Finite Group Databasebeta

← #10796346

Subgroups — #10796346

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2C22×#948C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5207C211yesmaximal, characteristic, complemented
5C22×#830C211yesmaximal, characteristic, complemented
6#71611C214yesmaximal, complemented
7#71610C214yesmaximal, complemented
8#71579C211yesmaximal, characteristic, complemented
9#71579C211yesmaximal, characteristic, complemented
10#20899—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C3×#525—24no—
14C3×#525—24no—
15C2×C3×#123C412yes—
16C2×C3×C22⋊Q8C412yes—
17C2×C3×#121C412yes—
18C2×C3×#120C412yes—
19C2×C3×C4×D4C2214yes—
20C2×C3×#117C2211yescharacteristic
21C2×#948—24no—
22C2×#948—24no—
23C2×#948—24no—
24C22×#208C2211yescharacteristic
25C22×#208C2211yescharacteristic
26C22×#208C2212yescomplemented
27C23×C3⋊C8C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C23×C3⋊C8C2212yescomplemented
30#5234—24no—
31#5207—24no—
32C2×#830—24no—
33C2×#830C2212yes—
34C2×#830C2212yes—
35C2×#830C2214yes—
36C2×#830C2214yescomplemented
37#4524C2212yescomplemented
38#4524C2214yescomplemented
39#4524C2212yescomplemented
40C22×#131—31no—
41C2×#525D311yescharacteristic
42C2×#408—31no—
43C2×#407—31no—
44C22×C22⋊C8—31no—
45#1870—34no—
46#1869—34no—
47#1729—31no—
48#1729—31no—
49C25×C3C2×C411yescharacteristic
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41D414yes—
55C2×C3×#41D414yes—
56C22×C3×Q8C2×C411yescharacteristic
57C22×C3×D4C2×C414yes—
58C22×C3×D4C2×C412yes—
59C22×C3×D4C2×C411yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4—22no—
62C23×C3×C4—22no—
63C23×C3×C4—22no—
64C23×C3×C4—22no—
65C23×C3×C4—22no—
66C23×C3×C4C2×C412yes—
67C23×C3×C4C2×C411yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C3×#123—28no—
72C3×C22⋊Q8—28no—
73C3×#121—28no—
74C3×#120—28no—
75C3×C4×D4—28no—
76C3×C4×D4—28no—
77C3×#117—22no—
78C3×#117—22no—
79C2×C3×C4⋊C4C2×C412yes—
80C2×C3×C4⋊C4C2×C414yes—
81C2×C3×C22⋊C4C2×C412yes—
82C2×C3×C22⋊C4C2×C414yes—
83C2×C3×C22⋊C4C2×C414yes—
84C2×C3×C42C2312yes—
85#948—232no—
86C2×#208—22no—
87C2×#208—22no—
88C2×#208—24no—
89C2×#208—22no—
90C2×#208—22no—
91C2×#208—22no—
92C2×#208—24no—
93C2×#208—24no—
94C2×#208D414yescomplemented
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—24no—
98C22×C3⋊C8—24no—
99C22×C3⋊C8—24no—
100C22×C3⋊C8—22no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8D414yescomplemented
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2312yes—
109#830—28no—
110#830—24no—
111#830—24no—
112#830—28no—
113C22×#41C2×D311yescharacteristic
114C24×C4C2×D311yescharacteristic
115C2×#131—64no—
116C2×#131—64no—
117C2×#131—64no—
118C22×M16—31no—
119C22×M16—31no—
120C22×M16—32no—
121C23×C8—31no—
122C23×C8—31no—
123C23×C8—32no—
124#525—24no—
125#525—24no—
126C2×#123Dic312yes—
127C2×C22⋊Q8Dic312yes—
128C2×#121Dic312yes—
129C2×#120Dic312yes—
130C2×C4×D4C2×D314yes—
131C2×#117C2×D311yescharacteristic
132#408—64no—
133#407—64no—
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—34no—
137C2×C22⋊C8—34no—
138C2×C22⋊C8—64no—
139#336—32no—
140#336—34no—
141#336—32no—
142C24×C3—22no—
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—42no—
147C24×C3C22×C412yes—
148C24×C3C22×C412yes—
149C24×C3C22×C411yescharacteristic
150C3×#41—28no—
151C3×#41—28no—
152C3×#41—28no—
153C3×#41—28no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C22⋊C414yes—
