Finite Group Databasebeta

← #10795144

Subgroups — #10795144

Order 768.

1694 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×C4×D3×D4C211yesmaximal, characteristic
2#79038C212yesmaximal, complemented
3C3×#1800C211yesmaximal, characteristic, complemented
4#71617C211yesmaximal, characteristic, complemented
5#70527C212yesmaximal, complemented
6#70079C212yesmaximal
7Dic3×#100C212yesmaximal, complemented
8#68990C212yesmaximal, complemented
9#68972C212yesmaximal, complemented
10#20726—31nomaximal, Sylow
11C22×D3×D4C2211yescharacteristic
12C23×C4×D3C412yes—
13C2×C3×C4×D4C2211yescharacteristic
14C2×D4×Dic3C2211yescharacteristic
15C2×C4×#67C412yes—
16C4×D3×D4—22no—
17C4×D3×D4—22no—
18C4×D3×D4—22no—
19C4×D3×D4—22no—
20C2×#893C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#67⋊C4C412yes—
23C2×D3×C22⋊C4C412yes—
24C2×C4×D12C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C3×#423C2212yescomplemented
27C2×C3×#100C2212yes—
28C2×#833C2212yes—
29C2×#823C2214yes—
30#5126C2214yes—
31Dic3×M16C2214yescomplemented
32#4924C2212yescomplemented
33#4835—22no—
34#4835—22no—
35#4833—22no—
36#4833—22no—
37C3×#343C2212yescomplemented
38#4554C2214yescomplemented
39#4530C2212yescomplemented
40C22×C4×D4—31no—
41#2920—32no—
42#1821—32no—
43#1800—32no—
44#1800—31no—
45#1800D311yescharacteristic, complemented
46#1783—32no—
47C4×#100—32no—
48C4×#100—32no—
49C24×D3C2×C412yes—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4C2×C412yes—
52C22×#67C2×C412yes—
53C23×Dic3C2×C412yes—
54C2×D3×D4—22no—
55C2×D3×D4—22no—
56C2×D3×D4—24no—
57C2×D3×D4—22no—
58C2×D3×D4—22no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—48no—
65C22×C4×D3C2×C411yescharacteristic
66C22×C4×D3C2×C412yes—
67C22×C4×D3C2×C412yes—
68C22×C4×D3C2×C412yes—
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C2×C3×M16C2314yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4C2×C412yes—
76C2×C3×C42C2311yescharacteristic
77C2×#217C2×C412yes—
78D4×Dic3—22no—
79D4×Dic3—22no—
80C4×#67—24no—
81C4×#67—24no—
82C2×(C2×D3)⋊C4C2×C412yes—
83C2×(C2×D3)⋊C4C2×C412yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4C2×C412yes—
86C2×C4×Dic3C2×C411yescharacteristic
87C2×C4×Dic3C2×C411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C2×#208C2314yes—
90#893—22no—
91#893—22no—
92D3×C4⋊C4—22no—
93D3×C4⋊C4—22no—
94#67⋊C4—24no—
95#67⋊C4—24no—
96D3×C22⋊C4—24no—
97D3×C22⋊C4—24no—
98C4×D12—22no—
99C4×D12—22no—
100C42×D3D412yes—
101C42×D3D412yes—
102C42×D3D412yes—
103C42×D3D412yes—
104C3×C4⋊C8—22no—
105C3×C4⋊C8—22no—
106C3×#100—24no—
107#833—24no—
108#823—28no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—24no—
111#815—24no—
112C8×Dic3—24no—
113C4⋊(C3⋊C8)—22no—
114C4⋊(C3⋊C8)—22no—
115C23×D4—31no—
116C24×C4—32no—
117C2×C4×D4—31no—
118C2×C4×D4—32no—
119C2×C4×D4—31no—
120C2×C4×D4—32no—
121C2×C4×D4—31no—
122C2×C4×D4—62no—
123C2×C4×D4—62no—
124C2×C4×D4—62no—
125C2×C4×D4—62no—
126C2×C4×D4C2×D311yescharacteristic
127C22×C4⋊C4—31no—
128C22×C22⋊C4—32no—
129C22×C42—31no—
130#432—62no—
131#432—62no—
132#431—62no—
133#431—62no—
134#423—32no—
135#423C2×D312yescomplemented
136C2×#100—32no—
137C2×#100—34no—
138C2×#100C2×D312yes—
139#407—34no—
140C4×M16—34no—
141#343—34no—
142#343—32no—
143#343C2×D312yescomplemented
144C24×C3C22×C412yes—
145C23×D3—24no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—24no—
149C23×D3—44no—
150C23×D3—44no—
151C23×D3C22×C411yescharacteristic
152C23×D3C22×C412yes—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4#4112yes—
156C2×C3×D4#4112yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—44no—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C22×C412yes—
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—24no—
165C2×#67—24no—
166C2×#67—24no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—44no—
170C22×Dic3C22×C411yescharacteristic
171C22×Dic3C22×C411yescharacteristic
172C22×Dic3C22×C412yes—
173C22×Dic3C22×C411yescharacteristic
174D3×D4—28no—
175D3×D4—28no—
176D3×D4—28no—
