Finite Group Databasebeta

← #10944288

Subgroups — #10944288

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×M16C211yesmaximal, characteristic
2C2×#5137C211yesmaximal, characteristic, complemented
3C2×#4887C211yesmaximal, characteristic, complemented
4C2×#4867C211yesmaximal, characteristic, complemented
5#74049C212yesmaximal
6C3×#1809C211yesmaximal, characteristic, complemented
7#70631C211yesmaximal, characteristic, complemented
8#70577C211yesmaximal, characteristic, complemented
9#69516C212yesmaximal, complemented
10#68884C212yesmaximal, complemented
11#68882C212yesmaximal, complemented
12#32937—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×#924C2212yes—
16C22×#208C2211yescharacteristic
17C2×D3×M16—24no—
18C2×D3×M16—24no—
19C2×D3×M16—24no—
20C22×Dic12C2211yescharacteristic, complemented
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24C22×D3×C8C2211yescharacteristic
25C2×#897C2212yes—
26C2×#897C2212yes—
27C2×C3×#107C2211yescharacteristic, complemented
28C2×C3×#106C2211yescharacteristic, complemented
29C3×#380C2212yes—
30#5137—24no—
31C2×#817C2211yescharacteristic, complemented
32C2×#816C2212yescomplemented
33#4987C2212yes—
34#4987C2212yes—
35C2×#810C2212yescomplemented
36C2×#810C2212yescomplemented
37#4955—22no—
38#4950—22no—
39#4887—24no—
40#4867—24no—
41#4668—22no—
42#4668—22no—
43C3×#340C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#470—31no—
48C2×#470—31no—
49C2×#467—31no—
50#2848—32no—
51#1894—32no—
52#1894—32no—
53#1892—32no—
54#1809—31no—
55#1809D311yescharacteristic, complemented
56#1807—31no—
57C24×D3D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3D411yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3D411yescharacteristic
67C22×C4×D3D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×Dic6C2312yes—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16D414yes—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×#217—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2312yes—
87#924—24no—
88#924—24no—
89C2×Dic3⋊C4—22no—
90C2×#208—22no—
91C2×#208—22no—
92C2×#208—22no—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95D3×M16—232no—
96C2×Dic12—22no—
97C2×Dic12—22no—
98C2×Dic12—22no—
99C2×ΓL(1,𝔽25)—22no—
100C2×ΓL(1,𝔽25)—22no—
101C2×ΓL(1,𝔽25)—22no—
102C2×ΓL(1,𝔽25)—22no—
103C2×ΓL(1,𝔽25)—22no—
104C2×ΓL(1,𝔽25)D414yes—
105C2×D3×C8—22no—
106C2×D3×C8—22no—
107C2×D3×C8—22no—
108C2×D3×C8—22no—
109C2×D3×C8—22no—
110C2×D3×C8D414yes—
111#897—24no—
112#897—24no—
113#897—24no—
114#897—24no—
115C3×#107—22no—
116C3×#106—22no—
117C3×#95—22no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#817—22no—
121#816—24no—
122#810—24no—
123#810—24no—
124C24×C4—31no—
125C22×Q16—31no—
126C22×M16—31no—
127C22×M16—31no—
128C22×M16—31no—
129C22×M16—64no—
130C22×M16—64no—
131C22×M16—64no—
132C22×M16C2×D311yescharacteristic
133C23×C8—31no—
134C23×C8—31no—
135C2×C22⋊Q8—32no—
136C2×C22⋊Q8—32no—
137C2×C22⋊Q8—32no—
138#470—64no—
139#470—64no—
140#467—64no—
141C2×#107C2×D311yescharacteristic, complemented
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic, complemented
144C2×#103—32no—
145C2×#103—32no—
146C2×#103—32no—
147#385—62no—
148#385—62no—
149#385—62no—
150#380—32no—
151#380—32no—
152#380C2×D312yes—
153#379—62no—
154#340—32no—
155#340—32no—
156#340C2×D312yescomplemented
157C24×C3C2×D411yescharacteristic
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—22no—
161C23×D3—24no—
162C23×D3—42no—
163C23×D3C2×D411yescharacteristic
164C23×D3C2×D411yescharacteristic
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C2×C4×D3—28no—
