Finite Group Databasebeta

← #10763287

Subgroups — #10763287

Order 768.

952 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#533C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic
3D3×#432C211yesmaximal, characteristic
4C3×#1537C211yesmaximal, characteristic, complemented
5#71823C211yesmaximal, characteristic, complemented
6#70815C211yesmaximal, characteristic, complemented
7#70815C211yesmaximal, characteristic, complemented
8#68365C214yesmaximal, complemented
9#68363C214yesmaximal, complemented
10#17672—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12#5728C411yescharacteristic
13#5728C411yescharacteristic
14D3×C4⋊Q8C411yescharacteristic
15D3×C4⋊Q8C411yescharacteristic
16#5688C2212yes—
17D3×#125C2212yes—
18D3×C22⋊Q8—21no—
19D3×C22⋊Q8—21no—
20#5622—21no—
21#5622—21no—
22#5578—21no—
23#5578—21no—
24C4×D3×Q8—21no—
25C4×D3×Q8—21no—
26D3×#117C412yes—
27#5517C412yes—
28C2×C42×D3C2211yescharacteristic
29#5470C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C3×#432C2211yescharacteristic
32#4926C2211yescharacteristic
33#4926C2211yescharacteristic
34#4837—21no—
35#4835—21no—
36#4833—21no—
37D3×C4⋊C8—21no—
38#4726—21no—
39D3×C22⋊C8—21no—
40#4719—21no—
41#4718—21no—
42#4707C2212yes—
43#4705C2212yes—
44#4705C2212yes—
45D3×#97C2212yes—
46C3×#330C2214yescomplemented
47#4543C2214yescomplemented
48#4465C2214yescomplemented
49#4465C2214yescomplemented
50C2×#533—31no—
51#2920—31no—
52C2×#432—31no—
53#1537—31no—
54#1537—31no—
55#1537—31no—
56#1537D311yescharacteristic, complemented
57#1531—34no—
58#1528—34no—
59C2×D3×Q8—21no—
60C2×D3×Q8—21no—
61C22×C4×D3C2×C411yescharacteristic
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C3×C4⋊Q8C2×C411yescharacteristic
65C3×C4⋊Q8C2×C411yescharacteristic
66C3×#125C2312yes—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8—21no—
71C3×#117C2×C412yes—
72C2×C3×C42C2311yescharacteristic
73#946—21no—
74#946—21no—
75Q8×Dic3—21no—
76Q8×Dic3—21no—
77#944—21no—
78#944—21no—
79#926C2×C412yes—
80#924—21no—
81#924—21no—
82C2×C4×Dic3C2×C411yescharacteristic
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×ΓL(1,𝔽25)—22no—
86C2×D3×C8—22no—
87#897—21no—
88#897—21no—
89#896—22no—
90#896—22no—
91#892C2×C412yes—
92#892C2×C412yes—
93D3×C4⋊C4—21no—
94D3×C4⋊C4—21no—
95D3×C4⋊C4—22no—
96D3×C4⋊C4—22no—
97D3×C4⋊C4C2×C412yes—
98D3×C4⋊C4C2×C412yes—
99#890C2×C412yes—
100#890C2×C412yes—
101#889—22no—
102#889—22no—
103#888—21no—
104#888—21no—
105#888C2×C412yes—
106#888C2×C412yes—
107#887—21no—
108#887—21no—
109#887—22no—
110#887—22no—
111D3×C22⋊C4—22no—
112#878—22no—
113#878—22no—
114#877—22no—
115#872—22no—
116C42×D3—22no—
117C42×D3C2×C411yescharacteristic
118C42×D3C2×C411yescharacteristic
119C42×D3D411yescharacteristic
120C42×D3D411yescharacteristic
121C42×D3D411yescharacteristic
122C42×D3D411yescharacteristic
123C42×D3C2311yescharacteristic
124C42×D3C2311yescharacteristic
125#870C2312yes—
126#869C2×C411yescharacteristic
127#869C2×C411yescharacteristic
128C4×Dic6—21no—
129C4×Dic6—21no—
130C4×Dic6—22no—
131C4×Dic6—22no—
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×C22⋊C8—21no—
135C3×C22⋊C8—21no—
136C3×#97C2312yes—
137C3×#97C2312yes—
138#830—21no—
139#830—21no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148#803C2312yes—
149#803C2312yes—
150#533—31no—
151#533—31no—
152#533—32no—
153#533—32no—
154#533—31no—
155#533—61no—
156#533—61no—
157#533—61no—
158#533—61no—
159#533C2×D311yescharacteristic
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×#125—32no—
163C2×C22⋊Q8—61no—
164C2×C22⋊Q8—61no—
165C2×C4×Q8—61no—
166C2×C4×Q8—61no—
167C2×#117—32no—
168C22×C42—31no—
169#432—31no—
170#432—31no—
171#432—32no—
172#432—32no—
173#432—61no—
174#432—61no—
175#432C2×D311yescharacteristic
