Finite Group Databasebeta

← 2−1+4×(C3⋊D3)⋊C3 #13164959

Subgroups — 2−1+4×(C3⋊D3)⋊C3 #13164959

Order 1728.

2925 conjugacy classes of subgroups; only the 958 normal subgroups are shown.

normal subgroups only

This group has 2925 conjugacy classes of subgroups — too many to draw. Only its 958 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
1H(3,𝔽3)×2−1+4C211yesmaximal, characteristic, complemented
2#11722639C2110yesmaximal, complemented
3#41×(C3⋊D3)⋊C3C2110yesmaximal, complemented
4#11722627C215yesmaximal, complemented
5C2×Q8×(C3⋊D3)⋊C3C215yesmaximal, complemented
6C3⋊D3×2−1+4C311yesmaximal, characteristic, complemented
7C3×D3×2−1+4—31nomaximal
8#41×H(3,𝔽3)C22110yescomplemented
9C2×Q8×H(3,𝔽3)C2215yescomplemented
10#88722C22110yescomplemented
11Q8×(C3⋊D3)⋊C3C22110yes—
12Q8×(C3⋊D3)⋊C3C22120yescomplemented
13#3830⋊C3C22130yescomplemented
14D4×(C3⋊D3)⋊C3C22110yes—
15#3828⋊C3C22130yescomplemented
16C2×C4×(C3⋊D3)⋊C3C22115yes—
17C2×#634⋊C3C22115yescomplemented
18C32×2−1+4—21no—
19C32×2−1+4D311yescharacteristic, complemented
20C32×2−1+4C2×C311yescharacteristic, complemented
21#64367C2×C3110yescomplemented
22#41×C3⋊D3C2×C3110yescomplemented
23#64364C2×C315yescomplemented
24C2×Q8×C3⋊D3C2×C315yescomplemented
25C3×#1010—310no—
26C3×D3×#41—310no—
27C3×#1007—35no—
28C2×C3×D3×Q8—35no—
29C22×(C3⋊D3)⋊C3C2315yes—
30Q8×H(3,𝔽3)C23110yes—
31D4×H(3,𝔽3)C23110yes—
32C2×C4×H(3,𝔽3)C23115yes—
33#638⋊C3C23120yes—
34C2×(C3⋊Dic3)⋊C3C23115yes—
35C4⋊((C3⋊D3)⋊C3)C23110yes—
36C4×(C3⋊D3)⋊C3C23110yes—
37C4×(C3⋊D3)⋊C3C23130yes—
38#634⋊C3C23130yes—
39C2×C3×2−1+4—91no—
40D3×2−1+4—31no—
41D3×2−1+4—91no—
42C32×#41—210no—
43C32×#41C2×D3110yescomplemented
44C32×#41C22×C3110yescomplemented
45C2×C32×Q8—25no—
46C2×C32×Q8C2×D315yescomplemented
47C2×C32×Q8C22×C315yescomplemented
48#3832C22×C3110yescomplemented
49Q8×C3⋊D3C22×C3110yes—
50Q8×C3⋊D3C22×C3120yescomplemented
51#3830C22×C3130yescomplemented
52D4×C3⋊D3C22×C3110yes—
53#3828C22×C3130yescomplemented
54C2×C4×C3⋊D3C22×C3115yes—
55C2×#634C22×C3115yescomplemented
56C3×#239—310no—
57C3×D3×Q8—310no—
58C3×D3×Q8—320no—
59C3×#237—330no—
60C3×D3×D4—310no—
61C3×#235—330no—
62C2×C3×C4×D3—315no—
63C2×C3×Dic6—315no—
64C22×H(3,𝔽3)C2415yes—
65C2×(C3⋊D3)⋊C3—210no—
66C2×(C3⋊D3)⋊C3C2415yes—
67C2×(C3⋊D3)⋊C3C2411yescharacteristic
68C4×H(3,𝔽3)C24110yes—
69(C3⋊Dic3)⋊C3C24110yes—
70C3×2−1+4—31no—
71C3×2−1+4—31no—
72C3×2−1+4—61no—
73C3×2−1+4—95no—
74C3×2−1+4—910no—
75C3×2−1+4C3×D311yescharacteristic
76C2×C3×#41—910no—
77C22×C3×Q8—95no—
78#1010—310no—
79#1010—910no—
80D3×#41—310no—
81D3×#41—910no—
82#1007—35no—
83#1007—95no—
84C2×D3×Q8—35no—
85C2×D3×Q8—95no—
