Finite Group Databasebeta

← #13140693

Subgroups — #13140693

Order 1728.

330 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 97 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×Q8×Dic9C211yesmaximal, characteristic
2#11720969C211yesmaximal, characteristic
3#11720582C211yesmaximal, characteristic
4C9×#835C211yesmaximal, characteristic
5#11720278C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic
8#10591544—31nomaximal
9#10588471—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6189—21no—
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×C4⋊Dic9—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4×Dic9—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21C3×Q8×Dic3—31no—
22#63918—31no—
23#63662—31no—
24C3×#835D311yescharacteristic
25#63574—31no—
26#63573—31no—
27Dic3⋊(C3⋊C8)—31no—
28Q8×Dic9D311yescharacteristic
29#63450—31no—
30C9×#103—31no—
31#63395—31no—
32#63395—61no—
33#63377—31no—
34Dic9⋊C8—31no—
35#63367—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9Q812yescomplemented
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45#6189—42no—
46C3⋊(C9⋊C8)—21no—
47C9×C3⋊C8—21no—
48#5020—91no—
49#4791—91no—
50C2×C32×Q8C2×D311yescharacteristic
51C2×#627—61no—
52C4⋊(C3⋊Dic3)—31no—
53C2×C3⋊(C3⋊C8)—31no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—61no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×C4×Dic3—31no—
58C3×C4×Dic3—61no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60Dic3⋊Dic3—61no—
61C2×Q8×C9—21no—
62C2×Q8×C9C2×D311yescharacteristic
63C2×Dic18—61no—
64C2×C8×C9—31no—
65C9×C4⋊C4—31no—
66C4⋊Dic9—21no—
67C4⋊Dic9—31no—
68C4⋊Dic9—61no—
69C4⋊Dic9C2×D311yescharacteristic
70Dic9⋊C4—61no—
71C4×Dic9—21no—
72C4×Dic9C2×D311yescharacteristic
73C2×C9⋊C8—31no—
74C2×C9⋊C8—61no—
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—22no—
77C3×C4×C9C2×Q811yescharacteristic
78C3×C4×C9#4111yescharacteristic
79C3⋊Dic9—41no—
80C9×Dic3—41no—
81C3×Dic9—21no—
82C3×Dic9—41no—
83C3×Dic9SD1612yes—
84C3×C4×Q8—91no—
85Q8×Dic3—31no—
86#888—91no—
87#869—91no—
88C3×#103—31no—
89#835—91no—
90#835—91no—
91#835—181no—
92#835D911yescharacteristic
93#817—91no—
94Dic3⋊C8—91no—
95#808—91no—
96#808—91no—
97#808—91no—
98C4⋊(C3⋊C8)—91no—
99C32×Q8—21no—
100C32×Q8Dic612yes—
101C2×C32×C4—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—61no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—31no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—61no—
108#627—122no—
109C3⋊(C3⋊C8)—61no—
110C3×C3⋊C8—21no—
111Q8×C9—21no—
112Q8×C9—22no—
113Q8×C9—41no—
114Q8×C9Dic612yescomplemented
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9—41no—
118C2×C4×C9—61no—
119C2×C4×C9C22×D311yescharacteristic
120C2×Dic9—21no—
121C2×Dic9—61no—
122C2×Dic9—121no—
123C2×Dic9#6711yescharacteristic
124C2×Dic9#6711yescharacteristic
125Dic18—122no—
126C8×C9—61no—
127C9⋊C8—61no—
128C9⋊C8—121no—
129#475—271noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9C2×SD1611yescharacteristic
132C2×C3×C9#13711yescharacteristic
133C2×C3×Q8—21no—
134C2×C3×Q8C2×D911yescharacteristic
135C2×C3×Q8D3211yescharacteristic
136C2×Dic6—181no—
137C2×Dic6—181no—
138C2×C3×C8—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C42—91no—
143C3×C42—181no—
144C4⋊Dic3—31no—
145C4⋊Dic3—61no—
146C4⋊Dic3—91no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149C4⋊Dic3—181no—
150C4⋊Dic3C2×D911yescharacteristic
151Dic3⋊C4—181no—
152C4×Dic3—31no—
153C4×Dic3—61no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—181no—
