Finite Group Databasebeta

← C32×C4×D3×Q8 #13178948

Subgroups — C32×C4×D3×Q8 #13178948

Order 1728.

2068 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×Q8C211yesmaximal, characteristic
2C33×C4×Q8C211yesmaximal, characteristic, complemented
3C32×Q8×Dic3C211yesmaximal, characteristic, complemented
4C32×D3×C4⋊C4C213yesmaximal
5C32×#887C213yesmaximal, complemented
6C32×C42×D3C213yesmaximal
7C32×C4×Dic6C213yesmaximal, complemented
8C2×C32×C4×Q8—31nomaximal
9C3×C4×D3×Q8C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C32×D3×Q8C418yesdirect product factor
12C2×C32×C4×D3C2213yes—
13C2×C32×C4×D3C2213yes—
14C2×C32×C4×D3C2211yescharacteristic
15C2×C32×Dic6C2213yes—
16C33×C4⋊C4C2213yes—
17C33×C42C2213yes—
18C32×C4⋊Dic3C2213yes—
19C32×Dic3⋊C4C2216yes—
20C32×C4×Dic3C2213yes—
21C32×C4×Dic3C2213yes—
22C32×C4×Dic3C2213yes—
23C22×C32×Q8—31no—
24C2×C3×D3×Q8C2×C314yes—
25C32×C4×Q8—24no—
26C32×C4×Q8—31no—
27C32×C4×Q8—33no—
28C32×C4×Q8—33no—
29C32×C4×Q8D311yescharacteristic, direct product factor
30C32×C4×Q8C2×C314yescomplemented
31C2×C32×C4⋊C4—33no—
32C2×C32×C42—33no—
33C3×Q8×Dic3C2×C314yescomplemented
34C3×D3×C4⋊C4C2×C3112yes—
35C3×#887C2×C3112yescomplemented
36C3×C42×D3C2×C3112yes—
37C3×C4×Dic6C2×C3112yescomplemented
38C22×C32×D3C2311yescharacteristic
39C33×Q8C2×C414yescomplemented
40C2×C33×C4C2313yes—
41C2×C33×C4C2313yes—
42C2×C33×C4C2311yescharacteristic
43C2×C32×Dic3C2313yes—
44C2×C32×Dic3C2313yes—
45C2×C32×Dic3C2311yescharacteristic
46C32×C4×D3—26no—
47C32×C4×D3C2×C4112yes—
48C32×C4×D3Q814yesdirect product factor
49C32×Dic6C2×C4112yescomplemented
50C2×C3×C4×Q8—34no—
51C4×D3×Q8C3211yescharacteristic, direct product factor
52C2×C32×Q8—24no—
53C2×C32×Q8—38no—
54C2×C32×Q8—33no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C22×C314yes—
57C22×C32×C4—33no—
58C22×C32×C4—31no—
59C22×C32×C4—33no—
60C3×D3×Q8C3×C4132yesdirect product factor
61C2×C3×C4×D3C22×C3112yes—
62C2×C3×C4×D3C22×C3112yes—
63C2×C3×C4×D3C22×C314yes—
64C2×C3×Dic6C22×C3112yes—
65C32×C4⋊C4—212no—
66C32×C4⋊C4—36no—
67C32×C4⋊C4—33no—
68C32×C4⋊C4C2×D313yes—
69C32×C4⋊C4C22×C3112yes—
70C32×C42—212no—
71C32×C42—33no—
72C32×C42—33no—
73C32×C42—33no—
74C32×C42C2×D313yes—
75C32×C42C22×C3112yes—
76C3×C4⋊Dic3C22×C3112yes—
77C3×Dic3⋊C4C22×C3124yes—
78C3×C4×Dic3C22×C3112yes—
79C3×C4×Dic3C22×C3112yes—
80C3×C4×Dic3C22×C3112yes—
81C22×C33C2411yescharacteristic
82C2×C32×D3C22×C412yes—
83C2×C32×D3C2×Q812yes—
84C2×C32×D3#4112yes—
85C33×C4—23no—
86C33×C4C22×C416yes—
87C33×C4C2×Q812yescomplemented
88C32×Dic3—23no—
