Finite Group Databasebeta

← #10690773

Subgroups — #10690773

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2847C211yesmaximal, characteristic
2C2×#5207C211yesmaximal, characteristic, complemented
3#72127C211yesmaximal, characteristic, complemented
4C2×#4537C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9117—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×C3×#121C411yescharacteristic, complemented
9C2×C3×#121C411yescharacteristic, complemented
10C2×C3×#121C2211yescharacteristic
11C2×C3×#120—21no—
12C22×#217—21no—
13C22×#208—21no—
14C3×#386—21no—
15C3×#386—21no—
16C3×#386—21no—
17C3×#380C411yescharacteristic, complemented
18C3×#380C411yescharacteristic, complemented
19C3×#380C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2212yes—
22#5207—24no—
23C2×#830—21no—
24C2×#830C2212yes—
25#4537—24no—
26C2×#407—31no—
27#2847D311yescharacteristic
28C2×#327—32no—
29C22⋊#95—31no—
30#1294—32no—
31C25×C3D411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4—21no—
34C22×C3×D4—21no—
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4—22no—
37C23×C3×C4—22no—
38C23×C3×C4—42no—
39C23×C3×C4—42no—
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—22no—
47C3×#121—22no—
48C3×#121—22no—
49C3×#121—42no—
50C3×#121—42no—
51C3×#120—44no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×#217—24no—
62C2×#217—42no—
63C2×#217—42no—
64C2×#208—24no—
65C2×#208—42no—
66C2×#208—42no—
67C22×C3⋊C8—22no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—42no—
74#830—24no—
75#830—22no—
76#830D414yescomplemented
77C24×C4C2×D311yescharacteristic
78C22×M16—61no—
79C2×#121Dic311yescharacteristic, complemented
80C2×#121Dic311yescharacteristic, complemented
81C2×#121C2×D311yescharacteristic
82C2×#120—21no—
83C22×C22⋊C4—61no—
84#407—64no—
85C2×C22⋊C8—32no—
86C2×C22⋊C8—61no—
87#386—21no—
88#386—21no—
89#386—21no—
90#380Dic311yescharacteristic, complemented
91#380Dic311yescharacteristic, complemented
92#380C2×D311yescharacteristic
93C2×#95—32no—
94C2×#95—61no—
95#327—64no—
96C24×C3—22no—
97C24×C3—22no—
98C24×C3—21no—
99C24×C3—21no—
100C24×C3—22no—
101C24×C3—22no—
102C24×C3—21no—
103C24×C3—41no—
104C24×C3—41no—
105C24×C3C22⋊C411yescharacteristic
106C24×C3C22⋊C411yescharacteristic
107C24×C3C2×D411yescharacteristic
108C2×C3×D4—22no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—48no—
144C22×C3×C4—48no—
145C22×C3×C4C22⋊C411yescharacteristic
146C22×C3×C4C22⋊C411yescharacteristic
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4112yes—
151C22×Dic3—24no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3—42no—
156C22×Dic3—42no—
157C22×Dic3—42no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—42no—
162C3×C22⋊C4—42no—
163C3×C22⋊C4—42no—
164C3×C22⋊C4—42no—
165C3×C22⋊C4—42no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168#217—48no—
169#208—48no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—42no—
172C2×C3⋊C8—42no—
173C25#6711yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C2×Dic311yescharacteristic
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—62no—
183C23×C4C2×Dic311yescharacteristic
184C23×C4C2×Dic311yescharacteristic
185C23×C4#6711yescharacteristic
186C23×C4#6711yescharacteristic
187C23×C4#6711yescharacteristic
188C23×C4C22×D311yescharacteristic
189C2×M16—64no—
190C2×M16—122no—
191C2×M16—122no—
192C22×C8—62no—
193#121—22no—
194#121—22no—
195#121—42no—
196#121—42no—
197#120—44no—
198C2×C4⋊C4C2×Dic311yescharacteristic
199C2×C4⋊C4C2×Dic311yescharacteristic
200C2×C4⋊C4C2×Dic311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—64no—
208C2×C22⋊C4—122no—
209C2×C22⋊C4—122no—
210C22⋊C8—34no—
211C22⋊C8—64no—
212C22⋊C8—62no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—62no—
217#95—62no—
218#95—122no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—44no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3#9912yes—
253C23×C3#10412yes—
254C23×C3C2×C22⋊C411yescharacteristic
255C23×C3#12112yes—
256C23×C3AGL(1,ℤ/8)12yes—