86C22×C3⋊D3C23×C315yes—
87C22×C3×D3—35no—
88C32×Q8—210no—
89C32×Q8C22×D3110yes—
90C32×Q8C23×C3110yes—
91C32×D4—210no—
92C32×D4C22×D3110yes—
93C32×D4C23×C3110yes—
94C2×C32×C4—215no—
95C2×C32×C4C22×D3115yes—
96C2×C32×C4C23×C3115yes—
97#638C23×C3120yes—
98C2×C3⋊Dic3C23×C3115yes—
99C4⋊(C3⋊D3)C23×C3110yes—
100C4×C3⋊D3C23×C3110yes—
101C4×C3⋊D3C23×C3130yes—
102#634C23×C3130yes—
103C3×#67—320no—
104C2×C3×Dic3—315no—
105C3×D12—310no—
106C3×C4×D3—310no—
107C3×C4×D3—330no—
108C3×Dic6—330no—
109C2×2−1+4—91noSylow
110C2×H(3,𝔽3)—25no—
111C2×H(3,𝔽3)C2511yescharacteristic
112(C3⋊D3)⋊C3—25no—
113(C3⋊D3)⋊C32−1+412yesdirect product factor
114C3×#41—310no—
115C3×#41—310no—
116C3×#41—610no—
117C3×#41—930no—
118C3×#41—930no—
119C3×#41—910no—
120C3×#41C2×C3×D3110yes—
121C2×C3×Q8—35no—
122C2×C3×Q8—35no—
123C2×C3×Q8—65no—
124C2×C3×Q8—915no—
125C2×C3×Q8—910no—
126C2×C3×Q8—920no—
127C2×C3×Q8C2×C3×D315yes—
128C2×C3×D4—910no—
129C22×C3×C4—915no—
130#239—310no—
131#239—910no—
132D3×Q8—310no—
133D3×Q8—320no—
134D3×Q8—910no—
135D3×Q8—920no—
136#237—330no—
137#237—930no—
138D3×D4—310no—
139D3×D4—910no—
140#235—330no—
141#235—930no—
142C2×C4×D3—315no—
143C2×C4×D3—915no—
144C2×Dic6—315no—
145C2×Dic6—915no—
146C22×C32—25no—
147C22×C32C23×D315yes—
148C22×C32C24×C315yes—
149C2×C3⋊D3—210no—
150C2×C3⋊D3C24×C315yes—
151C2×C3⋊D3C24×C311yescharacteristic
152C2×C3×D3—31no—
153C2×C3×D3—35no—
154C2×C3×D3—610no—
155C32×C4—210no—
156C32×C4C23×D3110yes—
157C32×C4C24×C3110yes—
158C3⋊Dic3C24×C3110yes—
159C3×Dic3—310no—
1602−1+4—910no—
1612−1+4—95no—
1622−1+4(C3⋊D3)⋊C311yescharacteristic, direct product factor
163C2×#41—910no—
164C22×Q8—95no—
165H(3,𝔽3)C2×2−1+411yescharacteristic, complemented, Sylow
166C23×C3—95no—
167C22×D3—35no—
168C22×D3—95no—
169C3×Q8—310no—
170C3×Q8—310no—
171C3×Q8—610no—
172C3×Q8—930no—
173C3×Q8C22×C3×D3110yes—
174C3×D4—310no—
175C3×D4—310no—
176C3×D4—610no—
177C3×D4—910no—
178C3×D4—920no—
179C3×D4C22×C3×D3110yes—
180C2×C3×C4—315no—
181C2×C3×C4—315no—
182C2×C3×C4—615no—
183C2×C3×C4—910no—
184C2×C3×C4—930no—
185C2×C3×C4—915no—
186C2×C3×C4C22×C3×D3115yes—
187#67—320no—
188#67—920no—
189C2×Dic3—315no—
190C2×Dic3—915no—
191D12—310no—
192D12—910no—
193C4×D3—310no—
194C4×D3—330no—
195C4×D3—910no—
196C4×D3—930no—
197Dic6—330no—
198Dic6—930no—
199C2×C32—25no—
200C2×C32—25no—
201C2×C32—21no—
202C2×C32—45no—
203C2×C32C24×D311yescharacteristic
204C2×C32C25×C311yescharacteristic
205C3⋊D3—25no—
206C3⋊D3C3×2−1+412yescomplemented
207C3×D3—32no—
208C3×D3—65no—
209#41—910no—
210#41—930no—
211#41—930no—
212#41C2×(C3⋊D3)⋊C3110yescomplemented
213C2×Q8—910no—
214C2×Q8—920no—
215C2×Q8—915no—
