Finite Group Databasebeta

← #13156725

Subgroups — #13156725

Order 1728.

1410 conjugacy classes of subgroups.

The subgroup lattice has 542 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×#88806C212yesmaximal, complemented
4C2×C3×#3722C211yesmaximal, characteristic
5C3×#63579C212yesmaximal, complemented
6#10591369—32nomaximal
7#10591355—31nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C2×Dic3×C3⋊D3—24no—
14C2×Dic3×C3⋊D3C414yescomplemented
15#88814—22no—
16#88806—24no—
17C3×#3722—22no—
18C22×C4×C3⋊D3—31no—
19C22×D3×Dic3—32no—
20C22×D3×Dic3—61no—
21C2×#3750—32no—
22C2×C3×(C2×D3)⋊C4—32no—
23C2×#3724—32no—
24C2×#3724—32no—
25C2×#3724—62no—
26C2×#3722—31no—
27C2×#3722D311yescharacteristic
28C2×(C2×D3)⋊Dic3—32no—
29C3×C3⋊#95—32no—
30C3×C3⋊#95—32no—
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⋊D3Q812yes—
38C2×C3⋊(C3⋊Dic3)C2×C412yes—
39C2×C3⋊(C3⋊Dic3)D414yes—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3C2×C412yes—
48Dic3×C3⋊D3—28no—
49#4942—91no—
50#4667—92no—
51C23×C3⋊D3C2×D311yescharacteristic
52C23×C3×D3—32no—
53C23×C3×D3—61no—
54C22×C32×C4—31no—
55C22×C32×C4—32no—
56C22×C3⋊Dic3—32no—
57C22×C3⋊Dic3—31no—
58C22×C3⋊Dic3—61no—
59C22×C3⋊Dic3—62no—
60C22×C3⋊Dic3—121no—
61C22×C3⋊Dic3C2×D312yes—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—64no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3—22no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—62no—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D312yes—
72C2×D3×Dic3—38no—
73C2×D3×Dic3—64no—
74C2×D3×Dic3—64no—
75C2×D3×Dic3—64no—
76C2×D3×Dic3—68no—
77#3750—64no—
78C3×(C2×D3)⋊C4—64no—
79#3724—64no—
80#3724—64no—
81#3724—68no—
82#3722—22no—
83#3722—62no—
84(C2×D3)⋊Dic3—64no—
85C22×C33C22×C411yescharacteristic
86C22×C33C2×D412yes—
87C22×C33C2×D411yescharacteristic
88C22×C33C2×Q811yescharacteristic
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⋊D3C4⋊C414yescomplemented
95C3⋊(C3⋊Dic3)C22⋊C414yescomplemented
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—
106C2×Dic3⋊C4—92no—
107C2×Dic3⋊C4—92no—
108C2×Dic3⋊C4—92no—
109C2×Dic3⋊C4—182no—
110C3×#95—92no—
111C3⋊#95—32no—
112C3⋊#95—32no—
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⋊D3Dic612yes—
121C22×C3⋊D3C4×D312yes—
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—62no—
137C2×C32×C4—62no—
138C2×C32×C4—62no—
139C2×C32×C4—62no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—32no—
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—34no—
148C2×C3⋊Dic3—64no—
149C2×C3⋊Dic3—62no—
150C2×C3⋊Dic3—64no—
151C2×C3⋊Dic3—64no—
152C2×C3⋊Dic3—64no—
153C2×C3⋊Dic3—122no—
154C2×C3⋊Dic3—124no—
155C4×C3⋊D3—68no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—42no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—122no—
177C2×C3×Dic3—122no—
178C2×C3×Dic3C4×D314yes—
179D3×Dic3—616no—
180D3×Dic3—616no—
181#380—271noSylow
182C2×C33C2×C22⋊C411yescharacteristic
183C2×C33C2×C4⋊C411yescharacteristic
184C2×C33#11711yescharacteristic
185C2×C33#12112yes—
186C2×C33C22⋊Q812yes—
187C3×C3⋊D3—24no—
188C24×C3—91no—
189C23×D3—32no—
190C23×D3—61no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—32no—
194C22×C3×C4—61no—
195C22×C3×C4—62no—
196C22×C3×C4—92no—
197C22×Dic3—32no—