176#432C2×D311yescharacteristic
177#431—61no—
178#431—61no—
179#423—61no—
180C2×C4⋊C8—61no—
181#407—61no—
182C2×C22⋊C8—61no—
183C4×M16—32no—
184C2×#97—32no—
185#330—34no—
186#330—34no—
187#330—34no—
188#330C2×D314yescomplemented
189C23×D3C22⋊C411yescharacteristic
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×Dic3C22⋊C411yescharacteristic
196C22×Dic3C22⋊C411yescharacteristic
197C22×Dic3C22⋊C411yescharacteristic
198D3×Q8—44no—
199D3×Q8—44no—
200C2×C4×D3—24no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3C22⋊C411yescharacteristic
206C2×C4×D3C22⋊C411yescharacteristic
207C2×C4×D3C22⋊C411yescharacteristic
208C2×C4×D3C22⋊C411yescharacteristic
209C2×C4×D3C22⋊C411yescharacteristic
210C2×C4×D3C22⋊C411yescharacteristic
211C2×C4×D3C22×C411yescharacteristic
212C2×C4×D3C22×C411yescharacteristic
213C2×C4×D3C2×D411yescharacteristic
214C2×C4×D3C2×D411yescharacteristic
215C2×C4×D3C2×D411yescharacteristic
216C2×C4×D3C2×D411yescharacteristic
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4C22×C412yes—
228C3×C4⋊C4C22×C412yes—
229C3×C22⋊C4—22no—
230C3×C42—22no—
231C3×C42C22×C411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2411yescharacteristic
235#217—22no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3C22×C412yes—
243C4⋊Dic3C22×C412yes—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252C4×Dic3—22no—
253C4×Dic3—22no—
254C4×Dic3—22no—
255C4×Dic3C22⋊C411yescharacteristic
256C4×Dic3C22⋊C411yescharacteristic
257C4×Dic3C22⋊C411yescharacteristic
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22×C411yescharacteristic
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269ΓL(1,𝔽25)—44no—
270D3×C8—44no—
271C22×Q8—61no—
272C22×Q8—61no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—31no—
276C2×M16—62no—
277C22×C8—62no—
278C4⋊Q8—32no—
279C4⋊Q8—32no—
280C4⋊Q8—31no—
281C4⋊Q8—31no—
282C4⋊Q8—61no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286C4⋊Q8C4×D311yescharacteristic
287C4⋊Q8C4×D311yescharacteristic
288#125—32no—
289#125—32no—
290#125—32no—
291#125—62no—
292#125—62no—
293#125C22×D312yes—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—62no—
297C22⋊Q8—62no—
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—61no—
301C22⋊Q8—61no—
302C22⋊Q8—61no—
303C22⋊Q8—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—61no—
311C4×Q8—61no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×Q8—61no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318#117—32no—
319#117—32no—
320#117—32no—
321#117—62no—
322#117—62no—
323#117C4×D312yes—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—62no—
327C2×C4⋊C4—62no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C22⋊C4—62no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—62no—
343C2×C42C22×D311yescharacteristic
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—62no—
347C4⋊C8—62no—
348C4⋊C8—61no—
349C4⋊C8—61no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354C22⋊C8—61no—
355C22⋊C8—61no—
356#97—32no—
357#97—32no—
358#97—62no—
359#97C22×D312yes—
360#97C22×D312yes—
361C4×C8—62no—
362C23×C3C2×C22⋊C411yescharacteristic
363C22×D3—22no—
364C22×D3—22no—
365C22×D3—41no—
366C22×D3C2×C22⋊C411yescharacteristic
367C22×D3C2×C22⋊C411yescharacteristic
368C3×Q8—42no—
369C3×Q8—42no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4C2×C22⋊C411yescharacteristic
376C2×C3×C4C2×C22⋊C411yescharacteristic
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C23×C411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—41no—
386C2×Dic3C2×C22⋊C411yescharacteristic
387C2×Dic3C2×C22⋊C411yescharacteristic
388C2×Dic3C2×C22⋊C411yescharacteristic
389C2×Dic3C2×C22⋊C411yescharacteristic
390C2×Dic3C2×C22⋊C411yescharacteristic
391C2×Dic3C2×C22⋊C411yescharacteristic
