Finite Group Databasebeta

← #13140706

Subgroups — #13140706

Order 1728.

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

normal subgroups only

This group has 340 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic
3C9×#835C211yesmaximal, characteristic
4C3×#63395C211yesmaximal, characteristic
5#11720278C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic
7#11720262C211yesmaximal, characteristic
8#10591615—31nomaximal
9#10588479—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C4⋊(C3⋊Dic9)—21no—
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C4×C3⋊Dic9—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21Q8×C3⋊Dic3—31no—
22#63872—31no—
23C3×#835—31no—
24C3×#835D311yescharacteristic
25#63575—31no—
26#63573—31no—
27#63572—31no—
28Q8×Dic9—31no—
29Q8×Dic9—61no—
30#63452—31no—
31C9×#103—31no—
32#63395D311yescharacteristic
33#63377—31no—
34Dic9⋊C8—31no—
35#63366—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9Q812yes—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C2×C9×Dic3—21no—
44C2×C3×Dic9—21no—
45C9×C3⋊C8—21no—
46C3×C9⋊C8—21no—
47#4799—91no—
48#4796—91no—
49C2×C32×Q8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—31no—
51C4⋊(C3⋊Dic3)—61no—
52C4×C3⋊Dic3—31no—
53C4×C3⋊Dic3—61no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8C2×D311yescharacteristic
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—61no—
60C2×Q8×C9—21no—
61C2×Q8×C9C2×D311yescharacteristic
62C2×C8×C9—31no—
63C9×C4⋊C4—31no—
64C4⋊Dic9—31no—
65C4⋊Dic9—61no—
66C4⋊Dic9—61no—
67C4⋊Dic9—61no—
68C4⋊Dic9—62no—
69C4⋊Dic9C2×D311yescharacteristic
70Dic9⋊C4—61no—
71C4×Dic9—31no—
72C4×Dic9—61no—
73C4×Dic9—61no—
74C4×Dic9—62no—
75C2×C9⋊C8C2×D311yescharacteristic
76C22×C3×C9C2×D411yescharacteristic
77C3×C4×C9—22no—
78C3×C4×C9C2×Q811yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9—21no—
81C3⋊Dic9—21no—
82C3⋊Dic9—41no—
83C9×Dic3—41no—
84C3×Dic9—41no—
85Q8×Dic3—91no—
86Q8×Dic3—91no—
87Q8×Dic3—181no—
88#890—91no—
89#890—91no—
90C3×#103—31no—
91C3×#103—91no—
92#835—31no—
93#835D911yescharacteristic
94#818—91no—
95#817—91no—
96Dic3⋊C8—91no—
97Dic3⋊C8—91no—
98#808—91no—
99#807—91no—
100C32×Q8—21no—
101C32×Q8Dic612yes—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—61no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—61no—
111Q8×C9—22no—
112Q8×C9—21no—
113Q8×C9—22no—
114Q8×C9Dic612yes—
115C2×C4×C9—22no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—61no—
119C2×C4×C9C22×D311yescharacteristic
120C2×Dic9—21no—
121C2×Dic9—31no—
122C2×Dic9—31no—
123C2×Dic9—61no—
124C2×Dic9—61no—
125C2×Dic9—62no—
126C2×Dic9—61no—
127C8×C9—61no—
128C9⋊C8—21no—
129#479—271noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9#13511yescharacteristic
132C2×C3×C9#13711yescharacteristic
133C2×C3×Q8—21no—
134C2×C3×Q8C2×D911yescharacteristic
135C2×C3×Q8D3211yescharacteristic
136C2×C3×C8—31no—
137C2×C3×C8—91no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—91no—
140C4⋊Dic3—31no—
141C4⋊Dic3—91no—
142C4⋊Dic3—91no—
143C4⋊Dic3—181no—
144C4⋊Dic3—181no—
145C4⋊Dic3—181no—
146C4⋊Dic3—181no—
147C4⋊Dic3—181no—
148C4⋊Dic3—182no—
149C4⋊Dic3C2×D911yescharacteristic
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152C4×Dic3—91no—
153C4×Dic3—91no—
154C4×Dic3—181no—
155C4×Dic3—181no—
156C4×Dic3—181no—
