Finite Group Databasebeta

← C2×C3×C4⋊C4×D9 #13152399

Subgroups — C2×C3×C4⋊C4×D9 #13152399

Order 1728.

1346 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C212yesmaximal
2C22×C3×C4×D9C211yesmaximal, characteristic
3C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
5C2×C3×Dic9⋊C4C212yesmaximal, complemented
6C3×C4⋊C4×D9C218yesmaximal, direct product factor
7C2×C3×D3×C4⋊C4—31nomaximal
8C2×C4⋊C4×D9C311yesmaximal, characteristic, direct product factor
9C23×C3×D9C2211yescharacteristic
10C22×C3×C4×C9C2212yes—
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×Dic9C2212yes—
13C22×C3×Dic9C2211yescharacteristic
14C2×C3×C4×D9—28no—
15C2×C3×C4×D9C418yescomplemented
16C2×C3×C4×D9C2218yes—
17C2×C3×C4×D9C2214yes—
18C3×C9×C4⋊C4C2214yescomplemented
19C3×C4⋊Dic9C2214yescomplemented
20C3×Dic9⋊C4C2218yescomplemented
21C22×C3×C4×D3—31no—
22C22×C3×C4×D3—32no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×C3×C4⋊Dic3—31no—
25C2×C3×Dic3⋊C4—32no—
26C3×D3×C4⋊C4—38no—
27C22×C4×D9C2×C312yes—
28C22×C4×D9C2×C311yescharacteristic
29C2×C9×C4⋊C4—21no—
30C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
31C2×C4⋊Dic9C2×C311yescharacteristic, complemented
32C2×Dic9⋊C4C2×C312yescomplemented
33C4⋊C4×D9C2×C318yesdirect product factor
34C23×C3×C9C2311yescharacteristic
35C22×C3×D9C2×C414yes—
36C22×C3×D9D414yes—
37C22×C3×D9Q814yes—
38C22×C3×D9C2312yes—
39C2×C3×C4×C9—24no—
40C2×C3×C4×C9C2×C414yescomplemented
41C2×C3×C4×C9C2314yes—
42C2×C3×C4×C9C2312yes—
43C2×C3×Dic9—24no—
44C2×C3×Dic9C2×C414yescomplemented
45C2×C3×Dic9C2314yes—
46C2×C3×Dic9C2312yes—
47C3×C4×D9—216no—
48C3×C4×D9C2×C4116yescomplemented
49C22×C3×C4⋊C4—91no—
50C2×D3×C4⋊C4—31no—
51C23×C3×D3—31no—
52C22×C32×C4C2×D312yes—
53C22×C32×C4C2×D311yescharacteristic
54C22×C3×Dic3—31no—
55C22×C3×Dic3—32no—
56C2×C3×C4×D3—38no—
57C2×C3×C4×D3—34no—
58C2×C3×C4×D3—38no—
59C2×C3×C4×D3—68no—
60C23×D9C22×C311yescharacteristic
61C32×C4⋊C4C2×D314yes—
62C3×C4⋊Dic3—34no—
63C3×Dic3⋊C4—38no—
64C22×C4×C9—21no—
65C22×C4×C9—22no—
66C22×C4×C9C22×C312yes—
67C22×C4×C9C22×C311yescharacteristic
68C22×Dic9C22×C312yes—
69C22×Dic9C22×C311yescharacteristic
70C2×C4×D9—28no—
71C2×C4×D9C3×C418yescomplemented
72C2×C4×D9C22×C318yes—
73C2×C4×D9C22×C314yes—
74C9×C4⋊C4—24no—
75C9×C4⋊C4C22×C314yescomplemented
76C4⋊Dic9C22×C314yescomplemented
77Dic9⋊C4C22×C318yescomplemented
78C22×C3×C9C22×C412yes—
79C22×C3×C9C2×D412yes—
80C22×C3×C9C2×Q812yes—
81C22×C3×C9C2411yescharacteristic
82C2×C3×D9C4⋊C4116yesdirect product factor
83C2×C3×D9C22×C414yes—
84C2×C3×D9C2×D414yes—
85C2×C3×D9C2×Q814yes—
86C3×C4×C9—24no—
87C3×C4×C9C22×C414yescomplemented
