Finite Group Databasebeta

← #13137726

Subgroups — #13137726

Order 1728.

360 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#63450C211yesmaximal, characteristic, complemented
2D12×Dic9C211yesmaximal, characteristic
3#11720526C211yesmaximal, characteristic, complemented
4C9×#809C211yesmaximal, characteristic
5#11720294C211yesmaximal, characteristic
6#11720278C211yesmaximal, characteristic, complemented
7Dic9⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590465—31nomaximal
9#10588441—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×Dic18—21no—
12C2×D3×Dic9—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C3×Dic9⋊C4—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21C3×#888—31no—
22Dic3×D12—31no—
23#63635—31no—
24C3×#809D311yescharacteristic
25#63573—31no—
26Dic3⋊(C3⋊C8)—31no—
27#63562—31no—
28D4×Dic9—31no—
29#63450D311yescharacteristic, complemented
30C9×#102—31no—
31#63392—31no—
32#63377—31no—
33Dic9⋊C8—31no—
34#63367—31no—
35#63367—61no—
36C22×D3×C9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3×Dic9—21no—
41C2×C3×Dic9D411yescharacteristic
42C2×C3×Dic9D411yescharacteristic
43C9×D12—21no—
44C9×D12Q812yescomplemented
45C3×Dic18—42no—
46D3×Dic9—42no—
47C3⋊(C9⋊C8)—21no—
48C9×C3⋊C8—21no—
49#5084—91no—
50#4761—91no—
51C2×C3×D12C2×D311yescharacteristic
52C2×C3×Dic6—61no—
53C2×D3×Dic3—61no—
54C32×C4⋊C4C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—31no—
56C2×C3⋊(C3⋊C8)—31no—
57C3×C4⋊Dic3—31no—
58C3×Dic3⋊C4—61no—
59C3×C4×Dic3—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61(C2×D3)⋊Dic3—61no—
62C2×D4×C9—31no—
63C22×Dic9—61no—
64C2×Dic18—21no—
65C2×C8×C9—31no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C2×D311yescharacteristic
68#3676—61no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9C2×D311yescharacteristic
72Dic9⋊C4—21no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C2×D3×C9—22no—
78C2×D3×C9—41no—
79C2×D3×C9—41no—
80C3×C4×C9—41no—
81C3×C4×C9C2×Q811yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—41no—
84C3×Dic9—21no—
85C3×Dic9—41no—
86C3×Dic9SD1612yes—
87C3×C4⋊Q8—91no—
88D4×Dic3—91no—
89#888—31no—
90C4×D12—91no—
91C3×#102—31no—
92#832—91no—
93#817—91no—
94Dic3⋊C8—91no—
95#809—91no—
96#809D911yescharacteristic
97#808—91no—
98#808—91no—
99#808—91no—
100#808—181no—
101C4⋊(C3⋊C8)—91no—
102C22×C3×D3—21no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—61no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—61no—
109C3×D12—21no—
110C3×D12Dic612yes—
111C3×Dic6—122no—
112D3×Dic3—122no—
113C23×C9—61no—
114C3⋊(C3⋊C8)—61no—
115C3×C3⋊C8—21no—
116D4×C9—32no—
117D4×C9—61no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—41no—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—61no—
124C2×Dic9—121no—
125C2×Dic9—122no—
126C2×Dic9#6711yescharacteristic
127C2×Dic9#6711yescharacteristic
128Dic18—42no—
129C8×C9—61no—
130C9⋊C8—62no—
131C9⋊C8—61no—
132#476—271noSylow
133C2×C3×C9C22⋊Q811yescharacteristic
134C2×C3×C9C2×SD1611yescharacteristic
135C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
136D3×C9—42no—
137C2×C3×Q8—181no—
138C2×C3×D4—31no—
139C22×Dic3—181no—
140C2×D12C2×D911yescharacteristic
141C2×C4×D3—181no—
142C2×Dic6—61no—
143C2×C3×C8—31no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—181no—
147C3×C4⋊C4C2×D911yescharacteristic
148C3×C4⋊C4D3211yescharacteristic
149C3×C42—91no—
150#217—181no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—31no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156Dic3⋊C4—61no—
157C4×Dic3—31no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—181no—
161C2×C3⋊C8C2×D911yescharacteristic
162C22×C32D3×D411yescharacteristic
