Finite Group Databasebeta

← #13171754

Subgroups — #13171754

Order 1728.

1762 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic3×C3⋊D3C211yesmaximal, characteristic
2#11723723C211yesmaximal, characteristic
3#11723598C212yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic
5C3×#63955C212yesmaximal, complemented
6#11722921C212yesmaximal, complemented
7#11722919C212yesmaximal, complemented
8#11722911C212yesmaximal, complemented
9Dic3×#624C212yesmaximal, complemented
10#10594969—31nomaximal
11D3×#877—32nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C32×Dic3C2211yescharacteristic
15C2×#6294C2212yes—
16C2×Dic3×C3⋊D3C418yescomplemented
17C2×Dic3×C3⋊D3C2211yescharacteristic
18C2×Dic3×C3⋊D3C2211yescharacteristic
19C2×Dic3×C3⋊D3C2211yescharacteristic
20C2×Dic3×C3⋊D3C2211yescharacteristic
21C2×C3×#624C2212yes—
22C3×#3757C2212yes—
23C32×#217C2212yes—
24#88814C2212yes—
25#88806C2214yescomplemented
26Dic3×C3⋊Dic3C2214yescomplemented
27C3×Dic3⋊Dic3C2214yescomplemented
28C3×#3722C2212yes—
29C3×Dic32C2214yescomplemented
30C22×C4×C3⋊D3—31no—
31C22×D3×Dic3—32no—
32C22×D3×Dic3—61no—
33#64086—32no—
34C3×#926—32no—
35C3×D3×C22⋊C4—32no—
36C3×#877—32no—
37#63974—31no—
38#63974D311yescharacteristic
39D3×#217—32no—
40#63957—32no—
41#63957—32no—
42#63957—62no—
43#63955D312yescomplemented
44#63900—32no—
45C4×#624—32no—
46#63878—34no—
47#63877—34no—
48D3×Dic3⋊C4—34no—
49C4×D3×Dic3—34no—
50#63874—32no—
51#63849—32no—
52C23×C33C2311yescharacteristic
53C22×C3×C3⋊D3—24no—
54C22×C3×C3⋊D3C2×C414yes—
55C22×C3×C3⋊D3C2311yescharacteristic
56C22×C3×C3⋊D3C2311yescharacteristic
57C2×C3⋊(C3⋊Dic3)C2×C414yescomplemented
58C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C2×C3×C3⋊Dic3C2314yes—
61C2×C32×Dic3C2×C414yescomplemented
62C2×C32×Dic3C2314yes—
63C2×C32×Dic3C2311yescharacteristic
64C2×C32×Dic3C2311yescharacteristic
65#6294—24no—
66Dic3×C3⋊D3C2×C414yes—
67Dic3×C3⋊D3C2×C414yes—
68Dic3×C3⋊D3C2×C414yescomplemented
69Dic3×C3⋊D3C2×C414yescomplemented
70C3×#624—24no—
71D3×#117—92no—
72C2×#877—91no—
73C23×C3⋊D3C2×D311yescharacteristic
74C23×C3×D3—32no—
75C23×C3×D3—61no—
76C22×C32×C4—31no—
77C22×C3⋊Dic3—32no—
78C22×C3⋊Dic3—31no—
79C22×C3⋊Dic3—61no—
80C22×C3⋊Dic3—62no—
81C22×C3⋊Dic3—121no—
82C2×C4×C3⋊D3—38no—
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—31no—
87C22×C3×Dic3—21no—
88C22×C3×Dic3C2×D312yes—
89C2×C3×C4×D3—34no—
90C2×#624—32no—
91C2×#624C2×D312yes—
92C2×D3×Dic3—316no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—34no—
98C2×D3×Dic3—68no—
