Finite Group Databasebeta

← D3×#1552 #10763554

Subgroups — D3×#1552 #10763554

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#533C211yesmaximal, characteristic
2D3×#432C211yesmaximal, characteristic
3C2×D3×C4⋊C8C211yesmaximal, characteristic
4C3×#1552C211yesmaximal, characteristic, complemented
5#71831C211yesmaximal, characteristic, complemented
6#70820C211yesmaximal, characteristic, complemented
7#70439C211yesmaximal, characteristic, complemented
8#68425C212yesmaximal, complemented
9#68424C212yesmaximal, complemented
10D3×#332C212yesmaximal, complemented
11D3×#332C212yesmaximal, complemented
12C2×#1552—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C22×D3×C8—21no—
15#5728C2211yescharacteristic
16#5728C2211yescharacteristic
17D3×C4⋊Q8C2211yescharacteristic
18D3×C4⋊Q8C2211yescharacteristic
19#5688C412yes—
20D3×#125C412yes—
21D3×C22⋊Q8—21no—
22D3×C22⋊Q8—21no—
23#5622—21no—
24#5622—21no—
25#5578—21no—
26#5578—21no—
27C4×D3×Q8—21no—
28C4×D3×Q8—21no—
29D3×#117C412yes—
30#5517C412yes—
31C2×C42×D3C2211yescharacteristic
32#5470C2211yescharacteristic
33C3×#432C2211yescharacteristic
34C2×C3×C4⋊C8C2211yescharacteristic
35C2×Dic3⋊C8—21no—
36#4926C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38#4835C2212yes—
39D3×C4⋊C8—21no—
40D3×C4⋊C8—21no—
41D3×C4⋊C8C2212yes—
42D3×C4⋊C8C2212yes—
43D3×C4⋊C8C2212yes—
44D3×C22⋊C8—21no—
45#4719—21no—
46#4705—21no—
47D3×#97—21no—
48C3×#332C2212yescomplemented
49C3×#332C2212yescomplemented
50#4547C2212yescomplemented
51#4547C2212yescomplemented
52#4486C2212yescomplemented
53#4486C2212yescomplemented
54#4482C2212yescomplemented
55#4482C2212yescomplemented
56C2×#533—31no—
57C2×#432—31no—
58C22×C4⋊C8—31no—
59#1552—31no—
60#1552—31no—
61#1552—32no—
62#1552—32no—
63#1552—31no—
64#1552D311yescharacteristic, direct product factor
65C2×#332—32no—
66C2×#332—32no—
67C2×D3×Q8—21no—
68C2×D3×Q8—21no—
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C3×C8—21no—
73C3×C4⋊Q8C2311yescharacteristic
74C3×C4⋊Q8C2311yescharacteristic
75C3×#125C2×C412yes—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8—21no—
78C3×C4×Q8—21no—
79C3×C4×Q8—21no—
80C3×#117C2×C412yes—
81C2×C3×C42C2311yescharacteristic
82#946—21no—
83#946—21no—
84Q8×Dic3—21no—
85Q8×Dic3—21no—
86#944—21no—
87#944—21no—
88#926C2×C412yes—
89#924—21no—
90#924—21no—
91C2×C4×Dic3C2×C411yescharacteristic
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C22×C3⋊C8—21no—
95C2×D3×C8—22no—
96C2×D3×C8—22no—
97C2×D3×C8—22no—
98C2×D3×C8—44no—
99#897—21no—
100#897—21no—
101#896—22no—
102#896—22no—
103#892C2×C412yes—
104#892C2×C412yes—
105D3×C4⋊C4—22no—
106D3×C4⋊C4—22no—
107D3×C4⋊C4—21no—
108D3×C4⋊C4—21no—
109D3×C4⋊C4C2×C412yes—
110D3×C4⋊C4C2×C412yes—
111#890C2×C412yes—
112#890C2×C412yes—
113#889—22no—
114#889—22no—
115#888—21no—
116#888—21no—
117#888C2×C412yes—
118#888C2×C412yes—
119#887—22no—
120#887—22no—
121#887—21no—
122#887—21no—
123D3×C22⋊C4—22no—
124#878—22no—
125#878—22no—
126#877—22no—
127#872—22no—
128C42×D3—22no—
129C42×D3C2×C411yescharacteristic
130C42×D3C2×C411yescharacteristic
131C42×D3D411yescharacteristic
132C42×D3D411yescharacteristic
133C42×D3D411yescharacteristic
134C42×D3D411yescharacteristic
135C42×D3C2311yescharacteristic
136C42×D3C2311yescharacteristic
