Finite Group Databasebeta

← #13138528

Subgroups — #13138528

Order 1728.

362 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#11721025C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic, complemented
3#11720537C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic
5C3×#63377C211yesmaximal, characteristic, complemented
6#11720294C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic, complemented
8#10590776—31nomaximal
9#10588441—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×D3×Dic9—21no—
12C2×#6189—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4⋊Dic9C2211yescharacteristic
19(C2×D3)⋊Dic9—21no—
20#88442—21no—
21#63919—31no—
22#63898—31no—
23#63641—31no—
24C3×#821D311yescharacteristic
25C3×#817—31no—
26#63574—31no—
27#63562—31no—
28D4×Dic9—31no—
29#63450—31no—
30C9×#102—31no—
31#63392—31no—
32#63377D311yescharacteristic, complemented
33Dic9⋊C8—31no—
34Dic9⋊C8—61no—
35#63367—31no—
36C22×D3×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9D411yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C2×C9×Dic3—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9—21no—
43C9×D12—21no—
44C9×D12Q812yescomplemented
45D3×Dic9—42no—
46#6189—42no—
47C3×C8×C9—21no—
48C3⋊(C9⋊C8)—21no—
49#4870—91no—
50#4761—91no—
51C2×C3×D12C2×D311yescharacteristic
52C2×#627—61no—
53C2×D3×Dic3—61no—
54C2×C32×C8C2×D311yescharacteristic
55C4×C3⋊Dic3—31no—
56C2×C3⋊(C3⋊C8)—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60#3724—61no—
61(C2×D3)⋊Dic3—61no—
62C2×D4×C9—31no—
63C22×Dic9—61no—
64C2×Dic18—61no—
65C2×C8×C9—21no—
66C2×C8×C9C2×D311yescharacteristic
67C9×C4⋊C4—31no—
68#3676—61no—
69C4⋊Dic9C2×D311yescharacteristic
70C4⋊Dic9C2×D311yescharacteristic
71Dic9⋊C4—61no—
72C4×Dic9—31no—
73C4×Dic9—61no—
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×C9C2×Q811yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—21no—
83C3⋊Dic9SD1612yes—
84C9×Dic3—41no—
85C3×Dic9—41no—
86C3×Dic9—41no—
87D4×Dic3—91no—
88#893—91no—
89#888—91no—
90#888—91no—
91C3×#106—91no—
92C3×#102—31no—
93#832—91no—
94#821D911yescharacteristic
95#817—31no—
96Dic3⋊C8—91no—
97Dic3⋊C8—91no—
98Dic3⋊C8—181no—
99#809—91no—
100#808—91no—
101#808—91no—
102C22×C3×D3—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—31no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C3×D12—21no—
110C3×D12Dic612yes—
111#627—122no—
112D3×Dic3—122no—
113C23×C9—61no—
114C32×C8—21no—
115C3⋊(C3⋊C8)—61no—
116D4×C9—32no—
117D4×C9—61no—
118C2×C4×C9—21no—
119C2×C4×C9—61no—
120C2×C4×C9C22×D311yescharacteristic
121C2×Dic9—21no—
122C2×Dic9—21no—
123C2×Dic9—31no—
124C2×Dic9—31no—
125C2×Dic9—61no—
126C2×Dic9—61no—
127C2×Dic9—122no—
128Dic18—122no—
129C8×C9—21no—
130C8×C9—41no—
131C9⋊C8—61no—
132C9⋊C8—121no—
133#476—271noSylow
134C2×C3×C9C22⋊Q811yescharacteristic
135C2×C3×C9C2×SD1611yescharacteristic
136C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
137D3×C9—42no—
138C2×C3×D4—31no—
139C22×Dic3—181no—
140C2×D12C2×D911yescharacteristic
141C2×C4×D3—181no—
142C2×Dic6—181no—
143C2×Dic6—181no—
144C2×C3×C8—21no—
145C2×C3×C8C2×D911yescharacteristic
146C2×C3×C8D3211yescharacteristic
147C3×C4⋊C4—31no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150#217—181no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3C2×D911yescharacteristic
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157C4×Dic3—91no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—181no—
163C22×C32D3×D411yescharacteristic
164C2×C3×D3—22no—
165C2×C3×D3—41no—
166C2×C3×D3—41no—
167C32×C4C2×Dic611yescharacteristic
168C32×C4#23711yescharacteristic
