Finite Group Databasebeta

← #10690919

Subgroups — #10690919

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82723C211yesmaximal, characteristic
2C2×D3×#95C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9147—31nomaximal, Sylow
9C23×C4×D3—21no—
10C23×C4×D3C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#938C412yescomplemented
13C2×#937—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5220—21no—
17#5212C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—22no—
22#4987C412yescomplemented
23#4956—22no—
24D3×#95—22no—
25D3×#95—22no—
26C3×#327—22no—
27#4537—22no—
28#2847—31no—
29C22×#95—31no—
30#1779—31no—
31C2×#327—31no—
32C2×#327D311yescharacteristic, complemented
33#1294—31no—
34#1294—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic, complemented
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—22no—
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C22×C4×D3—24no—
44C22×C4×D3—28no—
45C22×C4×D3—21no—
46C22×C4×D3—21no—
47C22×C4×D3—21no—
48C22×C4×D3—21no—
49C22×C4×D3—44no—
50C22×C4×D3—44no—
51C22×C4×D3—42no—
52C22×C4×D3—42no—
53C22×C4×D3C2×C412yes—
54C22×C4×D3D412yes—
55C22×C3×C8—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
61C2×C3×C22⋊C4—21no—
62C2×#217—21no—
63C2×#217—22no—
64#938—44no—
65#937—42no—
66#937—42no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3C2×C412yescomplemented
69C22×C3⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95—21no—
73C3×#95D412yescomplemented
74C3⋊#95—24no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—22no—
78C3⋊#95D412yescomplemented
79#830—21no—
80#830—21no—
81(C2×D3)⋊C8—44no—
82C24×C4—31no—
83C24×C4—61no—
84C2×#121—32no—
85C2×#121C2×D311yescharacteristic
86C2×#120—61no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—62no—
89C2×C22⋊C8C2×D311yescharacteristic
90#386—62no—
91#386—61no—
92#380—31no—
93#380—32no—
94C2×#95—31no—
95C2×#95—62no—
96C2×#95—61no—
97C2×#95—62no—
98C2×#95C2×D311yescharacteristic
99#327—22no—
100#327—62no—
101C24×C3—21no—
102C24×C3C2×D411yescharacteristic
103C23×D3—21no—
104C23×D3—21no—
105C23×D3—21no—
106C23×D3—21no—
107C23×D3—24no—
108C23×D3—24no—
109C23×D3C22⋊C412yes—
110C23×D3D812yes—
111C23×D3SD1612yes—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C2×#67—44no—
135C2×#67—44no—
136C2×#67—44no—
137C22×Dic3—24no—
138C22×Dic3—21no—
139C22×Dic3—22no—
140C22×Dic3—21no—
141C22×Dic3—22no—
142C22×Dic3—42no—
143C22×Dic3—42no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3C22⋊C412yes—
147C2×C4×D3—216no—
148C2×C4×D3—24no—
149C2×C4×D3—24no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—42no—
155C2×C4×D3—42no—
156C2×C4×D3—44no—
157C2×C4×D3—44no—
158C2×C4×D3—44no—
159C2×C4×D3—44no—
160C2×C4×D3—48no—
161C2×C4×D3—416no—
162C2×C4×D3—416no—
163C2×C4×D3—48no—
164C2×C3×C8—42no—
165C2×C3×C8—41no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170#217—44no—
171#217—41no—
172#217—41no—
173(C2×D3)⋊C4—44no—
174C4⋊Dic3—42no—
175C2×C3⋊C8—42no—
176C2×C3⋊C8—41no—
177C25—31no—
178C22×D4—21no—
179C22×D4—62no—
180C22×D4C4×D311yescharacteristic, complemented
181C23×C4—21no—
182C23×C4—31no—
183C23×C4—32no—
184C23×C4—32no—
185C23×C4—64no—
186C23×C4—68no—
187C23×C4—61no—
188C23×C4—61no—
189C23×C4—61no—
190C23×C4—61no—
191C23×C4—61no—
192C23×C4—124no—
193C23×C4—122no—
194C23×C4—124no—
195C23×C4—122no—
196C23×C4C22×D311yescharacteristic
197C22×C8—21no—
198C22×C8—61no—
199#121—21no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—124no—
204#120—122no—
205#120—122no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4C4×D311yescharacteristic, complemented
208C2×C22⋊C4—21no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—124no—
217#95—21no—
218#95—32no—
219#95—62no—
220#95—64no—
221#95—61no—
222#95—61no—
223#95#6712yescomplemented
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3C2×C22⋊C411yescharacteristic
232C23×C3#12011yescharacteristic
233C23×C3#12311yescharacteristic
234C23×C3C2×D811yescharacteristic
235C23×C3C2×SD1611yescharacteristic
236C23×C3AGL(1,ℤ/8)11yescharacteristic
237C23×C3#13711yescharacteristic
238C22×D3—24no—
239C22×D3—24no—
240C22×D3—28no—
241C22×D3—24no—
242C22×D3—24no—
243C22×D3—28no—
244C22×D3—22no—
245C22×D3—22no—
246C22×D3—22no—
247C22×D3—22no—
248C22×D3—22no—
249C22×D3—22no—
250C22×D3—22no—
251C22×D3—22no—
252C22×D3—28no—
253C22×D3—28no—
254C22×D3#9914yes—
255C22×D3#10214yes—
256C22×D3#10414yes—
257C3×D4—81no—
258C3×D4—81no—
259C3×D4—81no—
260C3×D4—81no—
261C3×D4—81no—
262C3×D4—81no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4C4×D411yescharacteristic
278C2×C3×C4C4×D411yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12111yescharacteristic
281#67—88no—
282C2×Dic3—24no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—42no—
287C2×Dic3—44no—
288C2×Dic3—44no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C4×D3—44no—