157C4×Dic3—182no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8C2×D911yescharacteristic
160C22×C32D3×D411yescharacteristic
161C32×C4—22no—
162C32×C4C2×Dic611yescharacteristic
163C32×C4#23711yescharacteristic
164C3⋊Dic3—61no—
165C3⋊Dic3—61no—
166C3⋊Dic3—121no—
167C3×Dic3—41no—
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9D3×D411yescharacteristic
171C4×C9—24no—
172C4×C9—22no—
173C4×C9—21no—
174C4×C9—21no—
175C4×C9—121no—
176C4×C9C2×Dic611yescharacteristic
177C4×C9#23711yescharacteristic
178Dic9—41no—
179Dic9—62no—
180Dic9—61no—
181Dic9—61no—
182Dic9—121no—
183Dic9—122no—
184Dic9—121no—
185#125—271no—
186C4×Q8—271no—
187#107—271no—
188#106—271no—
189C4⋊C8—271no—
190#103—31no—
191#103—91no—
192C3×C9#47911yescharacteristic, complemented, Sylow
193C3×Q8—22no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—22no—
197C3×Q8Dic1812yes—
198C3×Q8#62712yes—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4C22×D911yescharacteristic
206C2×C3×C4C2×D3211yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—61no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—182no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218C3×C8—61no—
219C3×C8—181no—
220C3⋊C8—21no—
221C3⋊C8—61no—
222C2×C32#87811yescharacteristic
223C2×C32#91611yescharacteristic
224C2×C32#91811yescharacteristic
225C2×C9—21no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9#87811yescharacteristic
229C2×C9#91511yescharacteristic
230C2×C9#91911yescharacteristic
231C2×Q8D3×D911yescharacteristic
232C2×C8—31no—
233C2×C8—91no—
234C4⋊C4—31no—
235C4⋊C4—91no—
236C4⋊C4—271no—
237C4⋊C4—541no—
238C4⋊C4—541no—
239C4⋊C4—541no—
240C42—271no—
241C42—541no—
242C22×C3—21no—
243C22×C3D4×D911yescharacteristic
244C22×C3#381111yescharacteristic
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—121no—
251C3×C4—361no—
252C3×C4C2×Dic1811yescharacteristic
253C3×C4#369911yescharacteristic
254C3×C4#380511yescharacteristic
255C3×C4C2×#62711yescharacteristic
256Dic3—41no—
257Dic3—121no—
258Dic3—182no—
259Dic3—181no—
260Dic3—181no—
261Dic3—181no—
262Dic3—181no—
263Dic3—361no—
264Dic3—362no—
265Dic3—361no—
266Dic3—361no—
267C32#479911yescharacteristic
268C9—21no—
269C9#479611yescharacteristic, complemented
270Q8—21no—
271Q8#618912yes—
272C2×C4—21no—
273C2×C4—61no—
274C2×C4—181no—
275C2×C4—271no—
276C2×C4—271no—
277C2×C4—541no—
278C2×C4C2×D3×D911yescharacteristic
279C8—61no—
280C8—181no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3#6344011yescharacteristic
285C2×C3#6347811yescharacteristic
286C2×C3#6348011yescharacteristic
287C2×C3#6395011yescharacteristic
288C2×C3#6395111yescharacteristic
289C2×C3#6398211yescharacteristic
290C22#8866811yescharacteristic
291C4—22no—
292C4—121no—
293C4—361no—
294C4—541no—
295C4—541no—
296C4—1081no—
297C4C2×#618911yescharacteristic
298C4#8866311yescharacteristic
299C3—21no—
300C3#1058847911yescharacteristic, complemented
301C3#1059161511yescharacteristic
302C2#1172121611yescharacteristic
303C2#1172122511yescharacteristic
304C2#1172126711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#479
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
192. C3×C9#479semidirect
269. C9#4796semidirect
300. C3#10588479semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
#117209372
C9×#8352
C3×#633952
#117202782
#117202732
#117202622
#105916153
#105884793