Finite Group Databasebeta

← C2×#11723404 #13176812

Subgroups — C2×#11723404 #13176812

Order 1728.

2348 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#88803C212yesmaximal
5C2×Dic3×C3⋊Dic3C212yesmaximal, complemented
6#11723404C218yesmaximal, direct product factor
7C2×#64101—31nomaximal
8C2×#63841—34nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C4×C3⋊D3C418yescomplemented
14C2×C3×C4×C3⋊D3C2214yes—
15C22×C32×Dic3C2212yes—
16C4⋊(C3⋊(C3⋊Dic3))C2214yescomplemented
17C32×C4⋊Dic3C2214yescomplemented
18#88803C2218yes—
19Dic3×C3⋊Dic3C2218yescomplemented
20C22×C4×C3⋊D3D311yescharacteristic
21C22×C3×C4×D3—34no—
22C2×C32×C4⋊C4—31no—
23C2×#3752—32no—
24C2×C4⋊(C3⋊Dic3)—31no—
25C2×C4⋊(C3⋊Dic3)—34no—
26C2×C4⋊(C3⋊Dic3)—64no—
27C2×C4×C3⋊Dic3—32no—
28#64101—38no—
29C2×C3×C4⋊Dic3D314yescomplemented
30C2×(C2×D3)⋊Dic3—38no—
31C2×Dic32—38no—
32#63841—332no—
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3—22no—
36C22×C3×C3⋊D3C2×C414yes—
37C22×C3×C3⋊D3C2312yes—
38C2×C33×C4C2×C414yes—
39C2×C33×C4C2312yes—
40C2×C3⋊(C3⋊Dic3)—24no—
41C2×C3⋊(C3⋊Dic3)C2314yes—
42C2×C3×C3⋊Dic3C2×C414yes—
43C2×C3×C3⋊Dic3C2312yes—
44C3×C4×C3⋊D3C2×C4116yescomplemented
45C2×C32×Dic3—24no—
46C2×C32×Dic3C2314yes—
47C2×#926—91no—
48C2×#892—94no—
49C23×C3⋊D3C2×D311yescharacteristic
50C23×C3×D3—34no—
51C22×C32×C4—24no—
52C22×C32×C4—32no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C2×D311yescharacteristic
55C22×C3⋊Dic3—32no—
56C22×C3⋊Dic3—38no—
57C22×C3⋊Dic3—68no—
58C22×C3⋊Dic3C2×D311yescharacteristic
59C2×C4×C3⋊D3Dic318yescomplemented
60C2×C4×C3⋊D3C2×D314yes—
61C22×C3×Dic3—34no—
62C22×C3×Dic3C2×D318yes—
63C2×C3×C4×D3—332no—
64C2×C3×C4×D3—316no—
65C32×C4⋊C4—34no—
66#3752—38no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—316no—
69C4⋊(C3⋊Dic3)—616no—
70C4×C3⋊Dic3—38no—
71C3×C4⋊Dic3C2×D3116yescomplemented
72(C2×D3)⋊Dic3—332no—
73Dic32—332no—
74C22×C33C22×C412yes—
75C22×C33#4112yes—
76C22×C33#4112yes—
77C22×C33C2411yescharacteristic
78C2×C3×C3⋊D3—28no—
79C2×C3×C3⋊D3—22no—
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3C22×C414yes—
82C33×C4C22×C414yes—
83C3⋊(C3⋊Dic3)—24no—
84C3×C3⋊Dic3C22×C414yes—
85C32×Dic3—24no—
86C23×C3×C4—91no—
87C22×C4×D3—34no—
88C2×C3×C4⋊C4—34no—
89C2×#217—92no—
90C2×(C2×D3)⋊C4—98no—
91#926—98no—
92C2×C4⋊Dic3—94no—
93C2×C4⋊Dic3—91no—
94C2×C4⋊Dic3—184no—
95C2×C4⋊Dic3C3⋊D311yescharacteristic, complemented
96C2×C4×Dic3—98no—
97C2×C4×Dic3—92no—
98#892—932no—
99C23×C32—24no—
100C23×C32C22×D314yes—
101C23×C32C22×D311yescharacteristic
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3C2×Dic314yes—
105C22×C3⋊D3C22×D312yes—
106C22×C3×D3—316no—
107C22×C3×D3—38no—
108C22×C3×D3—68no—
109C22×C3×D3—68no—
110C2×C32×C4—216no—
111C2×C32×C4—28no—
112C2×C32×C4—34no—
113C2×C32×C4—64no—
114C2×C32×C4C4×D3116yes—
115C2×C32×C4C2×Dic314yes—
116C2×C32×C4C22×D318yes—
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—316no—
