Finite Group Databasebeta

← C4×D3×Q8⋊C9 #13147206

Subgroups — C4×D3×Q8⋊C9 #13147206

Order 1728.

392 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×D3×Q8⋊C9C211yesmaximal, characteristic
2C3×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
3Dic3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×C4×D3×Q8C311yesmaximal, characteristic
5C2×C4×Q8⋊C9—31nomaximal
6C2×C4×D3×C9—41nomaximal
7C2×C3×Q8⋊C9C2211yescharacteristic
8D3×Q8⋊C9C412yesdirect product factor
9C2×C3×D3×Q8C2×C311yescharacteristic
10C32×C4×Q8C2×C311yescharacteristic
11C3×Q8×Dic3C2×C311yescharacteristic
12C3×D3×C4⋊C4—31no—
13C3×#887—31no—
14C3×C42×D3—31no—
15C3×C4×Dic6—31no—
16C22×Q8⋊C9—31no—
17C4×Q8⋊C9—21no—
18C4×Q8⋊C9—31no—
19C4×Q8⋊C9D311yescharacteristic, direct product factor
20C22×D3×C9—41no—
21C2×C3×C4×C9—41no—
22C2×C9×Dic3—41no—
23C4×D3×C9—44no—
24C3×Q8⋊C9C2×C411yescharacteristic, complemented
25C2×C3×C4×Q8—31no—
26C4×D3×Q8C911yescharacteristic, complemented
27C2×C32×Q8C22×C311yescharacteristic
28C3×D3×Q8—32no—
29C3×D3×Q8C3×C412yes—
30C2×C3×C4×D3—31no—
31C2×C3×C4×D3—31no—
32C2×C3×C4×D3A411yescharacteristic
33C2×C3×Dic6—31no—
34C32×C4⋊C4—31no—
35C32×C42—31no—
36C3×C4⋊Dic3—31no—
37C3×Dic3⋊C4—32no—
38C3×C4×Dic3—31no—
39C3×C4×Dic3—31no—
40C3×C4×Dic3—31no—
41C22×C4×C9—121no—
42C2×Q8⋊C9—21no—
43C2×Q8⋊C9—32no—
44C2×Q8⋊C9C2×D311yescharacteristic
45C22×C3×C9—41no—
46C2×D3×C9—42no—
47C2×D3×C9—42no—
48C2×D3×C9—42no—
49C3×C4×C9—42no—
50C9×Dic3—42no—
51C22×C3×Q8—31no—
52C2×D3×Q8C2×C911yescharacteristic
53C3×C4×Q8—21no—
54C3×C4×Q8—31no—
55C3×C4×Q8—91no—
56C3×C4×Q8—91no—
57C3×C4×Q8C2×C911yescharacteristic, complemented
58C3×C4×Q8C3×D311yescharacteristic
59C2×C3×C4⋊C4—91no—
60C2×C3×C42—91no—
61Q8×Dic3C2×C911yescharacteristic, complemented
62D3×C4⋊C4—31no—
63#887—31no—
64C42×D3—31no—
65C4×Dic6—31no—
66C22×C3×D3C2×A411yescharacteristic
67C32×Q8—31no—
68C32×Q8C2×C3×C411yescharacteristic
69C2×C32×C4—31no—
70C2×C32×C4—31no—
71C2×C32×C4C2×A411yescharacteristic
72C2×C3×Dic3—31no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3C2×A411yescharacteristic
75C3×C4×D3—32no—
76C3×C4×D3—32no—
77C3×C4×D3—62no—
78C3×C4×D3SL(2,𝔽3)14yes—
79C3×Dic6—32no—
80C3×Dic6—32no—
81C23×C9—121no—
82C2×C4×C9—41no—
83C2×C4×C9—81no—
84C2×C4×C9—124no—
85C2×C4×C9—121no—
86Q8⋊C9—21no—
87Q8⋊C9C4×D311yescharacteristic, direct product factor
88C2×C4×Q8—31noSylow
89C2×C3×C9—41no—
90C2×C3×C9—41no—
91C2×C3×C9—41no—
92D3×C9—44no—
93C2×C3×Q8—21no—
94C2×C3×Q8—32no—
95C2×C3×Q8—91no—
96C2×C3×Q8—92no—
97C2×C3×Q8C22×C911yescharacteristic
98C2×C3×Q8C2×C3×D311yescharacteristic
99C22×C3×C4—31no—
100C22×C3×C4—91no—
101C22×C3×C4—91no—
102D3×Q8—32no—
103D3×Q8C4×C912yescomplemented
104C2×C4×D3—31no—
105C2×C4×D3—31no—
106C2×C4×D3C22⋊C911yescharacteristic
107C2×Dic6—31no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—31no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—92no—
113C3×C42—31no—
114C3×C42—31no—
115C3×C42—61no—
116C3×C42—91no—
117C3×C42—91no—
118C3×C42—91no—
119C4⋊Dic3—31no—
120Dic3⋊C4—32no—
121C4×Dic3—31no—
122C4×Dic3—31no—
123C4×Dic3—31no—
124C22×C32C22×A411yescharacteristic
125C2×C3×D3C4×A412yes—
126C2×C3×D3C2×SL(2,𝔽3)12yes—
127C2×C3×D3#41⋊C312yes—
128C32×C4—31no—
129C32×C4—31no—
130C32×C4—61no—
131C32×C4C2×SL(2,𝔽3)12yes—
132C3×Dic3—32no—
133C3×Dic3—61no—
134C3×Dic3C2×SL(2,𝔽3)12yes—
135C22×C9—41no—
136C22×C9—81no—
137C22×C9—122no—
138C22×C9—122no—
139C22×C9—122no—
140C4×C9—42no—
141C4×C9—82no—
142C4×C9—122no—
143C22×Q8—31no—
144C4×Q8—31no—
145C4×Q8—91no—
146C4×Q8—91no—
147C4×Q8D3×C911yescharacteristic, complemented
148C2×C4⋊C4—91no—
149C2×C42—91no—
150C3×C9—41noSylow
151C23×C3—31no—
152C22×D3C2×C22⋊C911yescharacteristic