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—41no—
400C4×D3—41no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—44no—
404C4×D3—44no—
405C4×D3#10114yes—
406Dic6—44no—
407Dic6—42no—
408Dic6—42no—
409Dic6—44no—
410C3×C8—42no—
411C3×C8—42no—
412C3⋊C8—42no—
413C3⋊C8—42no—
414C24—31no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—62no—
418C2×Q8—62no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—124no—
422C2×Q8—124no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4C2×C4×D311yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D411yescharacteristic
446M16—124no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—62no—
450C2×C8—62no—
451C2×C8—124no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—32no—
457C4⋊C4—32no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4C2×C4×D312yes—
471C4⋊C4C2×C4×D312yes—
472C22⋊C4—22no—
473C22⋊C4—62no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C42—22no—
477C42—31no—
478C42—31no—
479C42—31no—
480C42—31no—
481C42—31no—
482C42—31no—
483C42—31no—
484C42—31no—
485C42—31no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—62no—
490C42—62no—
491C42—62no—
492C42C2×C4×D311yescharacteristic
493C42D3×D411yescharacteristic
494C42D3×D411yescharacteristic
495C42C23×D311yescharacteristic
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3C22×C22⋊C411yescharacteristic
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—21no—
503C2×D3—21no—
504C2×D3—44no—
505C2×D3C2×#10112yes—
506C2×D3#42112yes—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4C2×#10112yes—
515Dic3—21no—
516Dic3—21no—
517Dic3—21no—
518Dic3—21no—
519Dic3—42no—
520Dic3—42no—
521Dic3C2×#10112yes—
522Dic3#42112yes—
523C23—31no—
524C23—31no—
525C23—62no—
526C23—62no—
527C23—121no—
528C23D3×C22⋊C411yescharacteristic
529Q8—42no—
530Q8—42no—
531Q8—122no—
532Q8—122no—
533Q8—124no—
534Q8—124no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—31no—
539C2×C4—31no—
540C2×C4—34no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—31no—
544C2×C4—31no—
545C2×C4—41no—
546C2×C4—41no—
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—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—121no—
565C2×C4D3×C22⋊C411yescharacteristic
566C2×C4D3×C22⋊C411yescharacteristic
567C2×C4D3×C22⋊C411yescharacteristic
568C2×C4C22×C4×D311yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C8—42no—
572C8—42no—
573C8—122no—
574C8—122no—
575C2×C3—41no—
576C2×C3C22×#10111yescharacteristic
577C2×C3C2×#42111yescharacteristic
578C2×C3C2×#42111yescharacteristic
579D3—22no—
580D3—22no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—32no—
585C22—32no—
586C22—61no—
587C22—61no—
588C22—61no—
589C22—61no—
590C22—124no—
591C22C2×D3×C22⋊C411yescharacteristic
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—32no—
598C4—32no—
599C4—42no—
600C4—42no—
601C4—61no—
602C4—61no—
603C4—61no—
604C4—61no—
605C4—122no—
606C4—122no—
607C4D3×#10112yes—
608C3#1767211yescharacteristic, complemented, Sylow
609C2—41no—
610C2—62no—
611C2—62no—
612C2C2×D3×#10111yescharacteristic
613C2D3×#42111yescharacteristic
614C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1537C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1537D3
hypocenterC3#17672
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
4. C3×#1537C2semidirect
5. #71823C2semidirect
6. #70815C2semidirect
7. #70815C2semidirect
8. #68365C2semidirect
9. #68363C2semidirect
46. C3×#330C22semidirect
47. #4543C22semidirect
48. #4465C22semidirect
49. #4465C22semidirect
56. #1537D3semidirect
188. #330C2×D3semidirect
608. C3#17672semidirect

Maximal subgroups

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

subgroupindex
D3×#5332
#793312
D3×#4322
C3×#15372
#718232
#708152
#683652
#683632
#176723