183C2×C4×D3—28no—
184C2×C4×D3—28no—
185C2×C4×D3—28no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—21no—
191C2×C4×D3—21no—
192C2×C4×D3—21no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3C2×D412yes—
201C2×C4×D3C2×D412yes—
202C2×C4×D3#4112yes—
203C2×C4×D3#4112yes—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C3×M16—28no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8C2×D412yes—
214C2×C3×C8C2×D412yes—
215C2×C3×C8C2×D412yes—
216C2×C3×C8C2×D412yes—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C22⋊C4—44no—
222#217—44no—
223(C2×D3)⋊C4—44no—
224(C2×D3)⋊C4—44no—
225(C2×D3)⋊C4—44no—
226(C2×D3)⋊C4—44no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—44no—
230C4⋊Dic3—44no—
231Dic3⋊C4—44no—
232#208—28no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8C2×D412yes—
238C2×C3⋊C8C2×D412yes—
239Dic12—44no—
240ΓL(1,𝔽25)—28no—
241ΓL(1,𝔽25)—28no—
242D3×C8—28no—
243D3×C8—28no—
244C25—31no—
245C22×Q8—32no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4—62no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4C22×D311yescharacteristic
258C2×Q16—62no—
259C2×Q16—62no—
260C2×Q16—62no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—34no—
264C2×M16—62no—
265C2×M16—62no—
266C2×M16—62no—
267C2×M16—62no—
268C2×M16—632no—
269C2×M16—62no—
270C2×M16—62no—
271C2×M16—62no—
272C2×M16—62no—
273C2×M16D1214yes—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—34no—
277C22×C8—62no—
278C22×C8—62no—
279C22×C8—62no—
280C22×C8—62no—
281C22×C8—62no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C22⋊Q8—64no—
285C22⋊Q8—64no—
286C22⋊Q8—64no—
287C22⋊Q8—64no—
288C22⋊Q8—64no—
289C22⋊Q8—64no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D312yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—62no—
301C2×C22⋊C4—62no—
302#107—22no—
303#106—22no—
304#106—62no—
305#103—64no—
306#103—64no—
307#103—64no—
308#95—22no—
309#95—62no—
310#95—62no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3#12112yes—
316C23×C3C22×D411yescharacteristic
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—22no—
320C22×D3—28no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—48no—
326C22×D3—48no—
327C22×D3—44no—
328C22×D3—44no—
329C22×D3#12112yes—
330C22×D3#12112yes—
331C22×D3#13714yes—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4#12112yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×Dic3—28no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3#12112yes—
363C2×Dic3#12112yes—
364C4×D3—28no—
365C4×D3—28no—
366C4×D3—22no—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—22no—
370C4×D3—22no—
371C4×D3—22no—
372C4×D3—22no—
373C4×D3—22no—
374Dic6—42no—
375Dic6—42no—
376Dic6—42no—
377Dic6—42no—
378C3×C8—22no—
379C3×C8#12014yes—
380C3⋊C8—22no—
381C3⋊C8—22no—
382C24—31no—
383C24—31no—
384C24—62no—
385C24—62no—
386C24—64no—
387C24—62no—
388C24—122no—
389C24D3×D411yescharacteristic
390C2×Q8—62no—
391C2×Q8—62no—
392C2×Q8—62no—
393C2×Q8—62no—
394C2×Q8—62no—
395C2×Q8—62no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—68no—
413C22×C4—68no—
414C22×C4—68no—
415C22×C4—68no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4C2×D1212yes—
432C22×C4D3×D411yescharacteristic
433C22×C4#23912yes—
434C22×C4C23×D311yescharacteristic
435Q16—124no—
436M16—28no—
437M16—68no—
438M16—68no—
439M16—68no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—32no—
443C2×C8—32no—
444C2×C8—68no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—68no—
450C2×C8C2×D1212yes—
451C2×C8C2×D1212yes—
452C2×C8D3×D412yes—
