Finite Group Databasebeta

← C4×#88421 #13137221

Subgroups — C4×#88421 #13137221

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C9×D12C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic, complemented
3C2×#88421C211yesmaximal, characteristic
4C4×C3⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
5#11720294C211yesmaximal, characteristic, complemented
6#11720266C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8C4×#3713—31nomaximal
9C4×#3673—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C4×D3×C9—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C3×C4×D9—21no—
14C3×C42×C9C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C9×(C2×D3)⋊C4—21no—
18C3×(C2×D9)⋊C4—21no—
19C9×C4⋊Dic3C2211yescharacteristic
20C3×C4⋊Dic9C2211yescharacteristic
21#88421C414yesdirect product factor
22C3×C4×D12—31no—
23C3×C4×D12D311yescharacteristic
24C2×#3713—31no—
25C4×C3⋊(C3⋊C8)—31no—
26#63576—31no—
27#63562—31no—
28#63562—31no—
29C4×D4×C9—31no—
30C2×#3673—31no—
31C4×D36D311yescharacteristic, complemented
32#63392—31no—
33#63369—31no—
34#63366—31no—
35C4×C9⋊C8—31no—
36C4×C9⋊C8—61no—
37C22×D3×C9—21no—
38C22×C3×D9—21no—
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C9×D12—21no—
45C9×D12C2×C412yescomplemented
46C4×D3×C9—42no—
47C3×D36—21no—
48C3×D36C2×C412yescomplemented
49C3×C4×D9—42no—
50C3⋊(C9⋊C8)—21no—
51C3⋊(C9⋊C8)C2×C412yescomplemented
52C4×#213—91no—
53C4×#213—91no—
54C2×C3×D12—31no—
55C2×C3×D12C2×D311yescharacteristic
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—61no—
58C32×C42C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)—31no—
60C2×C3⋊(C3⋊C8)—31no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65#3713—34no—
66C2×D4×C9—31no—
67C22×C4×C9—61no—
68C2×D36C2×D311yescharacteristic
69C2×C4×D9—21no—
70C9×C4⋊C4—31no—
71C9×C22⋊C4—61no—
72C42×C9—21no—
73C42×C9C2×D311yescharacteristic
74#3673—34no—
75(C2×D9)⋊C4—21no—
76C4⋊Dic9C2×D311yescharacteristic
77C2×C9⋊C8—31no—
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80C2×C9⋊C8—61no—
81C22×C3×C9C2×D411yescharacteristic
82C2×D3×C9—22no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C2×C3×D9—22no—
86C2×C3×D9—41no—
87C2×C3×D9—41no—
88C3×C4×C9—21no—
89C3×C4×C9D812yescomplemented
90C3×C4×C9C22×C411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C9×Dic3—41no—
93C3×Dic9—41no—
94C3×C4×D4—31no—
95C3×C4×D4—91no—
96C2×#213—91no—
97C2×#213—91no—
98C4×D12—31no—
99C4×D12D911yescharacteristic, complemented
100#832—91no—
101#832—91no—
102#809—91no—
103#809—91no—
104#807—91no—
105#807—91no—
106C4×C3⋊C8—91no—
107C4×C3⋊C8—91no—
108C4×C3⋊C8—181no—
109C22×C3×D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4#6711yescharacteristic
112C2×C32×C4#6711yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3×Dic3—21no—
115C2×C3×Dic3—61no—
116C3×D12—21no—
117C3×D12—32no—
118C3×D12—61no—
119C3×D12C4×D312yes—
120C3×C4×D3—42no—
121C3×C4×D3—122no—
122C23×C9—61no—
123C22×D9—21no—
124C3⋊(C3⋊C8)—32no—
125C3⋊(C3⋊C8)—61no—
126D4×C9—32no—
127D4×C9—61no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—61no—
132C2×C4×C9—122no—
133C2×C4×C9#6711yescharacteristic
134C2×C4×C9#6711yescharacteristic
135C2×C4×C9C22×D311yescharacteristic
136C2×Dic9—21no—
137D36—21no—
138D36C4×D312yescomplemented
139C4×D9—42no—
140C9⋊C8—32no—
141C9⋊C8—62no—
142C9⋊C8—62no—
143C9⋊C8—61no—
144C4×D8—271noSylow
145C2×C3×C9C4×D411yescharacteristic
146C2×C3×C9C2×D811yescharacteristic
147C2×C3×C9#13511yescharacteristic
148D3×C9—42no—
149C3×D9—42no—
150C2×C3×D4—31no—
151C2×C3×D4—91no—
152C22×C3×C4—61no—
153C22×C3×C4—181no—
154C2×D12—31no—
155C2×D12C2×D911yescharacteristic
156C2×C4×D3—21no—
157C2×C4×D3—61no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—91no—
160C3×C22⋊C4—61no—
161C3×C22⋊C4—181no—
162C3×C42—21no—
163C3×C42C2×D911yescharacteristic
164C3×C42D3211yescharacteristic
165#213—94no—
166#213—94no—
167(C2×D3)⋊C4—21no—
168(C2×D3)⋊C4—61no—
169C4⋊Dic3—31no—
170C4⋊Dic3C2×D911yescharacteristic
171C2×C3⋊C8—91no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C2×C3⋊C8—181no—
177C22×C32C2×#6711yescharacteristic
178C2×C3×D3—22no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—62no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C32×C4—21no—
185C32×C4#21312yes—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4#23511yescharacteristic
188C3×Dic3—41no—
189C3×Dic3—121no—
190C22×C9—21no—
191C22×C9—62no—
192C22×C9—121no—
193C22×C9—121no—