257C23×C3AGL(1,ℤ/8)11yescharacteristic
258C23×C3#13711yescharacteristic
259C3×D4—82no—
260C3×D4—84no—
261C3×D4—84no—
262C3×D4—82no—
263C3×D4—82no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4#9912yes—
292C2×C3×C4C4×D412yes—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12311yescharacteristic
295C2×Dic3—44no—
296C2×Dic3—44no—
297C2×Dic3—44no—
298C2×Dic3—82no—
299C2×Dic3—82no—
300C2×Dic3—82no—
301C2×Dic3—82no—
302C3⋊C8—44no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—41no—
311C24—41no—
312C24#21711yescharacteristic
313C24#21711yescharacteristic
314C24C2×#6711yescharacteristic
315C2×D4—22no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—64no—
353C22×C4—122no—
354C22×C4—122no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4—122no—
358C22×C4—122no—
359C22×C4#21711yescharacteristic
360C22×C4#21711yescharacteristic
361C22×C4D3×D412yes—
362C22×C4#23712yes—
363C22×C4C22×Dic311yescharacteristic
364C22×C4C2×#6711yescharacteristic
365M16—128no—
366C2×C8—64no—
367C2×C8—122no—
368C2×C8—122no—
369C4⋊C4—22no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—128no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—48no—
395C22×C3—48no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3#38611yescharacteristic
412C22×C3C2×#9911yescharacteristic
413C22×C3C2×#9911yescharacteristic
414C22×C3#41711yescharacteristic
415C22×C3#41811yescharacteristic
416C22×C3C2×#10411yescharacteristic
417C22×C3#42111yescharacteristic
418C22×C3#46812yes—
419C22×C3#46912yes—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—22no—
424C3×C4—41no—
425C3×C4—81no—
426C3×C4—81no—
427C3×C4—81no—
428Dic3—84no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—24no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23#83412yes—
463C23#83712yes—
464C23#93212yes—
465C23#93912yes—
466C23#94911yescharacteristic
467C23#95111yescharacteristic
468C23C2×#21711yescharacteristic
469D4—82no—
470D4—84no—
471D4—84no—
472D4—82no—
473D4—82no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—124no—
502C2×C4—124no—
503C2×C4—124no—
504C2×C4—242no—
505C2×C4—242no—
506C2×C4—242no—
507C2×C4—242no—
508C2×C4#83412yes—
509C2×C4D4×Dic312yes—
510C2×C4#93511yescharacteristic
511C2×C4#93711yescharacteristic
512C8—124no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—42no—
516C2×C3—42no—
517C2×C3—161no—
518C2×C3#150411yescharacteristic
519C2×C3#150711yescharacteristic
520C2×C3#151011yescharacteristic
521C2×C3#151311yescharacteristic
522C2×C3#187111yescharacteristic
523C2×C3#189511yescharacteristic
524C2×C3#189811yescharacteristic
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—48no—
540C22—48no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22#503912yes—
557C22#504212yes—
558C22#521511yescharacteristic
559C22C2×#83411yescharacteristic
560C22C2×#83411yescharacteristic
561C22#522011yescharacteristic
562C22#523211yescharacteristic
563C22C2×#83711yescharacteristic
564C22#523611yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—244no—
574C3#911711yescharacteristic, complemented, Sylow
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2#7167011yescharacteristic
581C2#7167311yescharacteristic
582C2#7167411yescharacteristic
583C2#7168211yescharacteristic
584C2#7170811yescharacteristic
585C2#7170911yescharacteristic
586C2#7171311yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2847C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenter#2847D3
hypocenterC3#9117
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
2. C2×#5207C2semidirect
3. #72127C2semidirect
4. C2×#4537C2semidirect
5. #67624C2semidirect
8. C2×C3×#121C4semidirect
9. C2×C3×#121C4semidirect
17. C3×#380C4semidirect
18. C3×#380C4semidirect
76. #830D4semidirect
79. C2×#121Dic3semidirect
80. C2×#121Dic3semidirect
90. #380Dic3semidirect
91. #380Dic3semidirect
574. C3#9117semidirect

Maximal subgroups

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

subgroupindex
C3×#28472
C2×#52072
#721272
C2×#45372
#676242
#91173