163C2×C3×D3—22no—
164C2×C3×D3—41no—
165C2×C3×D3—41no—
166C32×C4—41no—
167C32×C4C2×Dic611yescharacteristic
168C32×C4#23711yescharacteristic
169C3⋊Dic3—121no—
170C3×Dic3—32no—
171C3×Dic3—61no—
172C3×Dic3—121no—
173C22×C9—21no—
174C22×C9—62no—
175C22×C9—121no—
176C22×C9—121no—
177C22×C9C2×#6711yescharacteristic
178C4×C9—21no—
179C4×C9—21no—
180C4×C9—41no—
181C4×C9—81no—
182C4×C9D3×Q811yescharacteristic
183C4×C9#23911yescharacteristic
184Dic9—21no—
185Dic9—41no—
186Dic9—121no—
187Dic9—241no—
188Dic9#21512yes—
189C4⋊Q8—91no—
190C4×D4—271no—
191#106—271no—
192#106—271no—
193C4⋊C8—271no—
194#102—31no—
195#102—271no—
196C3×C9#47611yescharacteristic, complemented, Sylow
197C23×C3—61no—
198C22×D3—21no—
199C3×Q8—362no—
200C3×D4—32no—
201C3×D4—61no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209C2×C3×C4C22×D911yescharacteristic
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—31no—
212C2×Dic3—31no—
213C2×Dic3—61no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—361no—
217C2×Dic3—362no—
218D12—21no—
219D12Dic1812yescomplemented
220C4×D3—362no—
221Dic6—122no—
222C3×C8—61no—
223C3⋊C8—21no—
224C3⋊C8—182no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C2×C32#87811yescharacteristic
228C2×C32#91111yescharacteristic
229C2×C32D3×SD1611yescharacteristic
230C3×D3—42no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—122no—
235C2×C9#93211yescharacteristic
236C2×C9C2×#21511yescharacteristic
237C2×C9#94611yescharacteristic
238C2×Q8—181no—
239C2×D4—31no—
240C22×C4—541no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C4⋊C4—271no—
246C4⋊C4D3×D911yescharacteristic
247C22⋊C4—541no—
248C42—91no—
249C22×C3—21no—
250C22×C3—62no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3D4×D911yescharacteristic
254C22×C3D3×#6711yescharacteristic
255C2×D3—22no—
256C2×D3—41no—
257C2×D3—41no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—92no—
264C3×C4—181no—
265C3×C4—361no—
266C3×C4C2×Dic1811yescharacteristic
267C3×C4#369911yescharacteristic
268C3×C4D3×Dic611yescharacteristic
269C3×C4#379611yescharacteristic
270Dic3—32no—
271Dic3—61no—
272Dic3—121no—
273Dic3—361no—
274Dic3—361no—
275Dic3—721no—
276C32#476111yescharacteristic
277C9—21no—
278C9#508411yescharacteristic, complemented
279C23—61no—
280Q8—362no—
281D4—32no—
282D4—61no—
283C2×C4—21no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×D911yescharacteristic
290C8—61no—
291C8—541no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—122no—
296C2×C3#6344011yescharacteristic
297C2×C3#6347311yescharacteristic
298C2×C3SD16×D911yescharacteristic
299C2×C3#6388711yescharacteristic
300C2×C3#6393211yescharacteristic
301C2×C3D3×#21511yescharacteristic
302D3—42no—
303C22—62no—
304C22—121no—
305C22—121no—
306C22D9×#6711yescharacteristic
307C4—41no—
308C4—92no—
309C4—181no—
310C4—361no—
311C4—1081no—
312C4D3×Dic1811yescharacteristic
313C4#8863911yescharacteristic
314C3—21no—
315C3#1058844111yescharacteristic, complemented
316C3#1059046511yescharacteristic
317C2—122no—
318C2#1172101811yescharacteristic
319C2#1172117611yescharacteristic
320C2D9×#21511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#476
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#63450C2semidirect
3. #11720526C2semidirect
6. #11720278C2semidirect
7. Dic9⋊(C3⋊C8)C2semidirect
29. #63450D3semidirect
44. C9×D12Q8semidirect
196. C3×C9#476semidirect
219. D12Dic18semidirect
278. C9#5084semidirect
315. C3#10588441semidirect

Maximal subgroups

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

subgroupindex
C3×#634502
D12×Dic92
#117205262
C9×#8092
#117202942
#117202782
Dic9⋊(C3⋊C8)2
#105904653
#105884413