Finite Group Databasebeta

← C4×#88424 #13137242

Subgroups — C4×#88424 #13137242

Order 1728.

436 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic
3C2×#88424C211yesmaximal, characteristic
4C3×C4×C9⋊C8C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic
6#11720269C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8C4×#3716—31nomaximal
9C4×#3672—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C4×D3×C9—21no—
13C3×C42×C9C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C4×C3⋊Dic9—21no—
17C9×(C2×D3)⋊C4—21no—
18C9×C4⋊Dic3C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88424C414yesdirect product factor
22C4×#634—31no—
23C3×C4×D12D311yescharacteristic
24C2×#3716—31no—
25C3×C4×C3⋊C8—31no—
26#63573—31no—
27#63567—31no—
28#63563—31no—
29C4×D4×C9—31no—
30C2×#3672—31no—
31C4×Dic18—31no—
32C4×Dic18—61no—
33#63392—31no—
34#63368—31no—
35#63367—31no—
36C4×C9⋊C8D311yescharacteristic, complemented
37C22×D3×C9—21no—
38C2×C3×C4×C9D411yescharacteristic
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226C2×C412yescomplemented
45C2×C9×Dic3—21no—
46C9×D12—21no—
47C9×D12C2×C412yes—
48C4×D3×C9—42no—
49C3×C9⋊C8—21no—
50C3×C9⋊C8C2×C412yescomplemented
51C4×#214—91no—
52C4×#204—91no—
53C2×#634—31no—
54C2×C3×D12C2×D311yescharacteristic
55C2×C3×C4×D3—21no—
56C32×C42C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)—31no—
58#3750—61no—
59C4×C3⋊Dic3—61no—
60C3×(C2×D3)⋊C4—21no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8—31no—
64#3716—34no—
65C2×D4×C9—31no—
66C22×C4×C9—61no—
67C2×Dic18—31no—
68C2×Dic18—61no—
69C9×C4⋊C4—31no—
70C9×C22⋊C4—61no—
71C42×C9—21no—
72C42×C9C2×D311yescharacteristic
73#3672—34no—
74C4⋊Dic9—31no—
75C4⋊Dic9—61no—
76Dic9⋊C4—62no—
77Dic9⋊C4—61no—
78C4×Dic9—62no—
79C4×Dic9—61no—
80C2×C9⋊C8C2×D311yescharacteristic
81C2×C9⋊C8C2×D311yescharacteristic
82C22×C3×C9C2×D411yescharacteristic
83C2×D3×C9—22no—
84C2×D3×C9—41no—
85C2×D3×C9—41no—
86C3×C4×C9—21no—
87C3×C4×C9SD1612yescomplemented
88C3×C4×C9C22×C411yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—22no—
91C3⋊Dic9—41no—
92C9×Dic3—41no—
93C3×C4×D4—31no—
94C2×#214—91no—
95C2×#204—91no—
96C4×D12D911yescharacteristic
97C4×Dic6—91no—
98C4×Dic6—91no—
99C4×Dic6—181no—
100C3×C4×C8—91no—
101#832—91no—
102#821—91no—
103#817—91no—
104#816—91no—
105#810—91no—
106#808—91no—
107C4×C3⋊C8—31no—
108C22×C3×D3—21no—
109C2×C32×C4#6711yescharacteristic
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114#634—32no—
115#634—61no—
116C2×C3×Dic3—21no—
117C3×D12—21no—
118C3×D12C4×D312yes—
119C3×C4×D3—42no—
120C23×C9—61no—
121C3×C3⋊C8—32no—
122C3×C3⋊C8—61no—
123D4×C9—32no—
124D4×C9—61no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—61no—
129C2×C4×C9—122no—
130C2×C4×C9D1211yescharacteristic
131C2×C4×C9D1211yescharacteristic
132C2×C4×C9C22×D311yescharacteristic
133C2×Dic9—62no—
134C2×Dic9—61no—
135C2×Dic9—62no—
136C2×Dic9—61no—
137Dic18—32no—
138Dic18—62no—
139Dic18—62no—
140Dic18—61no—
141C9⋊C8—21no—
142C9⋊C8C4×D312yescomplemented
143C4×SD16—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9C2×SD1611yescharacteristic
146C2×C3×C9#13511yescharacteristic
147D3×C9—42no—
148C2×C3×D4—31no—
149C22×C3×C4—61no—
150C2×D12C2×D911yescharacteristic
151C2×C4×D3—21no—
152C2×Dic6—91no—
153C2×Dic6—91no—
154C2×Dic6—181no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C3×C4⋊C4—31no—
158C3×C22⋊C4—61no—
159C3×C42—21no—
160C3×C42C2×D911yescharacteristic
161C3×C42D3211yescharacteristic
162#214—94no—
163(C2×D3)⋊C4—21no—
164C4⋊Dic3—91no—
165C4⋊Dic3—91no—
166C4⋊Dic3—181no—
167C4⋊Dic3C2×D911yescharacteristic
168Dic3⋊C4—182no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C4×Dic3—182no—
172C4×Dic3—181no—
173C4×Dic3—181no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176#204—94no—
177C22×C32C2×#6711yescharacteristic
178C2×C3×D3—22no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C32×C4—21no—
182C32×C4#21412yes—
183C32×C4C2×C4×D311yescharacteristic
184C32×C4#23511yescharacteristic
185C3⋊Dic3—62no—
186C3⋊Dic3—121no—
187C3×Dic3—41no—
188C22×C9—21no—
189C22×C9—62no—
190C22×C9—121no—
191C22×C9—121no—
