Finite Group Databasebeta

← Q8×A4×C3⋊D3 #13181465

Subgroups — Q8×A4×C3⋊D3 #13181465

Order 1728.

1326 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2C4×A4×C3⋊D3C213yesmaximal
3A4×#634C213yesmaximal, complemented
4C22×Q8×C3⋊D3C311yesmaximal, characteristic, complemented
5D3×Q8×A4—34nomaximal
6C2×A4×C3⋊D3C2211yescharacteristic
7C3×Q8×C3⋊D3—41nomaximal
8C32×C4×A4C2213yes—
9A4×C3⋊Dic3C2213yes—
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3—31no—
12C2×Q8×C3⋊D3—33no—
13C2×Q8×C3⋊D3—33no—
14C2×Q8×C3⋊D3—31no—
15C22×C4×C3⋊D3C2×C313yes—
16C22×#634C2×C313yescomplemented
17C3×Q8×A4—24no—
18C3×Q8×A4D314yescomplemented
19C4×D3×A4—312no—
20A4×Dic6—312no—
21C2×C32×A4C2311yescharacteristic
22A4×C3⋊D3Q812yesdirect product factor
23C33×Q8—41no—
24C3×C4×C3⋊D3—43no—
25C3×#634—43no—
26C22×D3×Q8—34no—
27C2×Q8×A4—91no—
28C23×C3⋊D3C22×C311yescharacteristic
29C2×D3×A4—34no—
30C2×C32×Q8—31no—
31C2×C32×Q8—33no—
32C22×C32×C4C22×C313yes—
33C22×C3⋊Dic3C22×C313yes—
34Q8×C3⋊D3—32no—
35Q8×C3⋊D3—36no—
36Q8×C3⋊D3—36no—
37Q8×C3⋊D3—31no—
38Q8×C3⋊D3—33no—
39Q8×C3⋊D3—33no—
40Q8×C3⋊D3A411yescharacteristic, direct product factor
41C2×C4×C3⋊D3—33no—
42C2×C4×C3⋊D3—33no—
43C2×C4×C3⋊D3—33no—
44C2×C4×C3⋊D3—33no—
45C2×#634—33no—
46C2×#634—33no—
47C2×#634—36no—
48C3×D3×Q8—124no—
49C3×C4×A4—212no—
50C3×C4×A4C2×D3112yes—
51Dic3×A4—312no—
52C2×C3×C3⋊D3—41no—
53C32×A4C2×Q811yescharacteristic, complemented
54C33×C4—43no—
55C3×C3⋊Dic3—43no—
56C22×C3×Q8C3×D314yescomplemented
57C2×D3×Q8—94no—
58C2×D3×Q8—912no—
59C2×D3×Q8—912no—
60C2×D3×Q8—94no—
61C22×C4×D3—312no—
62C22×Dic6—312no—
63Q8×A4—24no—
64Q8×A4—93no—
65Q8×A4C3⋊D311yescharacteristic, direct product factor
66C2×C4×A4—93no—
67C23×C32C23×C311yescharacteristic
68C22×C3⋊D3—31no—
69C22×C3⋊D3—32no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3C3×Q812yescomplemented
72C2×C3×A4—24no—
73C2×C3×A4C22×D314yes—
74D3×A4—38no—
75C32×Q8—32no—
76C32×Q8—33no—
77C32×Q8—44no—
78C32×Q8—84no—
79C32×Q8C2×A411yescharacteristic, complemented
80C2×C32×C4—33no—
81C2×C32×C4—33no—
82C2×C3⋊Dic3—33no—
83C2×C3⋊Dic3—33no—
84C4×C3⋊D3—36no—
85C4×C3⋊D3—33no—
86C4×C3⋊D3—33no—
87C4×C3⋊D3—33no—
88C4×C3⋊D3C2×A413yes—
89#634—36no—
90#634—33no—
91#634—36no—
92#634C2×A413yescomplemented
93C3×C4×D3—1212no—
94C3×Dic6—1212no—
95C23×Q8—91noSylow
96C2×C33—41no—
97C3×C3⋊D3—42no—
98C23×D3—34no—
99C22×A4—91no—
100C2×C3×Q8—312no—
101C2×C3×Q8—34no—
102C2×C3×Q8—361no—
103C22×C3×C4C2×C3×D3112yes—
104C22×Dic3—312no—
105D3×Q8—34no—
106D3×Q8—912no—
107D3×Q8—912no—
108D3×Q8—924no—
109D3×Q8—98no—
110D3×Q8—924no—
111D3×Q8—94no—
112C2×C4×D3—912no—
113C2×C4×D3—912no—
114C2×C4×D3—912no—
115C2×C4×D3—912no—
116C2×Dic6—912no—
117C2×Dic6—924no—
118C2×Dic6—912no—
119C4×A4—212no—
120C4×A4—93no—
121C4×A4C2×C3⋊D313yes—
122C22×C32—31no—
123C22×C32—31no—
124C22×C32C2×C3×Q811yescharacteristic, complemented
125C2×C3⋊D3—32no—
126C2×C3⋊D3—31no—
127C2×C3⋊D3—32no—
128C2×C3⋊D3—32no—
129C2×C3⋊D3—32no—
130C2×C3⋊D3C22×A411yescharacteristic
131C2×C3×D3—124no—
132C3×A4—24no—
133C3×A4D3×Q814yescomplemented
134C32×C4—33no—
135C32×C4—412no—
136C32×C4—812no—
137C32×C4C22×A413yes—
138C3⋊Dic3—33no—
139C3⋊Dic3C22×A413yes—
140C3×Dic3—1212no—
141C22×Q8—93no—
142C22×Q8—271no—
143C22×Q8—273no—
144C22×Q8—273no—
145C22×Q8—271no—
146C22×Q8C3×C3⋊D311yescharacteristic, complemented
147C23×C4—93no—
