Finite Group Databasebeta

← #13156729

Subgroups — #13156729

Order 1728.

1314 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic3×C3⋊D3C211yesmaximal, characteristic
2C2×#88814C211yesmaximal, characteristic
3C2×C3×#3722C211yesmaximal, characteristic
4#11722061C212yesmaximal, complemented
5C3×#63579C212yesmaximal, complemented
6#10591362—32nomaximal
7#10591219—31nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C2×Dic3×C3⋊D3—22no—
14C2×Dic3×C3⋊D3—22no—
15C2×Dic3×C3⋊D3C414yescomplemented
16#88814—22no—
17C3×#3722—22no—
18C22×C4×C3⋊D3—31no—
19C22×D3×Dic3—32no—
20C22×D3×Dic3—61no—
21C2×C3×(C2×D3)⋊C4—32no—
22C2×#3722—31no—
23C2×#3722D311yescharacteristic
24C2×(C2×D3)⋊Dic3—32no—
25#63658—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28#63579—32no—
29#63579—32no—
30#63579—62no—
31#63579D312yescomplemented
32C23×C33C2311yescharacteristic
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3C2×C412yes—
36C22×C3×C3⋊D3D412yes—
37C22×C3×C3⋊D3D412yes—
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)C2×C412yes—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—22no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3C2×C412yes—
50Dic3×C3⋊D3—28no—
51#4943—91no—
52#4664—92no—
53C23×C3⋊D3C2×D311yescharacteristic
54C23×C3×D3—32no—
55C23×C3×D3—61no—
56C22×C32×C4—31no—
57C22×C32×C4—32no—
58C22×C3⋊Dic3—32no—
59C22×C3⋊Dic3—31no—
60C22×C3⋊Dic3—61no—
61C22×C3⋊Dic3—62no—
62C22×C3⋊Dic3—121no—
63C22×C3⋊Dic3C2×D312yes—
64C2×C4×C3⋊D3—34no—
65C2×C4×C3⋊D3—62no—
66C2×C4×C3⋊D3—62no—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—62no—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C2×D312yes—
75C2×D3×Dic3—38no—
76C2×D3×Dic3—64no—
77C2×D3×Dic3—64no—
78C2×D3×Dic3—64no—
79C2×D3×Dic3—64no—
80C2×D3×Dic3—64no—
81C3×(C2×D3)⋊C4—64no—
82#3722—22no—
83#3722—62no—
84(C2×D3)⋊Dic3—64no—
85C22×C33C22×C411yescharacteristic
86C22×C33C2×D411yescharacteristic
87C22×C33C2×D411yescharacteristic
88C22×C33#4112yes—
89C22×C33#4112yes—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—24no—
93C2×C3×C3⋊D3—24no—
94C2×C3×C3⋊D3C22⋊C414yes—
95C3⋊(C3⋊Dic3)—22no—
96C3×C3⋊Dic3—42no—
97C32×Dic3—22no—
98C32×Dic3—42no—
99C23×Dic3—91no—
100C22×C4×D3—92no—
101C22×C4×D3—181no—
102C2×C3×C22⋊C4—91no—
103C2×#217—91no—
104C2×(C2×D3)⋊C4—32no—
105C2×(C2×D3)⋊C4—92no—
106C3×#95—92no—
107C3⋊#95—32no—
108C3⋊#95—32no—
109C3⋊#95—92no—
110C3⋊#95—92no—
111C3⋊#95—92no—
112C3⋊#95—182no—
113C23×C32—21no—
114C23×C32—22no—
115C23×C32—41no—
116C23×C32C22×D311yescharacteristic
117C23×C32C22×D312yes—
118C22×C3⋊D3—22no—
119C22×C3⋊D3—22no—
120C22×C3⋊D3C4×D312yes—
121C22×C3⋊D3D1212yes—
122C22×C3⋊D3#6712yes—
123C22×C3×D3—34no—
124C22×C3×D3—34no—
125C22×C3×D3—34no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—62no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C2×C32×C4—32no—
136C2×C32×C4—61no—
137C2×C32×C4—61no—
138C2×C32×C4—62no—
139C2×C32×C4—62no—
140C2×C32×C4—62no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—34no—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—62no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—62no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—122no—
155C2×C3⋊Dic3—122no—
156C2×C3⋊Dic3—122no—
157C2×C3⋊Dic3—122no—
158C2×C3⋊Dic3—122no—
159C4×C3⋊D3—68no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—42no—
173C2×C3×Dic3—42no—
174C2×C3×Dic3—64no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—122no—