99C2×D3×Dic3—61no—
100C2×D3×Dic3—61no—
101C2×D3×Dic3—61no—
102C2×D3×Dic3—61no—
103C32×C22⋊C4—32no—
104#3757C2×D312yes—
105#3750—34no—
106C4×C3⋊Dic3—34no—
107C3×#217—22no—
108C3×#217—32no—
109C3×#217—32no—
110C3×#217—62no—
111C3×#217C2×D312yes—
112C3×#217C2×D312yes—
113C3×(C2×D3)⋊C4—34no—
114C3×C4⋊Dic3—34no—
115C3×Dic3⋊C4—34no—
116C3×C4×Dic3—34no—
117C3×C4×Dic3—34no—
118#3724—34no—
119#3724—34no—
120#3724—64no—
121Dic3⋊Dic3C2×D314yescomplemented
122#3722—32no—
123#3722C2×D312yes—
124(C2×D3)⋊Dic3—34no—
125Dic32—34no—
126Dic32—34no—
127Dic32—64no—
128Dic32C2×D314yescomplemented
129C22×C33—22no—
130C22×C33C22×C412yes—
131C22×C33C2411yescharacteristic
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—28no—
134C2×C3×C3⋊D3C22×C412yes—
135C2×C3×C3⋊D3C22×C412yes—
136C2×C3×C3⋊D3#4114yes—
137C3⋊(C3⋊Dic3)C22×C412yes—
138C3⋊(C3⋊Dic3)C22×C412yes—
139C3×C3⋊Dic3—24no—
140C32×Dic3—24no—
141C32×Dic3C22×C412yes—
142C32×Dic3C22×C412yes—
143C23×Dic3—91no—
144C22×C4×D3—92no—
145C22×C4×D3—181no—
146C3×#117—92no—
147C2×C3×C22⋊C4—91no—
148C2×#217—91no—
149#926—32no—
150C2×Dic3⋊C4—92no—
151C2×C4×Dic3—92no—
152#892—94no—
153D3×C4⋊C4—94no—
154D3×C22⋊C4—32no—
155D3×C22⋊C4—92no—
156#877—32no—
157#877—92no—
158#877—92no—
159#877—92no—
160#877—92no—
161#877—92no—
162#877—182no—
163#872—94no—
164C42×D3—94no—
165C23×C32—21no—
166C23×C32—22no—
167C23×C32—41no—
168C23×C32C22×D312yes—
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3—24no—
171C22×C3⋊D3C4×D314yes—
172C22×C3⋊D3C22×D311yescharacteristic
173C22×C3⋊D3C22×D311yescharacteristic
174C22×C3×D3—32no—
175C22×C3×D3—38no—
176C22×C3×D3—32no—
177C22×C3×D3—64no—
178C22×C3×D3—61no—
179C22×C3×D3—61no—
180C22×C3×D3—68no—
181C22×C3×D3—68no—
182C2×C32×C4—34no—
183C2×C32×C4—31no—
184C2×C32×C4—31no—
185C2×C32×C4—34no—
186C2×C3⋊Dic3—38no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—32no—
192C2×C3⋊Dic3—62no—
193C2×C3⋊Dic3—62no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—68no—
196C2×C3⋊Dic3—61no—
197C2×C3⋊Dic3—61no—
198C2×C3⋊Dic3—124no—
199C2×C3⋊Dic3—121no—
200C2×C3⋊Dic3—121no—
201C2×C3⋊Dic3C22×D314yes—
202C4×C3⋊D3—34no—
203C4×C3⋊D3—34no—
204C4×C3⋊D3—34no—
205C4×C3⋊D3—34no—
206C2×C3×Dic3—24no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—24no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—64no—
213C2×C3×Dic3C4×D318yescomplemented
214C2×C3×Dic3C22×D314yes—
215C2×C3×Dic3C22×D314yes—
216C2×C3×Dic3C22×D312yes—
217C2×C3×Dic3C22×D312yes—
218C3×C4×D3—68no—
219#624—24no—
220#624—64no—
221D3×Dic3—38no—
222D3×Dic3—38no—