216C2×Q8C2×(C3⋊D3)⋊C315yescomplemented
217C2×D4—910no—
218C22×C4—915no—
219C22×C3—35no—
220C22×C3—35no—
221C22×C3—65no—
222C22×C3—91no—
223C22×C3—95no—
224C22×C3—1810no—
225C22×C3C23×C3×D315yes—
226C2×D3—31no—
227C2×D3—35no—
228C2×D3—610no—
229C2×D3—91no—
230C2×D3—95no—
231C2×D3—1810no—
232C3×C4—310no—
233C3×C4—310no—
234C3×C4—610no—
235C3×C4—910no—
236C3×C4C23×C3×D3110yes—
237Dic3—310no—
238Dic3—910no—
239C32—21no—
240C32D3×2−1+411yescharacteristic, complemented
241C32C2×C3×2−1+411yescharacteristic, complemented
242C23—95no—
243Q8—930no—
244Q8C22×(C3⋊D3)⋊C3110yes—
245D4—920no—
246D4—910no—
247D4C22×(C3⋊D3)⋊C3110yes—
248C2×C4—915no—
249C2×C4—910no—
250C2×C4—930no—
251C2×C4C22×(C3⋊D3)⋊C3115yes—
252C2×C3—25no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—65no—
256C2×C3—65no—
257C2×C3—61no—
258C2×C3—92no—
259C2×C3—125no—
260C2×C3—185no—
261C2×C3C24×C3×D311yescharacteristic
262D3—32no—
263D3—65no—
264D3—92no—
265D3—185no—
266C22—91no—
267C22—95no—
268C22—1810no—
269C22C23×(C3⋊D3)⋊C315yes—
270C4—910no—
271C4C23×(C3⋊D3)⋊C3110yes—
272C3—31no—
273C3—31no—
274C3—61no—
275C3C3×D3×2−1+411yescharacteristic
276C2—25no—
277C2—92no—
278C2—185no—
279C2C24×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C24×(C3⋊D3)⋊C3
commutator, cosocleC2×C32C25×C3
Frattini, socleC2×C3C24×C3×D3
FittingH(3,𝔽3)×2−1+4C2
hypercenter2−1+4(C3⋊D3)⋊C3
hypocenterC32C2×C3×2−1+4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×2−1+4C2semidirect
2. #11722639C2semidirect
3. #41×(C3⋊D3)⋊C3C2semidirect
4. #11722627C2semidirect
5. C2×Q8×(C3⋊D3)⋊C3C2semidirect
6. C3⋊D3×2−1+4C3semidirect
8. #41×H(3,𝔽3)C22semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
10. #88722C22semidirect
12. Q8×(C3⋊D3)⋊C3C22semidirect
13. #3830⋊C3C22semidirect
15. #3828⋊C3C22semidirect
17. C2×#634⋊C3C22semidirect
19. C32×2−1+4D3semidirect
20. C32×2−1+4C2×C3semidirect
21. #64367C2×C3semidirect
22. #41×C3⋊D3C2×C3semidirect
23. #64364C2×C3semidirect
24. C2×Q8×C3⋊D3C2×C3semidirect
43. C32×#41C2×D3semidirect
44. C32×#41C22×C3semidirect
46. C2×C32×Q8C2×D3semidirect
47. C2×C32×Q8C22×C3semidirect
48. #3832C22×C3semidirect
50. Q8×C3⋊D3C22×C3semidirect
51. #3830C22×C3semidirect
53. #3828C22×C3semidirect
55. C2×#634C22×C3semidirect
113. (C3⋊D3)⋊C32−1+4direct (N × H)
162. 2−1+4(C3⋊D3)⋊C3direct (N × H)
165. H(3,𝔽3)C2×2−1+4semidirect
206. C3⋊D3C3×2−1+4semidirect
212. #41C2×(C3⋊D3)⋊C3semidirect
216. C2×Q8C2×(C3⋊D3)⋊C3semidirect
240. C32D3×2−1+4semidirect
241. C32C2×C3×2−1+4semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×2−1+42
#117226392
#41×(C3⋊D3)⋊C32
#117226272
C2×Q8×(C3⋊D3)⋊C32
C3⋊D3×2−1+43
C3×D3×2−1+43