88C3×Dic9—24no—
89C3×Dic9C22×C414yescomplemented
90C23×C3×C4—91no—
91C23×C3×C4—92no—
92C22×C4×D3—31no—
93C22×C4×D3—32no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—92no—
96C2×C3×C4⋊C4—91no—
97C2×C3×C4⋊C4—98no—
98C2×C3×C4⋊C4D911yescharacteristic, direct product factor
99C2×C3×C4⋊C4C3×D311yescharacteristic
100C2×C4⋊Dic3—31no—
101C2×Dic3⋊C4—32no—
102D3×C4⋊C4—38no—
103C23×C32C22×D311yescharacteristic
104C22×C3×D3—34no—
105C22×C3×D3—34no—
106C22×C3×D3—34no—
107C22×C3×D3—32no—
108C2×C32×C4—24no—
109C2×C32×C4C4×D314yes—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D312yes—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—64no—
116C3×C4×D3—316no—
117C3×C4×D3—616no—
118C23×C9—21no—
119C23×C9C23×C311yescharacteristic
120C22×D9C2×C3×C414yes—
121C22×D9C3×D414yes—
122C22×D9C3×Q814yes—
123C22×D9C23×C312yes—
124C2×C4×C9—24no—
125C2×C4×C9—24no—
126C2×C4×C9—22no—
127C2×C4×C9—24no—
128C2×C4×C9—44no—
129C2×C4×C9C2×C3×C414yescomplemented
130C2×C4×C9C23×C314yes—
131C2×C4×C9C23×C312yes—
132C2×Dic9—24no—
133C2×Dic9C2×C3×C414yescomplemented
134C2×Dic9C23×C314yes—
135C2×Dic9C23×C312yes—
136C4×D9—216no—
137C4×D9C2×C3×C4116yescomplemented
138C22×C4⋊C4—91noSylow
139C2×C3×C9C2×C4⋊C414yescomplemented
140C2×C3×C9C23×C411yescharacteristic
141C2×C3×C9C22×D411yescharacteristic
142C2×C3×C9C22×Q811yescharacteristic
143C3×D9C2×C4⋊C418yesdirect product factor
144C24×C3—91no—
145C23×D3—31no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—91no—
149C22×C3×C4—98no—
150C22×C3×C4—94no—
151C22×C3×C4—92no—
152C22×C3×C4—98no—
153C22×C3×C4—188no—
154C22×C3×C4C2×D912yes—
155C22×C3×C4C2×D911yescharacteristic
156C22×C3×C4C2×C3×D312yes—
157C22×C3×C4C2×C3×D311yescharacteristic
158C22×Dic3—31no—
159C22×Dic3—32no—
160C2×C4×D3—38no—
161C2×C4×D3—34no—
162C2×C4×D3—38no—
163C2×C4×D3—68no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—98no—
166C3×C4⋊C4—94no—
167C3×C4⋊C4C2×D914yesdirect product factor
168C3×C4⋊C4C2×C3×D314yes—
169C4⋊Dic3—34no—
170Dic3⋊C4—38no—
171C22×C32C2×C4×D312yes—
172C22×C32D3×D412yes—
173C22×C32D3×Q812yes—
174C22×C32C23×D311yescharacteristic
175C2×C3×D3—316no—
176C2×C3×D3—34no—
177C2×C3×D3—34no—
178C2×C3×D3—34no—
179C32×C4—24no—
180C32×C4C2×C4×D314yes—
181C3×Dic3—34no—
182C3×Dic3—64no—
183C22×C9—21no—
184C22×C9—22no—
185C22×C9—22no—
186C22×C9—22no—
187C22×C9C22×C3×C412yes—
188C22×C9C2×C3×D412yes—
189C22×C9C2×C3×Q812yes—
190C22×C9C24×C311yescharacteristic
191C2×D9C3×C4⋊C4116yesdirect product factor
192C2×D9C22×C3×C414yes—
193C2×D9C2×C3×D414yes—
194C2×D9C2×C3×Q814yes—
195C4×C9—24no—
196C4×C9—24no—
197C4×C9—44no—