182C2×C3×Dic3—122no—
183C2×C3×Dic3C4×D314yes—
184D3×Dic3—616no—
185D3×Dic3—616no—
186#381—271noSylow
187C2×C33C2×C22⋊C411yescharacteristic
188C2×C33#11711yescharacteristic
189C2×C33#11711yescharacteristic
190C2×C33#12312yes—
191C2×C33#12312yes—
192C3×C3⋊D3—24no—
193C24×C3—91no—
194C23×D3—32no—
195C23×D3—61no—
196C22×C3×C4—32no—
197C22×C3×C4—32no—
198C22×C3×C4—32no—
199C22×C3×C4—61no—
200C22×C3×C4—62no—
201C22×C3×C4—92no—
202C22×Dic3—32no—
203C22×Dic3—32no—
204C22×Dic3—62no—
205C22×Dic3—92no—
206C22×Dic3—91no—
207C22×Dic3—94no—
208C22×Dic3—181no—
209C22×Dic3—182no—
210C22×Dic3—182no—
211C22×Dic3—182no—
212C22×Dic3—361no—
213C22×Dic3D3212yes—
214C22×Dic3D3211yescharacteristic
215C2×C4×D3—98no—
216C2×C4×D3—184no—
217C2×C4×D3—184no—
218C2×C4×D3—184no—
219C2×C4×D3—184no—
220C2×C4×D3—184no—
221C3×C22⋊C4—182no—
222#217—182no—
223(C2×D3)⋊C4—64no—
224(C2×D3)⋊C4—184no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—21no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32—42no—
241C22×C32—42no—
242C22×C32C2×C4×D311yescharacteristic
243C22×C32C2×C4×D312yes—
244C22×C32C2×D1211yescharacteristic
245C22×C32#23512yes—
246C22×C32#23512yes—
247C22×C32D3×D412yes—
248C22×C32D3×D412yes—
249C22×C32#23712yes—
250C22×C32#23712yes—
251C22×C32#23712yes—
252C22×C32#23912yes—
253C22×C32C2×#6711yescharacteristic
254C2×C3⋊D3—22no—
255C2×C3⋊D3—22no—
256C2×C3⋊D3—24no—
257C2×C3⋊D3—24no—
258C2×C3⋊D3(C2×D3)⋊C414yes—
259C2×C3×D3—38no—
260C2×C3×D3—64no—
261C2×C3×D3—64no—
262C2×C3×D3—68no—
263C2×C3×D3—68no—
264C2×C3×D3—64no—
265C2×C3×D3—64no—
266C2×C3×D3—64no—
267C2×C3×D3—64no—
268C2×C3×D3—64no—
269C2×C3×D3—64no—
270C2×C3×D3—64no—
271C32×C4—62no—
272C32×C4—122no—
273C3⋊Dic3—42no—
274C3⋊Dic3—62no—
275C3⋊Dic3—64no—
276C3⋊Dic3—64no—
277C3⋊Dic3—124no—
278C3⋊Dic3—124no—
279C3×Dic3—24no—
280C3×Dic3—24no—
281C3×Dic3—44no—
282C3×Dic3—42no—
283C3×Dic3—42no—
284C3×Dic3—124no—
285C3×Dic3—122no—
286C3×Dic3—122no—
287C23×C4—271no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—271no—
290#95—92no—
291#95—272no—
292C33#38111yescharacteristic, complemented, Sylow
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—41no—
296C23×C3—92no—
297C23×C3—92no—
298C23×C3—92no—
299C23×C3—182no—
300C23×C3—182no—
301C23×C3C2×D3212yes—
302C23×C3C2×D3211yescharacteristic
303C22×D3—34no—
304C22×D3—34no—
305C22×D3—34no—
306C22×D3—62no—
307C22×D3—62no—
308C22×D3—62no—
309C22×D3—64no—
310C22×D3—64no—
311C22×D3—62no—
312C22×D3—62no—
313C22×D3—62no—
314C22×D3—62no—
315C2×C3×C4—34no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—64no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—122no—
329C2×C3×C4—122no—
330C2×C3×C4—182no—
331C2×C3×C4—182no—
332C2×C3×C4—182no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—64no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—94no—
346C2×Dic3—92no—
347C2×Dic3—122no—
348C2×Dic3—122no—
349C2×Dic3—182no—
350C2×Dic3—184no—
351C2×Dic3—188no—
352C2×Dic3—182no—
353C2×Dic3—182no—
354C2×Dic3—182no—
355C2×Dic3—182no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—362no—
359C2×Dic3—362no—
360C2×Dic3—362no—
361C2×Dic3—362no—
362C2×Dic3—362no—
363C2×Dic3#62412yes—
364C4×D3—1816no—
365C4×D3—1816no—
366C2×C32—21no—
367C2×C32—22no—
368C2×C32—22no—
369C2×C32—21no—
370C2×C32—22no—
371C2×C32—22no—
372C2×C32—22no—
