Finite Group Databasebeta

← #13145745

Subgroups — #13145745

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3757C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal
3#10589187—31nomaximal
4#10589186—32nomaximal
5C24×C33C2211yescharacteristic
6C22×C3×C3⋊Dic3—22no—
7C22×C3×C3⋊Dic3C412yescomplemented
8C3×#3757—22no—
9C3×#3757—22no—
10C3×#3757C414yescomplemented
11C2×#3757D311yescharacteristic
12C2×#3757D312yes—
13C2×C3×#217—32no—
14C2×C3×#217—32no—
15C2×C3×#217—32no—
16C2×C3×#217—62no—
17C2×C3×#217—61no—
18C23×C33—42no—
19C23×C33C2×C411yescharacteristic
20C23×C33C2×C412yes—
21C23×C33D412yes—
22C23×C33D411yescharacteristic
23C23×C33Q811yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3—24no—
26C2×C3×C3⋊Dic3—44no—
27C2×C3×C3⋊Dic3—42no—
28#4527—92no—
29#4526—91no—
30C24×C32—21no—
31C24×C32—22no—
32C24×C32—41no—
33C24×C32C2×D312yes—
34C24×C32C2×D311yescharacteristic
35C22×C3⋊Dic3—22no—
36C22×C3⋊Dic3Dic312yescomplemented
37C22×C3×Dic3—34no—
38C22×C3×Dic3—62no—
39C22×C3×Dic3—64no—
40C22×C3×Dic3—62no—
41C22×C3×Dic3—62no—
42#3757—22no—
43#3757—22no—
44#3757Dic314yescomplemented
45C3×#217—34no—
46C3×#217—34no—
47C3×#217—64no—
48C3×#217—64no—
49C3×#217—64no—
50C3×#217—62no—
51C3×#217—62no—
52C3×#217—62no—
53C3×#217—62no—
54C22×C33—22no—
55C22×C33—24no—
56C22×C33—42no—
57C22×C33—41no—
58C22×C33—41no—
59C22×C33C4211yescharacteristic
60C22×C33C22⋊C412yes—
61C22×C33C22⋊C411yescharacteristic
62C22×C33C4⋊C412yes—
63C22×C33C4⋊C411yescharacteristic
64C3×C3⋊Dic3—44no—
65C3×C3⋊Dic3—42no—
66C2×C3×C22⋊C4—92no—
67C2×C3×C22⋊C4—91no—
68C2×#217—32no—
69C2×#217—32no—
70C2×#217—32no—
71C2×#217—62no—
72C2×#217—61no—
73C23×C32—22no—
74C23×C32—22no—
75C23×C32—22no—
76C23×C32—21no—
77C23×C32—22no—
78C23×C32—22no—
79C23×C32—21no—
80C23×C32—22no—
81C23×C32—22no—
82C23×C32—22no—
83C23×C32—22no—
84C23×C32—21no—
85C23×C32—41no—
86C23×C32—42no—
87C23×C32—42no—
88C23×C32—41no—
89C23×C32—41no—
90C23×C32—44no—
91C23×C32—42no—
92C23×C32—48no—
93C23×C32—44no—
94C23×C32—48no—
95C23×C32Dic612yes—
96C23×C32Dic611yescharacteristic
97C23×C32C4×D312yes—
98C23×C32C4×D312yes—
99C23×C32C4×D312yes—
100C23×C32D1212yes—
101C23×C32D1211yescharacteristic
102C23×C32C2×Dic311yescharacteristic
103C23×C32C2×Dic312yes—
104C23×C32#6712yes—
105C23×C32#6712yes—
106C23×C32#6712yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—24no—
109C2×C3⋊Dic3—44no—
110C2×C3⋊Dic3—42no—
111C2×C3×Dic3—68no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—128no—
117C2×C3×Dic3—124no—
118C2×C3×Dic3—124no—
119C2×C3×Dic3—124no—
120C2×C3×Dic3—124no—
121#342—271noSylow
122C2×C33—22no—
123C2×C33—21no—
124C2×C33—21no—
125C2×C33—22no—
126C2×C33#9511yescharacteristic
127C2×C33#9912yes—
128C24×C3—21no—
129C24×C3—22no—
130C24×C3—41no—
131C24×C3D3212yes—
132C24×C3D3211yescharacteristic
133C22×C3×C4—92no—
134C22×C3×C4—182no—
135C22×Dic3—34no—
136C22×Dic3—62no—
137C22×Dic3—64no—
138C22×Dic3—62no—
139C22×Dic3—62no—
140C3×C22⋊C4—94no—
141C3×C22⋊C4—182no—
142C3×C22⋊C4—182no—
143#217—34no—
144#217—34no—
145#217—64no—
146#217—64no—
147#217—64no—
148#217—62no—
149#217—62no—
150#217—62no—
151#217—62no—
152C22×C32—24no—
153C22×C32—22no—
154C22×C32—28no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32—42no—
173C22×C32—41no—
174C22×C32—42no—
175C22×C32—41no—
176C22×C32—44no—
177C22×C32—44no—
178C22×C32—44no—
179C22×C32—44no—
180C22×C32—48no—
181C22×C32—41no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32—44no—
189C22×C32—44no—
190C22×C32—44no—
191C22×C32—42no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—42no—
195C22×C32—48no—
196C22×C32—44no—
197C22×C32—44no—
198C22×C32C4×Dic311yescharacteristic
199C22×C32C4×Dic312yes—
200C22×C32Dic3⋊C412yes—
201C22×C32Dic3⋊C412yes—
202C22×C32Dic3⋊C412yes—
203C22×C32C4⋊Dic311yescharacteristic
204C22×C32C4⋊Dic312yes—
205C22×C32(C2×D3)⋊C412yes—
206C22×C32(C2×D3)⋊C412yes—
207C22×C32(C2×D3)⋊C412yes—
208C22×C32#21712yes—
209C22×C32#21711yescharacteristic
210C3⋊Dic3—44no—
211C3⋊Dic3—42no—
212C3×Dic3—128no—
213C3×Dic3—124no—
214C3×Dic3—124no—
215C3×Dic3—124no—
216C3×Dic3—124no—
217C2×C22⋊C4—92no—
218C2×C22⋊C4—91no—
