Finite Group Databasebeta

← #13138223

Subgroups — #13138223

Order 1728.

356 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 85 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#11720993C211yesmaximal, characteristic
2#11720927C211yesmaximal, characteristic, complemented
3#11720543C211yesmaximal, characteristic
4C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
5C3×#63377C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic
7#11720279C211yesmaximal, characteristic, complemented
8#10590769—31nomaximal
9#10588531—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×(C3⋊D9)⋊C4—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C4×C9×Dic3C2211yescharacteristic
15C3×C4⋊Dic9C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18#88445—21no—
19#88442—21no—
20Dic9⋊Dic3—21no—
21#63894—31no—
22#63869—31no—
23#63648—31no—
24C3×#817—31no—
25C3×Dic3⋊C8D311yescharacteristic
26#63575—31no—
27#63564—31no—
28C4×D36—31no—
29#63432—31no—
30C9×C4⋊C8—31no—
31#63380—31no—
32#63380—31no—
33#63380—61no—
34#63378—31no—
35#63377D311yescharacteristic, complemented
36C22×C3⋊D9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C4⋊(C3⋊D9)—21no—
40C4⋊(C3⋊D9)Q812yes—
41C2×C9×Dic3D411yescharacteristic
42C2×C9×Dic3D411yescharacteristic
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45(C3⋊D9)⋊C4—42no—
46C3×C8×C9—21no—
47C9×C3⋊C8—21no—
48#4871—91no—
49#4847—91no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×#624—61no—
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—31no—
56C3×C4×Dic3C2×D311yescharacteristic
57C2×C3×C3⋊C8C2×D311yescharacteristic
58#3724—61no—
59Dic3⋊Dic3—61no—
60#3722—61no—
61C2×D36—31no—
62C2×D36—61no—
63C2×C4×D9—61no—
64C2×C8×C9—21no—
65C2×C8×C9—31no—
66C2×C8×C9C2×D311yescharacteristic
67C42×C9—31no—
68(C2×D9)⋊C4—61no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9C2×D311yescharacteristic
72C4⋊Dic9C2×D311yescharacteristic
73Dic9⋊C4—61no—
74Dic9⋊C4—61no—
75C22×C3×C9C2×D411yescharacteristic
76C2×C3⋊D9—22no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—41no—
79C3×C4×C9C2×Q811yescharacteristic
80C3×C4×C9#4111yescharacteristic
81C3⋊Dic9—41no—
82C9×Dic3—21no—
83C9×Dic3—21no—
84C3×Dic9—41no—
85C3×Dic9—41no—
86#893—91no—
87#890—91no—
88C4×D12—91no—
89#870—91no—
90C3×#106—91no—
91C3×C4⋊C8—31no—
92#821—91no—
93#821—91no—
94#821—91no—
95#821—181no—
96#818—91no—
97#817—31no—
98Dic3⋊C8D911yescharacteristic, complemented
99#809—91no—
100#807—91no—
101C22×C3⋊D3—61no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—61no—
104C4⋊(C3⋊D3)—32no—
105C4⋊(C3⋊D3)—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3D1211yescharacteristic
109C2×C3×Dic3D1211yescharacteristic
110#624—122no—
111C22×D9—61no—
112C22×D9—121no—
113C32×C8—21no—
114C3×C3⋊C8—21no—
115C2×C4×C9—21no—
116C2×C4×C9—31no—
117C2×C4×C9—31no—
118C2×C4×C9C22×D311yescharacteristic
119C2×Dic9—21no—
120C2×Dic9—21no—
121C2×Dic9—61no—
122C2×Dic9—121no—
123D36—32no—
124D36—62no—
125D36—61no—
126D36—121no—
127C4×D9—122no—
128C8×C9—21no—
129C8×C9—41no—
130C8×C9—61no—
131#478—271noSylow
132C2×C3×C9C22⋊Q811yescharacteristic
133C2×C3×C9#13511yescharacteristic
134C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
135C3⋊D9—42no—
136C2×D12—91no—
137C2×D12—91no—
138C2×D12—181no—
139C2×C4×D3—181no—
140C2×C4×D3—181no—
141C2×C3×C8—21no—
142C2×C3×C8—31no—
143C2×C3×C8C2×D911yescharacteristic
144C2×C3×C8D3211yescharacteristic
145C3×C4⋊C4—91no—
146C3×C4⋊C4—91no—
147C3×C42—31no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—181no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—181no—
155C4⋊Dic3—181no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3C2×D911yescharacteristic
160C2×C3⋊C8C2×D911yescharacteristic
161C22×C32C2×D1211yescharacteristic