296C4×D3—44no—
297C4×D3—44no—
298C4×D3—44no—
299C4×D3—432no—
300C3×C8—81no—
301C3⋊C8—81no—
302C24—21no—
303C24—32no—
304C24—32no—
305C24—32no—
306C24—64no—
307C24—64no—
308C24—61no—
309C24—61no—
310C24—61no—
311C24—61no—
312C24D3×D411yescharacteristic
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—124no—
323C2×D4—124no—
324C2×D4—124no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—32no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—64no—
334C22×C4—64no—
335C22×C4—616no—
336C22×C4—64no—
337C22×C4—61no—
338C22×C4—62no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—122no—
346C22×C4—122no—
347C22×C4—124no—
348C22×C4—124no—
349C22×C4—124no—
350C22×C4—124no—
351C22×C4—1216no—
352C22×C4—128no—
353C22×C4—1216no—
354C22×C4—128no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4C2×C4×D311yescharacteristic
360C22×C4#23511yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4D3×D411yescharacteristic
363C22×C4#23711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—122no—
368C2×C8—121no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—122no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—124no—
375C22⋊C4—124no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3#38611yescharacteristic
392C22×C3C2×#9912yes—
393C22×C3C2×#10211yescharacteristic
394C22×C3#41711yescharacteristic
395C22×C3C2×#10412yes—
396C22×C3#44711yescharacteristic
397C22×C3#44811yescharacteristic
398C22×C3#48011yescharacteristic
399C22×C3#48111yescharacteristic
400C2×D3—28no—
401C2×D3—28no—
402C2×D3—28no—
403C2×D3—28no—
404C2×D3—24no—
405C2×D3—24no—
406C2×D3—24no—
407C2×D3—24no—
408C2×D3—24no—
409C2×D3—24no—
410C2×D3#32718yescomplemented
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—44no—
416C3×C4—81no—
417Dic3—44no—
418Dic3—42no—
419Dic3—82no—
420C23—22no—
421C23—22no—
422C23—34no—
423C23—34no—
424C23—34no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—68no—
431C23—68no—
432C23—64no—
433C23—64no—
434C23—64no—
435C23—68no—
436C23—68no—
437C23—64no—
438C23—62no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23—62no—
443C23—62no—
444C23—62no—
445C23—62no—
446C23D3×C22⋊C411yescharacteristic
447C23#88311yescharacteristic
448C23D3×D811yescharacteristic
449C23#91111yescharacteristic
450C23D3×SD1611yescharacteristic
451C23#91511yescharacteristic
452C23#93711yescharacteristic
453D4—81no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—248no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—64no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—124no—
479C2×C4—124no—
480C2×C4—124no—
481C2×C4—124no—
482C2×C4—1232no—
483C2×C4—122no—
484C2×C4—124no—
485C2×C4—124no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4#67⋊C411yescharacteristic
493C2×C4#88411yescharacteristic
494C2×C4C4×#6711yescharacteristic
495C2×C4#93811yescharacteristic
496C8—81no—
497C8—241no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—161no—
503C2×C3C2×#32712yes—
504C2×C3#150412yes—
505C2×C3#187411yescharacteristic
506C2×C3#189611yescharacteristic
507C2×C3#189811yescharacteristic
508D3—28no—
509C22—21no—
510C22—21no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—21no—
516C22—21no—
517C22—38no—
518C22—68no—
519C22—68no—
520C22—68no—
521C22—68no—
522C22—64no—
523C22—64no—
524C22—64no—
525C22—64no—
526C22—64no—
527C22—64no—
528C22—81no—
529C22—81no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22D3×#9912yes—
534C22D3×#10211yescharacteristic
535C22#477111yescharacteristic
536C22#477411yescharacteristic
537C22#477711yescharacteristic
538C22D3×#10412yes—
539C22#495611yescharacteristic
540C22#515611yescharacteristic
541C22#517011yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—81no—
548C4—124no—
549C4—122no—
550C4—242no—
551C3#914711yescharacteristic, complemented, Sylow
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—68no—
557C2—161no—
558C2D3×#32712yes—
559C2#6830212yes—
560C2#7003611yescharacteristic
561C2#7020011yescharacteristic
562C2#7038211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radical#82723C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9147
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×D3×#95C2semidirect
3. C2×C3×#327C2semidirect
4. C2×#4537C2semidirect
5. #69955C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
12. C2×#938C4semidirect
22. #4987C4semidirect
32. C2×#327D3semidirect
37. C22×C3×D4C2×C4semidirect
60. C2×C3×C4⋊C4C2×C4semidirect
68. C2×C4⋊Dic3C2×C4semidirect
73. C3×#95D4semidirect
78. C3⋊#95D4semidirect
180. C22×D4C4×D3semidirect
207. C2×C4⋊C4C4×D3semidirect
223. #95#67semidirect
410. C2×D3#327semidirect
551. C3#9147semidirect

Maximal subgroups

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

subgroupindex
#827232
C2×D3×#952
C2×C3×#3272
C2×#45372
#699552
#676752
#676122
#91473