Finite Group Databasebeta

← #104×C3⋊(C3⋊D3) #13163460

Subgroups — #104×C3⋊(C3⋊D3) #13163460

Order 1728.

2380 conjugacy classes of subgroups; only the 365 normal subgroups are shown.

normal subgroups only

This group has 2380 conjugacy classes of subgroups — too many to draw. Only its 365 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
1#41×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2M16×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C33×#104C211yesmaximal, characteristic, complemented
5#11722207C211yesmaximal, characteristic, complemented
6#11722195C211yesmaximal, characteristic, complemented
7#11722175C211yesmaximal, characteristic, complemented
8C3⋊D3×#104—313nomaximal
9C33×#41C2211yescharacteristic
10#89080C411yescharacteristic, complemented
11Q8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
12#89078—21no—
13#89078C411yescharacteristic, complemented
14D4×C3⋊(C3⋊D3)—21no—
15D4×C3⋊(C3⋊D3)C411yescharacteristic, complemented
16#89076—21no—
17#89076C2211yescharacteristic
18C2×C4×C3⋊(C3⋊D3)—21no—
19C2×C4×C3⋊(C3⋊D3)—21no—
20C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
21C33×M16C2211yescharacteristic, complemented
22C33×C42C2211yescharacteristic, complemented
23C4×C3⋊(C3⋊Dic3)—21no—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
25#88855C2211yescharacteristic, complemented
26#88850—21no—
27C8×C3⋊(C3⋊D3)—21no—
28#41×C3⋊D3—313no—
29M16×C3⋊D3—313no—
30C42×C3⋊D3—313no—
31C32×#104D3113yescomplemented
32#63664—313no—
33#63652—313no—
34#63632—313no—
35C22×C3⋊(C3⋊D3)—21no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C33×Q8C2×C411yescharacteristic, complemented
38C33×D4—21no—
39C33×D4C2×C411yescharacteristic, complemented
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43#6312—21no—
44#6312—21no—
45#6312—21no—
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic, complemented
51C4×C3⋊(C3⋊D3)—24no—
52C4×C3⋊(C3⋊D3)—21no—
53C4×C3⋊(C3⋊D3)—21no—
54C4×C3⋊(C3⋊D3)—21no—
55C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
56C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
57C4×C3⋊(C3⋊D3)D411yescharacteristic
58C4×C3⋊(C3⋊D3)D411yescharacteristic
59#6308—21no—
60#6308C2×C411yescharacteristic, complemented
61C33×C8—21no—
62C3⋊(C3⋊(C3⋊C8))—21no—
63D3×#104—913no—
64C32×#41C2×D3113yes—
65#3832—313no—
66Q8×C3⋊D3—313no—
67#3830—313no—
68#3830—613no—
69D4×C3⋊D3—313no—
70D4×C3⋊D3—613no—
71#3828—313no—
72#3828—613no—
73C2×C4×C3⋊D3—313no—
74C2×C4×C3⋊D3—613no—
75C2×C4×C3⋊D3—613no—
76C32×M16C2×D3113yescomplemented
77C32×C42C2×D3113yescomplemented
78C4×C3⋊Dic3—313no—
79C4×C3⋊Dic3—613no—
80#3748—313no—
81#3743—613no—
82C8×C3⋊D3—613no—
83C22×C33—21no—
84C22×C33C2×D411yescharacteristic
85C2×C3⋊(C3⋊D3)—22no—
86C2×C3⋊(C3⋊D3)—21no—
87C2×C3⋊(C3⋊D3)—42no—
88C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
89C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
90C33×C4—22no—
91C33×C4—21no—