162C2×C3⋊D3—62no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—121no—
165C32×C4#23711yescharacteristic
166C32×C4D3×Q811yescharacteristic
167C3⋊Dic3—121no—
168C3×Dic3—21no—
169C3×Dic3—21no—
170C3×Dic3—121no—
171C3×Dic3—121no—
172C22×C9—21no—
173C22×C9D3×D411yescharacteristic
174C2×D9—62no—
175C2×D9—122no—
176C2×D9—121no—
177C2×D9—121no—
178C2×D9—241no—
179C2×D9—241no—
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—61no—
183C4×C9—61no—
184C4×C9#23511yescharacteristic
185C4×C9D3×Q811yescharacteristic
186Dic9—41no—
187Dic9—41no—
188Dic9—121no—
189Dic9—241no—
190#125—271no—
191C4×D4—271no—
192#107—271no—
193#106—91no—
194C4⋊C8—31no—
195#102—271no—
196#102—271no—
197C3×C9#47811yescharacteristic, complemented, Sylow
198C22×D3—181no—
199C22×D3—181no—
200C22×D3—361no—
201C2×C3×C4—21no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4C22×D911yescharacteristic
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—361no—
213C2×Dic3D3611yescharacteristic
214C2×Dic3D3611yescharacteristic
215D12—92no—
216D12—92no—
217D12—182no—
218D12—181no—
219D12—181no—
220D12—361no—
221C4×D3—362no—
222C4×D3—362no—
223C3×C8—21no—
224C3×C8—21no—
225C3×C8—41no—
226C3×C8—61no—
227C3⋊C8—21no—
228C2×C32#89711yescharacteristic
229C2×C32#90411yescharacteristic
230C2×C32#90811yescharacteristic
231C3⋊D3—122no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9#89611yescharacteristic
236C2×C9#91411yescharacteristic
237C2×C9#91611yescharacteristic
238D9—122no—
239D9—242no—
240C2×D4—271no—
241C22×C4—541no—
242C2×C8—31no—
243C2×C8D3×D911yescharacteristic
244C4⋊C4—91no—
245C4⋊C4—91no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C4⋊C4—541no—
249C22⋊C4—541no—
250C42—31no—
251C22×C3—21no—
252C22×C3C2×D3611yescharacteristic
253C22×C3D3×D1211yescharacteristic
254C2×D3—182no—
255C2×D3—182no—
256C2×D3—362no—
257C2×D3—361no—
258C2×D3—361no—
259C2×D3—361no—
260C2×D3—361no—
261C2×D3—721no—
262C2×D3—721no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—361no—
268C3×C4—361no—
269C3×C4#369911yescharacteristic
270C3×C4Q8×D911yescharacteristic
271C3×C4#379711yescharacteristic
272C3×C4#380411yescharacteristic
273Dic3—21no—
274Dic3—21no—
275Dic3—121no—
276Dic3—121no—
277Dic3—361no—
278Dic3—361no—
279Dic3—721no—
280C32#484711yescharacteristic
281C9—21no—
282C9#487111yescharacteristic, complemented
283C23—541no—
284D4—272no—
285D4—541no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4—541no—
291C2×C4—1082no—
292C2×C4C2×D3×D911yescharacteristic
293C8—21no—
294C8—61no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3#6345911yescharacteristic
299C2×C3#6346711yescharacteristic
300C2×C3#6347011yescharacteristic
301C2×C3#6379411yescharacteristic
302C2×C3#6380611yescharacteristic
303C2×C3#6389511yescharacteristic
304D3—362no—
305D3—362no—
306D3—722no—
307C22—542no—
308C22—1081no—
309C22—1081no—
310C22D3×D3611yescharacteristic
311C4—61no—
312C4—61no—
313C4—361no—
314C4—361no—
315C4—1081no—
316C4(C3⋊D9)⋊Q811yescharacteristic
317C4#8866211yescharacteristic
318C3—21no—
319C3#1058853111yescharacteristic, complemented
320C3#1059076911yescharacteristic
321C2—1082no—
322C2#1172080411yescharacteristic
323C2#1172081011yescharacteristic
324C2#1172099511yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×C8D3×D9
hypocenterC3×C9#478
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. #11720927C2semidirect
4. C9×Dic3⋊C8C2semidirect
5. C3×#63377C2semidirect
7. #11720279C2semidirect
35. #63377D3semidirect
98. Dic3⋊C8D9semidirect
197. C3×C9#478semidirect
282. C9#4871semidirect
319. C3#10588531semidirect

Maximal subgroups

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

subgroupindex
#117209932
#117209272
#117205432
C9×Dic3⋊C82
C3×#633772
#117202842
#117202792
#105907693
#105885313