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—28no—
160C2×C3×D4—28no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—44no—
164C2×C3×D4C22⋊C418yes—
165C2×C3×D4C22⋊C414yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4—44no—
186C22×C3×C4C22×C414yes—
187C22×C3×C4C22×C412yes—
188C22×C3×C4C22×C412yes—
189C22×C3×C4C22×C411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C3×C4⋊C4—24no—
200C3×C4⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—28no—
204C3×C42—22no—
205C3×C42—22no—
206#208—216no—
207#208—28no—
208#208—44no—
209C2×C3⋊C8—216no—
210C2×C3⋊C8—28no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—44no—
224C2×C3⋊C8C2×D412yes—
225C2×C3⋊C8C2×D414yes—
226C2×C3⋊C8C2×D412yes—
227C25C2×Dic311yescharacteristic
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41#6714yes—
233C2×#41#6714yes—
234C22×Q8C2×Dic311yescharacteristic
235C22×D4C2×Dic314yes—
236C22×D4C2×Dic312yes—
237C22×D4C2×Dic311yescharacteristic
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×Dic312yes—
245C23×C4C2×Dic311yescharacteristic
246C23×C4C22×D312yes—
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249#131—632no—
250C2×M16—34no—
251C2×M16—64no—
252C2×M16—64no—
253C2×M16—62no—
254C2×M16—64no—
255C2×M16—62no—
256C2×M16—62no—
257C2×M16—62no—
258C2×M16—62no—
259C22×C8—34no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8—32no—
263C22×C8—32no—
264C22×C8—32no—
265C22×C8—64no—
266C22×C8—64no—
267C22×C8—62no—
268C22×C8—64no—
269C22×C8—62no—
270C22×C8—62no—
271C22×C8—62no—
272C22×C8—62no—
273#123—28no—
274C22⋊Q8—28no—
275#121—28no—
276#120—28no—
277C4×D4—28no—
278C4×D4—28no—
279#117—22no—
280#117—22no—
281C2×C4⋊C4C2×Dic312yes—
282C2×C4⋊C4C2×Dic314yes—
283C2×C22⋊C4C2×Dic312yes—
284C2×C22⋊C4C2×Dic314yes—
285C2×C22⋊C4C2×Dic314yes—
286C2×C42C22×D312yes—
287C22⋊C8—64no—
288C22⋊C8—64no—
289C22⋊C8—68no—
290C22⋊C8—68no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—28no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—24no—
303C23×C3—22no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3C2×C22⋊C414yes—
309C23×C3C2×C22⋊C412yes—
310C23×C3C4×D412yes—
311C23×C3C4×D412yes—
312C23×C3#13114yescomplemented
313C23×C3C23×C411yescharacteristic
314C3×Q8—28no—
315C3×D4—216no—
316C3×D4—28no—
317C3×D4—416no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—48no—
336C2×C3×C4C2×C22⋊C414yes—
337C2×C3×C4C2×C22⋊C412yes—
338C2×C3×C4C4×D412yes—
339C2×C3×C4C4×D412yes—
340C2×C3×C4#12014yes—
341C2×C3×C4#12114yes—
342C2×C3×C4#12114yes—
343C2×C3×C4#12314yes—
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C3⋊C8—22no—
351C3⋊C8—24no—
352C3⋊C8—22no—
353C3⋊C8—42no—
354C3⋊C8—42no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24C22×Dic312yes—
361C24C22×Dic312yes—
362C24C22×Dic311yescharacteristic
363#41—28no—
364#41—28no—
365#41—28no—
366#41—28no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8#21714yes—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—44no—
377C2×D4#21718yes—
378C2×D4#21714yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4D3×D412yes—
400C22×C4D3×D412yes—
401C22×C4#23712yes—
402C22×C4#23712yes—
403C22×C4C22×Dic314yes—
404C22×C4C22×Dic312yes—
405C22×C4C22×Dic312yes—
406C22×C4C22×Dic311yescharacteristic
407C22×C4C2×#6714yes—
408C22×C4C2×#6714yes—
409C22×C4C2×#6712yes—
410C22×C4C2×#6712yes—
411C22×C4C23×D311yescharacteristic
412M16—616no—
413M16—68no—
414M16—124no—
415C2×C8—32no—
