Finite Group Databasebeta

← D3×#5229 #11801727

Subgroups — D3×#5229 #11801727

Order 1152.

1949 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D3×Q8C211yesmaximal, characteristic
2C2×D3×C4⋊Dic3C211yesmaximal, characteristic
3C2×C4×D3×Dic3C211yesmaximal, characteristic
4#10592928C211yesmaximal, characteristic, complemented
5C3×#5229C211yesmaximal, characteristic, complemented
6D3×C3⋊#95C214yesmaximal
7#10591261C214yesmaximal, complemented
8#10591221C211yesmaximal, characteristic, complemented
9#10591220C211yesmaximal, characteristic, complemented
10C2×#5229—31nomaximal
11D3×#391—31nomaximal
12C22×C32×Q8C2211yescharacteristic
13C2×C3×D3×Q8—24no—
14C2×C3×D3×Q8—24no—
15C2×C3×D3×Q8C418yescomplemented
16C22×C3×C4×D3C2212yes—
17C22×C3×C4×D3C2211yescharacteristic
18C22×C3×Dic6C2212yes—
19C22×C3×Dic6C2211yescharacteristic
20C22×D3×Dic3C2212yes—
21C22×D3×Dic3C2212yes—
22C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C2×C4×C3⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×Dic3⋊Dic3C2212yes—
27C2×Dic32C2212yes—
28D3×C4⋊Dic3—24no—
29C4×D3×Dic3—24no—
30#63658C2214yes—
31C3×C3⋊#95C2214yes—
32#63579C2214yes—
33#63579C2214yes—
34C23×C3×Q8—31no—
35C22×D3×Q8D311yescharacteristic
36C22×C4⋊Dic3—31no—
37C22×C4×Dic3—31no—
38C2×D3×C4⋊C4—31no—
39C2×C42×D3—31no—
40C3×#391—31no—
41#5229—31no—
42#5229—31no—
43#5229—31no—
44#5229—31no—
45#5229—34no—
46#5229—61no—
47#5229D311yescharacteristic, direct product factor
48C2×C3⋊#95—34no—
49#4973—31no—
50#4929—31no—
51D3×#95—34no—
52#4646—34no—
53C23×C3×D3C2311yescharacteristic
54C2×C32×Q8—22no—
55C2×C32×Q8—22no—
56C2×C32×Q8C2×C414yescomplemented
57C22×C32×C4C2312yes—
58C22×C32×C4C2311yescharacteristic
59C22×C3⋊Dic3C2312yes—
60C22×C3⋊Dic3C2312yes—
61C22×C3×Dic3C2312yes—
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2311yescharacteristic
65C3×D3×Q8—232no—
66C2×C3×C4×D3—24no—
67C2×C3×C4×D3—24no—
68C2×C3×C4×D3C2×C418yes—
69C2×C3×C4×D3C2×C414yes—
70C2×C3×C4×D3D414yes—
71C2×C3×C4×D3D414yes—
72C2×C3×Dic6—22no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6—24no—
75C2×C3×Dic6—24no—
76C2×C3×Dic6C2×C414yescomplemented
77C2×C3×Dic6C2×C418yescomplemented
78C2×D3×Dic3—24no—
79C2×D3×Dic3—24no—
80C2×D3×Dic3—24no—
81C2×D3×Dic3—28no—
82C2×D3×Dic3Q818yescomplemented
83C4⋊(C3⋊Dic3)—22no—
84C4×C3⋊Dic3—22no—
85C3×C4⋊Dic3—22no—
86C3×C4×Dic3—22no—
87Dic3⋊Dic3—24no—
88Dic32—24no—
89C2×#391—91noSylow
90C22×C3×Q8—21no—
91C22×C3×Q8—38no—
92C22×C3×Q8—32no—
93C22×C3×Q8—31no—
94C22×C3×Q8—64no—
95C22×C3×Q8—64no—
96C22×C3×Q8C2×D311yescharacteristic
97C22×C3×Q8C2×D311yescharacteristic
98C23×C3×C4—31no—
99C23×C3×C4—32no—
100C23×Dic3—32no—
101C23×Dic3—32no—
102C2×D3×Q8—24no—
103C2×D3×Q8—24no—
104C2×D3×Q8Dic318yescomplemented
105C22×C4×D3—32no—
106C22×C4×D3—32no—
107C22×C4×D3C2×D312yes—
108C22×C4×D3C2×D311yescharacteristic
109C22×Dic6C2×D312yes—
110C22×Dic6C2×D311yescharacteristic
111C2×C3×C4⋊C4—31no—
112C2×C3×C42—31no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—31no—
115C2×C4⋊Dic3—32no—
116C2×C4⋊Dic3—61no—
117C2×C4⋊Dic3—64no—
118C2×C4⋊Dic3C2×D311yescharacteristic
119C2×Dic3⋊C4—32no—
