Finite Group Databasebeta

← C2×A4×C4⋊(C3⋊D3) #13181458

Subgroups — C2×A4×C4⋊(C3⋊D3) #13181458

Order 1728.

2030 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 2030 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×A4×C3⋊D3C212yesmaximal, complemented
2C2×C32×C4×A4C211yesmaximal, characteristic, complemented
3A4×C4⋊(C3⋊D3)C214yesmaximal, direct product factor
4C23×C4⋊(C3⋊D3)C311yesmaximal, characteristic, complemented
5C2×A4×D12—34nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×A4×C3⋊D3—24no—
8C2×A4×C3⋊D3C2214yescomplemented
9C2×C3×C4⋊(C3⋊D3)—41nomaximal
10C32×C4×A4C2212yescomplemented
11C24×C3⋊D3C2×C312yescomplemented
12C22×D3×A4—38no—
13C23×C32×C4C2×C311yescharacteristic, complemented
14C22×C4⋊(C3⋊D3)—31no—
15C22×C4⋊(C3⋊D3)—32no—
16C22×C4⋊(C3⋊D3)—31no—
17C22×C4⋊(C3⋊D3)—34no—
18C22×C4⋊(C3⋊D3)C2×C314yescomplemented
19C2×C3×C4×A4—24no—
20C2×C3×C4×A4D314yescomplemented
21A4×D12—316no—
22C22×C3×C3⋊D3—42no—
23C2×C32×A4D412yescomplemented
24C2×C32×A4C2311yescharacteristic
25A4×C3⋊D3—24no—
26C2×C33×C4—41no—
27C3×C4⋊(C3⋊D3)—44no—
28C23×D12—34no—
29C2×D4×A4—91no—
30C24×C32C22×C311yescharacteristic
31C23×C3⋊D3—24no—
32C23×C3⋊D3—32no—
33C23×C3⋊D3—32no—
34C23×C3⋊D3—34no—
35C23×C3⋊D3—64no—
36C23×C3⋊D3C22×C314yescomplemented
37C22×C3×A4—24no—
38C22×C3×A4C2×D314yes—
39C2×D3×A4—316no—
40C2×D3×A4—616no—
41C22×C32×C4—31no—
42C22×C32×C4—31no—
43C22×C32×C4—32no—
44C22×C32×C4C22×C312yescomplemented
45C2×C4⋊(C3⋊D3)—32no—
46C2×C4⋊(C3⋊D3)—38no—
47C2×C4⋊(C3⋊D3)—34no—
48C2×C4⋊(C3⋊D3)—32no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C4⋊(C3⋊D3)—34no—
51C2×C4⋊(C3⋊D3)—34no—
52C2×C4⋊(C3⋊D3)—38no—
53C2×C4⋊(C3⋊D3)—34no—
54C2×C4⋊(C3⋊D3)A411yescharacteristic, direct product factor
55C2×C3×D12—124no—
56C3×C4×A4—28no—
57C3×C4×A4C2×D318yescomplemented
58C22×C33—41no—
59C2×C3×C3⋊D3—44no—
60C2×C3×C3⋊D3—84no—
61C32×A4C2×D411yescharacteristic, complemented
62C33×C4—42no—
63C24×D3—38no—
64C23×A4—92no—
65C23×C3×C4C3×D314yescomplemented
66C22×D12—316no—
67C22×D12—94no—
68C22×D12—916no—
69C22×D12—98no—
70C22×D12—94no—
71D4×A4—94no—
72C2×C4×A4—24no—
73C2×C4×A4C3⋊D311yescharacteristic, complemented
74C23×C32—31no—
75C23×C32—32no—
76C23×C32—31no—
77C23×C32C3×D412yescomplemented
78C23×C32C23×C311yescharacteristic
79C22×C3⋊D3—24no—
80C22×C3⋊D3—34no—
81C22×C3⋊D3—32no—
82C22×C3⋊D3—34no—
83C22×C3⋊D3—34no—
84C22×C3⋊D3—34no—
85C22×C3⋊D3—64no—
86C22×C3⋊D3—64no—
87C22×C3⋊D3—68no—
88C22×C3⋊D3—68no—
89C22×C3⋊D3—64no—
90C22×C3⋊D3—64no—
91C22×C3⋊D3—64no—
92C22×C3⋊D3C2×A412yescomplemented
93C22×C3×D3—128no—
94C2×C3×A4—28no—
95C2×C3×A4—24no—
96C2×C3×A4D1218yescomplemented
97C2×C3×A4C22×D314yes—
98D3×A4—616no—
99C2×C32×C4—32no—
100C2×C32×C4—31no—
101C2×C32×C4—32no—
102C2×C32×C4—32no—
103C2×C32×C4—32no—
104C2×C32×C4—44no—
105C2×C32×C4—84no—
106C2×C32×C4C2×A411yescharacteristic, complemented
107C4⋊(C3⋊D3)—38no—
108C4⋊(C3⋊D3)—38no—
109C4⋊(C3⋊D3)—34no—
110C4⋊(C3⋊D3)C2×A414yesdirect product factor
111C3×D12—1216no—
112C23×D4—91noSylow
113C2×C33—42no—
114C2×C33—41no—
115C3×C3⋊D3—84no—
116C24×C3C2×C3×D314yes—
117C23×D3—316no—
118C23×D3—616no—
119C23×D3—98no—