373C2×C32—22no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—21no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—42no—
382C2×C32—42no—
383C2×C32#87212yes—
384C2×C32#87711yescharacteristic
385C2×C32#87711yescharacteristic
386C2×C32D3×C22⋊C412yes—
387C2×C32#88312yes—
388C2×C32#88312yes—
389C2×C32#88612yes—
390C2×C32#89212yes—
391C2×C32#89412yes—
392C2×C32#89412yes—
393C2×C32C2×(C2×D3)⋊C411yescharacteristic
394C2×C32#93512yes—
395C3⋊D3—24no—
396C3×D3—68no—
397C3×D3—68no—
398C24—91no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4—92no—
402C22×C4—274no—
403C22×C4—271no—
404C22×C4—542no—
405C22×C4—542no—
406C22⋊C4—182no—
407C22⋊C4—542no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—94no—
426C22×C3—182no—
427C22×C3—184no—
428C22×C3—184no—
429C22×C3—182no—
430C22×C3#379512yes—
431C22×C3#379612yes—
432C22×C3#379612yes—
433C22×C3#379812yes—
434C22×C3C4×D3212yes—
435C22×C3D3×D1212yes—
436C22×C3#380412yes—
437C22×C3#380512yes—
438C22×C3C2×#62411yescharacteristic
439C22×C3D3×#6712yes—
440C22×C3#381111yescharacteristic
441C22×C3#381111yescharacteristic
442C2×D3—38no—
443C2×D3—64no—
444C2×D3—64no—
445C2×D3—64no—
446C2×D3—68no—
447C2×D3—68no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—64no—
451C2×D3—64no—
452C2×D3—64no—
453C2×D3—64no—
454C3×C4—64no—
455C3×C4—64no—
456C3×C4—124no—
457C3×C4—122no—
458C3×C4—122no—
459C3×C4—362no—
460Dic3—22no—
461Dic3—42no—
462Dic3—124no—
463Dic3—122no—
464Dic3—122no—
465Dic3—184no—
466Dic3—184no—
467Dic3—182no—
468Dic3—364no—
469Dic3—364no—
470C32—22no—
471C32—21no—
472C32—41no—
473C32#466412yescomplemented
474C32#494311yescharacteristic, complemented
475C23—92no—
476C23—92no—
477C23—92no—
478C23—182no—
479C23—182no—
480C23D3311yescharacteristic
481C2×C4—32no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—182no—
488C2×C4—182no—
489C2×C4—182no—
490C2×C4—272no—
491C2×C4—548no—
492C2×C4—541no—
493C2×C4—541no—
494C2×C3—21no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—22no—
505C2×C3—22no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—42no—
510C2×C3—42no—
511C2×C3—184no—
512C2×C3#6384911yescharacteristic
513C2×C3#6385312yes—
514C2×C3#6385312yes—
515C2×C3#6387411yescharacteristic
516C2×C3#6387712yes—
517C2×C3#6387812yes—
518C2×C3#6388412yes—
519C2×C3#6389012yes—
520C2×C3#6389112yes—
521C2×C3D3×(C2×D3)⋊C412yes—
522C2×C3#6396912yes—
523C2×C3#6397411yescharacteristic
524D3—68no—
525D3—68no—
526C22—94no—
527C22—182no—
528C22—182no—
529C22—184no—
530C22—184no—
531C22D3×#62411yescharacteristic
532C22#8895211yescharacteristic
533C22#8895511yescharacteristic
534C22#8896012yes—
535C22#8896412yes—
536C4—62no—
537C4—122no—
538C4—362no—
539C4—542no—
540C3—21no—
541C3—22no—
542C3—41no—
543C3#1059121911yescharacteristic, complemented
544C3#1059136212yescomplemented
545C2—184no—
546C2#1172290411yescharacteristic
547C2#1172291911yescharacteristic
548C2#1172292111yescharacteristic
549C2#1172299112yes—
550C2#1172299712yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypocenterC33#381
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722061C2semidirect
5. C3×#63579C2semidirect
15. C2×Dic3×C3⋊D3C4semidirect
31. #63579D3semidirect
292. C33#381semidirect
473. C32#4664semidirect
474. C32#4943semidirect
543. C3#10591219semidirect
544. C3#10591362semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3×C3⋊D32
C2×#888142
C2×C3×#37222
#117220612
C3×#635792
#105913623
#105912193