Finite Group Databasebeta

← #13161836

Subgroups — #13161836

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic
3C2×#88813C211yesmaximal, characteristic
4C3×#63658C212yesmaximal
5#11722082C212yesmaximal
6#10591247—31nomaximal
7#10591245—32nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C3×#3750—22no—
14#88815—22no—
15#88813C414yescomplemented
16C2×#3750D311yescharacteristic
17C2×C3×Dic3⋊C4—32no—
18C2×C3×Dic3⋊C4—61no—
19C2×#3724—31no—
20C2×Dic3⋊Dic3—32no—
21C2×Dic32—32no—
22C2×Dic32—31no—
23#63658D312yes—
24C3×C3⋊#95—32no—
25C3×C3⋊#95—32no—
26C3×C3⋊#95—62no—
27#63579—32no—
28#63579—32no—
29#63579—32no—
30C23×C33C2311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C3×C3⋊Dic3C2×C412yes—
42C2×C3×C3⋊Dic3D412yes—
43C2×C3×C3⋊Dic3Q812yes—
44#4977—91no—
45#4650—92no—
46C22×C32×C4—21no—
47C22×C32×C4—22no—
48C22×C32×C4—41no—
49C22×C32×C4C2×D312yes—
50C22×C32×C4C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D312yes—
53C22×C3⋊Dic3C2×D312yes—
54C22×C3⋊Dic3C2×D311yescharacteristic
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3—62no—
64C22×C3×Dic3—62no—
65#3750—22no—
66C3×Dic3⋊C4—64no—
67C3×Dic3⋊C4—64no—
68#3724—62no—
69Dic3⋊Dic3—64no—
70Dic32—38no—
71Dic32—34no—
72C22×C33C22×C411yescharacteristic
73C22×C33C2×D411yescharacteristic
74C22×C33C2×Q811yescharacteristic
75C22×C33#4112yes—
76C22×C33#4112yes—
77C33×C4—41no—
78C3×C3⋊Dic3—24no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—42no—
81C3×C3⋊Dic3C4⋊C414yes—
82C2×C3×C4⋊C4—91no—
83C2×C4⋊Dic3—91no—
84C2×Dic3⋊C4—32no—
85C2×Dic3⋊C4—61no—
86C2×Dic3⋊C4—92no—
87C2×C4×Dic3—92no—
88C2×C4×Dic3—91no—
89C3×#95—92no—
90C3⋊#95—32no—
91C3⋊#95—32no—
92C3⋊#95—62no—
93C3⋊#95—92no—
94C3⋊#95—92no—
95C3⋊#95—92no—
96C23×C32—22no—
97C23×C32—21no—
98C23×C32—41no—
99C23×C32C22×D311yescharacteristic
100C23×C32C22×D312yes—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—24no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—42no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4—42no—
112C2×C32×C4—44no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3Dic612yes—
121C2×C3⋊Dic3C4×D314yes—
122C2×C3⋊Dic3C4×D312yes—
123C2×C3⋊Dic3#6712yes—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—122no—
149C2×C3×Dic3—122no—
150C2×C3×Dic3—122no—
151C2×C3×Dic3—122no—
152C2×C3×Dic3—122no—
153#382—271noSylow
154C2×C33C2×C4⋊C411yescharacteristic
155C2×C33#11712yes—
156C2×C33#12411yescharacteristic
157C2×C33#12411yescharacteristic
158C2×C33#12512yes—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—41no—
162C22×C3×C4—91no—
163C22×C3×C4—92no—
164C22×C3×C4—91no—
165C22×C3×C4—92no—
166C22×C3×C4D3211yescharacteristic
167C22×C3×C4D3212yes—
168C22×Dic3—32no—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—32no—
172C22×Dic3—32no—
173C22×Dic3—32no—
174C22×Dic3—61no—
175C22×Dic3—62no—
176C22×Dic3—62no—
177C22×Dic3—61no—
178C3×C4⋊C4—182no—
179C4⋊Dic3—182no—
180Dic3⋊C4—64no—
181Dic3⋊C4—64no—
182Dic3⋊C4—184no—
183C4×Dic3—98no—
184C4×Dic3—94no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32C2×Dic611yescharacteristic
203C22×C32C2×C4×D312yes—
204C22×C32C2×C4×D311yescharacteristic
205C22×C32#23512yes—
206C22×C32#23512yes—
207C22×C32D3×D412yes—
208C22×C32#23712yes—
209C22×C32#23712yes—
210C22×C32#23712yes—
211C22×C32D3×Q812yes—
212C22×C32#23912yes—
213C22×C32C2×#6711yescharacteristic
214C32×C4—42no—
215C32×C4—44no—
216C32×C4—44no—
217C32×C4—42no—
218C32×C4—41no—
219C3⋊Dic3—24no—
220C3⋊Dic3—42no—
221C3⋊Dic3—41no—
222C3⋊Dic3Dic3⋊C414yes—
223C3×Dic3—38no—
224C3×Dic3—64no—
225C3×Dic3—64no—
226C3×Dic3—64no—
227C3×Dic3—124no—
228C3×Dic3—124no—
229C3×Dic3—122no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C3×Dic3—122no—
233C2×C4⋊C4—91no—
234C2×C4⋊C4—271no—
235C2×C42—271no—
236#95—92no—