89C32×Dic3C22×C416yes—
90C32×Dic3C2×Q812yescomplemented
91C22×C3×Q8—34no—
92C2×D3×Q8C2×C3211yescharacteristic
93C3×C4×Q8—24no—
94C3×C4×Q8—34no—
95C3×C4×Q8—312no—
96C3×C4×Q8—312no—
97C3×C4×Q8C3×D314yesdirect product factor
98C3×C4×Q8C2×C3211yescharacteristic, complemented
99C2×C3×C4⋊C4—312no—
100C2×C3×C42—312no—
101Q8×Dic3C2×C3211yescharacteristic, complemented
102D3×C4⋊C4C2×C3213yes—
103#887C2×C3213yescomplemented
104C42×D3C2×C3213yes—
105C4×Dic6C2×C3213yescomplemented
106C23×C32—31no—
107C22×C3×D3C23×C314yes—
108C32×Q8—216no—
109C32×Q8—312no—
110C32×Q8C4×D314yesdirect product factor
111C32×Q8C2×C3×C4116yescomplemented
112C2×C32×C4—24no—
113C2×C32×C4—212no—
114C2×C32×C4—212no—
115C2×C32×C4—31no—
116C2×C32×C4—33no—
117C2×C32×C4—312no—
118C2×C32×C4—34no—
119C2×C32×C4—33no—
120C2×C32×C4—66no—
121C2×C32×C4C22×D313yes—
122C2×C32×C4C22×D313yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C3112yes—
125C2×C32×C4C23×C3112yes—
126C2×C32×C4C23×C314yes—
127C2×C3×Dic3C23×C3112yes—
128C2×C3×Dic3C23×C3112yes—
129C2×C3×Dic3C23×C314yes—
130C3×C4×D3—224no—
131C3×C4×D3C2×C3×C4148yes—
132C3×C4×D3C3×Q8116yesdirect product factor
133C3×Dic6C2×C3×C4148yescomplemented
134C2×C4×Q8—31noSylow
135C2×C33C23×C411yescharacteristic
136C2×C33C22×Q811yescharacteristic
137C2×C33C2×#4111yescharacteristic
138C32×D3C4×Q814yesdirect product factor
139C2×C3×Q8—24no—
140C2×C3×Q8—332no—
141C2×C3×Q8—312no—
142C2×C3×Q8C2×C3×D314yes—
143C2×C3×Q8C22×C3211yescharacteristic
144C22×C3×C4—34no—
145C22×C3×C4—312no—
146C22×C3×C4—312no—
147D3×Q8C32×C418yesdirect product factor
148C2×C4×D3C22×C3213yes—
149C2×C4×D3C22×C3213yes—
150C2×C4×D3C22×C3211yescharacteristic
151C2×Dic6C22×C3213yes—
152C3×C4⋊C4—212no—
153C3×C4⋊C4—324no—
154C3×C4⋊C4—312no—
155C3×C4⋊C4C2×C3×D3112yes—
156C3×C4⋊C4C22×C3213yes—
157C3×C42—212no—
158C3×C42—312no—
159C3×C42—312no—
160C3×C42—312no—
161C3×C42C2×C3×D3112yes—
162C3×C42C22×C3213yes—
163C4⋊Dic3C22×C3213yes—
164Dic3⋊C4C22×C3216yes—
165C4×Dic3C22×C3213yes—
166C4×Dic3C22×C3213yes—
167C4×Dic3C22×C3213yes—
168C22×C32—24no—
169C22×C32—32no—
170C22×C32—32no—
171C22×C32—32no—
172C22×C32C23×D311yescharacteristic
173C22×C32C24×C314yes—
174C2×C3×D3C22×C3×C418yes—
175C2×C3×D3C2×C3×Q818yes—
176C2×C3×D3C3×#4118yes—
177C32×C4—23no—
178C32×C4—224no—
179C32×C4—28no—
180C32×C4—212no—
181C32×C4—36no—
182C32×C4—32no—
183C32×C4—412no—
184C32×C4—63no—
185C32×C4C2×C4×D316yes—
186C32×C4D3×Q812yesdirect product factor
187C32×C4C22×C3×C4124yes—
188C32×C4C2×C3×Q818yescomplemented
189C3×Dic3—212no—
