Finite Group Databasebeta

← #13156743

Subgroups — #13156743

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88815C211yesmaximal, characteristic
2C2×#88806C212yesmaximal
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C2×C3×#3724C211yesmaximal, characteristic
5C2×C3×Dic3⋊Dic3C212yesmaximal
6#10591359—31nomaximal
7#10591244—32nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13#88815—22no—
14#88806—24no—
15Dic3×C3⋊Dic3C414yescomplemented
16C3×#3724—22no—
17C3×Dic3⋊Dic3—24no—
18C2×#3750—32no—
19C2×C4×C3⋊Dic3—31no—
20C2×C3×C4⋊Dic3—32no—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×Dic3⋊C4—32no—
23C2×#3724—31no—
24C2×#3724—32no—
25C2×#3724—32no—
26C2×#3724—62no—
27C2×#3724D311yescharacteristic
28C2×Dic3⋊Dic3—32no—
29C2×Dic3⋊Dic3D312yes—
30C2×Dic32—32no—
31C2×Dic32—61no—
32C23×C33C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C412yes—
34C2×C3⋊(C3⋊Dic3)D412yes—
35C2×C3⋊(C3⋊Dic3)Q812yes—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3C2×C412yes—
40C2×C3×C3⋊Dic3D412yes—
41C2×C3×C3⋊Dic3Q812yes—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C412yes—
46C2×C32×Dic3D412yes—
47C2×C32×Dic3Q812yes—
48#4975—91no—
49#4975—92no—
50C22×C32×C4—31no—
51C22×C32×C4—32no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—32no—
54C22×C3⋊Dic3—61no—
55C22×C3⋊Dic3—62no—
56C22×C3⋊Dic3—121no—
57C22×C3⋊Dic3C2×D312yes—
58C22×C3⋊Dic3C2×D311yescharacteristic
59C22×C3×Dic3—21no—
60C22×C3×Dic3—22no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—62no—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D312yes—
69#3750—64no—
70C4×C3⋊Dic3—34no—
71C3×C4⋊Dic3—64no—
72C3×Dic3⋊C4—64no—
73C3×Dic3⋊C4—64no—
74#3724—22no—
75#3724—64no—
76#3724—64no—
77#3724—68no—
78#3724—62no—
79Dic3⋊Dic3—24no—
80Dic3⋊Dic3—64no—
81Dic32—38no—
82Dic32—64no—
83C22×C33C22×C411yescharacteristic
84C22×C33C2×D411yescharacteristic
85C22×C33C2×D411yescharacteristic
86C22×C33C2×D411yescharacteristic
87C22×C33C2×Q811yescharacteristic
88C22×C33C2×Q811yescharacteristic
89C22×C33C2×Q811yescharacteristic
90C3⋊(C3⋊Dic3)C4⋊C414yescomplemented
91C3×C3⋊Dic3—42no—
92C3×C3⋊Dic3C4⋊C414yescomplemented
93C32×Dic3—42no—
94C32×Dic3C4⋊C414yescomplemented
95C2×C3×C4⋊C4—91no—
96C2×C3×C4⋊C4—92no—
97C2×C4⋊Dic3—32no—
98C2×C4⋊Dic3—91no—
99C2×Dic3⋊C4—32no—
100C2×Dic3⋊C4—32no—
101C2×Dic3⋊C4—92no—
102C2×Dic3⋊C4—92no—
103C2×Dic3⋊C4—92no—
104C2×Dic3⋊C4—92no—
105C2×Dic3⋊C4—182no—
106C2×C4×Dic3—92no—
107C2×C4×Dic3—91no—
108C2×C4×Dic3—181no—
109C23×C32—22no—
110C23×C32—21no—
111C23×C32—41no—
112C23×C32C22×D311yescharacteristic
113C23×C32C22×D312yes—
114C2×C32×C4—32no—
115C2×C32×C4—32no—
116C2×C32×C4—32no—
117C2×C32×C4—62no—
118C2×C32×C4—62no—
119C2×C32×C4—62no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—32no—
125C2×C3⋊Dic3—32no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—32no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—62no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—62no—
133C2×C3⋊Dic3—62no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—122no—
136C2×C3⋊Dic3—122no—
137C2×C3⋊Dic3—122no—
138C2×C3⋊Dic3Dic612yes—
139C2×C3⋊Dic3C4×D312yes—