120C2×C3⋊Dic3—616no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—616no—
123C2×C3⋊Dic3—1216no—
124C2×C3⋊Dic3C2×Dic314yes—
125C2×C3⋊Dic3C22×D312yes—
126C4×C3⋊D3C2×Dic3116yescomplemented
127C2×C3×Dic3—216no—
128C2×C3×Dic3—316no—
129C2×C3×Dic3—38no—
130C2×C3×Dic3C22×D3116yes—
131C3×C4×D3—364no—
132C2×#117—271noSylow
133C2×C33#11714yescomplemented
134C2×C33C23×C411yescharacteristic
135C2×C33C2×#4111yescharacteristic
136C2×C33C2×#4111yescharacteristic
137C3×C3⋊D3—24no—
138C24×C3—91no—
139C23×D3—34no—
140C22×C3×C4—24no—
141C22×C3×C4—38no—
142C22×C3×C4—91no—
143C22×C3×C4—98no—
144C22×C3×C4—94no—
145C22×C3×C4D3214yes—
146C22×C3×C4C2×C3⋊D311yescharacteristic
147C22×Dic3—34no—
148C22×Dic3—98no—
149C22×Dic3—92no—
150C22×Dic3—188no—
151C22×Dic3C2×C3⋊D312yes—
152C2×C4×D3—332no—
153C2×C4×D3—316no—
154C3×C4⋊C4—316no—
155#217—98no—
156(C2×D3)⋊C4—932no—
157C4⋊Dic3—916no—
158C4⋊Dic3—94no—
159C4⋊Dic3—1816no—
160C4⋊Dic3C2×C3⋊D314yescomplemented
161C4×Dic3—932no—
162C4×Dic3—98no—
163C22×C32—24no—
164C22×C32—28no—
165C22×C32—28no—
166C22×C32—28no—
167C22×C32C2×C4×D318yes—
168C22×C32#23512yes—
169C22×C32#23512yes—
170C22×C32#23718yes—
171C22×C32#23918yes—
172C22×C32C22×Dic312yes—
173C22×C32C23×D314yes—
174C22×C32C23×D311yescharacteristic
175C2×C3⋊D3—28no—
176C2×C3⋊D3—22no—
177C2×C3⋊D3—22no—
178C2×C3⋊D3C22×Dic314yes—
179C2×C3×D3—316no—
180C2×C3×D3—632no—
181C2×C3×D3—68no—
182C2×C3×D3—68no—
183C32×C4—216no—
184C32×C4—64no—
185C32×C4C2×C4×D3116yes—
186C32×C4C22×Dic314yes—
187C3⋊Dic3—64no—
188C3⋊Dic3—616no—
189C3⋊Dic3—1216no—
190C3⋊Dic3C22×Dic314yes—
191C3×Dic3—216no—
192C3×Dic3—316no—
193C23×C4—91no—
194#117—278no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—271no—
197C2×C22⋊C4—272no—
198C2×C42—272no—
199C33C2×#11711yescharacteristic, complemented, Sylow
200C23×C3—24no—
201C23×C3—92no—
202C23×C3—94no—
203C23×C3—182no—
204C23×C3—182no—
205C23×C3C2×D3214yes—
206C23×C3C22×C3⋊D311yescharacteristic
207C22×D3—38no—
208C22×D3—316no—
209C22×D3—68no—
210C22×D3—68no—
211C2×C3×C4—216no—
212C2×C3×C4—28no—
213C2×C3×C4—316no—
214C2×C3×C4—616no—
215C2×C3×C4—92no—
216C2×C3×C4—94no—
217C2×C3×C4—916no—
218C2×C3×C4D3×Dic3116yes—
219C2×C3×C4C4×C3⋊D314yes—
220C2×C3×C4C2×D3218yes—
221C2×C3×C4C22×C3⋊D312yes—
222C2×Dic3—24no—
223C2×Dic3—38no—
224C2×Dic3—316no—
225C2×Dic3—916no—
226C2×Dic3—94no—
227C2×Dic3—1816no—
228C2×Dic3—1816no—
229C2×Dic3—184no—
230C2×Dic3—3616no—
231C2×Dic3C22×C3⋊D314yes—
232C4×D3—364no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—24no—
236C2×C32—216no—
237C2×C32#892116yescomplemented
238C2×C32#92614yescomplemented
239C2×C32C22×C4×D314yes—
240C2×C32C2×#23511yescharacteristic
241C2×C32C2×#23511yescharacteristic
242C2×C32C2×#23714yes—
243C2×C32C2×#23914yes—
244C2×C32C23×Dic311yescharacteristic
245C3⋊D3—24no—
246C3×D3—616no—
247C24—91no—
248C22×C4—32no—
249C22×C4—91no—
250C22×C4—98no—