194C22×C9C2×#6711yescharacteristic
195C2×D9—22no—
196C2×D9—41no—
197C2×D9—41no—
198C4×C9—22no—
199C4×C9—21no—
200C4×C9—22no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—121no—
204C4×C9#21312yescomplemented
205C4×C9C2×C4×D311yescharacteristic
206C4×C9#23511yescharacteristic
207Dic9—41no—
208C2×D8—271no—
209C4×D4—31no—
210C4×D4—91no—
211#107—271no—
212#102—271no—
213#102—271no—
214C4×C8—271no—
215C3×C9C4×D811yescharacteristic, complemented, Sylow
216C23×C3—61no—
217C23×C3—181no—
218C22×D3—21no—
219C22×D3—61no—
220C3×D4—32no—
221C3×D4—61no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—61no—
228C2×C3×C4—122no—
229C2×C3×C4—181no—
230C2×C3×C4—362no—
231C2×C3×C4#61111yescharacteristic
232C2×C3×C4#61111yescharacteristic
233C2×C3×C4C22×D911yescharacteristic
234C2×C3×C4#62511yescharacteristic
235C2×C3×C4#62511yescharacteristic
236C2×C3×C4C2×D3211yescharacteristic
237C2×Dic3—21no—
238C2×Dic3—61no—
239D12—21no—
240D12—32no—
241D12—61no—
242D12C4×D912yescomplemented
243C4×D3—42no—
244C4×D3—122no—
245C3⋊C8—92no—
246C3⋊C8—92no—
247C3⋊C8—182no—
248C3⋊C8—182no—
249C3⋊C8—181no—
250C3⋊C8—181no—
251C2×C32C4×#6711yescharacteristic
252C2×C32C2×#21311yescharacteristic
253C2×C32#95011yescharacteristic
254C3×D3—42no—
255C3×D3—122no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—122no—
260C2×C9C4×#6711yescharacteristic
261C2×C9C2×#21311yescharacteristic
262C2×C9#95011yescharacteristic
263D9—42no—
264C2×D4—31no—
265C2×D4—91no—
266C22×C4—61no—
267C22×C4—181no—
268D8—274no—
269C2×C8—271no—
270C2×C8—271no—
271C4⋊C4—31no—
272C4⋊C4—91no—
273C22⋊C4—61no—
274C22⋊C4—181no—
275C42D3×D911yescharacteristic
276C22×C3—21no—
277C22×C3—62no—
278C22×C3—121no—
279C22×C3—121no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3C2×#61111yescharacteristic
284C22×C3C2×#62511yescharacteristic
285C2×D3—22no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—62no—
289C2×D3—121no—
290C2×D3—121no—
291C3×C4—22no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—22no—
297C3×C4—121no—
298C3×C4—361no—
299C3×C4#367312yescomplemented
300C3×C4C2×C4×D911yescharacteristic
301C3×C4#369711yescharacteristic
302C3×C4#371312yes—
303C3×C4#379811yescharacteristic
304C3×C4C4×D3211yescharacteristic
305Dic3—41no—
306Dic3—121no—
307C32C4×#21311yescharacteristic
308C9—21no—
309C9C4×#21311yescharacteristic, complemented
310C23—61no—
311C23—181no—
312D4—32no—
313D4—61no—
314D4—92no—
315D4—181no—
316C2×C4—61no—
317C2×C4—122no—
318C2×C4—181no—
319C2×C4—362no—
320C2×C4#619411yescharacteristic
321C2×C4#619411yescharacteristic
322C2×C4C2×D3×D911yescharacteristic
323C8—272no—
324C8—541no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—122no—
329C2×C3—362no—
330C2×C3C4×#61111yescharacteristic
331C2×C3C2×#367311yescharacteristic
332C2×C3#6351311yescharacteristic
333C2×C3C2×#371311yescharacteristic
334C2×C3#6382211yescharacteristic
335C2×C3C4×#62511yescharacteristic
336D3—42no—
337D3—122no—
338C22—62no—
339C22—121no—
340C22—121no—
341C22—182no—
342C22—361no—
343C22—361no—
344C22C2×#619411yescharacteristic
345C4—21no—
346C4—121no—
347C4—361no—
348C4#8842112yesdirect product factor
349C4#8864011yescharacteristic
350C4C4×D3×D911yescharacteristic
351C3—21no—
352C3C4×#367311yescharacteristic, complemented
353C3C4×#371311yescharacteristic
354C2—122no—
355C2—362no—
356C2C2×#8842111yescharacteristic
357C2#1172088111yescharacteristic
358C2C4×#619411yescharacteristic
subgroupG/H
centerC2×C4#6194
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C42×C9C22
socleC22×C32C2×#67
prime radicalC2×#88421C2
cosocleC2×C3×C4×C9C23
hypercenterC42D3×D9
hypocenterC3×C9C4×D8
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C9×D12C2semidirect
2. C3×C4×D36C2semidirect
4. C4×C3⋊(C9⋊C8)C2semidirect
5. #11720294C2semidirect
6. #11720266C2semidirect
7. #11720263C2semidirect
21. #88421C4direct (N × H)
31. C4×D36D3semidirect
45. C9×D12C2×C4semidirect
48. C3×D36C2×C4semidirect
51. C3⋊(C9⋊C8)C2×C4semidirect
89. C3×C4×C9D8semidirect
99. C4×D12D9semidirect
138. D36C4×D3semidirect
204. C4×C9#213semidirect
215. C3×C9C4×D8semidirect
242. D12C4×D9semidirect
299. C3×C4#3673semidirect
309. C9C4×#213semidirect
348. C4#88421direct (N × H)
352. C3C4×#3673semidirect

Maximal subgroups

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

subgroupindex
C4×C9×D122
C3×C4×D362
C2×#884212
C4×C3⋊(C9⋊C8)2
#117202942
#117202662
#117202632
C4×#37133
C4×#36733