223D3×Dic3—38no—
224D3×Dic3—38no—
225D3×Dic3—64no—
226D3×Dic3—64no—
227D3×Dic3—64no—
228D3×Dic3—64no—
229D3×Dic3—68no—
230C2×#117—271noSylow
231C2×C33—22no—
232C2×C33C23×C411yescharacteristic
233C2×C33C2×#4112yes—
234C3×C3⋊D3—22no—
235C3×C3⋊D3#11714yescomplemented
236C24×C3—91no—
237C23×D3—32no—
238C23×D3—61no—
239C22×C3×C4—32no—
240C22×C3×C4—61no—
241C22×C3×C4—92no—
242C22×Dic3—92no—
243C22×Dic3—92no—
244C22×Dic3—91no—
245C22×Dic3—98no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—91no—
249C22×Dic3—91no—
250C22×Dic3—182no—
251C22×Dic3—181no—
252C22×Dic3—361no—
253C22×Dic3D3211yescharacteristic
254C2×C4×D3—34no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C2×C4×D3—916no—
260C2×C4×D3—94no—
261C2×C4×D3—181no—
262C2×C4×D3—181no—
263C2×C4×D3—181no—
264C2×C4×D3—181no—
265C2×C4×D3—188no—
266C3×C4⋊C4—94no—
267C3×C22⋊C4—32no—
268C3×C22⋊C4—32no—
269C3×C22⋊C4—62no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C42—94no—
273#217—32no—
274#217—32no—
275#217—62no—
276#217—92no—
277#217D3212yes—
278(C2×D3)⋊C4—34no—
279(C2×D3)⋊C4—94no—
280C4⋊Dic3—34no—
281Dic3⋊C4—34no—
282Dic3⋊C4—94no—
283Dic3⋊C4—94no—
284Dic3⋊C4—94no—
285Dic3⋊C4—184no—
286C4×Dic3—34no—
287C4×Dic3—34no—
288C4×Dic3—94no—
289C4×Dic3—94no—
290C4×Dic3—94no—
291C4×Dic3—184no—
292C22×C32—22no—
293C22×C32—24no—
294C22×C32—24no—
295C22×C32—22no—
296C22×C32—21no—
297C22×C32—24no—
298C22×C32—22no—
299C22×C32—42no—
300C22×C32—41no—
301C22×C32—44no—
302C22×C32—44no—
303C22×C32C2×C4×D312yes—
304C22×C32C2×C4×D314yes—
305C22×C32C23×D311yescharacteristic
306C22×C32C23×D312yes—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—28no—
309C2×C3⋊D3C2×C4×D312yes—
310C2×C3⋊D3C2×C4×D312yes—
311C2×C3⋊D3#23714yes—
312C2×C3×D3—34no—
313C2×C3×D3—38no—
314C2×C3×D3—34no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—64no—
318C2×C3×D3—616no—
319C2×C3×D3—616no—
320C2×C3×D3—64no—
321C2×C3×D3—64no—
322C32×C4—32no—
323C32×C4—32no—
324C32×C4—64no—
325C3⋊Dic3—24no—
326C3⋊Dic3—34no—
327C3⋊Dic3—34no—
328C3⋊Dic3—32no—
329C3⋊Dic3—32no—
330C3⋊Dic3—62no—
331C3⋊Dic3—62no—
332C3⋊Dic3—64no—
333C3⋊Dic3—64no—
334C3⋊Dic3—122no—
335C3⋊Dic3—122no—
336C3×Dic3—24no—
337C3×Dic3—24no—
338C3×Dic3—22no—
339C3×Dic3—22no—
340C3×Dic3—44no—
341C3×Dic3—64no—
342C3×Dic3—64no—
343C3×Dic3—124no—
344C3×Dic3C2×C4×D314yes—
345C3×Dic3C2×C4×D314yes—
346C23×C4—271no—
347#117—92no—
348#117—272no—
349#117—272no—
350#117—272no—
351C2×C4⋊C4—272no—
352C2×C22⋊C4—91no—
353C2×C22⋊C4—271no—
354C2×C42—272no—
355C33C2×#11711yescharacteristic, complemented, Sylow