198C4×C9C22×C3×C414yescomplemented
199Dic9—24no—
200Dic9C22×C3×C414yescomplemented
201C23×C4—91no—
202C23×C4—92no—
203C2×C4⋊C4—92no—
204C2×C4⋊C4—91no—
205C2×C4⋊C4—98no—
206C2×C4⋊C4C3×D911yescharacteristic, direct product factor
207C3×C9C22×C4⋊C411yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3—94no—
210C23×C3—94no—
211C23×C3—94no—
212C23×C3—92no—
213C23×C3C22×D911yescharacteristic
214C23×C3C22×C3×D311yescharacteristic
215C22×D3—32no—
216C22×D3—34no—
217C22×D3—34no—
218C22×D3—34no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—22no—
223C2×C3×C4—24no—
224C2×C3×C4—44no—
225C2×C3×C4—94no—
226C2×C3×C4—92no—
227C2×C3×C4—916no—
228C2×C3×C4—94no—
229C2×C3×C4—1816no—
230C2×C3×C4—184no—
231C2×C3×C4C4×D914yescomplemented
232C2×C3×C4C22×D914yes—
233C2×C3×C4C22×D912yes—
234C2×C3×C4C3×C4×D314yes—
235C2×C3×C4C22×C3×D314yes—
236C2×C3×C4C22×C3×D312yes—
237C2×Dic3—32no—
238C2×Dic3—34no—
239C2×Dic3—34no—
240C2×Dic3—64no—
241C4×D3—316no—
242C4×D3—616no—
243C2×C32D3×C4⋊C414yes—
244C2×C32C22×C4×D311yescharacteristic
245C2×C32C2×D3×D411yescharacteristic
246C2×C32C2×D3×Q811yescharacteristic
247C3×D3—38no—
248C2×C9—24no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9C2×C3×C4⋊C414yescomplemented
253C2×C9C23×C3×C411yescharacteristic
254C2×C9C22×C3×D411yescharacteristic
255C2×C9C22×C3×Q811yescharacteristic
256D9C2×C3×C4⋊C418yesdirect product factor
257C24—91no—
258C22×C4—91no—
259C22×C4—98no—
260C22×C4—94no—
261C22×C4—92no—
262C22×C4—98no—
263C22×C4—188no—
264C22×C4C2×C3×D912yes—
265C22×C4C2×C3×D911yescharacteristic
266C4⋊C4—98no—
267C4⋊C4—94no—
268C4⋊C4C2×C3×D914yesdirect product factor
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—94no—
274C22×C3—94no—
275C22×C3—916no—
276C22×C3—94no—
277C22×C3C2×C4×D912yes—
278C22×C3D4×D912yes—
279C22×C3Q8×D912yes—
280C22×C3C23×D911yescharacteristic
281C22×C3C2×C3×C4×D312yes—
282C22×C3C3×D3×D412yes—
283C22×C3C3×D3×Q812yes—
284C22×C3C23×C3×D311yescharacteristic
285C2×D3—316no—
286C2×D3—34no—
287C2×D3—34no—
288C2×D3—34no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—44no—
293C3×C4—94no—
294C3×C4—184no—
295C3×C4C2×C4×D914yescomplemented
296C3×C4C2×C3×C4×D314yes—
297Dic3—34no—
298Dic3—64no—
299C32C2×D3×C4⋊C411yescharacteristic
300C9—21no—
301C9C22×C3×C4⋊C411yescharacteristic, complemented
302C23—92no—
303C23—94no—
304C23—94no—
305C23—94no—
306C23C22×C3×D911yescharacteristic
307C2×C4—24no—
308C2×C4—92no—
309C2×C4—94no—
310C2×C4—916no—
311C2×C4—94no—
312C2×C4—1816no—
313C2×C4—184no—
314C2×C4C3×C4×D914yescomplemented
315C2×C4C22×C3×D914yes—
316C2×C4C22×C3×D912yes—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—24no—
