Finite Group Databasebeta

← #10695437

Subgroups — #10695437

Order 768.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2626C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic, complemented
3C2×#4960C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic, complemented
6#69521C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9922—31nomaximal, Sylow
9C23×#67—21no—
10C23×D12C2211yescharacteristic
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×#215—21no—
14C22×(C2×D3)⋊C4—21no—
15C3×#399—21no—
16C3×#396—22no—
17C3×#393—21no—
18C3×#385—21no—
19C3×#380C2211yescharacteristic
20C3×#379—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5046—22no—
25#5046—22no—
26C2×#809—21no—
27C2×#809—22no—
28#4960—22no—
29#4960—22no—
30#4538—22no—
31#4525—22no—
32C2×#450—31no—
33#2626D311yescharacteristic, complemented
34C2×#392—31no—
35#1786—31no—
36C2×#328—31no—
37C2×#323—31no—
38#1294—31no—
39C25×C3D411yescharacteristic
40C24×D3—21no—
41C22×C3×Q8D411yescharacteristic, complemented
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—24no—
45C22×#67—44no—
46C22×#67—44no—
47C22×#67—42no—
48C22×#67—42no—
49C23×Dic3—21no—
50C22×D12—21no—
51C22×D12—21no—
52C22×D12—21no—
53C22×D12—21no—
54C22×D12—21no—
55C22×D12—44no—
56C22×D12D412yes—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4D411yescharacteristic, complemented
64C2×C3×C4⋊C4D412yescomplemented
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×#215—24no—
76C2×#215—44no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—44no—
79C22×C3⋊C8—21no—
80C22×C3⋊C8—21no—
81C3×#95—22no—
82C3×#95—21no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#830—21no—
86#830—21no—
87#830—21no—
88#830D412yescomplemented
89#809—24no—
90#809—44no—
91C23×D4—31no—
92C23×D4—61no—
93C22×SD16—61no—
94C2×C22⋊Q8C2×D311yescharacteristic
95C22×C22⋊C4—61no—
96C22×C22⋊C4C2×D311yescharacteristic
97#450—62no—
98#450—62no—
99C2×#102—61no—
100C2×#102—62no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#399—21no—
104#396—22no—
105#393—21no—
106#392—62no—
107#392—62no—
108#385—21no—
109#380C2×D311yescharacteristic
110#379—21no—
111C2×#95—31no—
112#328—62no—
113#323—62no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—42no—
121C24×C3—42no—
122C24×C3SD1612yescomplemented
123C24×C3C2×D411yescharacteristic
124C23×D3—24no—
125C23×D3—22no—
126C23×D3—42no—
127C23×D3—42no—
128C23×D3—42no—
129C23×D3—41no—
130C23×D3—41no—
131C23×D3—41no—
132C23×D3—41no—
133C23×D3—41no—
134C23×D3—41no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—42no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C2×#67—48no—
153C2×#67—48no—
154C2×#67—48no—
155C2×#67—416no—
156C2×#67—416no—
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—42no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C2×D12—24no—
163C2×D12—24no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—44no—
167C2×D12—44no—
168C2×D12—44no—
169C2×D12—42no—
170C2×D12—42no—
171C2×D12—44no—
172C2×D12—41no—
173C2×D12—41no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—41no—
179C3×C4⋊C4—82no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—41no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—82no—
189C3×C22⋊C4—82no—
190#215—48no—
191(C2×D3)⋊C4—48no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—42no—
194C2×C3⋊C8—42no—
195C2×C3⋊C8—41no—
196C25—61no—
197C25#6711yescharacteristic
198C22×Q8#6711yescharacteristic, complemented
199C22×D4—32no—
200C22×D4—64no—
201C22×D4—61no—
202C22×D4—61no—
203C22×D4—61no—
204C22×D4—61no—
205C22×D4—61no—
206C22×D4—124no—
207C22×D4—124no—
208C22×D4—122no—
209C22×D4—124no—
210C22×D4—122no—
211C23×C4—61no—
212C23×C4#6711yescharacteristic
213C23×C4C22×D311yescharacteristic
214C2×SD16—64no—
215C2×SD16—124no—
216C22×C8—61no—
217C22×C8—61no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4#6711yescharacteristic, complemented
225C2×C4⋊C4#6712yescomplemented
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—124no—
238#102—64no—
239#102—124no—
240C22⋊C8—32no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244#95—22no—
245#95—21no—
246#95—32no—
247#95—61no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—44no—
254C23×C3—44no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—82no—
264C23×C3—82no—
265C23×C3—81no—
266C23×C3—81no—
267C23×C3#12011yescharacteristic
268C23×C3#12011yescharacteristic
269C23×C3#12111yescharacteristic
270C23×C3C2×SD1611yescharacteristic
271C23×C3#13511yescharacteristic
272C23×C3AGL(1,ℤ/8)11yescharacteristic
273C23×C3#13711yescharacteristic
274C22×D3—24no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—44no—
278C22×D3—44no—
279C22×D3—44no—
280C22×D3—42no—
281C22×D3—44no—
282C22×D3—44no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—44no—
287C22×D3—82no—
288C22×D3—82no—
289C22×D3—82no—
290C22×D3—82no—
291C22×D3—82no—
292C22×D3—81no—
293C22×D3—81no—
294C22×D3—82no—
295C22×D3—81no—