356C23×C3—21no—
357C23×C3—22no—
358C23×C3—41no—
359C23×C3—91no—
360C23×C3—91no—
361C23×C3—94no—
362C23×C3—184no—
363C23×C3C2×D3212yes—
364C23×C3C2×D3211yescharacteristic
365C22×D3—32no—
366C22×D3—32no—
367C22×D3—38no—
368C22×D3—64no—
369C22×D3—61no—
370C22×D3—61no—
371C22×D3—68no—
372C22×D3—68no—
373C2×C3×C4—32no—
374C2×C3×C4—32no—
375C2×C3×C4—38no—
376C2×C3×C4—34no—
377C2×C3×C4—34no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—64no—
381C2×C3×C4—64no—
382C2×C3×C4—94no—
383C2×C3×C4—184no—
384C2×Dic3—34no—
385C2×Dic3—34no—
386C2×Dic3—64no—
387C2×Dic3—92no—
388C2×Dic3—91no—
389C2×Dic3—92no—
390C2×Dic3—91no—
391C2×Dic3—98no—
392C2×Dic3—94no—
393C2×Dic3—94no—
394C2×Dic3—94no—
395C2×Dic3—94no—
396C2×Dic3—94no—
397C2×Dic3—182no—
398C2×Dic3—182no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×Dic3—184no—
402C2×Dic3—188no—
403C2×Dic3—184no—
404C2×Dic3—361no—
405C2×Dic3—361no—
406C2×Dic3—364no—
407C2×Dic3#62414yescomplemented
408C2×Dic3C2×D3214yes—
409C2×Dic3C2×D3211yescharacteristic
410C2×Dic3C2×D3211yescharacteristic
411C4×D3—68no—
412C4×D3—98no—
413C4×D3—98no—
414C4×D3—98no—
415C4×D3—98no—
416C4×D3—184no—
417C4×D3—184no—
418C4×D3—184no—
419C4×D3—184no—
420C4×D3—188no—
421C2×C32—22no—
422C2×C32—24no—
423C2×C32—24no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—24no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—44no—
431C2×C32—44no—
432C2×C32C22×C4×D311yescharacteristic
433C2×C32C22×C4×D312yes—
434C2×C32C2×#23712yes—
435C2×C32D3×#4114yes—
436C3⋊D3—22no—
437C3⋊D3#87714yescomplemented
438C3×D3—38no—
439C3×D3—64no—
440C3×D3—64no—
441C3×D3—64no—
442C24—91no—
443C22×C4—31no—
444C22×C4—92no—
445C22×C4—271no—
446C22×C4—271no—
447C22×C4—271no—
448C22×C4—271no—
449C22×C4—278no—
450C22×C4—272no—
451C22×C4—271no—
452C4⋊C4—94no—
453C4⋊C4—274no—
454C22⋊C4—32no—
455C22⋊C4—92no—
456C22⋊C4—92no—
457C22⋊C4—272no—
458C42—94no—
459C42—274no—
460C22×C3—24no—
461C22×C3—22no—
462C22×C3—24no—
463C22×C3—22no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—41no—
470C22×C3—42no—
471C22×C3—94no—
472C22×C3—92no—
473C22×C3—92no—
474C22×C3—188no—
475C22×C3—182no—
476C22×C3C4×D3214yes—
477C22×C3C2×#62412yes—
478C22×C3C22×D3212yes—
479C22×C3C22×D3211yescharacteristic
480C2×D3—34no—
481C2×D3—38no—
482C2×D3—34no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—64no—
486C2×D3—616no—
487C2×D3—616no—
488C2×D3—64no—
489C2×D3—64no—
490C3×C4—34no—
491C3×C4—34no—
492C3×C4—62no—
493C3×C4—62no—
494C3×C4—64no—
495C3×C4—64no—
496C3×C4—124no—