177D3×D4—28no—
178C2×D12—22no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12#4114yes—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—22no—
185C2×C4×D3—24no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—42no—
193C2×C4×D3—48no—
194C2×C4×D3—48no—
195C2×C4×D3—48no—
196C2×C4×D3—48no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×C4×D3—44no—
200C2×C4×D3C22⋊C412yes—
201C2×C4×D3C22⋊C411yescharacteristic
202C2×C4×D3C22⋊C411yescharacteristic
203C2×C4×D3C22⋊C411yescharacteristic
204C2×C4×D3C22⋊C412yes—
205C2×C4×D3C22⋊C411yescharacteristic
206C2×C4×D3C2×D412yes—
207C2×C4×D3C2×D412yes—
208C3×M16—24no—
209C3×M16—24no—
210C2×C3×C8—24no—
211C3×C4⋊C4—22no—
212C3×C22⋊C4—24no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215#217—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218C4⋊Dic3—22no—
219Dic3⋊C4—24no—
220C4×Dic3C22⋊C412yes—
221C4×Dic3C22⋊C412yes—
222C4×Dic3C22⋊C412yes—
223C4×Dic3C22⋊C412yes—
224C4×Dic3C2×D412yes—
225C4×Dic3C2×D412yes—
226#208—24no—
227#208—24no—
228C2×C3⋊C8—24no—
229C25—32no—
230C22×D4—32no—
231C22×D4—31no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—64no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4C22×D311yescharacteristic
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—32no—
242C23×C4—32no—
243C23×C4—32no—
244C23×C4—32no—
245C23×C4—62no—
246C23×C4—62no—
247C23×C4—62no—
248C23×C4—62no—
249C23×C4—128no—
250C23×C4C4×D312yes—
251C2×M16—34no—
252C2×M16C22×D314yes—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—64no—
256C4×D4—64no—
257C4×D4—62no—
258C4×D4—62no—
259C4×D4—64no—
260C4×D4—64no—
261C4×D4—62no—
262C4×D4—62no—
263C4×D4—62no—
264C4×D4—62no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C22⋊C4—32no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4C4×D312yes—
276C2×C42—31no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42—32no—
280C2×C42—32no—
281C2×C42—32no—
282C2×C42—32no—
283C2×C42C22×D311yescharacteristic
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—62no—
287C4⋊C8—62no—
288#100—24no—
289#100—68no—
290#100—64no—
291C22⋊C8—64no—
292C22⋊C8—64no—
293#97—64no—
294C4×C8—64no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3#11714yes—
300C23×C3C23×C411yescharacteristic
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—28no—
306C22×D3—28no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—44no—
310C22×D3—416no—
311C22×D3—416no—
312C22×D3—44no—
313C22×D3—44no—
314C22×D3—44no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3C2×C22⋊C411yescharacteristic
318C22×D3C2×C22⋊C411yescharacteristic
319C22×D3#11714yes—
320C3×D4—24no—
321C3×D4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4C2×C22⋊C412yes—
329C2×C3×C4C2×C22⋊C411yescharacteristic
330C2×C3×C4C2×C22⋊C411yescharacteristic
331C2×C3×C4C22×D412yes—
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4112yes—
335#67—216no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—41no—
341C2×Dic3—44no—
342C2×Dic3—44no—
343C2×Dic3—42no—
344C2×Dic3C2×C22⋊C412yes—
345C2×Dic3C2×C22⋊C412yes—
346C2×Dic3C2×C22⋊C411yescharacteristic
347C2×Dic3C2×C22⋊C411yescharacteristic
348D12—24no—
349D12—24no—
350C4×D3—22no—
351C4×D3—24no—
352C4×D3—22no—
353C4×D3—22no—
354C4×D3—22no—
355C4×D3—22no—
356C4×D3—22no—
357C4×D3—22no—
358C4×D3—42no—
359C4×D3—44no—
360C4×D3—44no—
361C4×D3—44no—
362C4×D3—44no—
363C4×D3—44no—
364C4×D3—44no—
365C4×D3#10014yescomplemented
366C3×C8—44no—
367C3⋊C8—44no—
368C24—31no—
369C24—32no—
370C24—62no—
371C24—62no—
372C24—64no—
373C24—64no—
374C24—124no—
375C24—124no—
376C24C2×C4×D312yes—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—34no—
380C2×D4—64no—
381C2×D4—68no—
382C2×D4—68no—
383C2×D4—68no—
384C2×D4—68no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4#23712yes—
391C2×D4#23912yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—32no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—44no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—124no—
420C22×C4—128no—
421C22×C4—122no—