153C3×Q8—21no—
154C3×Q8—31no—
155C3×Q8—31no—
156C3×Q8—61no—
157C3×Q8—92no—
158C3×Q8—92no—
159C3×Q8C2×C4×C911yescharacteristic, complemented
160C3×Q8C3×C4×D311yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—34no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—61no—
169C2×C3×C4—61no—
170C2×C3×C4—92no—
171C2×C3×C4—92no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—182no—
175C2×C3×C4C2×C22⋊C911yescharacteristic
176C2×C3×C4D3×A411yescharacteristic
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3C2×C22⋊C911yescharacteristic
180C4×D3—32no—
181C4×D3—32no—
182C4×D3—62no—
183C4×D3Q8⋊C914yesdirect product factor
184Dic6—32no—
185Dic6—32no—
186C2×C32C2×C4×A411yescharacteristic
187C2×C32C22×SL(2,𝔽3)11yescharacteristic
188C2×C32C2×#41⋊C311yescharacteristic
189C3×D3C4×SL(2,𝔽3)14yes—
190C2×C9—41no—
191C2×C9—41no—
192C2×C9—41no—
193C2×C9—81no—
194C2×C9—81no—
195C2×C9—81no—
196C2×C9—124no—
197C2×Q8—32no—
198C2×Q8—91no—
199C2×Q8—92no—
200C2×Q8C2×D3×C911yescharacteristic
201C22×C4—31no—
202C22×C4—91no—
203C22×C4—91no—
204C4⋊C4—31no—
205C4⋊C4—91no—
206C4⋊C4—92no—
207C42—31no—
208C42—91no—
209C42—91no—
210C42—91no—
211C22×C3—21no—
212C22×C3—32no—
213C22×C3—32no—
214C22×C3—32no—
215C22×C3C22×C22⋊C911yescharacteristic
216C22×C3C2×D3×A411yescharacteristic
217C2×D3C4×C22⋊C912yes—
218C2×D3C2×Q8⋊C912yes—
219C2×D3#41⋊C912yes—
220C3×C4—22no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—32no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—61no—
229C3×C4—61no—
230C3×C4—92no—
231C3×C4—121no—
232C3×C4—181no—
233C3×C4C2×Q8⋊C912yescomplemented
234C3×C4D3×SL(2,𝔽3)12yes—
235Dic3—32no—
236Dic3—61no—
237Dic3C2×Q8⋊C912yescomplemented
238C32C2×C4×SL(2,𝔽3)11yescharacteristic
239C9—41no—
240C9—81no—
241C23—31no—
242Q8—31no—
243Q8—92no—
244Q8—92no—
245Q8C4×D3×C911yescharacteristic, complemented
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—34no—
249C2×C4—31no—
250C2×C4—92no—
251C2×C4—92no—
252C2×C4—91no—
253C2×C4—91no—
254C2×C4—182no—
255C2×C4D3×C22⋊C911yescharacteristic
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—34no—
260C2×C3C2×C4×C22⋊C911yescharacteristic
261C2×C3C22×Q8⋊C911yescharacteristic
262C2×C3C2×#41⋊C911yescharacteristic
263C2×C3C4×D3×A411yescharacteristic
264C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
265C2×C3D3×#41⋊C311yescharacteristic
266D3C4×Q8⋊C914yesdirect product factor
267C22—32no—
268C22—32no—
269C22—32no—
270C22C2×D3×C22⋊C911yescharacteristic
271C4—31no—
272C4—31no—
273C4—32no—
274C4—61no—
275C4—92no—
276C4—181no—
277C4D3×Q8⋊C912yesdirect product factor
278C3—21no—
279C3C2×C4×Q8⋊C911yescharacteristic, complemented
280C3C4×D3×SL(2,𝔽3)11yescharacteristic
281C2—34no—
282C2C4×D3×C22⋊C911yescharacteristic
283C2C2×D3×Q8⋊C911yescharacteristic
284C2D3×#41⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FrattiniC22×C3C2×D3×A4
FittingC32×C4×Q8C2×C3
socleC22×C32C22×A4
prime radicalC22×C3×D3C2×A4
cosocleC2×C32×Q8C22×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q8⋊C9C2semidirect
3. Dic3×Q8⋊C9C2semidirect
8. D3×Q8⋊C9C4direct (N × H)
19. C4×Q8⋊C9D3direct (N × H)
24. C3×Q8⋊C9C2×C4semidirect
26. C4×D3×Q8C9semidirect
57. C3×C4×Q8C2×C9semidirect
61. Q8×Dic3C2×C9semidirect
87. Q8⋊C9C4×D3direct (N × H)
103. D3×Q8C4×C9semidirect
147. C4×Q8D3×C9semidirect
159. C3×Q8C2×C4×C9semidirect
183. C4×D3Q8⋊C9direct (N × H)
233. C3×C4C2×Q8⋊C9semidirect
237. Dic3C2×Q8⋊C9semidirect
245. Q8C4×D3×C9semidirect
266. D3C4×Q8⋊C9direct (N × H)
277. C4D3×Q8⋊C9direct (N × H)
279. C3C2×C4×Q8⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Q8⋊C92
C3×C4×Q8⋊C92
Dic3×Q8⋊C92
C3×C4×D3×Q83
C2×C4×Q8⋊C93
C2×C4×D3×C94