497C3×C4—184no—
498Dic3—24no—
499Dic3—64no—
500Dic3—64no—
501Dic3—94no—
502Dic3—94no—
503Dic3—92no—
504Dic3—92no—
505Dic3—124no—
506Dic3—182no—
507Dic3—182no—
508Dic3—184no—
509Dic3—184no—
510Dic3—362no—
511Dic3—362no—
512Dic3C2×#62412yes—
513Dic3C2×#62412yes—
514C32—21no—
515C32—22no—
516C32—41no—
517C32C2×#87711yescharacteristic, complemented
518C32D3×#11712yescomplemented
519C23—91no—
520C23—91no—
521C23—94no—
522C23—184no—
523C23D3311yescharacteristic
524C2×C4—31no—
525C2×C4—31no—
526C2×C4—34no—
527C2×C4—34no—
528C2×C4—94no—
529C2×C4—184no—
530C2×C4—274no—
531C2×C4—274no—
532C2×C4—274no—
533C2×C4—274no—
534C2×C4—271no—
535C2×C4—271no—
536C2×C4—274no—
537C2×C4—544no—
538C2×C3—24no—
539C2×C3—22no—
540C2×C3—24no—
541C2×C3—24no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—22no—
545C2×C3—44no—
546C2×C3—44no—
547C2×C3—42no—
548C2×C3—41no—
549C2×C3—94no—
550C2×C3—182no—
551C2×C3C2×C4×D3212yes—
552C2×C3#6430612yes—
553C2×C3D3×#23714yes—
554C2×C3C22×#62411yescharacteristic
555D3—38no—
556D3—64no—
557D3—64no—
558D3—64no—
559C22—22no—
560C22—94no—
561C22—92no—
562C22—92no—
563C22—188no—
564C22—182no—
565C22D3×#62412yes—
566C22C2×D3311yescharacteristic
567C4—32no—
568C4—32no—
569C4—64no—
570C4—184no—
571C4—272no—
572C4—272no—
573C3—21no—
574C3—22no—
575C3—41no—
576C3D3×#87712yescomplemented
577C3#1059496911yescharacteristic, complemented
578C2—22no—
579C2—94no—
580C2—182no—
581C2#1172440712yes—
582C2C2×D3×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC23×C3×C3⋊D3C22
hypercenterC23D33
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11723598C2semidirect
5. C3×#63955C2semidirect
6. #11722921C2semidirect
7. #11722919C2semidirect
8. #11722911C2semidirect
9. Dic3×#624C2semidirect
16. C2×Dic3×C3⋊D3C4semidirect
25. #88806C22semidirect
26. Dic3×C3⋊Dic3C22semidirect
27. C3×Dic3⋊Dic3C22semidirect
29. C3×Dic32C22semidirect
43. #63955D3semidirect
57. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
61. C2×C32×Dic3C2×C4semidirect
68. Dic3×C3⋊D3C2×C4semidirect
69. Dic3×C3⋊D3C2×C4semidirect
121. Dic3⋊Dic3C2×D3semidirect
128. Dic32C2×D3semidirect
213. C2×C3×Dic3C4×D3semidirect
235. C3×C3⋊D3#117semidirect
355. C33C2×#117semidirect
407. C2×Dic3#624semidirect
437. C3⋊D3#877semidirect
517. C32C2×#877semidirect
518. C32D3×#117semidirect
576. C3D3×#877semidirect
577. C3#10594969semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3×C3⋊D32
#117237232
#117235982
C3×#639742
C3×#639552
#117229212
#117229192
#117229112
Dic3×#6242
#105949693
D3×#8773