Finite Group Databasebeta

← #13140839

Subgroups — #13140839

Order 1728.

366 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 117 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
2C2×C9×#216C211yesmaximal, characteristic
3Dic6×Dic9C211yesmaximal, characteristic
4#11720582C211yesmaximal, characteristic
5#11720297C211yesmaximal, characteristic
6#11720278C211yesmaximal, characteristic
7#11720274C211yesmaximal, characteristic
8#10591576—31nomaximal
9#10588864—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×#216C414yescomplemented
15C3×C4⋊Dic9—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4×Dic9—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic9—21no—
21Dic3×Dic9—21no—
22C3×Q8×Dic3—31no—
23C2×C3×#216D311yescharacteristic
24Dic3×Dic6—31no—
25#63662—31no—
26#63573—31no—
27#63565—31no—
28#63554—31no—
29C2×C9×Q16—31no—
30Q8×Dic9—31no—
31Q8×Dic9D311yescharacteristic
32#63395—31no—
33#63395—31no—
34#63395—61no—
35#63377—31no—
36#63375—31no—
37C3×Q8×C9—21no—
38C3×Q8×C9C2×C412yes—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C9×Dic6—21no—
47C9×Dic6C2×C412yes—
48C3⋊(C9⋊C8)—21no—
49C9×C3⋊C8C2×C412yes—
50#5184—91no—
51#5030—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C3×Dic6C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—31no—
55C2×C3⋊(C3⋊C8)—31no—
56C3×#216Dic314yes—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—61no—
59C3×C4×Dic3—31no—
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63Dic32—61no—
64C2×Q8×C9—21no—
65C2×Q8×C9—31no—
66C2×Q8×C9C2×D311yescharacteristic
67C9×Q16—34no—
68C2×C8×C9—31no—
69C4⋊Dic9—21no—
70C4⋊Dic9—31no—
71C4⋊Dic9—61no—
72C4⋊Dic9—61no—
73C4⋊Dic9C2×D311yescharacteristic
74C4×Dic9—21no—
75C4×Dic9—61no—
76C4×Dic9C2×D311yescharacteristic
77C2×C9⋊C8—31no—
78C2×C9⋊C8—61no—
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9—22no—
81C3×C4×C9C22×C411yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—41no—
84C9×Dic3—22no—
85C3×Dic9—21no—
86C3×Dic9—21no—
87C3×Dic9—41no—
88C2×C3×Q16—31no—
89C3×C4×Q8—91no—
90Q8×Dic3—31no—
91Q8×Dic3—91no—
92C2×#216D911yescharacteristic
93C4×Dic6—91no—
94#835—91no—
95#835—91no—
96#835—91no—
97#835—181no—
98#817—91no—
99#815—91no—
100#810—91no—
101#808—91no—
102#803—91no—
103C32×Q8—21no—
104C32×Q8C2×Dic312yes—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—61no—
112C3×Dic6—21no—
113C3×Dic6C2×Dic312yes—
114C3⋊(C3⋊C8)—61no—
115C3×C3⋊C8C2×Dic312yes—
116Q8×C9—21no—
117Q8×C9—22no—
118Q8×C9—32no—
119Q8×C9—41no—
120Q8×C9—61no—
121Q8×C9C4×D312yes—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—41no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—21no—
128C2×Dic9—61no—
129C2×Dic9—121no—
130C2×Dic9#6711yescharacteristic
131C2×Dic9#6711yescharacteristic
132C8×C9—32no—
133C9⋊C8—61no—
134C9⋊C8—121no—
135#441—271noSylow
136C2×C3×C9C4×D411yescharacteristic
137C2×C3×C9#13711yescharacteristic
138C2×C3×C9#13711yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—31no—
141C2×C3×Q8C2×D911yescharacteristic
142C2×C3×Q8D3211yescharacteristic
143C2×Dic6C2×D911yescharacteristic
144C3×Q16—34no—
145C2×C3×C8—31no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C42—91no—
149C3×C42—181no—
150#216Dic914yescomplemented
151C4⋊Dic3—31no—
152C4⋊Dic3—61no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157Dic3⋊C4—181no—
158C4×Dic3—31no—
159C4×Dic3—61no—
160C4×Dic3—181no—
161C4×Dic3—181no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—181no—