237#95—272no—
238C33#38211yescharacteristic, complemented, Sylow
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3C2×D3212yes—
243C23×C3C2×D3211yescharacteristic
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—44no—
255C2×C3×C4—42no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—94no—
260C2×C3×C4—182no—
261C2×C3×C4—182no—
262C2×C3×C4—182no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4—181no—
267C2×Dic3—34no—
268C2×Dic3—34no—
269C2×Dic3—34no—
270C2×Dic3—34no—
271C2×Dic3—34no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—64no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—64no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—62no—
291C2×Dic3—122no—
292C2×Dic3—122no—
293C2×Dic3—122no—
294C2×Dic3—122no—
295C2×Dic3—122no—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—22no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—22no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—41no—
309C2×C32—42no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—42no—
313C2×C32#87212yes—
314C2×C32#87712yes—
315C2×C32#88512yes—
316C2×C32#88512yes—
317C2×C32#88912yes—
318C2×C32#88912yes—
319C2×C32#89012yes—
320C2×C32D3×C4⋊C412yes—
321C2×C32#89212yes—
322C2×C32C2×Dic3⋊C411yescharacteristic
323C2×C32#93611yescharacteristic
324C2×C32#94711yescharacteristic
325C22×C4—91no—
326C22×C4—92no—
327C22×C4—92no—
328C22×C4—91no—
329C22×C4#111311yescharacteristic
330C4⋊C4—182no—
331C4⋊C4—542no—
332C42—274no—
333C22×C3—21no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—41no—
346C22×C3—42no—
347C22×C3—41no—
348C22×C3—42no—
349C22×C3—41no—
350C22×C3D3×Dic612yes—
351C22×C3#379512yes—
352C22×C3#379612yes—
353C22×C3#379711yescharacteristic
354C22×C3#379912yes—
355C22×C3C4×D3212yes—
356C22×C3C4×D3211yescharacteristic
357C22×C3#380412yes—
358C22×C3#380412yes—
359C22×C3#380512yes—
360C22×C3D3×#6712yes—
361C22×C3#381111yescharacteristic
362C3×C4—42no—
363C3×C4—44no—
364C3×C4—44no—
365C3×C4—41no—
366C3×C4—42no—
367C3×C4—94no—
368C3×C4—184no—
369C3×C4—362no—
370C3×C4—361no—
371Dic3—38no—
372Dic3—64no—
373Dic3—64no—
374Dic3—64no—
375Dic3—124no—
376Dic3—124no—
377Dic3—122no—
378Dic3—122no—
379Dic3—122no—
380Dic3—122no—
381C32—22no—
382C32—21no—
383C32—41no—
384C32#465012yescomplemented
385C32#497711yescharacteristic, complemented
386C23C2×#111311yescharacteristic
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—94no—
393C2×C4—182no—
394C2×C4—182no—
395C2×C4—182no—
396C2×C4—182no—
397C2×C4—182no—
398C2×C4—181no—
399C2×C4—182no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—42no—
416C2×C3—41no—
417C2×C3#6383612yes—
418C2×C3#6384512yes—
419C2×C3#6384612yes—
420C2×C3#6384712yes—
421C2×C3D3×Dic3⋊C412yes—
422C2×C3#6387712yes—
423C2×C3#6387812yes—
424C2×C3#6388311yescharacteristic
425C2×C3#6390712yes—
426C2×C3#6391311yescharacteristic
427C2×C3#6395311yescharacteristic
428C2×C3#6395912yes—
429C22#8904011yescharacteristic
430C22#8904112yes—
431C22C4×#111311yescharacteristic
432C22#8904612yes—
433C22#8904911yescharacteristic
434C4—41no—
435C4—94no—
436C4—184no—
437C4—362no—
438C4—361no—
439C3—22no—
440C3—21no—
441C3—41no—
442C3#1059124512yescomplemented
443C3#1059124711yescharacteristic, complemented
444C2#1172366912yes—
445C2#1172367011yescharacteristic
446C2#1172368211yescharacteristic
447C2#1172368312yes—
448C2#1172371811yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4#1113
hypocenterC33#382
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
15. #88813C4semidirect
238. C33#382semidirect
384. C32#4650semidirect
385. C32#4977semidirect
442. C3#10591245semidirect
443. C3#10591247semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C2×#888152
C2×#888132
C3×#636582
#117220822
#105912473
#105912453