148C33—41noSylow
149C23×C3C22×C3×D314yes—
150C22×D3—38no—
151C22×D3—94no—
152C22×D3—94no—
153C22×D3—98no—
154C2×A4—24no—
155C2×A4—92no—
156C2×A4C22×C3⋊D311yescharacteristic
157C3×Q8—38no—
158C3×Q8—312no—
159C3×Q8—41no—
160C3×Q8—84no—
161C3×Q8—363no—
162C3×Q8D3×A414yescomplemented
163C2×C3×C4—312no—
164C2×C3×C4—312no—
165C2×C3×C4—363no—
166C2×Dic3—912no—
167C2×Dic3—912no—
168C4×D3—312no—
169C4×D3—912no—
170C4×D3—912no—
171C4×D3—924no—
172C4×D3—912no—
173Dic6—312no—
174Dic6—924no—
175Dic6—924no—
176Dic6—912no—
177C2×C32—31no—
178C2×C32—31no—
179C2×C32—44no—
180C2×C32—84no—
181C2×C32C23×A411yescharacteristic
182C3⋊D3—32no—
183C3⋊D3Q8×A412yesdirect product factor
184C3×D3—128no—
185C24—91no—
186C2×Q8—33no—
187C2×Q8—31no—
188C2×Q8—91no—
189C2×Q8—273no—
190C2×Q8—273no—
191C2×Q8—276no—
192C2×Q8—273no—
193C2×Q8—273no—
194C2×Q8—276no—
195C2×Q8—272no—
196C2×Q8—276no—
197C2×Q8—271no—
198C22×C4—93no—
199C22×C4—273no—
200C22×C4—273no—
201C22×C4—273no—
202C22×C4—273no—
203C22×C4C2×C3×C3⋊D313yes—
204C22×C3—34no—
205C22×C3—34no—
206C22×C3—361no—
207C22×C3C3×D3×Q814yescomplemented
208C2×D3—34no—
209C2×D3—98no—
210C2×D3—98no—
211C2×D3—98no—
212C2×D3—94no—
213C2×D3—98no—
214A4—24no—
215A4Q8×C3⋊D311yescharacteristic, direct product factor
216C3×C4—312no—
217C3×C4—43no—
218C3×C4—812no—
219C3×C4—363no—
220C3×C4C2×D3×A4112yes—
221Dic3—312no—
222Dic3—912no—
223C32—44no—
224C32—84no—
225C32C2×Q8×A411yescharacteristic, complemented
226C23—92no—
227C23—271no—
228C23—271no—
229C23—272no—
230C23C22×C3×C3⋊D311yescharacteristic
231Q8—32no—
232Q8—33no—
233Q8—93no—
234Q8—276no—
235Q8—276no—
236Q8—273no—
237Q8A4×C3⋊D311yescharacteristic, direct product factor
238C2×C4—33no—
239C2×C4—33no—
240C2×C4—93no—
241C2×C4—273no—
242C2×C4—273no—
243C2×C4—276no—
244C2×C4—273no—
245C2×C4—273no—
246C2×C4—273no—
247C2×C3—34no—
248C2×C3—34no—
249C2×C3—41no—
250C2×C3—84no—
251C2×C3—362no—
252C2×C3C22×D3×A414yes—
253D3—38no—
254D3—98no—
255C22—31no—
256C22—31no—
257C22—91no—
258C22—272no—
259C22—272no—
260C22—272no—
261C22—272no—
262C22—271no—
263C22C3×Q8×C3⋊D311yescharacteristic, complemented
264C4—33no—
265C4—93no—
266C4—273no—
267C4C2×A4×C3⋊D313yes—
268C3—41no—
269C3—84no—
270C3D3×Q8×A414yescomplemented
271C2—31no—
272C2—31no—
273C2—92no—
274C2—272no—
275C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4×C3⋊D3
commutator, socle, cosocleC23×C32C23×C3
FittingC22×C32×Q8C2×C3
prime radicalC2×A4×C3⋊D3C22
hypercenterQ8A4×C3⋊D3
hypocenterC22×C32C2×C3×Q8
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
3. A4×#634C2semidirect
4. C22×Q8×C3⋊D3C3semidirect
10. C22×C32×Q8C2×C3semidirect
16. C22×#634C2×C3semidirect
18. C3×Q8×A4D3semidirect
22. A4×C3⋊D3Q8direct (N × H)
40. Q8×C3⋊D3A4direct (N × H)
53. C32×A4C2×Q8semidirect
56. C22×C3×Q8C3×D3semidirect
65. Q8×A4C3⋊D3direct (N × H)
71. C22×C3⋊D3C3×Q8semidirect
79. C32×Q8C2×A4semidirect
92. #634C2×A4semidirect
124. C22×C32C2×C3×Q8semidirect
133. C3×A4D3×Q8semidirect
146. C22×Q8C3×C3⋊D3semidirect
162. C3×Q8D3×A4semidirect
183. C3⋊D3Q8×A4direct (N × H)
207. C22×C3C3×D3×Q8semidirect
215. A4Q8×C3⋊D3direct (N × H)
225. C32C2×Q8×A4semidirect
237. Q8A4×C3⋊D3direct (N × H)
263. C22C3×Q8×C3⋊D3semidirect
270. C3D3×Q8×A4semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C4×A4×C3⋊D32
A4×#6342
C22×Q8×C3⋊D33
D3×Q8×A43
C3×Q8×C3⋊D34