137#870C2×C412yes—
138#869C2311yescharacteristic
139#869C2311yescharacteristic
140C4×Dic6—21no—
141C4×Dic6—21no—
142C4×Dic6—22no—
143C4×Dic6—22no—
144C3×C4⋊C8—21no—
145C3×C4⋊C8C2312yes—
146C3×C4⋊C8C2312yes—
147C3×C22⋊C8—21no—
148C3×#97—21no—
149#830—21no—
150(C2×D3)⋊C8—22no—
151#815—22no—
152Dic3⋊C8—22no—
153Dic3⋊C8—22no—
154Dic3⋊C8—22no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2312yes—
157C4⋊(C3⋊C8)C2312yes—
158#803—21no—
159C23×C8—61no—
160#533—31no—
161#533—31no—
162#533—32no—
163#533—32no—
164#533—31no—
165#533—61no—
166#533—61no—
167#533—61no—
168#533—61no—
169#533C2×D311yescharacteristic
170C2×C4⋊Q8—31no—
171C2×C4⋊Q8—31no—
172C2×#125—32no—
173C2×C22⋊Q8—61no—
174C2×C22⋊Q8—61no—
175C2×C4×Q8—61no—
176C2×C4×Q8—61no—
177C2×#117—32no—
178C22×C42—31no—
179#432—31no—
180#432—32no—
181#432—61no—
182#432—61no—
183#432C2×D311yescharacteristic
184C2×C4⋊C8—31no—
185C2×C4⋊C8—32no—
186C2×C4⋊C8—32no—
187C2×C4⋊C8—32no—
188C2×C4⋊C8—61no—
189C2×C4⋊C8—61no—
190C2×C4⋊C8—61no—
191C2×C4⋊C8C2×D311yescharacteristic
192C2×C22⋊C8—61no—
193C2×#97—61no—
194#332—32no—
195#332—32no—
196#332—32no—
197#332—32no—
198#332—32no—
199#332—32no—
200#332C2×D312yescomplemented
201#332C2×D312yescomplemented
202C23×D3C22⋊C411yescharacteristic
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C22×C3×C4C22×C411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×Dic3C22⋊C411yescharacteristic
209C22×Dic3C22⋊C411yescharacteristic
210C22×Dic3C22⋊C411yescharacteristic
211D3×Q8—44no—
212D3×Q8—42no—
213D3×Q8—42no—
214C2×C4×D3—24no—
215C2×C4×D3—24no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3C22⋊C411yescharacteristic
221C2×C4×D3C22⋊C411yescharacteristic
222C2×C4×D3C22⋊C411yescharacteristic
223C2×C4×D3C22⋊C411yescharacteristic
224C2×C4×D3C22⋊C411yescharacteristic
225C2×C4×D3C22⋊C411yescharacteristic
226C2×C4×D3C22×C411yescharacteristic
227C2×C4×D3C22×C411yescharacteristic
228C2×C4×D3C2×D411yescharacteristic
229C2×C4×D3C2×D411yescharacteristic
230C2×C4×D3C2×D411yescharacteristic
231C2×C4×D3C2×D411yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C2×C3×C8—42no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C22×C412yes—
244C3×C4⋊C4C22×C412yes—
245C3×C22⋊C4—22no—
246C3×C42—22no—
247C3×C42C22×C411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42C2411yescharacteristic
251#217—22no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—22no—
257C4⋊Dic3—22no—
258C4⋊Dic3C22×C412yes—
259C4⋊Dic3C22×C412yes—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262Dic3⋊C4—22no—
263Dic3⋊C4—22no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268C4×Dic3—22no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3C22⋊C411yescharacteristic
272C4×Dic3C22⋊C411yescharacteristic
273C4×Dic3C22⋊C411yescharacteristic
274C4×Dic3C22⋊C411yescharacteristic
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22×C411yescharacteristic
278C4×Dic3C22×C411yescharacteristic
279C4×Dic3C2×D411yescharacteristic
280C4×Dic3C2×D411yescharacteristic
281C4×Dic3C2×D411yescharacteristic
282C4×Dic3C2×D411yescharacteristic
283C2×C3⋊C8—22no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8—42no—
286D3×C8—44no—
287D3×C8—44no—
288D3×C8—44no—
289C22×Q8—61no—
290C22×Q8—61no—
291C23×C4—31no—
292C23×C4—31no—
293C23×C4—31no—