120C23×D3—916no—
121C23×D3—98no—
122C23×D3—1816no—
123C22×A4—24no—
124C22×A4—94no—
125C22×A4—184no—
126C22×A4C2×C3⋊D311yescharacteristic
127C2×C3×D4—361no—
128C22×C3×C4—34no—
129C22×C3×C4—34no—
130C22×C3×C4—38no—
131C22×C3×C4C2×C3×D318yescomplemented
132C2×D12—34no—
133C2×D12—916no—
134C2×D12—916no—
135C2×D12—98no—
136C2×D12—932no—
137C2×D12—98no—
138C2×D12—916no—
139C2×D12—94no—
140C2×D12—932no—
141C2×D12—916no—
142C4×A4—28no—
143C4×A4C2×C3⋊D312yescomplemented
144C22×C32—32no—
145C22×C32—31no—
146C22×C32—32no—
147C22×C32—32no—
148C22×C32—32no—
149C22×C32—31no—
150C22×C32—31no—
151C22×C32—44no—
152C22×C32—84no—
153C22×C32C2×C3×D411yescharacteristic, complemented
154C22×C32C22×A411yescharacteristic
155C2×C3⋊D3—24no—
156C2×C3⋊D3—34no—
157C2×C3⋊D3—64no—
158C2×C3⋊D3—64no—
159C2×C3⋊D3—68no—
160C2×C3⋊D3—68no—
161C2×C3⋊D3—64no—
162C2×C3⋊D3—64no—
163C2×C3⋊D3—64no—
164C2×C3⋊D3C22×A414yescomplemented
165C2×C3×D3—1216no—
166C2×C3×D3—2416no—
167C3×A4—24no—
168C3×A4C2×D1214yescomplemented
169C32×C4—32no—
170C32×C4—48no—
171C32×C4—88no—
172C32×C4C22×A412yescomplemented
173C25—92no—
174C22×D4—94no—
175C22×D4—271no—
176C22×D4—271no—
177C22×D4—274no—
178C22×D4—272no—
179C23×C4C3×C3⋊D311yescharacteristic, complemented
180C33—41noSylow
181C23×C3—34no—
182C23×C3—38no—
183C23×C3—34no—
184C23×C3—362no—
185C23×C3C3×D1218yescomplemented
186C23×C3C22×C3×D314yes—
187C22×D3—38no—
188C22×D3—616no—
189C22×D3—916no—
190C22×D3—916no—
191C22×D3—916no—
192C22×D3—98no—
193C22×D3—916no—
194C22×D3—1816no—
195C22×D3—1816no—
196C22×D3—1832no—
197C22×D3—1832no—
198C22×D3—1816no—
199C22×D3—1816no—
200C22×D3—1816no—
201C2×A4—28no—
202C2×A4—24no—
203C2×A4—184no—
204C2×A4C4⋊(C3⋊D3)12yesdirect product factor
205C2×A4C22×C3⋊D311yescharacteristic
206C3×D4—364no—
207C2×C3×C4—38no—
208C2×C3×C4—34no—
209C2×C3×C4—38no—
210C2×C3×C4—38no—
211C2×C3×C4—38no—
212C2×C3×C4—41no—
213C2×C3×C4—84no—
214C2×C3×C4D3×A414yescomplemented
215D12—316no—
216D12—932no—
217D12—932no—
218D12—916no—
219C2×C32—32no—
220C2×C32—31no—
221C2×C32—31no—
222C2×C32—48no—
223C2×C32—44no—
224C2×C32—88no—
225C2×C32—84no—
226C2×C32D4×A412yescomplemented
227C2×C32C23×A411yescharacteristic
228C3⋊D3—24no—
229C3⋊D3—64no—
230C3×D3—2416no—
231C24—94no—
232C24—184no—
233C24—272no—
234C24—274no—
235C24—272no—
236C24—544no—
237C24C2×C3×C3⋊D311yescharacteristic
238C2×D4—91no—
239C2×D4—274no—
240C2×D4—274no—
241C2×D4—272no—
242C2×D4—278no—
243C2×D4—272no—
244C2×D4—274no—
245C2×D4—274no—
246C2×D4—278no—
247C2×D4—271no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4C2×C3×C3⋊D312yescomplemented
252C22×C3—38no—
253C22×C3—34no—
254C22×C3—38no—
255C22×C3—38no—
256C22×C3—38no—
257C22×C3—34no—
258C22×C3—34no—
259C22×C3—41no—
260C22×C3—84no—
261C22×C3—364no—
262C22×C3—724no—
263C22×C3C2×C3×D1214yescomplemented
264C22×C3C2×D3×A414yes—
265C2×D3—316no—
266C2×D3—616no—
267C2×D3—916no—
268C2×D3—1816no—
269C2×D3—1816no—
270C2×D3—1832no—
271C2×D3—1832no—
272C2×D3—1816no—
273C2×D3—1816no—
274C2×D3—1816no—
275A4—24no—
276A4C2×C4⋊(C3⋊D3)11yescharacteristic, direct product factor
277C3×C4—38no—
278C3×C4—42no—
279C3×C4—88no—
