—
164C2×C3×D4—28no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—44no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—44no—
175C2×C3×D4—84no—
176C2×C3×D4—84no—
177C2×C3×D4C2×D414yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4#4112yes—
192C22×C3×C4#4112yes—
193C22×C3×C4#4112yes—
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—42no—
198C22×Dic3—82no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—28no—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—24no—
206C3×C42—24no—
207#217—42no—
208#217—84no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4×Dic3—42no—
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—21no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C22×Q8#6711yescharacteristic
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4#6712yes—
240C22×D4#6711yescharacteristic
241C22×D4#6714yes—
242C22×D4C22×D312yes—
243C23×C4—61no—
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—28no—
255#124—28no—
256#121—24no—
257#121—24no—
258#121—24no—
259#121—24no—
260#120—216no—
261#117—22no—
262#117—62no—
263#117—62no—
264#117#6714yes—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—64no—
277C2×C42—21no—
278C2×C42—61no—
279#107—64no—
280#106—64no—
281#102—68no—
282#102—68no—
283C22⋊C8—68no—
284#95—64no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—48no—
298C23×C3—84no—
299C23×C3—84no—
300C23×C3—88no—
301C23×C3—88no—
302C23×C3#12012yes—
303C23×C3#12112yes—
304C23×C3#12312yes—
305C23×C3C22×D411yescharacteristic
306C3×Q8—44no—
307C3×D4—24no—
308C3×D4—24no—
309C3×D4—48no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—48no—
313C3×D4—44no—
314C3×D4—88no—
315C3×D4—88no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12012yes—
328C2×C3×C4#12114yes—
329C2×C3×C4#12114yes—
330C2×C3×C4#12112yes—
331C2×C3×C4#12314yes—
332C2×C3×C4#12312yes—
333C2×C3×C4#12414yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—44no—
344C2×Dic3—42no—
345C2×Dic3—84no—
346C3⋊C8—24no—
347C3⋊C8—42no—
348C24—22no—
349C24—24no—
350C24—42no—
351C24—42no—
352C24—82no—
353C24—82no—
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—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—28no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4C2×#6714yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—122no—
397C22×C4—242no—
398C22×C4D3×D412yes—
399C22×C4#23712yes—
400C22×C4#23712yes—
401C22×C4#23712yes—
402C22×C4C2×#6712yes—
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6712yes—
405C22×C4C2×#6712yes—
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—21no—
416C4⋊C4—21no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—62no—
420C4⋊C4—62no—
421C4⋊C4—122no—
422C4⋊C4C2×#6712yes—
423C4⋊C4C2×#6712yes—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—24no—
427C22⋊C4—122no—
428C22⋊C4—244no—
429C42—24no—
430C42—122no—
431C22×C3—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3—44no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3—42no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—816no—
444C22×C3—84no—
445C22×C3—84no—
446C22×C3#39414yes—
447C22×C3C2×#12011yescharacteristic
448C22×C3C2×#12111yescharacteristic
449C22×C3C2×#12311yescharacteristic
450C22×C3#57612yes—
451C22×C3#57712yes—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4#39414yes—
456C3×C4C2×#12111yescharacteristic
457C3×C4C2×#12111yescharacteristic
458C3×C4C2×#12311yescharacteristic
459C3×C4C2×#12411yescharacteristic
460Dic3—82no—
461Dic3—82no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—48no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—84no—
476C23—84no—
477C23—88no—
478C23—88no—
479C23#93512yes—
480C23#93912yes—
481C23#95312yes—
482C23C22×#6711yescharacteristic
483Q8—44no—
484D4—24no—
485D4—24no—
486D4—48no—
487D4—48no—
488D4—44no—
489D4—42no—
490D4—42no—
491D4—88no—
492D4—88no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—124no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—244no—
510C2×C4#93512yes—
511C2×C4#93514yes—
512C2×C4#93614yes—
513C2×C4#93814yes—
514C2×C4#93912yes—
515C2×C4#93914yes—
516C2×C4#95312yes—
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×#23711yescharacteristic
519C2×C4C2×#23711yescharacteristic
520C2×C4C2×#23711yescharacteristic
521C2×C4C22×#6711yescharacteristic
522C2×C4C22×#6711yescharacteristic
523C8—64no—
524C8—122no—
525C2×C3—24no—
526C2×C3—82no—
527C2×C3—84no—
528C2×C3C2×#39411yescharacteristic
529C2×C3#300811yescharacteristic
530C2×C3#300911yescharacteristic
531C2×C3#306212yes—
532C2×C3#309612yes—
533C22—22no—
534C22—22no—
535C22—24no—
536C22—24no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—816no—
546C22—84no—
547C22—84no—
548C22#522214yes—
549C22C2×#93511yescharacteristic
550C22C2×#93911yescharacteristic
551C22#583512yes—
552C22#583612yes—
553C22C2×#95311yescharacteristic
554C4—42no—
555C4—42no—
556C4—42no—
557C4—242no—
558C4—242no—
559C4#522214yes—
560C4C2×#93511yescharacteristic
561C4C2×#93611yescharacteristic
562C4C2×#93811yescharacteristic
563C4C2×#93911yescharacteristic
564C3#2348311yescharacteristic, complemented, Sylow
565C2—24no—
566C2—82no—
567C2—84no—
568C2#8337612yes—
569C2#8337912yes—
570C2C2×#522211yescharacteristic
571C2#8410111yescharacteristic
572C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3#23483
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. C2×#5234C2semidirect
4. C2×#5215C2semidirect
5. C22×#832C2semidirect
6. C2×#4995C2semidirect
7. #71680C2semidirect
8. #71679C2semidirect
9. #71580C2semidirect
33. #5212C22semidirect
34. C2×#830C22semidirect
36. C2×#808C22semidirect
37. C2×#807C22semidirect
38. #4529C22semidirect
39. #4529C22semidirect
99. C2×#208D4semidirect
564. C3#23483semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#840872
C2×#52342
C2×#52152
C22×#8322
C2×#49952
#716802
#716792
#715802
#234833