Finite Group Databasebeta

← #10695436

Subgroups — #10695436

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
1C2×C3×#450C211yesmaximal, characteristic, complemented
2C2×#5223C211yesmaximal, characteristic
3#82822C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5C2×#4538C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9922—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67—21no—
11C22×C3×SD16—21no—
12C2×C3×C22⋊Q8C2211yescharacteristic
13C22×#217—21no—
14C22×(C2×D3)⋊C4C2211yescharacteristic
15C3×#450—22no—
16C3×#450—22no—
17C2×C3×#102—21no—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5231—21no—
20#5223—22no—
21#5223—22no—
22C2×#832—22no—
23#5212C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4994—22no—
27#4993—21no—
28#4955—21no—
29#4946—21no—
30#4538—22no—
31#4468—22no—
32C2×#450D311yescharacteristic, complemented
33#2626—31no—
34C2×#392—31no—
35#1786—31no—
36C2×#328—31no—
37C2×#323—31no—
38#1294—31no—
39C25×C3—21no—
40C24×D3D411yescharacteristic
41C22×C3×Q8D411yescharacteristic, complemented
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—44no—
48C22×C3×D4D412yes—
49C23×C3×C4C2311yescharacteristic
50C22×#67—24no—
51C22×#67—44no—
52C22×#67—42no—
53C22×#67—44no—
54C22×#67—42no—
55C23×Dic3—21no—
56C23×Dic3D411yescharacteristic
57C2×C3×SD16—24no—
58C2×C3×SD16—44no—
59C22×C3×C8—21no—
60C3×C22⋊Q8—21no—
61C3×C22⋊Q8—21no—
62C3×C22⋊Q8—21no—
63C3×C22⋊Q8—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic, complemented
66C2×C3×C22⋊C4—21no—
67C2×#217—24no—
68C2×#217—21no—
69C2×#217—44no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3D412yescomplemented
79C2×Dic3⋊C4—22no—
80C22×C3⋊C8—21no—
81C3×#102—24no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8D412yescomplemented
84#832—44no—
85C3⋊#95—22no—
86C3⋊#95—21no—
87C3⋊#95—21no—
88C3⋊#95D412yes—
89#830—21no—
90#830—21no—
91C23×D4—61no—
92C23×D4C2×D311yescharacteristic
93C22×SD16—21no—
94C2×C22⋊Q8C2×D311yescharacteristic
95C22×C22⋊C4—31no—
96C22×C22⋊C4—61no—
97#450—22no—
98#450—22no—
99C2×#102—21no—
100C2×#102—62no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#399—61no—
104#396—62no—
105#393—61no—
106#392—62no—
107#392—62no—
108#385—61no—
109#380—31no—
110#379—61no—
111C2×#95—31no—
112#328—62no—
113#323—62no—
114C24×C3—24no—
115C24×C3—22no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3—42no—
119C24×C3—41no—
120C24×C3—41no—
121C24×C3—41no—
122C24×C3—41no—
123C24×C3—41no—
124C24×C3—41no—
125C24×C3C2×D411yescharacteristic
126C23×D3—21no—
127C23×D3—21no—
128C23×D3—21no—
129C23×D3—21no—
130C23×D3—21no—
131C23×D3—21no—
132C23×D3—42no—
133C23×D3—42no—
134C23×D3SD1612yescomplemented
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—42no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—41no—
149C2×C3×D4—41no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×#67—48no—
160C2×#67—48no—
161C2×#67—48no—
162C2×#67—416no—
163C2×#67—416no—
164C22×Dic3—24no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—42no—
171C22×Dic3—41no—
172C22×Dic3—41no—
173C22×Dic3—42no—
174C3×SD16—48no—
175C2×C3×C8—22no—
176C2×C3×C8—42no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—41no—
183#217—48no—
184#217—41no—
185#217—41no—
186(C2×D3)⋊C4—41no—
187(C2×D3)⋊C4—41no—
188(C2×D3)⋊C4—41no—
189(C2×D3)⋊C4—41no—
190(C2×D3)⋊C4—82no—
191(C2×D3)⋊C4—82no—
192C4⋊Dic3—42no—
193Dic3⋊C4—82no—
194C2×C3⋊C8—42no—
195C2×C3⋊C8—41no—
196C25—21no—
197C25—31no—
198C22×Q8#6711yescharacteristic, complemented
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—44no—
205C22×D4—64no—
206C22×D4—124no—
207C22×D4—124no—
208C22×D4—122no—
209C22×D4—122no—
210C22×D4#6712yes—
211C23×C4—31no—
212C23×C4—61no—
213C23×C4C22×D311yescharacteristic
214C2×SD16—24no—
215C2×SD16—44no—
216C22×C8—21no—
217C22×C8—61no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4#6711yescharacteristic, complemented
226C2×C22⋊C4—21no—
227C2×C22⋊C4—64no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—61no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—124no—
238#102—24no—
239#102—124no—
240C22⋊C8—21no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8#6712yescomplemented
244#95—32no—
245#95—62no—
246#95—61no—
247#95—61no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—42no—
252C23×C3—44no—
253C23×C3—44no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—44no—
257C23×C3—44no—
258C23×C3—44no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—44no—
263C23×C3—82no—
264C23×C3—82no—
265C23×C3—82no—
266C23×C3—82no—
267C23×C3—82no—
268C23×C3—81no—
269C23×C3—81no—
270C23×C3—82no—
271C23×C3—81no—
272C23×C3—81no—
273C23×C3#12011yescharacteristic
274C23×C3#12011yescharacteristic
275C23×C3#12111yescharacteristic
276C23×C3C2×SD1611yescharacteristic
277C23×C3#13511yescharacteristic
278C23×C3AGL(1,ℤ/8)11yescharacteristic
279C23×C3#13711yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—44no—