92C33×C4C22×C411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3⋊(C3⋊Dic3)—22no—
95C3⋊(C3⋊Dic3)—21no—
96C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
97C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
98D3×#41—913no—
99D3×M16—913no—
100C42×D3—913no—
101C3×#104C3⋊D3113yescomplemented
102#837—913no—
103#825—913no—
104#805—913no—
105C22×C3⋊D3—313no—
106C22×C3⋊D3—613no—
107C32×Q8C4×D3113yescomplemented
108C32×D4—213no—
109C32×D4C4×D3113yescomplemented
110C2×C32×C4—213no—
111C2×C32×C4—213no—
112C2×C32×C4C22×D3113yes—
113#638—613no—
114#638—613no—
115#638—613no—
116C2×C3⋊Dic3—313no—
117C2×C3⋊Dic3—613no—
118C2×C3⋊Dic3—613no—
119C4⋊(C3⋊D3)—313no—
120C4⋊(C3⋊D3)—613no—
121C4×C3⋊D3—313no—
122C4×C3⋊D3—313no—
123C4×C3⋊D3—313no—
124C4×C3⋊D3—313no—
125C4×C3⋊D3—613no—
126C4×C3⋊D3—613no—
127C4×C3⋊D3—652no—
128C4×C3⋊D3—613no—
129#634—313no—
130#634—613no—
131C32×C8—213no—
132C3⋊(C3⋊C8)—613no—
133C2×#104—271noSylow
134C2×C33—21no—
135C2×C33—41no—
136C2×C33C2×C22⋊C411yescharacteristic
137C3⋊(C3⋊D3)—21no—
138C3⋊(C3⋊D3)—41no—
139C3⋊(C3⋊D3)#10412yesdirect product factor
140C3×#41C2×C3⋊D3113yes—
141#239—913no—
142D3×Q8—913no—
143#237—913no—
144#237—1813no—
145D3×D4—913no—
146D3×D4—1813no—
147#235—913no—
148#235—1813no—
149C2×C4×D3—913no—
150C2×C4×D3—1813no—
151C2×C4×D3—1813no—
152C3×M16C2×C3⋊D3113yescomplemented
153C3×C42C2×C3⋊D3113yescomplemented
154C4×Dic3—913no—
155C4×Dic3—1813no—
156#208—913no—
157ΓL(1,𝔽25)—1813no—
158D3×C8—1813no—
159C22×C32—213no—
160C22×C32D3×D4113yes—
161C2×C3⋊D3—313no—
162C2×C3⋊D3—313no—
163C2×C3⋊D3—626no—
164C2×C3⋊D3—613no—
165C2×C3⋊D3—1226no—
166C32×C4—226no—
167C32×C4—213no—
168C32×C4C2×C4×D3113yes—
169C32×C4D3×D4113yes—
170C3⋊Dic3—313no—
171C3⋊Dic3—313no—
172C3⋊Dic3—626no—
173C3⋊Dic3—613no—
174C2×#41—271no—
175C2×M16—271no—
176C2×C42—271no—
177#104—271no—
178#104—271no—
179#104—271no—
180#104C3⋊(C3⋊D3)11yescharacteristic, direct product factor
181C33C2×#10411yescharacteristic, complemented, Sylow
182C22×D3—913no—
183C22×D3—1813no—
184C3×Q8C4×C3⋊D3113yescomplemented
185C3×D4—213no—
186C3×D4C4×C3⋊D3113yescomplemented
187C2×C3×C4—213no—
188C2×C3×C4—213no—
189C2×C3×C4C22×C3⋊D3113yes—
190#67—1813no—
191#67—1813no—
192#67—1813no—
193C2×Dic3—913no—
194C2×Dic3—1813no—
195C2×Dic3—1813no—
196D12—913no—
197D12—1813no—
198C4×D3—913no—
199C4×D3—913no—
200C4×D3—913no—
201C4×D3—913no—
202C4×D3—1813no—
203C4×D3—1813no—
204C4×D3—1852no—
205C4×D3—1813no—
206Dic6—913no—
207Dic6—1813no—
208C3×C8—213no—
209C3⋊C8—1813no—
210C2×C32—213no—
211C2×C32—413no—
212C2×C32D3×C22⋊C4113yes—
213C3⋊D3—326no—
214C3⋊D3—613no—
215C3⋊D3—1213no—
216#41—271no—
217#41—271no—
218#41—271no—
219#41—541no—
220#41—541no—
221#41C2×C3⋊(C3⋊D3)11yescharacteristic
222C2×Q8—271no—
223C2×D4—271no—
224C2×D4—541no—
225C22×C4—271no—
226C22×C4—541no—
227C22×C4—541no—
228M16—271no—
229M16—541no—
230M16C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
231C2×C8—541no—
232C42—271no—