120C2×C4×Dic3—31no—
121C2×C4×Dic3—31no—
122C2×C4×Dic3—32no—
123C2×C4×Dic3—32no—
124C2×C4×Dic3—61no—
125C2×C4×Dic3—64no—
126C2×C4×Dic3C2×D311yescharacteristic
127D3×C4⋊C4—64no—
128C42×D3—64no—
129C3×#95—34no—
130C3⋊#95—34no—
131C3⋊#95—34no—
132C3⋊#95—34no—
133C3⋊#95—34no—
134C3⋊#95—34no—
135C3⋊#95—34no—
136C3⋊#95—64no—
137C3⋊#95C2×D314yes—
138C23×C32C2411yescharacteristic
139C22×C3×D3C22×C412yes—
140C22×C3×D3C2×D412yes—
141C22×C3×D3C2×D412yes—
142C22×C3×D3C2×Q814yes—
143C22×C3×D3#4114yes—
144C32×Q8—28no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4C22×C414yes—
148C2×C32×C4C22×C412yes—
149C2×C32×C4C2×D412yes—
150C2×C32×C4C2×D412yes—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—24no—
155C2×C3⋊Dic3C2×Q814yescomplemented
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3C22×C414yes—
163C2×C3×Dic3C22×C412yes—
164C2×C3×Dic3C2×D412yes—
165C2×C3×Dic3C2×D412yes—
166C2×C3×Dic3C2×Q814yescomplemented
167C3×C4×D3—216no—
168C3×C4×D3C22⋊C4116yes—
169C3×Dic6—28no—
170C3×Dic6—216no—
171D3×Dic3—216no—
172D3×Dic3—48no—
173C23×Q8—31no—
174C22×C4⋊C4—91no—
175C22×C42—91no—
176#391—31no—
177#391—94no—
178#391—91no—
179#391—91no—
180#391—91no—
181C2×#95—94no—
182C24×C3—31no—
183C23×D3C22×D311yescharacteristic
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—24no—
189C2×C3×Q8—38no—
190C2×C3×Q8—34no—
191C2×C3×Q8—42no—
192C2×C3×Q8—42no—
193C2×C3×Q8—64no—
194C2×C3×Q8—64no—
195C2×C3×Q8—632no—
196C2×C3×Q8—62no—
197C2×C3×Q8—62no—
198C2×C3×Q8C4×D314yescomplemented
199C2×C3×Q8C2×Dic314yescomplemented
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—31no—
203C22×C3×C4—34no—
204C22×C3×C4—34no—
205C22×C3×C4—34no—
206C22×C3×C4—32no—
207C22×C3×C4—32no—
208C22×C3×C4—38no—
209C22×C3×C4—32no—
210C22×C3×C4—64no—
211C22×C3×C4—64no—
212C22×C3×C4C22×D312yes—
213C22×C3×C4C22×D312yes—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C22×D311yescharacteristic
216C22×Dic3—32no—
217C22×Dic3—32no—
218C22×Dic3—38no—
219C22×Dic3—32no—
220C22×Dic3—32no—
221C22×Dic3—62no—
222C22×Dic3—62no—
223C22×Dic3—64no—
224C22×Dic3—64no—
225C22×Dic3—64no—
226C22×Dic3—68no—
227C22×Dic3C22×D312yes—
228C22×Dic3C22×D312yes—
229C22×Dic3C22×D312yes—
230C22×Dic3C22×D311yescharacteristic
231D3×Q8—232no—
232C2×C4×D3—24no—
233C2×C4×D3—24no—
234C2×C4×D3—38no—
235C2×C4×D3—64no—
236C2×C4×D3—64no—
237C2×C4×D3—68no—
238C2×C4×D3—64no—
239C2×C4×D3C2×Dic318yes—
240C2×C4×D3C2×Dic314yes—
241C2×C4×D3#6714yes—
242C2×C4×D3#6714yes—
243C2×Dic6—22no—
244C2×Dic6—22no—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6C2×Dic314yescomplemented
248C2×Dic6C2×Dic318yescomplemented
249C3×C4⋊C4—62no—
250C3×C42—62no—
251C4⋊Dic3—22no—
252C4⋊Dic3—64no—
253C4⋊Dic3—62no—
254C4⋊Dic3—62no—
255C4⋊Dic3—122no—
256Dic3⋊C4—64no—
257C4×Dic3—22no—
258C4×Dic3—64no—
259C4×Dic3—62no—
260C4×Dic3—62no—
261C4×Dic3—64no—
262C4×Dic3—122no—
263C22×C32C23×C411yescharacteristic
264C22×C32C22×D411yescharacteristic
265C22×C32C22×D411yescharacteristic
266C22×C32C22×Q812yes—
267C22×C32C2×#4112yes—
268C2×C3×D3C2×C22⋊C414yes—
269C2×C3×D3C4×Q818yes—
270C2×C3×D3C22⋊Q818yes—
271C2×C3×D3#12414yes—
272C2×C3×D3C4⋊Q814yes—