198C22×Dic3—32no—
199C22×Dic3—62no—
200C22×Dic3—92no—
201C22×Dic3—91no—
202C22×Dic3—94no—
203C22×Dic3—182no—
204C22×Dic3—181no—
205C22×Dic3—184no—
206C22×Dic3—361no—
207C22×Dic3D3212yes—
208C22×Dic3D3211yescharacteristic
209C2×C4×D3—98no—
210C2×C4×D3—184no—
211C2×C4×D3—184no—
212C2×C4×D3—184no—
213C2×C4×D3—188no—
214C3×C22⋊C4—182no—
215#217—182no—
216(C2×D3)⋊C4—64no—
217(C2×D3)⋊C4—184no—
218Dic3⋊C4—184no—
219Dic3⋊C4—184no—
220Dic3⋊C4—184no—
221Dic3⋊C4—188no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—41no—
235C22×C32—42no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—42no—
239C22×C32C2×Dic611yescharacteristic
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×C4×D312yes—
242C22×C32#23512yes—
243C22×C32#23512yes—
244C22×C32D3×D412yes—
245C22×C32D3×D412yes—
246C22×C32D3×D412yes—
247C22×C32D3×D412yes—
248C22×C32#23712yes—
249C22×C32D3×Q812yes—
250C22×C32C2×#6711yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—24no—
253C2×C3⋊D3—22no—
254C2×C3⋊D3—24no—
255C2×C3⋊D3Dic3⋊C414yescomplemented
256C2×C3×D3—38no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C2×C3×D3—68no—
260C2×C3×D3—64no—
261C2×C3×D3—68no—
262C2×C3×D3—64no—
263C2×C3×D3—64no—
264C2×C3×D3—64no—
265C2×C3×D3—64no—
266C2×C3×D3—64no—
267C2×C3×D3—64no—
268C32×C4—62no—
269C32×C4—122no—
270C3⋊Dic3—38no—
271C3⋊Dic3—34no—
272C3⋊Dic3—42no—
273C3⋊Dic3—64no—
274C3⋊Dic3—68no—
275C3⋊Dic3—124no—
276C3×Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—44no—
279C3×Dic3—42no—
280C3×Dic3—42no—
281C3×Dic3—124no—
282C3×Dic3—122no—
283C3×Dic3—122no—
284C23×C4—271no—
285C2×C4⋊C4—272no—
286C2×C22⋊C4—91no—
287C2×C22⋊C4—271no—
288#95—92no—
289C33#38011yescharacteristic, complemented, Sylow
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—41no—
293C23×C3—92no—
294C23×C3—92no—
295C23×C3—92no—
296C23×C3—182no—
297C23×C3—182no—
298C23×C3C2×D3212yes—
299C23×C3C2×D3211yescharacteristic
300C22×D3—34no—
301C22×D3—34no—
302C22×D3—34no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—64no—
307C22×D3—64no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—62no—
312C2×C3×C4—34no—
313C2×C3×C4—62no—
314C2×C3×C4—64no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—64no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—122no—
325C2×C3×C4—122no—
326C2×C3×C4—182no—
327C2×C3×C4—182no—
328C2×C3×C4—182no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—64no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—94no—
341C2×Dic3—94no—
342C2×Dic3—92no—
343C2×Dic3—94no—
344C2×Dic3—94no—
345C2×Dic3—122no—
346C2×Dic3—122no—
347C2×Dic3—184no—
348C2×Dic3—182no—
349C2×Dic3—184no—
350C2×Dic3—184no—
351C2×Dic3—184no—
352C2×Dic3—188no—
353C2×Dic3—362no—
354C2×Dic3—364no—
355C2×Dic3#62412yes—
356C4×D3—1816no—
357C4×D3—1816no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—22no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—22no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—22no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—42no—
374C2×C32—42no—
375C2×C32#87212yes—
376C2×C32#87711yescharacteristic
377C2×C32#87812yes—
378C2×C32D3×C22⋊C411yescharacteristic
379C2×C32D3×C22⋊C412yes—
380C2×C32#88412yes—
381C2×C32#88412yes—
382C2×C32D3×C4⋊C412yes—