422C22×C4—128no—
423C22×C4—128no—
424C22×C4—128no—
425C22×C4—124no—
426C22×C4—124no—
427C22×C4—124no—
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×C4×D312yes—
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C23×D311yescharacteristic
432M16—24no—
433M16—24no—
434M16—64no—
435M16—64no—
436C2×C8—24no—
437C2×C8—64no—
438C4⋊C4—22no—
439C4⋊C4—64no—
440C4⋊C4—62no—
441C22⋊C4—24no—
442C22⋊C4—64no—
443C22⋊C4—64no—
444C22⋊C4—64no—
445C42—32no—
446C42—32no—
447C42—32no—
448C42—32no—
449C42—32no—
450C42—32no—
451C42D3×D412yes—
452C42D3×D412yes—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—24no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3C22×C22⋊C411yescharacteristic
461C22×C3C2×#11711yescharacteristic
462C22×C3C2×#11711yescharacteristic
463C2×D3—24no—
464C2×D3—24no—
465C2×D3—21no—
466C2×D3—21no—
467C2×D3—21no—
468C2×D3—21no—
469C2×D3—24no—
470C2×D3—416no—
471C2×D3—416no—
472C2×D3—44no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—44no—
476C2×D3C2×#10012yes—
477C2×D3#41312yes—
478C3×C4—21no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4C2×#10012yes—
482C3×C4#52512yes—
483C3×C4#52512yes—
484Dic3—21no—
485Dic3—22no—
486Dic3—22no—
487Dic3—42no—
488Dic3C2×#10012yes—
489C23—22no—
490C23—22no—
491C23—31no—
492C23—31no—
493C23—34no—
494C23—42no—
495C23—42no—
496C23—62no—
497C23—62no—
498C23—68no—
499C23—68no—
500C23—61no—
501C23—61no—
502C23—62no—
503C23—62no—
504C23—122no—
505C23—122no—
506C23—124no—
507C23—124no—
508C23—124no—
509C23—1216no—
510C23—1216no—
511C23—124no—
512C23#89214yes—
513C23C22×C4×D311yescharacteristic
514D4—24no—
515D4—24no—
516D4—616no—
517D4—64no—
518D4—64no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—31no—
522C2×C4—34no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—31no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—122no—
543C2×C4—124no—
544C2×C4—124no—
545C2×C4—124no—
546C2×C4—124no—
547C2×C4—124no—
548C2×C4—124no—
549C2×C4—121no—
550C2×C4—124no—
551C2×C4—124no—
552C2×C4—122no—
553C2×C4D3×C22⋊C412yes—
554C2×C4D3×C22⋊C411yescharacteristic
555C2×C4D3×C22⋊C411yescharacteristic
556C2×C4C2×D3×D412yes—
557C2×C4C2×#23711yescharacteristic
558C2×C4C2×#23911yescharacteristic
559C2×C4D3×#4112yes—
560C8—44no—
561C8—124no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—44no—
565C2×C3#228011yescharacteristic
566C2×C3C22×#10011yescharacteristic
567C2×C3C2×#41311yescharacteristic
568D3—22no—
569D3—22no—
570D3—44no—
571C22—21no—
572C22—21no—
573C22—24no—
574C22—32no—
575C22—32no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—64no—
581C22—64no—
582C22—61no—
583C22—61no—
584C22—61no—
585C22—61no—
586C22—64no—
587C22—124no—
588C22—124no—
589C22—1216no—
590C22—1216no—
591C22—122no—
592C22—122no—
593C22C2×D3×C22⋊C411yescharacteristic
594C22C2×#89211yescharacteristic
595C22D3×#11711yescharacteristic
596C4—21no—
597C4—22no—
598C4—32no—
599C4—42no—
600C4—61no—
601C4—62no—
602C4—62no—
603C4—122no—
604C4D3×#10012yes—
605C4#560612yes—
606C4#562712yes—
607C3#2072611yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—44no—
611C2—62no—
612C2—62no—
613C2—124no—
614C2#7386211yescharacteristic
615C2C2×D3×#10011yescharacteristic
616C2D3×#41311yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1800C2
socleC22×C3C22×C22⋊C4
prime radicalC22×D3×D4C22
cosocleC22×C3×C4C24
hypercenter#1800D3
hypocenterC3#20726
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #79038C2semidirect
3. C3×#1800C2semidirect
4. #71617C2semidirect
5. #70527C2semidirect
7. Dic3×#100C2semidirect
8. #68990C2semidirect
9. #68972C2semidirect
26. C3×#423C22semidirect
31. Dic3×M16C22semidirect
32. #4924C22semidirect
37. C3×#343C22semidirect
38. #4554C22semidirect
39. #4530C22semidirect
45. #1800D3semidirect
135. #423C2×D3semidirect
143. #343C2×D3semidirect
365. C4×D3#100semidirect
607. C3#20726semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D42
#790382
C3×#18002
#716172
#705272
#700792
Dic3×#1002
#689902
#689722
#207263