Finite Group Databasebeta

← #10690193

Subgroups — #10690193

Order 768.

1296 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5223C211yesmaximal, characteristic
2#74005C211yesmaximal, characteristic, complemented
3C3×#1786C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9017—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67—21no—
11C22×#217—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C2×D3×C22⋊C4—21no—
14C2×C3×#102—22no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17#5223—22no—
18#5223C414yescomplemented
19C2×#832—22no—
20#5212C2211yescharacteristic
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#4989—22no—
24#4942—21no—
25#4668—22no—
26#4468—22no—
27C2×#392—31no—
28#2315—31no—
29#1786—31no—
30#1786D311yescharacteristic, complemented
31C2×#323—31no—
32#1294—31no—
33#1294—31no—
34C25×C3—21no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—44no—
42C22×C3×D4C2×C412yescomplemented
43C23×C3×C4C2311yescharacteristic
44C22×#67—24no—
45C22×#67—44no—
46C22×#67—44no—
47C22×#67—42no—
48C22×#67—42no—
49C23×Dic3—21no—
50C23×Dic3D411yescharacteristic
51C22×C4×D3—22no—
52C22×C3×C8—21no—
53C2×C3×C4⋊C4D412yescomplemented
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—24no—
57C2×#217—44no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4C2×C412yes—
64C2×(C2×D3)⋊C4D412yes—
65C2×C4⋊Dic3D412yescomplemented
66C2×Dic3⋊C4—22no—
67C22×C3⋊C8—21no—
68D3×C22⋊C4—42no—
69D3×C22⋊C4—42no—
70C3×#102—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74#832—44no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95C2×C412yescomplemented
79#830—21no—
80#830—21no—
81C23×D4—61no—
82C23×D4C2×D311yescharacteristic
83C22×C22⋊C4—31no—
84C22×C22⋊C4—61no—
85C22×C22⋊C4—61no—
86C2×#102—22no—
87C2×#102—62no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8C2×D311yescharacteristic
90#392—34no—
91#392—62no—
92#385—62no—
93#385—62no—
94#380—31no—
95#380—61no—
96#380C2×D311yescharacteristic
97C2×#95—31no—
98#323—62no—
99C24×C3—24no—
100C24×C3—22no—
101C24×C3—42no—
102C24×C3—42no—
103C24×C3—42no—
104C24×C3—41no—
105C24×C3—41no—
106C24×C3—41no—
107C24×C3—41no—
108C24×C3—41no—
109C24×C3—41no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—42no—
116C23×D3—42no—
117C23×D3C22⋊C412yes—
118C23×D3D812yes—
119C23×D3SD1612yes—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—44no—
127C2×C3×D4—44no—
128C2×C3×D4—44no—
129C2×C3×D4—44no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C22×C3×C4—22no—
133C22×C3×C4—42no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—48no—
141C2×#67—48no—
142C2×#67—416no—
143C2×#67—416no—
144C2×#67—48no—
145C22×Dic3—24no—
146C22×Dic3—22no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—42no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—42no—
154C22×Dic3C22⋊C412yes—
155C2×C4×D3—44no—
156C2×C4×D3—82no—
157C2×C4×D3—82no—
158C2×C3×C8—42no—
159C2×C3×C8—41no—
160C3×C4⋊C4—42no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163#217—48no—
164#217—41no—
165#217—41no—
166(C2×D3)⋊C4—22no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—22no—
170(C2×D3)⋊C4—44no—
171C4⋊Dic3—42no—
172Dic3⋊C4—82no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—41no—
175C25—21no—
176C25—31no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—44no—
183C22×D4—64no—
184C22×D4—124no—
185C22×D4—124no—
186C22×D4—122no—
187C22×D4—122no—
188C22×D4C4×D312yescomplemented
189C23×C4—31no—
190C23×C4—62no—
191C23×C4—61no—
192C23×C4C22×D311yescharacteristic
193C22×C8—21no—
194C22×C8—61no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4#6712yescomplemented
198C2×C22⋊C4—21no—
199C2×C22⋊C4—32no—
200C2×C22⋊C4—32no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—122no—
209C2×C22⋊C4—122no—
210C2×C22⋊C4—124no—
211#102—44no—
212#102—124no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217#95—21no—
218#95—32no—
219#95—62no—
220#95—61no—
221#95—61no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—24no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—44no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—44no—
237C23×C3—82no—
238C23×C3—82no—
239C23×C3—82no—
240C23×C3—82no—
241C23×C3—82no—
242C23×C3—81no—
243C23×C3—81no—
244C23×C3—82no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3C2×C22⋊C411yescharacteristic
248C23×C3#12011yescharacteristic
249C23×C3#12311yescharacteristic
250C23×C3C2×D811yescharacteristic
251C23×C3C2×SD1611yescharacteristic
252C23×C3AGL(1,ℤ/8)11yescharacteristic
253C23×C3#13711yescharacteristic
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—22no—
258C22×D3—22no—
259C22×D3—22no—
260C22×D3—42no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—44no—
265C22×D3—44no—
266C22×D3—82no—
267C22×D3—82no—
268C22×D3—81no—
269C22×D3—81no—
270C22×D3#10214yescomplemented
271C3×D4—24no—
272C3×D4—24no—
273C3×D4—44no—