383C2×C32#89612yes—
384C2×C32#89612yes—
385C2×C32C2×Dic3⋊C411yescharacteristic
386C2×C32#93912yes—
387C3⋊D3—24no—
388C3×D3—68no—
389C3×D3—68no—
390C24—91no—
391C22×C4—31no—
392C22×C4—32no—
393C22×C4—92no—
394C22×C4—274no—
395C22×C4—271no—
396C22×C4—544no—
397C4⋊C4—544no—
398C22⋊C4—182no—
399C22⋊C4—542no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—42no—
415C22×C3—41no—
416C22×C3—42no—
417C22×C3—94no—
418C22×C3—184no—
419C22×C3—182no—
420C22×C3—184no—
421C22×C3—182no—
422C22×C3D3×Dic612yes—
423C22×C3#379512yes—
424C22×C3#379711yescharacteristic
425C22×C3#379812yes—
426C22×C3C4×D3212yes—
427C22×C3#380211yescharacteristic
428C22×C3#380212yes—
429C22×C3#380412yes—
430C22×C3C2×#62411yescharacteristic
431C22×C3D3×#6712yes—
432C22×C3#381112yes—
433C22×C3#381112yes—
434C2×D3—38no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—68no—
439C2×D3—68no—
440C2×D3—64no—
441C2×D3—64no—
442C2×D3—64no—
443C2×D3—64no—
444C2×D3—64no—
445C2×D3—64no—
446C3×C4—64no—
447C3×C4—64no—
448C3×C4—124no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—362no—
452Dic3—22no—
453Dic3—42no—
454Dic3—98no—
455Dic3—94no—
456Dic3—124no—
457Dic3—122no—
458Dic3—122no—
459Dic3—184no—
460Dic3—188no—
461Dic3—364no—
462C32—21no—
463C32—22no—
464C32—41no—
465C32#466712yescomplemented
466C32#494211yescharacteristic, complemented
467C23—92no—
468C23—92no—
469C23—92no—
470C23—182no—
471C23—182no—
472C23D3311yescharacteristic
473C2×C4—32no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—182no—
479C2×C4—182no—
480C2×C4—182no—
481C2×C4—272no—
482C2×C4—274no—
483C2×C4—548no—
484C2×C3—21no—
485C2×C3—22no—
486C2×C3—21no—
487C2×C3—22no—
488C2×C3—21no—
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—22no—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—42no—
500C2×C3—42no—
501C2×C3—184no—
502C2×C3#6386512yes—
503C2×C3#6387411yescharacteristic
504C2×C3D3×Dic3⋊C412yes—
505C2×C3#6387712yes—
506C2×C3#6388512yes—
507C2×C3#6389612yes—
508C2×C3#6389912yes—
509C2×C3#6390011yescharacteristic
510C2×C3#6391212yes—
511C2×C3#6392112yes—
512C2×C3#6397112yes—
513C2×C3#6397411yescharacteristic
514D3—68no—
515D3—68no—
516C22—94no—
517C22—182no—
518C22—182no—
519C22—184no—
520C22—184no—
521C22D3×#62411yescharacteristic
522C22#8895311yescharacteristic
523C22#8895412yes—
524C22#8895711yescharacteristic
525C22#8896012yes—
526C4—62no—
527C4—122no—
528C4—274no—
529C4—362no—
530C3—21no—
531C3—22no—
532C3—41no—
533C3#1059135511yescharacteristic, complemented
534C3#1059136912yescomplemented
535C2—184no—
536C2#1172290611yescharacteristic
537C2#1172291111yescharacteristic
538C2#1172291911yescharacteristic
539C2#1172296712yes—
540C2#1172298312yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypocenterC33#380
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×#88806C2semidirect
5. C3×#63579C2semidirect
14. C2×Dic3×C3⋊D3C4semidirect
31. #63579D3semidirect
94. C2×C3×C3⋊D3C4⋊C4semidirect
95. C3⋊(C3⋊Dic3)C22⋊C4semidirect
255. C2×C3⋊D3Dic3⋊C4semidirect
289. C33#380semidirect
465. C32#4667semidirect
466. C32#4942semidirect
533. C3#10591355semidirect
534. C3#10591369semidirect

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×#888062
C2×C3×#37222
C3×#635792
#105913693
#105913553