Finite Group Databasebeta

← C2×#11723955 #13179884

Subgroups — C2×#11723955 #13179884

Order 1728.

4172 conjugacy classes of subgroups; only the 675 normal subgroups are shown.

normal subgroups only

This group has 4172 conjugacy classes of subgroups — too many to draw. Only its 675 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×#89080C211yesmaximal, characteristic, complemented
2C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×#88862C211yesmaximal, characteristic, complemented
5C2×#88860C211yesmaximal, characteristic, complemented
6#11723955C218yesmaximal, direct product factor
7C2×#88852C211yesmaximal, characteristic, complemented
8C2×#88850C211yesmaximal, characteristic, complemented
9C2×#64123—313nomaximal
10C23×C3⋊(C3⋊D3)—21no—
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C33×D4C2211yescharacteristic
13C2×#6312—21no—
14#89080—22no—
15#89080C2214yescomplemented
16D4×C3⋊(C3⋊D3)—22no—
17D4×C3⋊(C3⋊D3)C2214yescomplemented
18C2×C4⋊(C3⋊(C3⋊D3))—21no—
19C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
20C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
21C2×C4×C3⋊(C3⋊D3)—21no—
22C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
23C33×SD16C2214yescomplemented
24C2×C33×C8C2211yescharacteristic, complemented
25#88862C2214yescomplemented
26#88860C2214yescomplemented
27C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
28#88852C2214yescomplemented
29#88850C2214yescomplemented
30C2×#3832—313no—
31C2×D4×C3⋊D3—313no—
32C2×C32×SD16D3113yescomplemented
33C2×#3755—313no—
34C2×#3753—313no—
35#64123—3104no—
36C2×#3745—313no—
37C2×#3743—313no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)—24no—
42C22×C3⋊(C3⋊D3)—44no—
43C22×C3⋊(C3⋊D3)D411yescharacteristic
44C33×Q8—21no—
45C33×Q8C2312yescomplemented
46C33×D4—21no—
47C33×D4C2312yes—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50#6312—24no—
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—24no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))—21no—
55C4⋊(C3⋊(C3⋊D3))C2312yes—
56C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
57C4×C3⋊(C3⋊D3)—24no—
58C4×C3⋊(C3⋊D3)D412yes—
59C4×C3⋊(C3⋊D3)C2312yes—
60C33×C8C2312yescomplemented
61C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
62C2×#914—913no—
63C23×C3⋊D3—613no—
64C2×C32×Q8C2×D3113yescomplemented
65C2×C32×D4C2×D3113yes—
66C2×#638—613no—
67#3832—352no—
68#3832—626no—
69D4×C3⋊D3—352no—
70D4×C3⋊D3—626no—
71C2×C4⋊(C3⋊D3)—313no—
72C2×C4⋊(C3⋊D3)—313no—
73C2×C4⋊(C3⋊D3)—613no—
74C2×C4×C3⋊D3—313no—
75C2×C4×C3⋊D3—613no—
76C32×SD16C2×D3152yescomplemented
77C2×C32×C8C2×D3113yescomplemented
78#3755—352no—
79#3753—352no—
80C2×C3⋊(C3⋊C8)—313no—
81#3745—352no—
82#3743—352no—
83C22×C33—22no—
84C22×C33—42no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—22no—
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—22no—
89C2×C3⋊(C3⋊D3)—48no—
90C2×C3⋊(C3⋊D3)—42no—
91C2×C3⋊(C3⋊D3)—42no—
92C2×C3⋊(C3⋊D3)C2×D412yes—