140C2×C3⋊Dic3#6712yes—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—64no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—122no—
167C2×C3×Dic3—122no—
168C2×C3×Dic3Dic614yes—
169C2×C3×Dic3C4×D314yes—
170C2×C3×Dic3#6714yes—
171#384—271noSylow
172C2×C33C2×C4⋊C411yescharacteristic
173C2×C33C2×C4⋊C411yescharacteristic
174C2×C33C2×C4⋊C411yescharacteristic
175C2×C33#12711yescharacteristic
176C2×C33C4⋊Q811yescharacteristic
177C2×C33C4⋊Q811yescharacteristic
178C2×C33C4⋊Q811yescharacteristic
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—32no—
182C22×C3×C4—61no—
183C22×C3×C4—62no—
184C22×C3×C4—91no—
185C22×C3×C4—92no—
186C22×Dic3—32no—
187C22×Dic3—32no—
188C22×Dic3—32no—
189C22×Dic3—61no—
190C22×Dic3—62no—
191C22×Dic3—92no—
192C22×Dic3—91no—
193C22×Dic3—181no—
194C22×Dic3—182no—
195C22×Dic3—361no—
196C22×Dic3D3212yes—
197C22×Dic3D3211yescharacteristic
198C3×C4⋊C4—184no—
199C3×C4⋊C4—182no—
200C4⋊Dic3—64no—
201C4⋊Dic3—182no—
202Dic3⋊C4—64no—
203Dic3⋊C4—64no—
204Dic3⋊C4—184no—
205Dic3⋊C4—184no—
206Dic3⋊C4—184no—
207Dic3⋊C4—184no—
208Dic3⋊C4—188no—
209C4×Dic3—98no—
210C4×Dic3—94no—
211C4×Dic3—184no—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32C2×Dic611yescharacteristic
234C22×C32C2×Dic612yes—
235C22×C32C2×C4×D311yescharacteristic
236C22×C32C2×C4×D312yes—
237C22×C32D3×D412yes—
238C22×C32D3×D412yes—
239C22×C32D3×D411yescharacteristic
240C22×C32D3×D411yescharacteristic
241C22×C32D3×Q811yescharacteristic
242C22×C32D3×Q812yes—
243C22×C32D3×Q811yescharacteristic
244C22×C32D3×Q812yes—
245C22×C32C2×#6712yes—
246C22×C32C2×#6711yescharacteristic
247C32×C4—34no—
248C32×C4—122no—
249C3⋊Dic3—34no—
250C3⋊Dic3—38no—
251C3⋊Dic3—42no—
252C3⋊Dic3—68no—
253C3⋊Dic3—64no—
254C3⋊Dic3—124no—
255C3⋊Dic3Dic3⋊C414yescomplemented
256C3×Dic3—24no—
257C3×Dic3—38no—
258C3×Dic3—44no—
259C3×Dic3—42no—
260C3×Dic3—42no—
261C3×Dic3—64no—
262C3×Dic3—124no—
263C3×Dic3—122no—
264C3×Dic3—122no—
265C3×Dic3Dic3⋊C418yescomplemented
266C2×C4⋊C4—91no—
267C2×C4⋊C4—92no—
268C2×C4⋊C4—272no—
269C2×C4⋊C4—271no—
270C2×C42—271no—
271C33#38411yescharacteristic, complemented, Sylow
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—41no—
275C23×C3C2×D3212yes—
276C23×C3C2×D3211yescharacteristic
277C2×C3×C4—34no—
278C2×C3×C4—34no—
279C2×C3×C4—34no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—64no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—92no—
291C2×C3×C4—92no—
292C2×C3×C4—92no—
293C2×C3×C4—122no—
294C2×C3×C4—122no—
295C2×C3×C4—182no—
296C2×C3×C4—182no—
297C2×C3×C4—182no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—34no—
302C2×Dic3—34no—
303C2×Dic3—34no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—64no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314C2×Dic3—94no—
315C2×Dic3—92no—
316C2×Dic3—94no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—94no—
320C2×Dic3—122no—
321C2×Dic3—122no—
322C2×Dic3—182no—
323C2×Dic3—184no—
324C2×Dic3—184no—
325C2×Dic3—182no—
326C2×Dic3—182no—
327C2×Dic3—184no—
328C2×Dic3—362no—
329C2×Dic3—362no—
330C2×Dic3—362no—
331C2×Dic3#62412yes—
332C2×Dic3#62512yes—
333C2×Dic3#62712yes—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—22no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32#88811yescharacteristic
