Finite Group Databasebeta

← #13145731

Subgroups — #13145731

Order 1728.

786 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C22×C3×C3⋊(C3⋊C8)C212yesmaximal
3#10589156—31nomaximal
4#10589155—32nomaximal
5C22×C33×C4C2211yescharacteristic
6C22×C3×C3⋊Dic3C412yes—
7C3×C4×C3⋊Dic3—22no—
8C2×C3×C3⋊(C3⋊C8)—22no—
9C2×C3×C3⋊(C3⋊C8)—22no—
10C2×C3×C3⋊(C3⋊C8)C414yescomplemented
11C2×C4×C3⋊Dic3D311yescharacteristic
12C22×C3⋊(C3⋊C8)D312yes—
13C2×C3×C4×Dic3—32no—
14C2×C3×C4×Dic3—61no—
15C22×C3×C3⋊C8—32no—
16C22×C3×C3⋊C8—32no—
17C22×C3×C3⋊C8—62no—
18C23×C33C2×C411yescharacteristic
19C2×C33×C4C2×C412yes—
20C2×C33×C4D411yescharacteristic
21C2×C33×C4D412yes—
22C2×C33×C4Q811yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C814yescomplemented
26C3×C3⋊(C3⋊C8)—24no—
27#4550—92no—
28#4524—91no—
29C22×C32×C4—22no—
30C22×C32×C4—21no—
31C22×C32×C4—41no—
32C22×C32×C4C2×D311yescharacteristic
33C22×C32×C4C2×D312yes—
34C22×C3⋊Dic3Dic312yes—
35C22×C3×Dic3—34no—
36C22×C3×Dic3—62no—
37C4×C3⋊Dic3—22no—
38C2×C3⋊(C3⋊C8)—22no—
39C2×C3⋊(C3⋊C8)—22no—
40C2×C3⋊(C3⋊C8)Dic314yescomplemented
41C3×C4×Dic3—64no—
42C3×C4×Dic3—62no—
43C3×C4×Dic3—62no—
44C2×C3×C3⋊C8—34no—
45C2×C3×C3⋊C8—34no—
46C2×C3×C3⋊C8—64no—
47C2×C3×C3⋊C8—64no—
48C2×C3×C3⋊C8—64no—
49C2×C3×C3⋊C8—62no—
50C2×C3×C3⋊C8—62no—
51C2×C3×C3⋊C8—62no—
52C2×C3×C3⋊C8—62no—
53C22×C33C4211yescharacteristic
54C22×C33C22⋊C411yescharacteristic
55C22×C33C4⋊C411yescharacteristic
56C22×C33C2×C812yes—
57C22×C33M1612yes—
58C33×C4C22⋊C412yes—
59C33×C4C4⋊C412yes—
60C3×C3⋊Dic3—24no—
61C22×C3×C8—92no—
62C2×C3×C42—91no—
63C2×C4×Dic3—32no—
64C2×C4×Dic3—61no—
65C22×C3⋊C8—32no—
66C22×C3⋊C8—32no—
67C22×C3⋊C8—62no—
68C23×C32—22no—
69C23×C32—21no—
70C23×C32—41no—
71C23×C32C4×D312yes—
72C23×C32C2×Dic311yescharacteristic
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4—22no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C32×C4—41no—
87C2×C32×C4Dic611yescharacteristic
88C2×C32×C4Dic612yes—
89C2×C32×C4C4×D312yes—
90C2×C32×C4C4×D312yes—
91C2×C32×C4D1212yes—
92C2×C32×C4D1211yescharacteristic
93C2×C32×C4C2×Dic312yes—
94C2×C32×C4#6712yes—
95C2×C32×C4#6712yes—
96C2×C32×C4#6712yes—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3C3⋊C814yescomplemented
100C2×C3×Dic3—38no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—64no—
106C3⋊(C3⋊C8)—24no—
107C3×C3⋊C8—68no—
108C3×C3⋊C8—64no—
109C3×C3⋊C8—64no—
110#336—271noSylow
111C2×C33#9511yescharacteristic
112C2×C33C4×C811yescharacteristic
113C2×C33#9711yescharacteristic
114C2×C33C22⋊C812yes—
115C2×C33C4⋊C812yes—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—92no—
120C22×C3×C4D3211yescharacteristic
121C22×C3×C4D3212yes—
122C22×Dic3—34no—
123C22×Dic3—62no—
124C2×C3×C8—94no—
125C2×C3×C8—182no—
126C2×C3×C8—182no—
127C3×C42—182no—
128C4×Dic3—64no—
129C4×Dic3—62no—
130C4×Dic3—62no—
131C2×C3⋊C8—34no—
132C2×C3⋊C8—34no—
133C2×C3⋊C8—64no—
134C2×C3⋊C8—64no—
135C2×C3⋊C8—64no—
136C2×C3⋊C8—62no—
137C2×C3⋊C8—62no—
138C2×C3⋊C8—62no—
139C2×C3⋊C8—62no—
140C22×C32—24no—
141C22×C32—22no—
142C22×C32—24no—
143C22×C32—22no—
144C22×C32—22no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—21no—
150C22×C32—41no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32D3×C814yes—
156C22×C32ΓL(1,𝔽25)14yes—
157C22×C32C2×C3⋊C812yes—
158C22×C32#20812yes—
159C22×C32C4×Dic311yescharacteristic
160C22×C32C4×Dic312yes—
161C22×C32Dic3⋊C412yes—
162C22×C32C4⋊Dic311yescharacteristic
163C22×C32(C2×D3)⋊C412yes—
164C22×C32#21711yescharacteristic
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4Dic3⋊C412yes—
174C32×C4Dic3⋊C412yes—
175C32×C4C4⋊Dic312yes—
176C32×C4(C2×D3)⋊C412yes—
177C32×C4(C2×D3)⋊C412yes—
178C32×C4#21712yes—
179C3⋊Dic3—24no—
180C3×Dic3—68no—
181C3×Dic3—68no—
182C22×C8—92no—
183C2×C42—91no—