273C32×C4—24no—
274C32×C4C2×C22⋊C414yes—
275C3⋊Dic3—24no—
276C3⋊Dic3—42no—
277C3×Dic3—24no—
278C3×Dic3—24no—
279C3×Dic3—42no—
280C3×Dic3C2×C22⋊C414yes—
281C22×Q8—38no—
282C22×Q8—32no—
283C22×Q8—31no—
284C22×Q8—64no—
285C22×Q8—64no—
286C22×Q8D3211yescharacteristic
287C23×C4—31no—
288C23×C4—32no—
289C23×C4—92no—
290C23×C4—92no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—92no—
293C2×C4⋊C4—91no—
294C2×C4⋊C4—184no—
295C2×C42—31no—
296C2×C42—92no—
297C2×C42—91no—
298C2×C42—184no—
299#95—34no—
300#95—94no—
301#95—94no—
302#95—94no—
303C23×C3—21no—
304C23×C3—32no—
305C23×C3—32no—
306C23×C3—32no—
307C23×C3—34no—
308C23×C3—34no—
309C23×C3C23×D311yescharacteristic
310C23×C3C23×D311yescharacteristic
311C22×D3D3×Q814yes—
312C22×D3#23914yes—
313C22×D3C22×Dic312yes—
314C22×D3C2×#6712yes—
315C22×D3C2×#6712yes—
316C3×Q8—28no—
317C3×Q8—28no—
318C3×Q8—48no—
319C3×Q8—616no—
320C3×Q8—68no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—316no—
333C2×C3×C4—34no—
334C2×C3×C4—34no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—616no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—64no—
343C2×C3×C4—62no—
344C2×C3×C4C2×C4×D314yes—
345C2×C3×C4C2×C4×D312yes—
346C2×C3×C4D3×D412yes—
347C2×C3×C4D3×D412yes—
348C2×C3×C4C22×Dic314yes—
349C2×C3×C4C22×Dic312yes—
350C2×C3×C4C2×#6712yes—
351C2×C3×C4C2×#6712yes—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—34no—
359C2×Dic3—34no—
360C2×Dic3—64no—
361C2×Dic3—616no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—64no—
368C2×Dic3—64no—
369C2×Dic3—62no—
370C2×Dic3—122no—
371C2×Dic3—122no—
372C2×Dic3—122no—
373C2×Dic3—124no—
374C2×Dic3—128no—
375C2×Dic3D3×Q814yescomplemented
376C2×Dic3C22×Dic314yes—
377C2×Dic3C22×Dic312yes—
378C2×Dic3C2×#6712yes—
379C2×Dic3C2×#6712yes—
380C4×D3—216no—
381C4×D3—616no—
382C4×D3—128no—
383C4×D3#217116yes—
384Dic6—28no—
385Dic6—216no—
386C2×C32C22×C22⋊C411yescharacteristic
387C2×C32C2×C4×Q812yes—
388C2×C32C2×C22⋊Q812yes—
389C2×C32C2×#12411yescharacteristic
390C2×C32C2×C4⋊Q811yescharacteristic
391C3×D3#39118yescomplemented
392C24—31no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—38no—
396C2×Q8—34no—
397C2×Q8—64no—
398C2×Q8—64no—
399C2×Q8—632no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8D3×Dic314yescomplemented
403C22×C4—31no—
404C22×C4—34no—
405C22×C4—34no—
406C22×C4—34no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—38no—
410C22×C4—32no—
411C22×C4—64no—
412C22×C4—64no—
413C22×C4—98no—
414C22×C4—92no—
415C22×C4—92no—
416C22×C4—184no—
417C22×C4—184no—
418C22×C4—184no—
419C22×C4—188no—
420C22×C4C2×D3212yes—
421C22×C4C2×D3211yescharacteristic
422C4⋊C4—62no—
423C4⋊C4—184no—
424C4⋊C4—182no—
425C42—62no—
426C42—184no—
427C42—182no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—34no—
434C22×C3—34no—
435C22×C3—34no—
436C22×C3—38no—
437C22×C3—38no—
438C22×C3C22×C4×D311yescharacteristic
439C22×C3C2×D3×D411yescharacteristic
440C22×C3C2×D3×D411yescharacteristic
441C22×C3C2×D3×Q812yes—
442C22×C3C2×D3×Q812yes—
443C22×C3C2×#23912yes—
444C22×C3D3×#4112yes—
445C22×C3C23×Dic311yescharacteristic
446C22×C3C22×#6711yescharacteristic