165C2×C3⋊C8C2×D911yescharacteristic
166C22×C32D3×D411yescharacteristic
167C32×C4—22no—
168C32×C4#23711yescharacteristic
169C32×C4C22×Dic311yescharacteristic
170C3⋊Dic3—121no—
171C3×Dic3—22no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—121no—
175C22×C9—21no—
176C22×C9C2×#6711yescharacteristic
177C4×C9—22no—
178C4×C9—21no—
179C4×C9—21no—
180C4×C9—42no—
181C4×C9—62no—
182C4×C9C2×C4×D311yescharacteristic
183C4×C9#23511yescharacteristic
184Dic9—21no—
185Dic9—21no—
186Dic9—41no—
187Dic9—121no—
188Dic9—241no—
189C2×Q16—31no—
190C4×Q8—91no—
191C4×Q8—271no—
192#106—271no—
193#103—271no—
194#103—271no—
195#97—271no—
196C3×C9#44111yescharacteristic, complemented, Sylow
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—22no—
200C3×Q8—32no—
201C3×Q8—41no—
202C3×Q8—61no—
203C3×Q8C2×Dic912yes—
204C3×Q8D3×Dic312yes—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—61no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—181no—
213C2×C3×C4C22×D911yescharacteristic
214C2×C3×C4C2×D3211yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—31no—
217C2×Dic3—31no—
218C2×Dic3—61no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221C2×Dic3—361no—
222Dic6—21no—
223Dic6C2×Dic912yes—
224C3×C8—32no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C3⋊C8—361no—
228C3⋊C8C2×Dic912yes—
229C2×C32#91811yescharacteristic
230C2×C32#91811yescharacteristic
231C2×C32D4×Dic311yescharacteristic
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9C4×#6711yescharacteristic
236C2×C9#94211yescharacteristic
237C2×C9#95111yescharacteristic
238C2×Q8—31no—
239C2×Q8D3×D911yescharacteristic
240Q16—34no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C42—91no—
248C42—181no—
249C42—541no—
250C22×C3—21no—
251C22×C3D4×D911yescharacteristic
252C22×C3D3×#6711yescharacteristic
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—42no—
258C3×C4—62no—
259C3×C4—181no—
260C3×C4—181no—
261C3×C4—361no—
262C3×C4#369911yescharacteristic
263C3×C4C22×Dic911yescharacteristic
264C3×C4C2×D3×Dic311yescharacteristic
265C3×C4#380411yescharacteristic
266Dic3—22no—
267Dic3—61no—
268Dic3—61no—
269Dic3—121no—
270Dic3—361no—
271Dic3—361no—
272Dic3—721no—
273C32#518411yescharacteristic
274C9—21no—
275C9#503011yescharacteristic, complemented
276Q8—21no—
277Q8—32no—
278Q8—61no—
279Q8D3×Dic912yes—
280C2×C4—21no—
281C2×C4—61no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×D911yescharacteristic
287C8—32no—
288C8—541no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3#6348011yescharacteristic
293C2×C3#6348011yescharacteristic
294C2×C3D4×Dic911yescharacteristic
295C2×C3#6394311yescharacteristic
296C2×C3#6394811yescharacteristic
297C2×C3Dic3×#6711yescharacteristic
298C22D9×#6711yescharacteristic
299C4—22no—
300C4—62no—
301C4—181no—
302C4—181no—
303C4—361no—
304C4—1081no—
305C4C2×D3×Dic911yescharacteristic
306C4#8866211yescharacteristic
307C3—21no—
308C3#1058886411yescharacteristic, complemented
309C3#1059157611yescharacteristic
310C2#1172121911yescharacteristic
311C2#1172122211yescharacteristic
312C2#67×Dic911yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#441
LCSG ▷ 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
14. C9×#216C4semidirect
150. #216Dic9semidirect
196. C3×C9#441semidirect
275. C9#5030semidirect
308. C3#10588864semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×Dic92
C2×C9×#2162
Dic6×Dic92
#117205822
#117202972
#117202782
#117202742
#105915763
#105888643