284C22×D3—44no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—41no—
293C22×D3—82no—
294C22×D3—82no—
295C22×D3—81no—
296C22×D3—81no—
297C3×Q8—42no—
298C3×Q8—81no—
299C3×D4—24no—
300C3×D4—24no—
301C3×D4—44no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—44no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12711yescharacteristic
319#67—432no—
320C2×Dic3—24no—
321C2×Dic3—44no—
322C2×Dic3—44no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—81no—
332C3×C8—42no—
333C3⋊C8—81no—
334C24—24no—
335C24—22no—
336C24—32no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24—61no—
347C24—61no—
348C24—61no—
349C24—61no—
350C24—61no—
351C24—61no—
352C24—122no—
353C24—122no—
354C24D3×D411yescharacteristic
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—128no—
371C2×D4—128no—
372C2×D4—128no—
373C2×D4—1216no—
374C2×D4—1216no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—122no—
385C22×C4—121no—
386C22×C4—121no—
387C22×C4—122no—
388C22×C4D3×D411yescharacteristic
389C22×C4D3×D411yescharacteristic
390C22×C4#23711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394SD16—48no—
395C2×C8—22no—
396C2×C8—42no—
397C2×C8—122no—
398C2×C8—121no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—122no—
404C4⋊C4—242no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—128no—
408C22⋊C4—121no—
409C22⋊C4—121no—
410C22⋊C4—121no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C22⋊C4—242no—
415C22⋊C4—242no—
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—44no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—84no—
430C22×C3—84no—
431C22×C3—84no—
432C22×C3—84no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3#39211yescharacteristic
436C22×C3#44811yescharacteristic
437C22×C3#44911yescharacteristic
438C22×C3#45011yescharacteristic
439C22×C3#45211yescharacteristic
440C22×C3#45312yes—
441C22×C3B(4,𝔽2)12yes—
442C22×C3#46511yescharacteristic
443C22×C3#46811yescharacteristic
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—42no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—81no—
451C2×D3—81no—
452C2×D3—82no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—81no—
458C3×C4—81no—
459Dic3—44no—
460Dic3—82no—
461Dic3—81no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—62no—
478C23—62no—
479C23—62no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—82no—
490C23—124no—
491C23—124no—
492C23—122no—
493C23—122no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—122no—
498C23—121no—
499C23—122no—
500C23—242no—
501C23—242no—
502C23—241no—
503C23—241no—
504C23D3×SD1611yescharacteristic
505C23#91411yescharacteristic
506C23#91511yescharacteristic
507C23#91611yescharacteristic
508C23#93711yescharacteristic
509C23#93711yescharacteristic
510C23#93811yescharacteristic
511Q8—42no—
512Q8—81no—
513D4—24no—
514D4—24no—
515D4—42no—
516D4—44no—
517D4—42no—
518D4—44no—
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—42no—
530C2×C4—64no—
531C2×C4—124no—
532C2×C4—124no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—241no—
542C2×C4#93911yescharacteristic
543C2×C4#93911yescharacteristic
544C2×C4#94011yescharacteristic
545C2×C4#95311yescharacteristic
546C8—42no—
547C8—241no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—84no—
553C2×C3#159612yes—
554C2×C3#160012yes—
555C2×C3#200011yescharacteristic
556C2×C3#200211yescharacteristic
557C2×C3#202111yescharacteristic
558D3—82no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—82no—
577C22—82no—
578C22—124no—
579C22—124no—
580C22—122no—
581C22—242no—
582C22—242no—
583C22—242no—
584C22—242no—
585C22—241no—
586C22—241no—
587C22#503212yes—
588C22#516911yescharacteristic
589C22#517011yescharacteristic
590C22#517111yescharacteristic
591C22#517211yescharacteristic
592C22#517311yescharacteristic
593C22#517411yescharacteristic
594C22#519812yes—
595C22#522311yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—81no—
601C4—81no—
602C4—124no—
603C4—242no—
604C4—241no—
605C3#992211yescharacteristic, complemented, Sylow
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—84no—
611C2—242no—
612C2#7097612yes—
613C2#7097712yes—
614C2#7167111yescharacteristic
615C2#7205411yescharacteristic
616C2#7209411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#450C2
socleC23×C3#120
prime radicalC2×#5223C2
hypercenterC2×#450D3
hypocenterC3#9922
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C2×C3×#450C2semidirect
3. #82822C2semidirect
4. #72128C2semidirect
5. C2×#4538C2semidirect
6. C2×#4468C2semidirect
7. #67675C2semidirect
32. C2×#450D3semidirect
41. C22×C3×Q8D4semidirect
65. C2×C3×C4⋊C4D4semidirect
78. C2×C4⋊Dic3D4semidirect
83. C3×C22⋊C8D4semidirect
134. C23×D3SD16semidirect
198. C22×Q8#67semidirect
225. C2×C4⋊C4#67semidirect
243. C22⋊C8#67semidirect
605. C3#9922semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4502
C2×#52232
#828222
#721282
C2×#45382
C2×#44682
#676752
#99223