169C3⋊Dic3—32no—
170C3⋊Dic3—61no—
171C3×Dic3—41no—
172C3×Dic3—121no—
173C3×Dic3—121no—
174C22×C9—21no—
175C22×C9—62no—
176C22×C9—121no—
177C22×C9—121no—
178C22×C9D3×D411yescharacteristic
179C4×C9—21no—
180C4×C9—21no—
181C4×C9—121no—
182C4×C9#23511yescharacteristic
183C4×C9D3×Q811yescharacteristic
184Dic9—32no—
185Dic9—41no—
186Dic9—41no—
187Dic9—62no—
188Dic9—61no—
189Dic9—121no—
190C4⋊Q8—271no—
191C4×D4—271no—
192#106—91no—
193#106—271no—
194C4⋊C8—271no—
195#102—31no—
196#102—271no—
197C3×C9#47611yescharacteristic, complemented, Sylow
198C23×C3—61no—
199C22×D3—21no—
200C3×D4—32no—
201C3×D4—61no—
202C2×C3×C4—21no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4C22×D911yescharacteristic
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—181no—
216C2×Dic3—181no—
217C2×Dic3—362no—
218D12—21no—
219D12Dic1812yescomplemented
220C4×D3—362no—
221Dic6—362no—
222Dic6—362no—
223C3×C8—21no—
224C3×C8—21no—
225C3×C8—41no—
226C3⋊C8—181no—
227C3⋊C8—181no—
228C3⋊C8—361no—
229C2×C32#87811yescharacteristic
230C2×C32#91111yescharacteristic
231C2×C32D3×SD1611yescharacteristic
232C3×D3—42no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—122no—
237C2×C9#89611yescharacteristic
238C2×C9D3×SD1611yescharacteristic
239C2×C9#91411yescharacteristic
240C2×Q8—541no—
241C2×D4—31no—
242C22×C4—541no—
243C2×C8—271no—
244C2×C8D3×D911yescharacteristic
245C4⋊C4—31no—
246C4⋊C4—91no—
247C4⋊C4—91no—
248C4⋊C4—541no—
249C22⋊C4—541no—
250C42—271no—
251C22×C3—21no—
252C22×C3—62no—
253C22×C3—121no—
254C22×C3—121no—
255C22×C3D4×D911yescharacteristic
256C22×C3#380211yescharacteristic
257C2×D3—22no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—121no—
263C3×C4—361no—
264C3×C4—361no—
265C3×C4C2×Dic1811yescharacteristic
266C3×C4#369911yescharacteristic
267C3×C4D3×Dic611yescharacteristic
268C3×C4#380411yescharacteristic
269Dic3—41no—
270Dic3—92no—
271Dic3—92no—
272Dic3—121no—
273Dic3—121no—
274Dic3—182no—
275Dic3—181no—
276Dic3—181no—
277Dic3—361no—
278C32#476111yescharacteristic
279C9—21no—
280C9#487011yescharacteristic, complemented
281C23—61no—
282Q8—1082no—
283D4—32no—
284D4—61no—
285C2×C4—61no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—271no—
289C2×C4—271no—
290C2×C4—1082no—
291C2×C4C2×D3×D911yescharacteristic
292C8—21no—
293C8—541no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—122no—
298C2×C3#6344011yescharacteristic
299C2×C3#6347311yescharacteristic
300C2×C3SD16×D911yescharacteristic
301C2×C3#6379611yescharacteristic
302C2×C3#6379811yescharacteristic
303C2×C3#6389911yescharacteristic
304D3—42no—
305C22—62no—
306C22—121no—
307C22—121no—
308C22#8864611yescharacteristic
309C4—121no—
310C4—272no—
311C4—361no—
312C4—361no—
313C4—541no—
314C4D3×Dic1811yescharacteristic
315C4#8866211yescharacteristic
316C3—21no—
317C3#1058844111yescharacteristic, complemented
318C3#1059077611yescharacteristic
319C2—122no—
320C2#1172081711yescharacteristic
321C2#1172081911yescharacteristic
322C2#1172102611yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
hypercenterC2×C8D3×D9
hypocenterC3×C9#476
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720971C2semidirect
3. #11720537C2semidirect
5. C3×#63377C2semidirect
7. #11720265C2semidirect
32. #63377D3semidirect
44. C9×D12Q8semidirect
197. C3×C9#476semidirect
219. D12Dic18semidirect
280. C9#4870semidirect
317. C3#10588441semidirect

Maximal subgroups

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

subgroupindex
#117210252
#117209712
#117205372
C9×#8212
C3×#633772
#117202942
#117202652
#105907763
#105884413