251C22×C4—94no—
252C22×C4—272no—
253C22×C4D3×C3⋊D311yescharacteristic
254C4⋊C4—34no—
255C4⋊C4—274no—
256C22⋊C4—278no—
257C42—278no—
258C22×C3—24no—
259C22×C3—28no—
260C22×C3—28no—
261C22×C3—28no—
262C22×C3—94no—
263C22×C3—188no—
264C22×C3—182no—
265C22×C3—182no—
266C22×C3#379518yes—
267C22×C3#379918yes—
268C22×C3C2×D3×Dic318yes—
269C22×C3C2×C4×C3⋊D312yes—
270C22×C3#383012yes—
271C22×C3#383212yes—
272C22×C3C22×D3214yes—
273C22×C3C23×C3⋊D311yescharacteristic
274C2×D3—316no—
275C2×D3—68no—
276C2×D3—68no—
277C2×D3—632no—
278C3×C4—216no—
279C3×C4—616no—
280C3×C4—94no—
281C3×C4C2×D3×Dic3116yes—
282C3×C4C2×C4×C3⋊D314yes—
283Dic3—24no—
284Dic3—316no—
285Dic3—1816no—
286Dic3—184no—
287Dic3—3616no—
288C32—24no—
289C32C2×#89214yescomplemented
290C32C2×#92611yescharacteristic, complemented
291C23—92no—
292C23—94no—
293C23—182no—
294C23—182no—
295C23C2×D3×C3⋊D311yescharacteristic
296C2×C4—34no—
297C2×C4—64no—
298C2×C4—92no—
299C2×C4—94no—
300C2×C4—916no—
301C2×C4—274no—
302C2×C4—544no—
303C2×C4Dic3×C3⋊D314yes—
304C2×C4C2×D3×C3⋊D312yes—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—216no—
309C2×C3—184no—
310C2×C3#63841116yescomplemented
311C2×C3#6410114yescomplemented
312C2×C3C2×#379514yes—
313C2×C3C2×#379914yes—
314C2×C3C22×D3×Dic314yes—
315C2×C3C22×C4×C3⋊D311yescharacteristic
316C2×C3C2×#383011yescharacteristic
317C2×C3C2×#383211yescharacteristic
318D3—616no—
319C22—94no—
320C22—182no—
321C22—182no—
322C22—188no—
323C22#8901412yes—
324C22#8901812yes—
325C22C2×Dic3×C3⋊D312yes—
326C22C22×D3×C3⋊D311yescharacteristic
327C4—64no—
328C4—94no—
329C4—544no—
330C4C2×Dic3×C3⋊D314yes—
331C3—24no—
332C3C2×#6384114yescomplemented
333C3C2×#6410111yescharacteristic, complemented
334C2—184no—
335C2#1172340414yesdirect product factor
336C2C2×#8901411yescharacteristic
337C2C2×#8901811yescharacteristic
338C2C22×Dic3×C3⋊D311yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C33C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊(C3⋊Dic3))C2semidirect
3. C2×C32×C4⋊Dic3C2semidirect
5. C2×Dic3×C3⋊Dic3C2semidirect
6. #11723404C2direct (N × H)
13. C2×C3×C4×C3⋊D3C4semidirect
16. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
17. C32×C4⋊Dic3C22semidirect
19. Dic3×C3⋊Dic3C22semidirect
29. C2×C3×C4⋊Dic3D3semidirect
44. C3×C4×C3⋊D3C2×C4semidirect
59. C2×C4×C3⋊D3Dic3semidirect
71. C3×C4⋊Dic3C2×D3semidirect
95. C2×C4⋊Dic3C3⋊D3semidirect
126. C4×C3⋊D3C2×Dic3semidirect
133. C2×C33#117semidirect
160. C4⋊Dic3C2×C3⋊D3semidirect
199. C33C2×#117semidirect
237. C2×C32#892semidirect
238. C2×C32#926semidirect
289. C32C2×#892semidirect
290. C32C2×#926semidirect
310. C2×C3#63841semidirect
311. C2×C3#64101semidirect
332. C3C2×#63841semidirect
333. C3C2×#64101semidirect
335. C2#11723404direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C4⋊(C3⋊(C3⋊Dic3))2
C2×C32×C4⋊Dic32
C2×#888032
C2×Dic3×C3⋊Dic32
#117234042
C2×#641013
C2×#638413