447C22×C3C22×#6711yescharacteristic
448C2×D3#94414yes—
449C2×D3Q8×Dic318yes—
450C2×D3#94618yes—
451C2×D3#94714yes—
452C2×D3C2×#21714yes—
453C3×C4—24no—
454C3×C4—24no—
455C3×C4—24no—
456C3×C4—34no—
457C3×C4—44no—
458C3×C4—64no—
459C3×C4—64no—
460C3×C4—122no—
461C3×C4D3×C22⋊C414yes—
462C3×C4C2×#21714yes—
463Dic3—24no—
464Dic3—24no—
465Dic3—42no—
466Dic3—64no—
467Dic3—64no—
468Dic3—124no—
469Dic3—122no—
470Dic3—122no—
471Dic3—242no—
472Dic3C2×#21714yes—
473C32C2×#39111yescharacteristic, complemented, Sylow
474C23—34no—
475C23—34no—
476C23—32no—
477C23—32no—
478C23—32no—
479C23C22×D3211yescharacteristic
480Q8—28no—
481Q8—616no—
482Q8—68no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—316no—
489C2×C4—34no—
490C2×C4—34no—
491C2×C4—616no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—64no—
497C2×C4—62no—
498C2×C4—94no—
499C2×C4—1816no—
500C2×C4—182no—
501C2×C4—182no—
502C2×C4—184no—
503C2×C4—182no—
504C2×C4—368no—
505C2×C4C2×D3×Dic314yes—
506C2×C4C2×D3×Dic312yes—
507C2×C4D3×#6712yes—
508C2×C4D3×#6712yes—
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—38no—
515C2×C3C2×D3×C22⋊C411yescharacteristic
516C2×C3C4×D3×Q812yes—
517C2×C3D3×C22⋊Q812yes—
518C2×C3D3×#12411yescharacteristic
519C2×C3D3×C4⋊Q811yescharacteristic
520C2×C3C2×#94411yescharacteristic
521C2×C3C2×Q8×Dic312yes—
522C2×C3C2×#94612yes—
523C2×C3C2×#94711yescharacteristic
524C2×C3C22×#21711yescharacteristic
525D3#522918yesdirect product factor
526C22—34no—
527C22—38no—
528C22—38no—
529C22—34no—
530C22—34no—
531C22D32×Q812yes—
532C22D3×#23912yes—
533C22C22×D3×Dic311yescharacteristic
534C22C2×D3×#6711yescharacteristic
535C22C2×D3×#6711yescharacteristic
536C4—24no—
537C4—34no—
538C4—64no—
539C4—64no—
540C4—122no—
541C4—184no—
542C4—362no—
543C4D3×#21714yes—
544C3—21no—
545C3D3×#39111yescharacteristic, complemented
546C3C2×#522911yescharacteristic, complemented
547C2—38no—
548C2D3×#94411yescharacteristic
549C2D3×Q8×Dic312yes—
550C2D3×#94612yes—
551C2D3×#94711yescharacteristic
552C2C2×D3×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC22×C32×Q8C22
socle, cosocleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC22×Q8D32
hypocenterC32C2×#391
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #10592928C2semidirect
5. C3×#5229C2semidirect
7. #10591261C2semidirect
8. #10591221C2semidirect
9. #10591220C2semidirect
15. C2×C3×D3×Q8C4semidirect
47. #5229D3direct (N × H)
56. C2×C32×Q8C2×C4semidirect
76. C2×C3×Dic6C2×C4semidirect
77. C2×C3×Dic6C2×C4semidirect
82. C2×D3×Dic3Q8semidirect
104. C2×D3×Q8Dic3semidirect
155. C2×C3⋊Dic3C2×Q8semidirect
166. C2×C3×Dic3C2×Q8semidirect
198. C2×C3×Q8C4×D3semidirect
199. C2×C3×Q8C2×Dic3semidirect
247. C2×Dic6C2×Dic3semidirect
248. C2×Dic6C2×Dic3semidirect
375. C2×Dic3D3×Q8semidirect
391. C3×D3#391semidirect
402. C2×Q8D3×Dic3semidirect
473. C32C2×#391semidirect
525. D3#5229direct (N × H)
545. C3D3×#391semidirect
546. C3C2×#5229semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D3×Q82
C2×D3×C4⋊Dic32
C2×C4×D3×Dic32
#105929282
C3×#52292
D3×C3⋊#952
#105912612
#105912212
#105912202
C2×#52293
D3×#3913