356C2×C32#88812yes—
357C2×C32#88811yescharacteristic
358C2×C32#88812yes—
359C2×C32D3×C4⋊C411yescharacteristic
360C2×C32D3×C4⋊C412yes—
361C2×C32D3×C4⋊C411yescharacteristic
362C2×C32D3×C4⋊C412yes—
363C2×C32C2×Dic3⋊C411yescharacteristic
364C2×C32C2×Dic3⋊C412yes—
365C2×C32#94012yes—
366C2×C32#94011yescharacteristic
367C2×C32#94411yescharacteristic
368C2×C32#94412yes—
369C22×C4—31no—
370C22×C4—32no—
371C22×C4—91no—
372C22×C4—92no—
373C22×C4—271no—
374C4⋊C4—184no—
375C4⋊C4—182no—
376C4⋊C4—544no—
377C4⋊C4—542no—
378C42—274no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3D3×Dic612yes—
401C22×C3D3×Dic612yes—
402C22×C3#379711yescharacteristic
403C22×C3#379712yes—
404C22×C3#379711yescharacteristic
405C22×C3C4×D3212yes—
406C22×C3#380211yescharacteristic
407C22×C3#380211yescharacteristic
408C22×C3C2×#62411yescharacteristic
409C22×C3C2×#62511yescharacteristic
410C22×C3C2×#62711yescharacteristic
411C22×C3D3×#6712yes—
412C22×C3D3×#6712yes—
413C22×C3#381112yes—
414C3×C4—38no—
415C3×C4—64no—
416C3×C4—94no—
417C3×C4—124no—
418C3×C4—122no—
419C3×C4—122no—
420C3×C4—362no—
421Dic3—38no—
422Dic3—42no—
423Dic3—64no—
424Dic3—98no—
425Dic3—94no—
426Dic3—124no—
427Dic3—122no—
428Dic3—122no—
429Dic3—184no—
430Dic3—188no—
431Dic3—364no—
432Dic3#372414yescomplemented
433C32—22no—
434C32—21no—
435C32—41no—
436C32#497511yescharacteristic, complemented
437C32#497512yescomplemented
438C23D3311yescharacteristic
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—272no—
452C2×C4—272no—
453C2×C4—272no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3#6383912yes—
476C2×C3#6383912yes—
477C2×C3#6384012yes—
478C2×C3#6387311yescharacteristic
479C2×C3D3×Dic3⋊C412yes—
480C2×C3D3×Dic3⋊C412yes—
481C2×C3#6388312yes—
482C2×C3#6390011yescharacteristic
483C2×C3#6390011yescharacteristic
484C2×C3#6391411yescharacteristic
485C2×C3#6391911yescharacteristic
486C2×C3#6391911yescharacteristic
487C2×C3C2×#372411yescharacteristic
488C2×C3#6397912yes—
489C22D3×#62411yescharacteristic
490C22D3×#62511yescharacteristic
491C22#8895311yescharacteristic
492C22#8895411yescharacteristic
493C22D3×#62711yescharacteristic
494C22#8895711yescharacteristic
495C22#8895811yescharacteristic
496C4—34no—
497C4—94no—
498C4—122no—
499C4—274no—
500C4—362no—
501C3—22no—
502C3—21no—
503C3—41no—
504C3#1059124412yescomplemented
505C3#1059135911yescharacteristic, complemented
506C2D3×#372411yescharacteristic
507C2#1172291111yescharacteristic
508C2#1172291211yescharacteristic
509C2#1172302911yescharacteristic
510C2#1172303211yescharacteristic
511C2#1172303511yescharacteristic
512C2#1172303711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#384
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
15. Dic3×C3⋊Dic3C4semidirect
90. C3⋊(C3⋊Dic3)C4⋊C4semidirect
92. C3×C3⋊Dic3C4⋊C4semidirect
94. C32×Dic3C4⋊C4semidirect
255. C3⋊Dic3Dic3⋊C4semidirect
265. C3×Dic3Dic3⋊C4semidirect
271. C33#384semidirect
432. Dic3#3724semidirect
436. C32#4975semidirect
437. C32#4975semidirect
504. C3#10591244semidirect
505. C3#10591359semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
C2×#888062
C2×Dic3×C3⋊Dic32
C2×C3×#37242
C2×C3×Dic3⋊Dic32
#105913593
#105912443