192C22×C9C2×D1211yescharacteristic
193C4×C9—21no—
194C4×C9—22no—
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—41no—
198C4×C9—121no—
199C4×C9#20412yescomplemented
200C4×C9C2×C4×D311yescharacteristic
201C4×C9#23511yescharacteristic
202Dic9—64no—
203Dic9—62no—
204Dic9—122no—
205Dic9—121no—
206C2×SD16—271no—
207C4×Q8—271no—
208C4×D4—31no—
209#106—271no—
210#103—271no—
211#102—271no—
212C4×C8—91no—
213C3×C9C4×SD1611yescharacteristic, complemented, Sylow
214C23×C3—61no—
215C22×D3—21no—
216C3×D4—32no—
217C3×D4—61no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—61no—
222C2×C3×C4—122no—
223C2×C3×C4#61111yescharacteristic
224C2×C3×C4#61111yescharacteristic
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4#62611yescharacteristic
227C2×C3×C4#62611yescharacteristic
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—21no—
230C2×Dic3—182no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233C2×Dic3—182no—
234C2×Dic3—181no—
235C2×Dic3—181no—
236D12—21no—
237D12C4×D912yes—
238C4×D3—42no—
239Dic6—92no—
240Dic6—92no—
241Dic6—182no—
242Dic6—182no—
243Dic6—181no—
244Dic6—181no—
245C3×C8—92no—
246C3×C8—181no—
247C3⋊C8—32no—
248C3⋊C8—61no—
249C2×C32C4×#6711yescharacteristic
250C2×C32C2×#21411yescharacteristic
251C2×C32#95011yescharacteristic
252C3×D3—42no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—122no—
257C2×C9C4×D1211yescharacteristic
258C2×C9C2×#20411yescharacteristic
259C2×C9#90411yescharacteristic
260C2×Q8—271no—
261C2×D4—31no—
262C22×C4—61no—
263SD16—274no—
264C2×C8—91no—
265C2×C8—91no—
266C4⋊C4—31no—
267C4⋊C4—271no—
268C4⋊C4—541no—
269C22⋊C4—61no—
270C42—541no—
271C42D3×D911yescharacteristic
272C22×C3—21no—
273C22×C3—62no—
274C22×C3—121no—
275C22×C3—121no—
276C22×C3C2×#61111yescharacteristic
277C22×C3C2×#62611yescharacteristic
278C2×D3—22no—
279C2×D3—41no—
280C2×D3—41no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4—121no—
288C3×C4#367212yescomplemented
289C3×C4C2×C4×D911yescharacteristic
290C3×C4#369711yescharacteristic
291C3×C4#371612yes—
292C3×C4#379811yescharacteristic
293C3×C4C4×D3211yescharacteristic
294Dic3—41no—
295Dic3—184no—
296Dic3—182no—
297Dic3—182no—
298Dic3—362no—
299Dic3—361no—
300Dic3—361no—
301C32C4×#21411yescharacteristic
302C9—21no—
303C9C4×#20411yescharacteristic, complemented
304C23—61no—
305Q8—272no—
306Q8—541no—
307D4—32no—
308D4—61no—
309C2×C4—61no—
310C2×C4—122no—
311C2×C4—541no—
312C2×C4—541no—
313C2×C4#619511yescharacteristic
314C2×C4#619511yescharacteristic
315C2×C4C2×D3×D911yescharacteristic
316C8—92no—
317C8—181no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—122no—
322C2×C3C4×#61111yescharacteristic
323C2×C3C2×#367211yescharacteristic
324C2×C3#6351311yescharacteristic
325C2×C3C2×#371611yescharacteristic
326C2×C3#6382911yescharacteristic
327C2×C3C4×#62611yescharacteristic
328D3—42no—
329C22—62no—
330C22—121no—
331C22—121no—
332C22C2×#619511yescharacteristic
333C4—21no—
334C4—121no—
335C4—542no—
336C4—1081no—
337C4#8842412yesdirect product factor
338C4#8864011yescharacteristic
339C4C4×D3×D911yescharacteristic
340C3—21no—
341C3C4×#367211yescharacteristic, complemented
342C3C4×#371611yescharacteristic
343C2—122no—
344C2C2×#8842411yescharacteristic
345C2#1172088911yescharacteristic
346C2C4×#619511yescharacteristic
subgroupG/H
centerC2×C4#6195
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C42×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×D12C2×D3
cosocleC2×C3×C4×C9C23
hypercenterC42D3×D9
hypocenterC3×C9C4×SD16
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×#6226C2semidirect
4. C3×C4×C9⋊C8C2semidirect
6. #11720269C2semidirect
7. #11720264C2semidirect
21. #88424C4direct (N × H)
36. C4×C9⋊C8D3semidirect
44. #6226C2×C4semidirect
50. C3×C9⋊C8C2×C4semidirect
87. C3×C4×C9SD16semidirect
142. C9⋊C8C4×D3semidirect
199. C4×C9#204semidirect
213. C3×C9C4×SD16semidirect
288. C3×C4#3672semidirect
303. C9C4×#204semidirect
337. C4#88424direct (N × H)
341. C3C4×#3672semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C4×C9×D122
C2×#884242
C3×C4×C9⋊C82
#117202952
#117202692
#117202642
C4×#37163
C4×#36723