93C33×C4—22no—
94C33×C4C2×D411yescharacteristic
95C33×C4C2411yescharacteristic
96C3⋊(C3⋊Dic3)C2×D412yes—
97C2×#239—913no—
98C2×D3×D4—913no—
99C2×C3×SD16C3⋊D3113yescomplemented
100C2×#215—913no—
101C2×#213—913no—
102#914—9104no—
103C2×D24—913no—
104C2×ΓL(1,𝔽25)—913no—
105C23×C32—213no—
106C22×C3⋊D3—313no—
107C22×C3⋊D3—652no—
108C22×C3⋊D3—613no—
109C22×C3⋊D3—613no—
110C22×C3⋊D3—1252no—
111C32×Q8—213no—
112C32×Q8C22×D3126yescomplemented
113C32×D4—213no—
114C32×D4C22×D3126yes—
115C2×C32×C4—213no—
116C2×C32×C4C22×D3113yes—
117#638—652no—
118C2×C3⋊Dic3—313no—
119C4⋊(C3⋊D3)—326no—
120C4⋊(C3⋊D3)—326no—
121C4⋊(C3⋊D3)—652no—
122C4⋊(C3⋊D3)—613no—
123C4⋊(C3⋊D3)—613no—
124C4×C3⋊D3—326no—
125C4×C3⋊D3—326no—
126C4×C3⋊D3—652no—
127C32×C8C22×D3126yescomplemented
128C3⋊(C3⋊C8)—326no—
129C2×AGL(1,ℤ/8)—271noSylow
130C2×C33—42no—
131C2×C33AGL(1,ℤ/8)12yescomplemented
132C2×C33C22×D411yescharacteristic
133C3⋊(C3⋊D3)—22no—
134C3⋊(C3⋊D3)—42no—
135C3⋊(C3⋊D3)—42no—
136C23×D3—1813no—
137C2×C3×Q8C2×C3⋊D3113yescomplemented
138C2×C3×D4C2×C3⋊D3113yes—
139C2×#67—1813no—
140#239—952no—
141#239—1826no—
142D3×D4—952no—
143D3×D4—1826no—
144C2×D12—913no—
145C2×D12—913no—
146C2×D12—1813no—
147C2×C4×D3—913no—
148C2×C4×D3—1813no—
149C3×SD16C2×C3⋊D3152yescomplemented
150C2×C3×C8C2×C3⋊D3113yescomplemented
151#215—952no—
152#213—952no—
153C2×C3⋊C8—913no—
154D24—952no—
155ΓL(1,𝔽25)—952no—
156C22×C32—226no—
157C22×C32—426no—
158C22×C32D3×D4113yes—
159C2×C3⋊D3—326no—
160C2×C3⋊D3—626no—
161C2×C3⋊D3—626no—
162C2×C3⋊D3—626no—
163C2×C3⋊D3—12104no—
164C2×C3⋊D3—1226no—
165C2×C3⋊D3—1226no—
166C32×C4—226no—
167C32×C4D3×D4113yes—
168C32×C4C23×D3113yes—
169C3⋊Dic3—326no—
170C2×#41—271no—
171C22×D4—271no—
172AGL(1,ℤ/8)—278no—
173C2×SD16—271no—
174C2×SD16C3⋊(C3⋊D3)11yescharacteristic, complemented
175C2×D8—271no—
176C2×D8—271no—
177C2×M16—271no—
178C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
179C23×C3—213no—
180C22×D3—913no—
181C22×D3—1852no—
182C22×D3—1813no—
183C22×D3—1813no—
184C22×D3—3652no—
185C3×Q8—213no—
186C3×Q8C22×C3⋊D3126yescomplemented
187C3×D4—213no—
188C3×D4C22×C3⋊D3126yes—
189C2×C3×C4—213no—
190C2×C3×C4C22×C3⋊D3113yes—
191#67—1852no—
192C2×Dic3—913no—
193D12—926no—
194D12—926no—
195D12—1852no—
196D12—1813no—
197D12—1813no—
198C4×D3—926no—
199C4×D3—926no—
200C4×D3—1852no—
201C3×C8C22×C3⋊D3126yescomplemented
202C3⋊C8—926no—
203C2×C32—426no—
204C2×C32#914126yescomplemented
205C2×C32C2×D3×D4113yes—
206C3⋊D3—626no—
207C3⋊D3—1226no—
208C3⋊D3—1226no—
209C24—541no—
210#41—274no—
211#41—542no—
212C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
213C2×D4—274no—
214C2×D4—271no—
215C2×D4—271no—
216C2×D4—541no—
217C2×D4—542no—
218C2×D4—541no—
219C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
220C22×C4—271no—
221C22×C4—541no—
222SD16—274no—
223SD16C2×C3⋊(C3⋊D3)14yescomplemented
224D8—274no—
225D8—274no—
226M16—274no—
227C2×C8—271no—
228C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