280C3×C4C2×D3×A418yescomplemented
281C32—44no—
282C32—84no—
283C32C2×D4×A411yescharacteristic, complemented
284C23—31no—
285C23—32no—
286C23—31no—
287C23—92no—
288C23—184no—
289C23—274no—
290C23—274no—
291C23—274no—
292C23—274no—
293C23—272no—
294C23—544no—
295C23—544no—
296C23—544no—
297C23—548no—
298C23—544no—
299C23—544no—
300C23—548no—
301C23C3×C4⋊(C3⋊D3)12yescomplemented
302C23C22×C3×C3⋊D311yescharacteristic
303D4—94no—
304D4—278no—
305D4—278no—
306D4—274no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—32no—
310C2×C4—32no—
311C2×C4—32no—
312C2×C4A4×C3⋊D311yescharacteristic, complemented
313C2×C3—34no—
314C2×C3—38no—
315C2×C3—34no—
316C2×C3—42no—
317C2×C3—41no—
318C2×C3—88no—
319C2×C3—84no—
320C2×C3—724no—
321C2×C3A4×D1218yescomplemented
322C2×C3C22×D3×A414yes—
323D3—616no—
324D3—1816no—
325C22—32no—
326C22—31no—
327C22—32no—
328C22—32no—
329C22—32no—
330C22—31no—
331C22—31no—
332C22—94no—
333C22—184no—
334C22—274no—
335C22—544no—
336C22—544no—
337C22—548no—
338C22—544no—
339C22—544no—
340C22—548no—
341C22—544no—
342C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
343C22C2×A4×C3⋊D311yescharacteristic
344C4—32no—
345C4C2×A4×C3⋊D312yescomplemented
346C3—41no—
347C3—84no—
348C3C2×A4×D1214yescomplemented
349C2—31no—
350C2—32no—
351C2—31no—
352C2—184no—
353C2—544no—
354C2A4×C4⋊(C3⋊D3)12yesdirect product factor
355C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC22C2×A4×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC23×C32×C4C2×C3
socleC24×C32C22×C3
hypercenterC2×C4A4×C3⋊D3
hypocenterC22×C32C2×C3×D4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×A4×C3⋊D3C2semidirect
2. C2×C32×C4×A4C2semidirect
3. A4×C4⋊(C3⋊D3)C2direct (N × H)
4. C23×C4⋊(C3⋊D3)C3semidirect
8. C2×A4×C3⋊D3C22semidirect
10. C32×C4×A4C22semidirect
11. C24×C3⋊D3C2×C3semidirect
13. C23×C32×C4C2×C3semidirect
18. C22×C4⋊(C3⋊D3)C2×C3semidirect
20. C2×C3×C4×A4D3semidirect
23. C2×C32×A4D4semidirect
36. C23×C3⋊D3C22×C3semidirect
44. C22×C32×C4C22×C3semidirect
54. C2×C4⋊(C3⋊D3)A4direct (N × H)
57. C3×C4×A4C2×D3semidirect
61. C32×A4C2×D4semidirect
65. C23×C3×C4C3×D3semidirect
73. C2×C4×A4C3⋊D3semidirect
77. C23×C32C3×D4semidirect
92. C22×C3⋊D3C2×A4semidirect
96. C2×C3×A4D12semidirect
106. C2×C32×C4C2×A4semidirect
110. C4⋊(C3⋊D3)C2×A4direct (N × H)
131. C22×C3×C4C2×C3×D3semidirect
143. C4×A4C2×C3⋊D3semidirect
153. C22×C32C2×C3×D4semidirect
164. C2×C3⋊D3C22×A4semidirect
168. C3×A4C2×D12semidirect
172. C32×C4C22×A4semidirect
179. C23×C4C3×C3⋊D3semidirect
185. C23×C3C3×D12semidirect
204. C2×A4C4⋊(C3⋊D3)direct (N × H)
214. C2×C3×C4D3×A4semidirect
226. C2×C32D4×A4semidirect
251. C22×C4C2×C3×C3⋊D3semidirect
263. C22×C3C2×C3×D12semidirect
276. A4C2×C4⋊(C3⋊D3)direct (N × H)
280. C3×C4C2×D3×A4semidirect
283. C32C2×D4×A4semidirect
301. C23C3×C4⋊(C3⋊D3)semidirect
312. C2×C4A4×C3⋊D3semidirect
321. C2×C3A4×D12semidirect
342. C22C2×C3×C4⋊(C3⋊D3)semidirect
345. C4C2×A4×C3⋊D3semidirect
348. C3C2×A4×D12semidirect
354. C2A4×C4⋊(C3⋊D3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4×C3⋊D32
C2×C32×C4×A42
A4×C4⋊(C3⋊D3)2
C23×C4⋊(C3⋊D3)3
C2×A4×D123
C2×C3×C4⋊(C3⋊D3)4