190C3×Dic3C22×C3×C4124yes—
191C3×Dic3C2×C3×Q818yescomplemented
192C22×Q8—31no—
193C4×Q8—31no—
194C4×Q8—33no—
195C4×Q8—33no—
196C4×Q8C32×D311yescharacteristic, direct product factor
197C2×C4⋊C4—33no—
198C2×C42—33no—
199C33C2×C4×Q811yescharacteristic, complemented, Sylow
200C23×C3—34no—
201C22×D3C23×C3211yescharacteristic
202C3×Q8—216no—
203C3×Q8—348no—
204C3×Q8C3×C4×D3116yesdirect product factor
205C3×Q8C2×C32×C414yescomplemented
206C2×C3×C4—24no—
207C2×C3×C4—212no—
208C2×C3×C4—212no—
209C2×C3×C4—34no—
210C2×C3×C4—312no—
211C2×C3×C4—348no—
212C2×C3×C4—316no—
213C2×C3×C4—312no—
214C2×C3×C4—624no—
215C2×C3×C4C22×C3×D3112yes—
216C2×C3×C4C22×C3×D3112yes—
217C2×C3×C4C22×C3×D314yes—
218C2×C3×C4C23×C3213yes—
219C2×C3×C4C23×C3213yes—
220C2×C3×C4C23×C3211yescharacteristic
221C2×Dic3C23×C3213yes—
222C2×Dic3C23×C3213yes—
223C2×Dic3C23×C3211yescharacteristic
224C4×D3—26no—
225C4×D3C2×C32×C4112yes—
226C4×D3C32×Q814yesdirect product factor
227Dic6C2×C32×C4112yescomplemented
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—34no—
232C2×C32C22×C4×D311yescharacteristic
233C2×C32C2×D3×Q811yescharacteristic
234C2×C32D3×#4111yescharacteristic
235C2×C32C23×C3×C414yes—
236C2×C32C22×C3×Q814yes—
237C2×C32C2×C3×#4114yes—
238C3×D3C3×C4×Q8116yesdirect product factor
239C2×Q8—38no—
240C2×Q8—33no—
241C2×Q8C2×C32×D311yescharacteristic
242C22×C4—31no—
243C22×C4—33no—
244C22×C4—33no—
245C4⋊C4—36no—
246C4⋊C4—33no—
247C4⋊C4C2×C32×D313yes—
248C42—33no—
249C42—33no—
250C42—33no—
251C42C2×C32×D313yes—
252C22×C3—24no—
253C22×C3—38no—
254C22×C3—38no—
255C22×C3—38no—
256C22×C3C23×C3×D314yes—
257C22×C3C24×C3211yescharacteristic
258C2×D3C22×C32×C412yes—
259C2×D3C2×C32×Q812yes—
260C2×D3C32×#4112yes—
261C3×C4—212no—
262C3×C4—23no—
263C3×C4—224no—
264C3×C4—28no—
265C3×C4—324no—
266C3×C4—38no—
267C3×C4—412no—
268C3×C4—612no—
269C3×C4C2×C3×C4×D3124yes—
270C3×C4C3×D3×Q818yesdirect product factor
271C3×C4C22×C32×C416yes—
272C3×C4C2×C32×Q812yescomplemented
273Dic3—23no—
274Dic3C22×C32×C416yes—
275Dic3C2×C32×Q812yescomplemented
276C32—24no—
277C32C4×D3×Q811yescharacteristic, direct product factor
278C32C2×C3×C4×Q814yescomplemented
279C23—31no—
280Q8—312no—
281Q8C32×C4×D314yesdirect product factor
282C2×C4—31no—
283C2×C4—33no—
284C2×C4—312no—
285C2×C4—34no—
286C2×C4—33no—
287C2×C4—66no—
288C2×C4C22×C32×D313yes—
289C2×C4C22×C32×D313yes—
290C2×C4C22×C32×D311yescharacteristic
291C2×C3—24no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—316no—
295C2×C3C22×C3×C4×D314yes—
296C2×C3C2×C3×D3×Q814yes—
297C2×C3C3×D3×#4114yes—