416C2×C8—32no—
417C2×C8—34no—
418C2×C8—616no—
419C2×C8—68no—
420C2×C8—61no—
421C2×C8—62no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—62no—
430C2×C8—62no—
431C2×C8—62no—
432C2×C8—124no—
433C4⋊C4—24no—
434C4⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—28no—
437C22⋊C4—28no—
438C42—22no—
439C42—22no—
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—28no—
446C22×C3—24no—
447C22×C3—24no—
448C22×C3—24no—
449C22×C3—416no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3#38614yes—
458C22×C3C22×C22⋊C411yescharacteristic
459C22×C3C2×C4×D411yescharacteristic
460C22×C3C2×C4×D411yescharacteristic
461C22×C3C2×#13112yes—
462C22×C3#56812yes—
463C3×C4—24no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4#38614yes—
467C3×C4C2×#12011yescharacteristic
468C3×C4C2×#12111yescharacteristic
469C3×C4C2×#12111yescharacteristic
470C3×C4C2×#12311yescharacteristic
471C23—22no—
472C23—22no—
473C23—28no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—48no—
485C23—48no—
486C23—44no—
487C23—44no—
488C23D4×Dic312yes—
489C23D4×Dic312yes—
490C23#94814yescomplemented
491C23C2×#21714yes—
492C23C2×#21712yes—
493C23C23×Dic311yescharacteristic
494Q8—28no—
495D4—216no—
496D4—28no—
497D4—416no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4D4×Dic312yes—
517C2×C4D4×Dic312yes—
518C2×C4#93514yes—
519C2×C4#93714yes—
520C2×C4#93914yes—
521C2×C4#93914yes—
522C2×C4C2×#21714yes—
523C2×C4C2×#21712yes—
524C2×C4C2×D3×D411yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×#23711yescharacteristic
527C2×C4C2×#23711yescharacteristic
528C2×C4C22×#6711yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—62no—
531C8—62no—
532C8—64no—
533C8—122no—
534C8—122no—
535C2×C3—24no—
536C2×C3—44no—
537C2×C3—44no—
538C2×C3C2×#38611yescharacteristic
539C2×C3#288012yes—
540C2×C3#292911yescharacteristic
541C2×C3#293011yescharacteristic
542C2×C3#293312yes—
543C22—24no—
544C22—22no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—24no—
551C22—24no—
552C22—416no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22#522014yes—
561C22C2×D4×Dic311yescharacteristic
562C22C2×D4×Dic311yescharacteristic
563C22C2×#94812yes—
564C22#581912yes—
565C22C22×#21711yescharacteristic
566C4—24no—
567C4—42no—
568C4—44no—
569C4#522014yes—
570C4C2×#93511yescharacteristic
571C4C2×#93711yescharacteristic
572C4C2×#93911yescharacteristic
573C4C2×#93911yescharacteristic
574C3#2089911yescharacteristic, complemented, Sylow
575C2—24no—
576C2—44no—
577C2—44no—
578C2#8291611yescharacteristic
579C2#8291711yescharacteristic
580C2#8291812yes—
581C2C2×#522011yescharacteristic
582C2#8407512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
Fitting, prime radicalC2×C3×#525C2
cosocleC22×C3×C4C24
hypercenterC2×#525D3
hypocenterC3#20899
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. C22×#948C2semidirect
3. C2×#5234C2semidirect
4. C2×#5207C2semidirect
5. C22×#830C2semidirect
6. #71611C2semidirect
7. #71610C2semidirect
8. #71579C2semidirect
9. #71579C2semidirect
26. C22×#208C22semidirect
29. C23×C3⋊C8C22semidirect
36. C2×#830C22semidirect
37. #4524C22semidirect
38. #4524C22semidirect
39. #4524C22semidirect
94. C2×#208D4semidirect
103. C22×C3⋊C8D4semidirect
312. C23×C3#131semidirect
490. C23#948semidirect
574. C3#20899semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5252
C22×#9482
C2×#52342
C2×#52072
C22×#8302
#716112
#716102
#715792
#208993