219C33#34211yescharacteristic, complemented, Sylow
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—21no—
232C23×C3—44no—
233C23×C3—42no—
234C23×C3—44no—
235C23×C3—48no—
236C23×C3—48no—
237C23×C3—41no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3D3×Dic312yes—
243C23×C3D3×Dic312yes—
244C23×C3D3×Dic312yes—
245C23×C3#62411yescharacteristic
246C23×C3#62412yes—
247C23×C3#62512yes—
248C23×C3#62511yescharacteristic
249C23×C3#62612yes—
250C23×C3#62612yes—
251C23×C3#62612yes—
252C23×C3#62712yes—
253C23×C3#62711yescharacteristic
254C2×C3×C4—184no—
255C2×C3×C4—182no—
256C2×C3×C4—364no—
257C2×C3×C4—362no—
258C2×Dic3—68no—
259C2×Dic3—64no—
260C2×Dic3—64no—
261C2×Dic3—64no—
262C2×Dic3—64no—
263C2×Dic3—128no—
264C2×Dic3—124no—
265C2×Dic3—124no—
266C2×Dic3—124no—
267C2×Dic3—124no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—22no—
271C2×C32—22no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—22no—
281C2×C32—22no—
282C2×C32—22no—
283C2×C32—21no—
284C2×C32—24no—
285C2×C32—22no—
286C2×C32—41no—
287C2×C32—42no—
288C2×C32—42no—
289C2×C32—42no—
290C2×C32—42no—
291C2×C32—42no—
292C2×C32—42no—
293C2×C32—42no—
294C2×C32—44no—
295C2×C32—42no—
296C2×C32—42no—
297C2×C32—44no—
298C2×C32#80614yes—
299C2×C32C3⋊#9511yescharacteristic
300C2×C32C3⋊#9512yes—
301C2×C32#83412yes—
302C24#111311yescharacteristic
303C22×C4—92no—
304C22×C4—182no—
305C22⋊C4—94no—
306C22⋊C4—182no—
307C22⋊C4—182no—
308C22×C3—24no—
309C22×C3—28no—
310C22×C3—24no—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—44no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—48no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—41no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—41no—
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—44no—
346C22×C3—48no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—41no—
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—41no—
354C22×C3Dic3212yes—
355C22×C3Dic3211yescharacteristic
356C22×C3(C2×D3)⋊Dic312yes—
357C22×C3(C2×D3)⋊Dic312yes—
358C22×C3(C2×D3)⋊Dic312yes—
359C22×C3#372212yes—
360C22×C3#372211yescharacteristic
361C22×C3Dic3⋊Dic312yes—
362C22×C3Dic3⋊Dic312yes—
363C22×C3Dic3⋊Dic312yes—
364C22×C3#372412yes—
365C22×C3#372411yescharacteristic
366C3×C4—364no—
367C3×C4—362no—
368Dic3—128no—
369Dic3—124no—
370Dic3—124no—
371Dic3—124no—
372Dic3—124no—
373C32—22no—
374C32—21no—
375C32—41no—
376C32#452611yescharacteristic, complemented
377C32#452712yescomplemented
378C23—42no—
379C23#629411yescharacteristic
380C23#629412yes—
381C23#629512yes—
382C23#629511yescharacteristic
383C23#629611yescharacteristic
384C2×C4—184no—
385C2×C4—182no—
386C2×C4—364no—
387C2×C4—362no—
388C2×C3—24no—
389C2×C3—22no—
390C2×C3—22no—
391C2×C3—24no—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—22no—
398C2×C3—22no—
399C2×C3—21no—
400C2×C3—22no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—24no—
406C2×C3—42no—
407C2×C3—42no—
408C2×C3—42no—
409C2×C3—42no—
410C2×C3—42no—
411C2×C3—42no—
412C2×C3—44no—
413C2×C3—42no—
414C2×C3—42no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—42no—
418C2×C3#6357912yes—
419C2×C3#6357911yescharacteristic
420C2×C3#6358014yes—
421C2×C3#6358112yes—
422C22—22no—
423C22—24no—
424C22—42no—
425C22—41no—
426C22—41no—
427C22#8881311yescharacteristic
428C22#8881412yes—
429C22#8881411yescharacteristic
430C22#8881511yescharacteristic
431C22#8881512yes—
432C4—364no—
433C4—362no—
434C3—21no—
435C3—22no—
436C3—41no—
437C3#1058918612yescomplemented
438C3#1058918711yescharacteristic, complemented
439C2—22no—
440C2—21no—
441C2—21no—
442C2—22no—
443C2#1172208211yescharacteristic
444C2#1172208312yes—
subgroupG/H
centerC22#88814
commutatorC22×C33C42
Frattini, hypercenterC24#1113
Fitting, prime radical, cosocleC24×C33C22
socleC22×C33C22⋊C4
hypocenterC33#342
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
7. C22×C3×C3⋊Dic3C4semidirect
10. C3×#3757C4semidirect
36. C22×C3⋊Dic3Dic3semidirect
44. #3757Dic3semidirect
219. C33#342semidirect
376. C32#4526semidirect
377. C32#4527semidirect
437. C3#10589186semidirect
438. C3#10589187semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37572
#105891873
#105891863