Finite Group Databasebeta

← D12×Q8⋊C9 #13147207

Subgroups — D12×Q8⋊C9 #13147207

Order 1728.

380 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 81 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⋊C9C212yesmaximal, complemented
2C3×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
3C3×Q8×D12C311yesmaximal, characteristic
4D4×Q8⋊C9—31nomaximal
5C2×C9×D12—41nomaximal
6C2×C3×Q8⋊C9C2211yescharacteristic
7D3×Q8⋊C9—22no—
8C2×C3×D3×Q8C2×C312yes—
9C32×C4×Q8C2×C311yescharacteristic
10C3×#897—32no—
11C3×C4×D12—31no—
12C3×#869—31no—
13C22×Q8⋊C9—32no—
14C4×Q8⋊C9—21no—
15C4×Q8⋊C9D311yescharacteristic, complemented
16C22×D3×C9—42no—
17C2×C3×C4×C9—41no—
18C9×D12—44no—
19C3×Q8⋊C9D411yescharacteristic, complemented
20C3×D4×Q8—31no—
21Q8×D12C911yescharacteristic, complemented
22C2×C32×Q8C22×C311yescharacteristic
23C3×D3×Q8—22no—
24C3×D3×Q8—62no—
25C2×C3×D12A411yescharacteristic
26C2×C3×C4×D3—32no—
27C2×C3×Dic6—32no—
28C32×C4⋊C4—31no—
29C32×C42—31no—
30C3×(C2×D3)⋊C4—32no—
31C3×C4⋊Dic3—32no—
32C3×C4⋊Dic3—31no—
33C2×D4×C9—121no—
34C2×Q8⋊C9—21no—
35C2×Q8⋊C9—62no—
36C2×Q8⋊C9C2×D311yescharacteristic
37C22×C3×C9—41no—
38C2×D3×C9—44no—
39C2×D3×C9—82no—
40C2×D3×C9—82no—
41C3×C4×C9—42no—
42C22×C3×Q8—32no—
43C2×D3×Q8C2×C912yescomplemented
44C3×C4⋊Q8—91no—
45C3×C22⋊Q8—92no—
46C3×C4×Q8—21no—
47C3×C4×Q8C2×C911yescharacteristic, complemented
48C3×C4×Q8C3×D311yescharacteristic
49C3×C4×D4—91no—
50#897—32no—
51C4×D12—31no—
52#869—31no—
53C22×C3×D3C2×A412yes—
54C32×Q8—31no—
55C32×Q8C3×D411yescharacteristic
56C2×C32×C4—31no—
57C2×C32×C4—31no—
58C2×C32×C4C2×A411yescharacteristic
59C2×C3×Dic3—32no—
60C3×D12SL(2,𝔽3)14yes—
61C3×C4×D3—62no—
62C3×C4×D3—62no—
63C3×Dic6—62no—
64C3×Dic6—62no—
65C23×C9—122no—
66D4×C9—124no—
67C2×C4×C9—41no—
68C2×C4×C9—81no—
69Q8⋊C9—21no—
70Q8⋊C9D1211yescharacteristic, direct product factor
71D4×Q8—31noSylow
72C2×C3×C9—41no—
73C2×C3×C9—41no—
74C2×C3×C9—41no—
75D3×C9—84no—
76C2×C3×Q8—21no—
77C2×C3×Q8—62no—
78C2×C3×Q8—92no—
79C2×C3×Q8—182no—
80C2×C3×Q8C22×C911yescharacteristic
81C2×C3×Q8C2×C3×D311yescharacteristic
82C2×C3×D4—31no—
83C22×C3×C4—92no—
84D3×Q8—22no—
85D3×Q8—62no—
86C2×D12C22⋊C911yescharacteristic
87C2×C4×D3—32no—
88C2×Dic6—32no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—61no—
92C3×C4⋊C4—92no—
93C3×C4⋊C4—91no—
94C3×C22⋊C4—92no—
95C3×C42—31no—
96C3×C42—31no—
97C3×C42—61no—
98(C2×D3)⋊C4—32no—
99C4⋊Dic3—32no—
100C4⋊Dic3—31no—
101C22×C32C22×A411yescharacteristic
102C2×C3×D3—22no—
103C2×C3×D3—22no—
104C2×C3×D3C2×SL(2,𝔽3)14yes—
105C32×C4—31no—
106C32×C4—31no—
107C32×C4—61no—
108C32×C4C2×SL(2,𝔽3)12yes—
109C3×Dic3—62no—
110C22×C9—41no—
111C22×C9—81no—
112C22×C9—124no—
113C22×C9—242no—
114C22×C9—242no—
115C4×C9—42no—
116C4×C9—82no—
117C22×Q8—32no—
118C4⋊Q8—91no—
119C22⋊Q8—92no—
120C4×Q8D3×C911yescharacteristic, complemented
121C4×D4—91no—
122C3×C9—41noSylow
123C23×C3—32no—
124C22×D3C2×C22⋊C912yes—
125C3×Q8—21no—
126C3×Q8—31no—
127C3×Q8—31no—
128C3×Q8—61no—
129C3×Q8—182no—
130C3×Q8—182no—