274C3×D4—44no—
275C3×D4—42no—
276C3×D4—42no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4C4×D411yescharacteristic
287C2×C3×C4C4×D411yescharacteristic
288C2×C3×C4#12111yescharacteristic
289C2×C3×C4#12111yescharacteristic
290#67—432no—
291C2×Dic3—24no—
292C2×Dic3—22no—
293C2×Dic3—44no—
294C2×Dic3—44no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—81no—
303C4×D3—88no—
304C3×C8—81no—
305C3⋊C8—81no—
306C24—24no—
307C24—22no—
308C24—32no—
309C24—32no—
310C24—32no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—61no—
321C24—61no—
322C24—61no—
323C24—61no—
324C24—122no—
325C24—122no—
326C24D3×D411yescharacteristic
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—128no—
340C2×D4—128no—
341C2×D4—128no—
342C2×D4—1216no—
343C2×D4—1216no—
344C22×C4—22no—
345C22×C4—32no—
346C22×C4—42no—
347C22×C4—64no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—62no—
352C22×C4—124no—
353C22×C4—122no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—122no—
357C22×C4—242no—
358C22×C4—242no—
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—42no—
370C4⋊C4—122no—
371C4⋊C4—242no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—62no—
375C22⋊C4—62no—
376C22⋊C4—62no—
377C22⋊C4—62no—
378C22⋊C4—124no—
379C22⋊C4—128no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—44no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—84no—
396C22×C3—84no—
397C22×C3—84no—
398C22×C3—84no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3#38611yescharacteristic
402C22×C3C2×#10211yescharacteristic
403C22×C3#41711yescharacteristic
404C22×C3#44711yescharacteristic
405C22×C3#44811yescharacteristic
406C22×C3#45312yes—
407C22×C3B(4,𝔽2)12yes—
408C22×C3#48011yescharacteristic
409C22×C3#48111yescharacteristic
410C2×D3—24no—
411C2×D3—44no—
412C2×D3—44no—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—44no—
416C2×D3—82no—
417C2×D3—82no—
418C2×D3—82no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—82no—
424Dic3—44no—
425Dic3—42no—
426Dic3—82no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—34no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—62no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—82no—
457C23—81no—
458C23—81no—
459C23—124no—
460C23—124no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—242no—
466C23—242no—
467C23—241no—
468C23—241no—
469C23D3×C22⋊C411yescharacteristic
470C23#88311yescharacteristic
471C23D3×D811yescharacteristic
472C23#91111yescharacteristic
473C23D3×SD1611yescharacteristic
474C23#91511yescharacteristic
475C23#93711yescharacteristic
476D4—24no—
477D4—24no—
478D4—44no—
479D4—44no—
480D4—42no—
481D4—42no—
482D4—1232no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—124no—
495C2×C4—124no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—248no—
504C2×C4—241no—
505C2×C4#67⋊C411yescharacteristic
506C2×C4#88411yescharacteristic
507C2×C4C4×#6711yescharacteristic
508C2×C4#93811yescharacteristic
509C8—81no—
510C8—241no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—84no—
516C2×C3#159512yes—
517C2×C3#159612yes—
518C2×C3#187411yescharacteristic
519C2×C3#188711yescharacteristic
520C2×C3#188911yescharacteristic
521D3—44no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—64no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22—82no—
539C22—84no—
540C22—84no—
541C22—82no—
542C22—124no—
543C22—124no—
544C22—124no—
545C22—122no—
546C22—122no—
547C22—242no—
548C22—242no—
549C22—242no—
550C22D3×#10211yescharacteristic
551C22#477111yescharacteristic
552C22#477411yescharacteristic
553C22#477711yescharacteristic
554C22#495611yescharacteristic
555C22#503212yes—
556C22#515611yescharacteristic
557C22#517011yescharacteristic
558C22#519812yes—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—82no—
564C4—124no—
565C4—122no—
566C4—242no—
567C3#901711yescharacteristic, complemented, Sylow
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—84no—
573C2—124no—
574C2#6949911yescharacteristic
575C2#7020011yescharacteristic
576C2#7032311yescharacteristic
577C2#7096812yes—
578C2#7097612yes—
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1786C2
socleC23×C3#120
prime radicalC2×#5223C2
cosocleC23×C3×C4C23
hypercenter#1786D3
hypocenterC3#9017
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #74005C2semidirect
3. C3×#1786C2semidirect
4. #72128C2semidirect
5. C2×#4468C2semidirect
6. #67675C2semidirect
7. #67624C2semidirect
18. #5223C4semidirect
30. #1786D3semidirect
42. C22×C3×D4C2×C4semidirect
53. C2×C3×C4⋊C4D4semidirect
65. C2×C4⋊Dic3D4semidirect
78. C3⋊#95C2×C4semidirect
188. C22×D4C4×D3semidirect
197. C2×C4⋊C4#67semidirect
270. C22×D3#102semidirect
567. C3#9017semidirect

Maximal subgroups

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

subgroupindex
C2×#52232
#740052
C3×#17862
#721282
C2×#44682
#676752
#676242
#90173