184C33#33611yescharacteristic, complemented, Sylow
185C23×C3—22no—
186C23×C3—21no—
187C23×C3—41no—
188C23×C3D3×Dic312yes—
189C23×C3#62411yescharacteristic
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—94no—
205C2×C3×C4—182no—
206C2×C3×C4—182no—
207C2×C3×C4D3×Dic312yes—
208C2×C3×C4D3×Dic312yes—
209C2×C3×C4#62412yes—
210C2×C3×C4#62511yescharacteristic
211C2×C3×C4#62512yes—
212C2×C3×C4#62612yes—
213C2×C3×C4#62612yes—
214C2×C3×C4#62612yes—
215C2×C3×C4#62711yescharacteristic
216C2×C3×C4#62712yes—
217C2×Dic3—38no—
218C2×Dic3—64no—
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—64no—
222C2×Dic3—64no—
223C3×C8—184no—
224C3⋊C8—68no—
225C3⋊C8—64no—
226C3⋊C8—64no—
227C2×C32—22no—
228C2×C32—21no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—24no—
232C2×C32—22no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—22no—
236C2×C32—21no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—42no—
241C2×C32—42no—
242C2×C32C4×C3⋊C811yescharacteristic
243C2×C32#80311yescharacteristic
244C2×C32C4⋊(C3⋊C8)12yes—
245C2×C32C8×Dic312yes—
246C2×C32Dic3⋊C814yes—
247C2×C32#81512yes—
248C2×C32(C2×D3)⋊C814yes—
249C2×C32#83012yes—
250C2×C32C3⋊#9511yescharacteristic
251C2×C32C3⋊#9512yes—
252C22×C4—92no—
253C22×C4#111311yescharacteristic
254C2×C8—94no—
255C2×C8—182no—
256C2×C8—182no—
257C42—182no—
258C22×C3—22no—
259C22×C3—24no—
260C22×C3—22no—
261C22×C3—24no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3D3×C3⋊C814yes—
274C22×C3#371012yes—
275C22×C3#371114yes—
276C22×C3#371212yes—
277C22×C3Dic3212yes—
278C22×C3Dic3211yescharacteristic
279C22×C3(C2×D3)⋊Dic312yes—
280C22×C3#372211yescharacteristic
281C22×C3Dic3⋊Dic312yes—
282C22×C3#372411yescharacteristic
283C3×C4—22no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—42no—
290C3×C4—42no—
291C3×C4—184no—
292C3×C4(C2×D3)⋊Dic312yes—
293C3×C4(C2×D3)⋊Dic312yes—
294C3×C4#372212yes—
295C3×C4Dic3⋊Dic312yes—
296C3×C4Dic3⋊Dic312yes—
297C3×C4#372412yes—
298Dic3—68no—
299Dic3—68no—
300C32—22no—
301C32—21no—
302C32—41no—
303C32#452411yescharacteristic, complemented
304C32#455012yescomplemented
305C23#629411yescharacteristic
306C2×C4—94no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4#629412yes—
310C2×C4#629511yescharacteristic
311C2×C4#629512yes—
312C2×C4#629611yescharacteristic
313C8—184no—
314C2×C3—21no—
315C2×C3—22no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—24no—
320C2×C3—22no—
321C2×C3—24no—
322C2×C3—21no—
323C2×C3—22no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—42no—
327C2×C3—42no—
328C2×C3—41no—
329C2×C3Dic3×C3⋊C812yes—
330C2×C3#6355311yescharacteristic
331C2×C3#6355412yes—
332C2×C3#6355511yescharacteristic
333C2×C3(C2×D3)⋊(C3⋊C8)14yes—
334C2×C3#6355712yes—
335C2×C3Dic3⋊(C3⋊C8)14yes—
336C2×C3#6357212yes—
337C2×C3#6357912yes—
338C2×C3#6357911yescharacteristic
339C22#8880812yes—
340C22#8880912yes—
341C22#8881311yescharacteristic
342C22#8881411yescharacteristic
343C22#8881511yescharacteristic
344C4—184no—
345C4#8881412yes—
346C4#8881512yes—
347C3—21no—
348C3—22no—
349C3—41no—
350C3#1058915512yescomplemented
351C3#1058915611yescharacteristic, complemented
352C2#1172207011yescharacteristic
353C2#1172207111yescharacteristic
354C2#1172207212yes—
355C2#1172207812yes—
356C2#1172208211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22×C4#1113
commutatorC2×C33C4×C8
Fitting, cosocleC22×C33×C4C22
socle, prime radicalC23×C33C2×C4
hypocenterC33#336
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
25. C2×C3×C3⋊Dic3C8semidirect
40. C2×C3⋊(C3⋊C8)Dic3semidirect
99. C2×C3⋊Dic3C3⋊C8semidirect
184. C33#336semidirect
303. C32#4524semidirect
304. C32#4550semidirect
350. C3#10589155semidirect
351. C3#10589156semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C22×C3×C3⋊(C3⋊C8)2
#105891563
#105891553