294C22×C8—21no—
295C22×C8—62no—
296C22×C8—62no—
297C22×C8—62no—
298C22×C8—61no—
299C22×C8—124no—
300C4⋊Q8—32no—
301C4⋊Q8—32no—
302C4⋊Q8—31no—
303C4⋊Q8—31no—
304C4⋊Q8—61no—
305C4⋊Q8—61no—
306C4⋊Q8—61no—
307C4⋊Q8—61no—
308C4⋊Q8C22×D311yescharacteristic
309C4⋊Q8C22×D311yescharacteristic
310#125—32no—
311#125—32no—
312#125—32no—
313#125—62no—
314#125—62no—
315#125C4×D312yes—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—62no—
319C22⋊Q8—62no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322C22⋊Q8—61no—
323C22⋊Q8—61no—
324C22⋊Q8—61no—
325C22⋊Q8—61no—
326C22⋊Q8—61no—
327C22⋊Q8—61no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—62no—
331C4×Q8—62no—
332C4×Q8—62no—
333C4×Q8—62no—
334C4×Q8—61no—
335C4×Q8—61no—
336C4×Q8—61no—
337C4×Q8—61no—
338C4×Q8—61no—
339C4×Q8—61no—
340#117—32no—
341#117—32no—
342#117—32no—
343#117—62no—
344#117—62no—
345#117C4×D312yes—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—62no—
349C2×C4⋊C4—62no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C22⋊C4—62no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—62no—
365C2×C42C22×D311yescharacteristic
366C4⋊C8—21no—
367C4⋊C8—32no—
368C4⋊C8—32no—
369C4⋊C8—62no—
370C4⋊C8—62no—
371C4⋊C8—62no—
372C4⋊C8—61no—
373C4⋊C8C22×D312yes—
374C4⋊C8C22×D312yes—
375C22⋊C8—21no—
376C22⋊C8—62no—
377C22⋊C8—61no—
378#97—21no—
379#97—62no—
380#97—61no—
381C23×C3C2×C22⋊C411yescharacteristic
382C22×D3—22no—
383C22×D3—22no—
384C22×D3#10114yes—
385C22×D3C2×C22⋊C411yescharacteristic
386C22×D3C2×C22⋊C411yescharacteristic
387C3×Q8—42no—
388C3×Q8—41no—
389C3×Q8—41no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4C2×C22⋊C411yescharacteristic
397C2×C3×C4C2×C22⋊C411yescharacteristic
398C2×C3×C4C2×C22⋊C411yescharacteristic
399C2×C3×C4C23×C411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3C2×C22⋊C411yescharacteristic
408C2×Dic3C2×C22⋊C411yescharacteristic
409C2×Dic3C2×C22⋊C411yescharacteristic
410C2×Dic3C2×C22⋊C411yescharacteristic
411C2×Dic3C2×C22⋊C411yescharacteristic
412C2×Dic3C2×C22⋊C411yescharacteristic
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—22no—
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—21no—
421C4×D3—21no—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—44no—
425C4×D3—44no—
426C4×D3#13514yes—
427C4×D3AGL(1,ℤ/8)12yes—
428C4×D3#13712yes—
429Dic6—42no—
430Dic6—41no—
431Dic6—41no—
432Dic6—44no—
433Dic6—44no—
434C3×C8—42no—
435C3×C8—42no—
436C3⋊C8—42no—
437C3⋊C8—42no—
438C24—31no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—62no—
442C2×Q8—62no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—124no—
446C2×Q8—122no—
447C2×Q8—122no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—62no—
464C22×C4—64no—
465C22×C4—62no—
466C22×C4—64no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4C2×C4×D311yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4D3×D411yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—42no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—124no—
478C2×C8—124no—
479C2×C8—124no—
480C2×C8—122no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—32no—
486C4⋊C4—32no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4C2×C4×D312yes—
500C4⋊C4C2×C4×D312yes—
501C22⋊C4—22no—
502C22⋊C4—62no—
503C22⋊C4—62no—