157C2×C3⋊C8C2×D911yescharacteristic
158C22×C32D3×D411yescharacteristic
159C32×C4—22no—
160C32×C4C2×Dic611yescharacteristic
161C32×C4#23711yescharacteristic
162C3⋊Dic3—121no—
163C3×Dic3—32no—
164C3×Dic3—41no—
165C3×Dic3—61no—
166C3×Dic3—121no—
167C22×C9—21no—
168C22×C9C2×#6711yescharacteristic
169C4×C9—22no—
170C4×C9—21no—
171C4×C9—21no—
172C4×C9—42no—
173C4×C9—121no—
174C4×C9C2×Dic611yescharacteristic
175C4×C9#23511yescharacteristic
176Dic9—21no—
177Dic9—41no—
178Dic9—121no—
179Dic9—241no—
180Dic9#21512yes—
181C4⋊Q8—271no—
182C4×Q8—91no—
183#106—271no—
184#106—271no—
185C4⋊C8—271no—
186#103—31no—
187#103—271no—
188C3×C9#47511yescharacteristic, complemented, Sylow
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—22no—
192C3×Q8—41no—
193C3×Q8Dic1812yescomplemented
194C3×Q8#62712yes—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—61no—
200C2×C3×C4—91no—
201C2×C3×C4—91no—
202C2×C3×C4—181no—
203C2×C3×C4C22×D911yescharacteristic
204C2×C3×C4C2×D3211yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—31no—
207C2×Dic3—31no—
208C2×Dic3—61no—
209C2×Dic3—181no—
210C2×Dic3—181no—
211C2×Dic3—361no—
212Dic6—362no—
213Dic6—362no—
214C3×C8—61no—
215C3⋊C8—21no—
216C3⋊C8—181no—
217C3⋊C8—181no—
218C3⋊C8—361no—
219C2×C32#87811yescharacteristic
220C2×C32D3×SD1611yescharacteristic
221C2×C32#91811yescharacteristic
222C2×C9—21no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9#92411yescharacteristic
226C2×C9C2×#21511yescharacteristic
227C2×C9#95111yescharacteristic
228C2×Q8—541no—
229C2×Q8D3×D911yescharacteristic
230C2×C8—31no—
231C2×C8—271no—
232C4⋊C4—31no—
233C4⋊C4—91no—
234C4⋊C4—181no—
235C4⋊C4—271no—
236C4⋊C4—541no—
237C42—91no—
238C42—181no—
239C22×C3—21no—
240C22×C3D4×D911yescharacteristic
241C22×C3D3×#6711yescharacteristic
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—42no—
247C3×C4—92no—
248C3×C4—121no—
249C3×C4—181no—
250C3×C4—361no—
251C3×C4C2×Dic1811yescharacteristic
252C3×C4#369911yescharacteristic
253C3×C4#380411yescharacteristic
254C3×C4C2×#62711yescharacteristic
255Dic3—32no—
256Dic3—41no—
257Dic3—61no—
258Dic3—121no—
259Dic3—361no—
260Dic3—361no—
261Dic3—721no—
262C32#479111yescharacteristic
263C9—21no—
264C9#502011yescharacteristic, complemented
265Q8—21no—
266Q8—1082no—
267Q8#618912yescomplemented
268C2×C4—21no—
269C2×C4—61no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—181no—
273C2×C4—541no—
274C2×C4C2×D3×D911yescharacteristic
275C8—61no—
276C8—541no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3#6344011yescharacteristic
281C2×C3SD16×D911yescharacteristic
282C2×C3#6348011yescharacteristic
283C2×C3D3×#21511yescharacteristic
284C2×C3#6394311yescharacteristic
285C2×C3#6396411yescharacteristic
286C22D9×#6711yescharacteristic
287C4—22no—
288C4—92no—
289C4—121no—
290C4—181no—
291C4—361no—
292C4—1081no—
293C4C2×#618911yescharacteristic
294C4#8866211yescharacteristic
295C3—21no—
296C3#1058847111yescharacteristic, complemented
297C3#1059154411yescharacteristic
298C2D9×#21511yescharacteristic
299C2#1172121911yescharacteristic
300C2#1172125311yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#475
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
37. C3×Q8×C9Q8semidirect
114. Q8×C9Dic6semidirect
188. C3×C9#475semidirect
193. C3×Q8Dic18semidirect
264. C9#5020semidirect
267. Q8#6189semidirect
296. C3#10588471semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×Dic92
#117209692
#117205822
C9×#8352
#117202782
Dic9⋊(C3⋊C8)2
#117202652
#105915443
#105884713