298C2×C3C23×C32×C411yescharacteristic
299C2×C3C22×C32×Q811yescharacteristic
300C2×C3C2×C32×#4111yescharacteristic
301D3C32×C4×Q814yesdirect product factor
302C22—32no—
303C22—32no—
304C22—32no—
305C22C23×C32×D311yescharacteristic
306C4—23no—
307C4—36no—
308C4—32no—
309C4—63no—
310C4C2×C32×C4×D316yes—
311C4C32×D3×Q812yesdirect product factor
312C3—24no—
313C3C3×C4×D3×Q814yesdirect product factor
314C3C2×C32×C4×Q811yescharacteristic, complemented
315C2—34no—
316C2C22×C32×C4×D311yescharacteristic
317C2C2×C32×D3×Q811yescharacteristic
318C2C32×D3×#4111yescharacteristic
subgroupG/H
centerC2×C32×C4C22×D3
commutatorC2×C3C23×C32×C4
FrattiniC22C23×C32×D3
FittingC33×C4×Q8C2
socleC22×C33C24
prime radicalC22×C32×D3C23
cosocleC22×C3C24×C32
hypercenterC32×C4×Q8D3
hypocenterC3C2×C32×C4×Q8
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32×C4 ◁ C32×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C33×C4×Q8C2semidirect
3. C32×Q8×Dic3C2semidirect
5. C32×#887C2semidirect
7. C32×C4×Dic6C2semidirect
9. C3×C4×D3×Q8C3direct (N × H)
11. C32×D3×Q8C4direct (N × H)
29. C32×C4×Q8D3direct (N × H)
30. C32×C4×Q8C2×C3semidirect
33. C3×Q8×Dic3C2×C3semidirect
35. C3×#887C2×C3semidirect
37. C3×C4×Dic6C2×C3semidirect
39. C33×Q8C2×C4semidirect
48. C32×C4×D3Q8direct (N × H)
49. C32×Dic6C2×C4semidirect
51. C4×D3×Q8C32direct (N × H)
60. C3×D3×Q8C3×C4direct (N × H)
87. C33×C4C2×Q8semidirect
90. C32×Dic3C2×Q8semidirect
97. C3×C4×Q8C3×D3direct (N × H)
98. C3×C4×Q8C2×C32semidirect
101. Q8×Dic3C2×C32semidirect
103. #887C2×C32semidirect
105. C4×Dic6C2×C32semidirect
110. C32×Q8C4×D3direct (N × H)
111. C32×Q8C2×C3×C4semidirect
132. C3×C4×D3C3×Q8direct (N × H)
133. C3×Dic6C2×C3×C4semidirect
138. C32×D3C4×Q8direct (N × H)
147. D3×Q8C32×C4direct (N × H)
186. C32×C4D3×Q8direct (N × H)
188. C32×C4C2×C3×Q8semidirect
191. C3×Dic3C2×C3×Q8semidirect
196. C4×Q8C32×D3direct (N × H)
199. C33C2×C4×Q8semidirect
204. C3×Q8C3×C4×D3direct (N × H)
205. C3×Q8C2×C32×C4semidirect
226. C4×D3C32×Q8direct (N × H)
227. Dic6C2×C32×C4semidirect
238. C3×D3C3×C4×Q8direct (N × H)
270. C3×C4C3×D3×Q8direct (N × H)
272. C3×C4C2×C32×Q8semidirect
275. Dic3C2×C32×Q8semidirect
277. C32C4×D3×Q8direct (N × H)
278. C32C2×C3×C4×Q8semidirect
281. Q8C32×C4×D3direct (N × H)
301. D3C32×C4×Q8direct (N × H)
311. C4C32×D3×Q8direct (N × H)
313. C3C3×C4×D3×Q8direct (N × H)
314. C3C2×C32×C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C33×C4×Q82
C32×Q8×Dic32
C32×D3×C4⋊C42
C32×#8872
C32×C42×D32
C32×C4×Dic62
C2×C32×C4×Q83
C3×C4×D3×Q83