321C2×C3—98no—
322C2×C3C4⋊C4×D914yesdirect product factor
323C2×C3C22×C4×D911yescharacteristic
324C2×C3C2×D4×D911yescharacteristic
325C2×C3C2×Q8×D911yescharacteristic
326C2×C3C3×D3×C4⋊C414yes—
327C2×C3C22×C3×C4×D311yescharacteristic
328C2×C3C2×C3×D3×D411yescharacteristic
329C2×C3C2×C3×D3×Q811yescharacteristic
330D3—38no—
331C22—94no—
332C22—94no—
333C22—94no—
334C22—916no—
335C22C2×C3×C4×D912yes—
336C22C3×D4×D912yes—
337C22C3×Q8×D912yes—
338C22C23×C3×D911yescharacteristic
339C4—24no—
340C4—94no—
341C4—184no—
342C4C2×C3×C4×D914yescomplemented
343C3—21no—
344C3C2×C4⋊C4×D911yescharacteristic, direct product factor
345C3C2×C3×D3×C4⋊C411yescharacteristic
346C2—98no—
347C2C3×C4⋊C4×D914yesdirect product factor
348C2C22×C3×C4×D911yescharacteristic
349C2C2×C3×D4×D911yescharacteristic
350C2C2×C3×Q8×D911yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC22×C9C24×C3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C22×C3×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C9×C4⋊C4C2semidirect
4. C2×C3×C4⋊Dic9C2semidirect
5. C2×C3×Dic9⋊C4C2semidirect
6. C3×C4⋊C4×D9C2direct (N × H)
8. C2×C4⋊C4×D9C3direct (N × H)
15. C2×C3×C4×D9C4semidirect
18. C3×C9×C4⋊C4C22semidirect
19. C3×C4⋊Dic9C22semidirect
20. C3×Dic9⋊C4C22semidirect
30. C2×C9×C4⋊C4C2×C3semidirect
31. C2×C4⋊Dic9C2×C3semidirect
32. C2×Dic9⋊C4C2×C3semidirect
33. C4⋊C4×D9C2×C3direct (N × H)
40. C2×C3×C4×C9C2×C4semidirect
44. C2×C3×Dic9C2×C4semidirect
48. C3×C4×D9C2×C4semidirect
71. C2×C4×D9C3×C4semidirect
75. C9×C4⋊C4C22×C3semidirect
76. C4⋊Dic9C22×C3semidirect
77. Dic9⋊C4C22×C3semidirect
82. C2×C3×D9C4⋊C4direct (N × H)
87. C3×C4×C9C22×C4semidirect
89. C3×Dic9C22×C4semidirect
98. C2×C3×C4⋊C4D9direct (N × H)
129. C2×C4×C9C2×C3×C4semidirect
133. C2×Dic9C2×C3×C4semidirect
137. C4×D9C2×C3×C4semidirect
139. C2×C3×C9C2×C4⋊C4semidirect
143. C3×D9C2×C4⋊C4direct (N × H)
167. C3×C4⋊C4C2×D9direct (N × H)
191. C2×D9C3×C4⋊C4direct (N × H)
198. C4×C9C22×C3×C4semidirect
200. Dic9C22×C3×C4semidirect
206. C2×C4⋊C4C3×D9direct (N × H)
207. C3×C9C22×C4⋊C4semidirect
231. C2×C3×C4C4×D9semidirect
252. C2×C9C2×C3×C4⋊C4semidirect
256. D9C2×C3×C4⋊C4direct (N × H)
268. C4⋊C4C2×C3×D9direct (N × H)
295. C3×C4C2×C4×D9semidirect
301. C9C22×C3×C4⋊C4semidirect
314. C2×C4C3×C4×D9semidirect
322. C2×C3C4⋊C4×D9direct (N × H)
342. C4C2×C3×C4×D9semidirect
344. C3C2×C4⋊C4×D9direct (N × H)
347. C2C3×C4⋊C4×D9direct (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×D92
C2×C3×C9×C4⋊C42
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C3×C4⋊C4×D92
C2×C3×D3×C4⋊C43
C2×C4⋊C4×D93