229C22×C3—226no—
230C22×C3—426no—
231C22×C3D4×C3⋊D3113yes—
232C2×D3—926no—
233C2×D3—1826no—
234C2×D3—1826no—
235C2×D3—1826no—
236C2×D3—36104no—
237C2×D3—3626no—
238C2×D3—3626no—
239C3×C4—226no—
240C3×C4D4×C3⋊D3113yes—
241C3×C4C23×C3⋊D3113yes—
242Dic3—926no—
243C32C2×#914113yescomplemented
244C23—21no—
245C23—271no—
246C23—544no—
247C23—541no—
248C23—541no—
249C23—1084no—
250Q8—21no—
251Q8C22×C3⋊(C3⋊D3)12yescomplemented
252D4—21no—
253D4—272no—
254D4—272no—
255D4—544no—
256D4—544no—
257D4—541no—
258D4—541no—
259D4C22×C3⋊(C3⋊D3)12yes—
260C2×C4—21no—
261C2×C4—271no—
262C2×C4—272no—
263C2×C4—272no—
264C2×C4—544no—
265C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
266C8—272no—
267C8C22×C3⋊(C3⋊D3)12yescomplemented
268C2×C3—426no—
269C2×C3#64123126yescomplemented
270C2×C3C2×D4×C3⋊D3113yes—
271D3—1826no—
272D3—3626no—
273D3—3626no—
274C22—22no—
275C22—42no—
276C22—272no—
277C22—542no—
278C22—542no—
279C22—542no—
280C22—1088no—
281C22—1082no—
282C22—1082no—
283C22D4×C3⋊(C3⋊D3)11yescharacteristic
284C4—22no—
285C4—272no—
286C4D4×C3⋊(C3⋊D3)11yescharacteristic
287C4C23×C3⋊(C3⋊D3)11yescharacteristic
288C3C2×#64123113yescomplemented
289C2—42no—
290C2—542no—
291C2—1082no—
292C2—1082no—
293C2#1172395512yesdirect product factor
294C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#89080C2semidirect
2. C2×D4×C3⋊(C3⋊D3)C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×#88862C2semidirect
5. C2×#88860C2semidirect
6. #11723955C2direct (N × H)
7. C2×#88852C2semidirect
8. C2×#88850C2semidirect
11. C2×C33×Q8C22semidirect
15. #89080C22semidirect
17. D4×C3⋊(C3⋊D3)C22semidirect
20. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
23. C33×SD16C22semidirect
24. C2×C33×C8C22semidirect
25. #88862C22semidirect
26. #88860C22semidirect
27. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
28. #88852C22semidirect
29. #88850C22semidirect
32. C2×C32×SD16D3semidirect
45. C33×Q8C23semidirect
56. C4⋊(C3⋊(C3⋊D3))C23semidirect
60. C33×C8C23semidirect
61. C3⋊(C3⋊(C3⋊C8))C23semidirect
64. C2×C32×Q8C2×D3semidirect
76. C32×SD16C2×D3semidirect
77. C2×C32×C8C2×D3semidirect
99. C2×C3×SD16C3⋊D3semidirect
112. C32×Q8C22×D3semidirect
127. C32×C8C22×D3semidirect
131. C2×C33AGL(1,ℤ/8)semidirect
137. C2×C3×Q8C2×C3⋊D3semidirect
149. C3×SD16C2×C3⋊D3semidirect
150. C2×C3×C8C2×C3⋊D3semidirect
174. C2×SD16C3⋊(C3⋊D3)semidirect
178. C33C2×AGL(1,ℤ/8)semidirect
186. C3×Q8C22×C3⋊D3semidirect
201. C3×C8C22×C3⋊D3semidirect
204. C2×C32#914semidirect
212. C2×Q8C2×C3⋊(C3⋊D3)semidirect
223. SD16C2×C3⋊(C3⋊D3)semidirect
228. C2×C8C2×C3⋊(C3⋊D3)semidirect
243. C32C2×#914semidirect
251. Q8C22×C3⋊(C3⋊D3)semidirect
267. C8C22×C3⋊(C3⋊D3)semidirect
269. C2×C3#64123semidirect
288. C3C2×#64123semidirect
293. C2#11723955direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890802
C2×D4×C3⋊(C3⋊D3)2
C2×C33×SD162
C2×#888622
C2×#888602
#117239552
C2×#888522
C2×#888502
C2×#641233