131C3×Q8D4×C911yescharacteristic, complemented
132C3×Q8C3×D1211yescharacteristic
133C3×D4—34no—
134C2×C3×C4—21no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—92no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4C2×C22⋊C911yescharacteristic
145C2×C3×C4D3×A411yescharacteristic
146C2×Dic3—32no—
147D12Q8⋊C914yesdirect product factor
148C4×D3—62no—
149C4×D3—62no—
150Dic6—62no—
151Dic6—62no—
152C2×C32D4×A411yescharacteristic
153C2×C32C22×SL(2,𝔽3)11yescharacteristic
154C2×C322−1+4⋊C311yescharacteristic
155C3×D3—24no—
156C2×C9—41no—
157C2×C9—41no—
158C2×C9—41no—
159C2×C9—81no—
160C2×C9—81no—
161C2×C9—81no—
162C2×C9—244no—
163C2×Q8—62no—
164C2×Q8—92no—
165C2×Q8—182no—
166C2×Q8C2×D3×C911yescharacteristic
167C2×D4—31no—
168C22×C4—92no—
169C4⋊C4—31no—
170C4⋊C4—92no—
171C4⋊C4—91no—
172C22⋊C4—92no—
173C42—31no—
174C22×C3—21no—
175C22×C3—34no—
176C22×C3—62no—
177C22×C3—62no—
178C22×C3C22×C22⋊C911yescharacteristic
179C22×C3C2×D3×A411yescharacteristic
180C2×D3—22no—
181C2×D3—22no—
182C2×D3C2×Q8⋊C914yescomplemented
183C3×C4—22no—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—31no—
187C3×C4—31no—
188C3×C4—61no—
189C3×C4—61no—
190C3×C4—61no—
191C3×C4—61no—
192C3×C4—121no—
193C3×C4—182no—
194C3×C4C2×Q8⋊C912yescomplemented
195C3×C4D3×SL(2,𝔽3)12yes—
196Dic3—62no—
197C32D4×SL(2,𝔽3)11yescharacteristic
198C9—41no—
199C9—81no—
200C23—32no—
201Q8—31no—
202Q8—182no—
203Q8—182no—
204Q8C9×D1211yescharacteristic, complemented
205D4—34no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—92no—
209C2×C4—182no—
210C2×C4—182no—
211C2×C4D3×C22⋊C911yescharacteristic
212C2×C3—21no—
213C2×C3—21no—
214C2×C3—21no—
215C2×C3—64no—
216C2×C3D4×C22⋊C911yescharacteristic
217C2×C3C22×Q8⋊C911yescharacteristic
218C2×C32−1+4⋊C911yescharacteristic
219C2×C3A4×D1211yescharacteristic
220C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
221C2×C3#1010⋊C311yescharacteristic
222D3—24no—
223C22—34no—
224C22—62no—
225C22—62no—
226C22C2×D3×C22⋊C911yescharacteristic
227C4—31no—
228C4—31no—
229C4—61no—
230C4—182no—
231C4D3×Q8⋊C912yescomplemented
232C3—21no—
233C3D4×Q8⋊C911yescharacteristic, complemented
234C3SL(2,𝔽3)×D1211yescharacteristic
235C2—64no—
236C2D12×C22⋊C911yescharacteristic
237C2C2×D3×Q8⋊C911yescharacteristic
238C2#1010⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC2×C3×Q8C22×C9
FittingC32×C4×Q8C2×C3
socleC22×C32C22×A4
prime radicalC2×C3×D12A4
cosocleC2×C32×Q8C22×C3
hypercenterC2×C3×C4D3×A4
hypocenterC3×Q8D4×C9
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×Q8⋊C9C2semidirect
2. C3×C4×Q8⋊C9C2semidirect
15. C4×Q8⋊C9D3semidirect
19. C3×Q8⋊C9D4semidirect
21. Q8×D12C9semidirect
43. C2×D3×Q8C2×C9semidirect
47. C3×C4×Q8C2×C9semidirect
70. Q8⋊C9D12direct (N × H)
120. C4×Q8D3×C9semidirect
131. C3×Q8D4×C9semidirect
147. D12Q8⋊C9direct (N × H)
182. C2×D3C2×Q8⋊C9semidirect
194. C3×C4C2×Q8⋊C9semidirect
204. Q8C9×D12semidirect
231. C4D3×Q8⋊C9semidirect
233. C3D4×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
C3×Q8×D123
D4×Q8⋊C93
C2×C9×D124