233C42—541no—
234C42C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
235C22×C3—213no—
236C22×C3D4×C3⋊D3113yes—
237C2×D3—913no—
238C2×D3—913no—
239C2×D3—1826no—
240C2×D3—1813no—
241C2×D3—3626no—
242C3×C4—226no—
243C3×C4—213no—
244C3×C4C2×C4×C3⋊D3113yes—
245C3×C4D4×C3⋊D3113yes—
246Dic3—913no—
247Dic3—913no—
248Dic3—1826no—
249Dic3—1813no—
250C32D3×#104113yescomplemented
251C23—271no—
252C23—541no—
253Q8—271no—
254Q8—541no—
255Q8C4×C3⋊(C3⋊D3)11yescharacteristic, complemented
256D4—21no—
257D4—271no—
258D4—541no—
259D4—541no—
260D4—541no—
261D4—541no—
262D4C4×C3⋊(C3⋊D3)11yescharacteristic, complemented
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—271no—
266C2×C4—271no—
267C2×C4—271no—
268C2×C4—271no—
269C2×C4—271no—
270C2×C4—541no—
271C2×C4—541no—
272C2×C4—544no—
273C2×C4—541no—
274C2×C4—541no—
275C2×C4—541no—
276C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
277C8—21no—
278C8—541no—
279C2×C3—213no—
280C2×C3—413no—
281C2×C3C22⋊C4×C3⋊D3113yes—
282D3—926no—
283D3—1813no—
284D3—3613no—
285C22—21no—
286C22—271no—
287C22—271no—
288C22—542no—
289C22—541no—
290C22—1082no—
291C22D4×C3⋊(C3⋊D3)11yescharacteristic
292C4—22no—
293C4—21no—
294C4—271no—
295C4—271no—
296C4—542no—
297C4—541no—
298C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
299C4D4×C3⋊(C3⋊D3)11yescharacteristic
300C3C3⋊D3×#104113yescomplemented
301C2—21no—
302C2—41no—
303C2—272no—
304C2—541no—
305C2—1081no—
306C2C22⋊C4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC4D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#104C2
socleC2×C33C2×C22⋊C4
prime radical#41×C3⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenter#104C3⋊(C3⋊D3)
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊(C3⋊D3)C2semidirect
3. C42×C3⋊(C3⋊D3)C2semidirect
4. C33×#104C2semidirect
5. #11722207C2semidirect
6. #11722195C2semidirect
7. #11722175C2semidirect
10. #89080C4semidirect
11. Q8×C3⋊(C3⋊D3)C4semidirect
13. #89078C4semidirect
15. D4×C3⋊(C3⋊D3)C4semidirect
21. C33×M16C22semidirect
22. C33×C42C22semidirect
24. C4×C3⋊(C3⋊Dic3)C22semidirect
25. #88855C22semidirect
31. C32×#104D3semidirect
37. C33×Q8C2×C4semidirect
39. C33×D4C2×C4semidirect
50. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
60. #6308C2×C4semidirect
76. C32×M16C2×D3semidirect
77. C32×C42C2×D3semidirect
101. C3×#104C3⋊D3semidirect
107. C32×Q8C4×D3semidirect
109. C32×D4C4×D3semidirect
139. C3⋊(C3⋊D3)#104direct (N × H)
152. C3×M16C2×C3⋊D3semidirect
153. C3×C42C2×C3⋊D3semidirect
180. #104C3⋊(C3⋊D3)direct (N × H)
181. C33C2×#104semidirect
184. C3×Q8C4×C3⋊D3semidirect
186. C3×D4C4×C3⋊D3semidirect
230. M16C2×C3⋊(C3⋊D3)semidirect
234. C42C2×C3⋊(C3⋊D3)semidirect
250. C32D3×#104semidirect
255. Q8C4×C3⋊(C3⋊D3)semidirect
262. D4C4×C3⋊(C3⋊D3)semidirect
300. C3C3⋊D3×#104semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊(C3⋊D3)2
M16×C3⋊(C3⋊D3)2
C42×C3⋊(C3⋊D3)2
C33×#1042
#117222072
#117221952
#117221752
C3⋊D3×#1043