296C22×D3—81no—
297C3×Q8—42no—
298C3×Q8—81no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—81no—
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12711yescharacteristic
320#67—432no—
321C2×Dic3—24no—
322C2×Dic3—44no—
323C2×Dic3—44no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326D12—24no—
327D12—24no—
328D12—42no—
329D12—44no—
330D12—42no—
331D12—44no—
332C3⋊C8—42no—
333C3⋊C8—81no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—42no—
341C24—42no—
342C24—64no—
343C24—62no—
344C24—122no—
345C24—122no—
346C24—122no—
347C24—121no—
348C24—121no—
349C24—121no—
350C24—121no—
351C24—121no—
352C24—121no—
353C24#21512yescomplemented
354C24C2×#6711yescharacteristic
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—122no—
363C2×D4—122no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—128no—
369C2×D4—128no—
370C2×D4—128no—
371C2×D4—1216no—
372C2×D4—1216no—
373C2×D4—121no—
374C2×D4—121no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—122no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4D3×D411yescharacteristic
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4#23711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394SD16—128no—
395C2×C8—62no—
396C2×C8—122no—
397C2×C8—122no—
398C2×C8—121no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—82no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—128no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—42no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3#39211yescharacteristic
434C22×C3#44811yescharacteristic
435C22×C3#44911yescharacteristic
436C22×C3#45011yescharacteristic
437C22×C3#45211yescharacteristic
438C22×C3#45312yes—
439C22×C3B(4,𝔽2)12yes—
440C22×C3#46511yescharacteristic
441C22×C3#46811yescharacteristic
442C2×D3—44no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—82no—
446C2×D3—82no—
447C2×D3—84no—
448C2×D3—84no—
449C2×D3—84no—
450C2×D3—84no—
451C2×D3—82no—
452C2×D3—82no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—82no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—81no—
461Dic3—44no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—64no—
478C23—82no—
479C23—81no—
480C23—81no—
481C23—82no—
482C23—122no—
483C23—122no—
484C23—124no—
485C23—122no—
486C23—124no—
487C23—124no—
488C23—124no—
489C23—124no—
490C23—122no—
491C23—124no—
492C23—122no—
493C23—122no—
494C23—242no—
495C23—242no—
496C23—242no—
497C23—242no—
498C23—242no—
499C23—241no—
500C23—241no—
501C23—242no—
502C23—241no—
503C23—241no—
504C23#93711yescharacteristic
505C23#93911yescharacteristic
506C23C2×#21511yescharacteristic
507C23#94211yescharacteristic
508C23#94911yescharacteristic
509C23#95011yescharacteristic
510C23#95311yescharacteristic
511Q8—42no—
512Q8—81no—
513D4—64no—
514D4—64no—
515D4—122no—
516D4—124no—
517D4—122no—
518D4—124no—
519D4—1232no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—64no—
537C2×C4—81no—
538C2×C4—122no—
539C2×C4—124no—
540C2×C4—124no—
541C2×C4—122no—
542C2×C4#93711yescharacteristic
543C2×C4#93811yescharacteristic
544C2×C4#93911yescharacteristic
545C2×C4#94011yescharacteristic
546C8—122no—
547C8—241no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—82no—
553C2×C3#159612yes—
554C2×C3#160012yes—
555C2×C3#200011yescharacteristic
556C2×C3#200211yescharacteristic
557C2×C3#202111yescharacteristic
558D3—84no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—124no—
577C22—124no—
578C22—124no—
579C22—242no—
580C22—242no—
581C22—244no—
582C22—244no—
583C22—242no—
584C22—242no—
585C22—244no—
586C22—244no—
587C22#504611yescharacteristic
588C22#504711yescharacteristic
589C22#504811yescharacteristic
590C22#504911yescharacteristic
591C22#522311yescharacteristic
592C22#522711yescharacteristic
593C22#523811yescharacteristic
594C22#524112yes—
595C22#524412yes—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—82no—
604C4—124no—
605C3#992211yescharacteristic, complemented, Sylow
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—244no—
612C2#7169911yescharacteristic
613C2#7185111yescharacteristic
614C2#7185611yescharacteristic
615C2#7188412yes—
616C2#7189212yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2626C2
socleC23×C3#120
prime radicalC2×#4960C2
hypercenter#2626D3
hypocenterC3#9922
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
1. C3×#2626C2semidirect
2. C2×#5046C2semidirect
4. C2×#4538C2semidirect
5. C2×#4525C2semidirect
6. #69521C2semidirect
7. #67624C2semidirect
33. #2626D3semidirect
41. C22×C3×Q8D4semidirect
63. C2×C3×C4⋊C4D4semidirect
64. C2×C3×C4⋊C4D4semidirect
88. #830D4semidirect
122. C24×C3SD16semidirect
198. C22×Q8#67semidirect
224. C2×C4⋊C4#67semidirect
225. C2×C4⋊C4#67semidirect
353. C24#215semidirect
605. C3#9922semidirect

Maximal subgroups

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

subgroupindex
C3×#26262
C2×#50462
C2×#49602
C2×#45382
C2×#45252
#695212
#676242
#99223