453C2×C8D3×D412yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—124no—
461C4⋊C4—124no—
462C4⋊C4—124no—
463C22⋊C4—44no—
464C22⋊C4—124no—
465C22⋊C4—124no—
466C22⋊C4—124no—
467C22⋊C4—124no—
468C22⋊C4—124no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—42no—
474C22×C3C2×#12111yescharacteristic
475C22×C3C2×#12111yescharacteristic
476C22×C3C2×#12111yescharacteristic
477C22×C3C2×#13712yes—
478C22×C3#57512yes—
479C2×D3—22no—
480C2×D3—22no—
481C2×D3—24no—
482C2×D3—24no—
483C2×D3—22no—
484C2×D3—24no—
485C2×D3—416no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—42no—
489C2×D3#47014yescomplemented
490C3×C4—22no—
491C3×C4—82no—
492C3×C4—82no—
493C3×C4C2×#12011yescharacteristic
494C3×C4#52511yescharacteristic
495C3×C4#52511yescharacteristic
496C3×C4#52511yescharacteristic
497Dic3—22no—
498Dic3—22no—
499Dic3—82no—
500Dic3—82no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—34no—
506C23—32no—
507C23—32no—
508C23—68no—
509C23—61no—
510C23—62no—
511C23—61no—
512C23—61no—
513C23—62no—
514C23—62no—
515C23—61no—
516C23—128no—
517C23—124no—
518C23—128no—
519C23—124no—
520C23#89512yes—
521C23C2×D3×D411yescharacteristic
522Q8—122no—
523Q8—122no—
524Q8—122no—
525Q8—122no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—68no—
541C2×C4—68no—
542C2×C4—68no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4#89512yes—
562C2×C4C22×D1211yescharacteristic
563C2×C4C2×D3×D411yescharacteristic
564C2×C4C2×D3×D411yescharacteristic
565C2×C4C2×#23911yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C8—22no—
569C8—62no—
570C8—62no—
571C8#88214yes—
572C2×C3—42no—
573C2×C3#284711yescharacteristic
574C2×C3C2×#47011yescharacteristic
575C2×C3#305411yescharacteristic
576C2×C3#305612yes—
577C2×C3#305612yes—
578D3—24no—
579D3—42no—
580C22—22no—
581C22—24no—
582C22—34no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—62no—
587C22—64no—
588C22—64no—
589C22—62no—
590C22—62no—
591C22—64no—
592C22—1216no—
593C22—121no—
594C22—122no—
595C22—121no—
596C22C2×#89511yescharacteristic
597C22D3×#12111yescharacteristic
598C22D3×#12111yescharacteristic
599C22#577312yes—
600C22D3×#13712yes—
601C4—22no—
602C4—62no—
603C4—62no—
604C4—82no—
605C4—82no—
606C4—242no—
607C4—242no—
608C4C2×#88211yescharacteristic
609C4#552411yescharacteristic
610C4#562711yescharacteristic
611C4#562711yescharacteristic
612C3#3293711yescharacteristic, complemented, Sylow
613C2—42no—
614C2—64no—
615C2—122no—
616C2#7404811yescharacteristic
617C2#7921212yes—
618C2#8080612yes—
619C2#8086311yescharacteristic
620C2D3×#47011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1809C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenter#1809D3
hypocenterC3#32937
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×#5137C2semidirect
3. C2×#4887C2semidirect
4. C2×#4867C2semidirect
6. C3×#1809C2semidirect
7. #70631C2semidirect
8. #70577C2semidirect
9. #69516C2semidirect
10. #68884C2semidirect
11. #68882C2semidirect
20. C22×Dic12C22semidirect
27. C2×C3×#107C22semidirect
28. C2×C3×#106C22semidirect
31. C2×#817C22semidirect
32. C2×#816C22semidirect
35. C2×#810C22semidirect
36. C2×#810C22semidirect
43. C3×#340C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
55. #1809D3semidirect
141. C2×#107C2×D3semidirect
143. C2×#106C2×D3semidirect
156. #340C2×D3semidirect
489. C2×D3#470semidirect
612. C3#32937semidirect

Maximal subgroups

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

subgroupindex
C22×D3×M162
C2×#51372
C2×#48872
C2×#48672
#740492
C3×#18092
#706312
#705772
#695162
#688842
#688822
#329373