504C22⋊C4—62no—
505C42—22no—
506C42—31no—
507C42—31no—
508C42—31no—
509C42—31no—
510C42—31no—
511C42—31no—
512C42—31no—
513C42—31no—
514C42—31no—
515C42—31no—
516C42—31no—
517C42—31no—
518C42—62no—
519C42—62no—
520C42—62no—
521C42C2×C4×D311yescharacteristic
522C42D3×D411yescharacteristic
523C42D3×D411yescharacteristic
524C42C23×D311yescharacteristic
525C22×C3—21no—
526C22×C3—21no—
527C22×C3C2×#10112yes—
528C22×C3C22×C22⋊C411yescharacteristic
529C2×D3—21no—
530C2×D3—21no—
531C2×D3—28no—
532C2×D3C2×#10112yes—
533C2×D3C2×#10112yes—
534C2×D3#41612yes—
535C2×D3#41712yes—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4C2×#13512yes—
543C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
544C3×C4C2×#13711yescharacteristic
545Dic3—21no—
546Dic3—21no—
547Dic3—21no—
548Dic3—21no—
549Dic3—42no—
550Dic3—42no—
551Dic3#41612yes—
552Dic3#41712yes—
553C23—31no—
554C23—31no—
555C23—34no—
556C23—62no—
557C23—62no—
558C23D3×C22⋊C411yescharacteristic
559Q8—42no—
560Q8—41no—
561Q8—41no—
562Q8—124no—
563Q8—124no—
564Q8—122no—
565Q8—121no—
566Q8—121no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—31no—
574C2×C4—31no—
575C2×C4—34no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—31no—
579C2×C4—31no—
580C2×C4—31no—
581C2×C4—31no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—124no—
599C2×C4—124no—
600C2×C4D3×C22⋊C411yescharacteristic
601C2×C4D3×C22⋊C411yescharacteristic
602C2×C4D3×C22⋊C411yescharacteristic
603C2×C4C22×C4×D311yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×D3×D411yescharacteristic
606C8—42no—
607C8—42no—
608C8—122no—
609C8—122no—
610C2×C3—22no—
611C2×C3C22×#10111yescharacteristic
612C2×C3C2×#41611yescharacteristic
613C2×C3C2×#41711yescharacteristic
614D3—22no—
615D3#155214yesdirect product factor
616C22—21no—
617C22—21no—
618C22—32no—
619C22—32no—
620C22—32no—
621C22—32no—
622C22—68no—
623C22—61no—
624C22—61no—
625C22D3×#10112yes—
626C22C2×D3×C22⋊C411yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—32no—
632C4—32no—
633C4—42no—
634C4—42no—
635C4—61no—
636C4—61no—
637C4—61no—
638C4—61no—
639C4—122no—
640C4—122no—
641C4D3×#13512yes—
642C4D3×AGL(1,ℤ/8)11yescharacteristic
643C4D3×#13711yescharacteristic
644C3C2×#155211yescharacteristic, complemented, Sylow
645C2—22no—
646C2—34no—
647C2—62no—
648C2C2×D3×#10111yescharacteristic
649C2D3×#41611yescharacteristic
650C2D3×#41711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1552C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1552D3
hypocenterC3C2×#1552
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
4. C3×#1552C2semidirect
5. #71831C2semidirect
6. #70820C2semidirect
7. #70439C2semidirect
8. #68425C2semidirect
9. #68424C2semidirect
10. D3×#332C2semidirect
11. D3×#332C2semidirect
48. C3×#332C22semidirect
49. C3×#332C22semidirect
50. #4547C22semidirect
51. #4547C22semidirect
52. #4486C22semidirect
53. #4486C22semidirect
54. #4482C22semidirect
55. #4482C22semidirect
64. #1552D3direct (N × H)
200. #332C2×D3semidirect
201. #332C2×D3semidirect
615. D3#1552direct (N × H)
644. C3C2×#1552semidirect

Maximal subgroups

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

subgroupindex
D3×#5332
D3×#4322
C2×D3×C4⋊C82
C3×#15522
#718312
#708202
#704392
#684252
#684242
D3×#3322
C2×#15523