Finite Group Databasebeta

← #13163410

Subgroups — #13163410

Order 1728.

2268 conjugacy classes of subgroups; only the 419 normal subgroups are shown.

normal subgroups only

This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 419 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×#89078C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722186C211yesmaximal, characteristic, complemented
7#11722180C211yesmaximal, characteristic, complemented
8#10592470—313nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×#6312—21no—
11C22×C3⋊(C3⋊Dic3)—21no—
12#89078—22no—
13#89078C414yescomplemented
14C2×C4×C3⋊(C3⋊D3)—21no—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857—21no—
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22#88850—22no—
23C2×#3830—313no—
24C2×#3743—313no—
25C4⋊C4×C3⋊D3—313no—
26C32×#102D3113yescomplemented
27#63659—313no—
28#63643—313no—
29#63637—313no—
30C23×C33—21no—
31C22×C3⋊(C3⋊D3)D411yescharacteristic
32C33×D4—21no—
33C33×D4C2×C412yescomplemented
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36#6312—24no—
37C2×C3⋊(C3⋊Dic3)—21no—
38C2×C3⋊(C3⋊Dic3)—24no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C4×C3⋊(C3⋊D3)—42no—
42C4×C3⋊(C3⋊D3)C2×C412yes—
43C4×C3⋊(C3⋊D3)D412yes—
44#6308—21no—
45#6308C2×C412yescomplemented
46C33×C8—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48#4771—913no—
49C2×C32×D4C2×D3113yes—
50C2×#638—613no—
51C22×C3⋊Dic3—613no—
52#3830—352no—
53#3830—626no—
54C2×C4×C3⋊D3—313no—
55C2×C4×C3⋊D3—613no—
56C2×#634—313no—
57C2×C32×C8C2×D3113yes—
58C32×C4⋊C4C2×D3113yes—
59C4⋊(C3⋊Dic3)—313no—
60#3750—613no—
61C2×C3⋊(C3⋊C8)—313no—
62#3743—626no—
63C22×C33—22no—
64C22×C33—41no—
65C22×C33—41no—
66C22×C33C2×D411yescharacteristic
67C2×C3⋊(C3⋊D3)—21no—
68C2×C3⋊(C3⋊D3)—21no—
69C2×C3⋊(C3⋊D3)C22⋊C412yes—
70C33×C4—41no—
71C33×C4C22×C411yescharacteristic
72C33×C4C2×D411yescharacteristic
73C3⋊(C3⋊Dic3)—22no—
74C3⋊(C3⋊Dic3)—41no—
75C3⋊(C3⋊Dic3)C22⋊C412yes—
76C2×#237—913no—
77C2×ΓL(1,𝔽25)—913no—
78D3×C4⋊C4—913no—
79C3×#102C3⋊D3113yescomplemented
80#832—913no—
81#816—913no—
82#810—913no—
83C23×C32—213no—
84C22×C3⋊D3—313no—
85C32×D4—213no—
86C32×D4C4×D3126yescomplemented
87C2×C32×C4—213no—
88C2×C32×C4C22×D3113yes—
89#638—652no—
90C2×C3⋊Dic3—313no—
91C2×C3⋊Dic3—652no—
92C2×C3⋊Dic3—613no—
93C2×C3⋊Dic3—613no—
94C4×C3⋊D3—326no—
95C4×C3⋊D3—326no—
96C4×C3⋊D3—1226no—
97#634—326no—
98#634—613no—
99C32×C8—213no—
100C3⋊(C3⋊C8)—613no—
101#417—271noSylow
102C2×C33—42no—
103C2×C33C2×C22⋊C411yescharacteristic
104C2×C33AGL(1,ℤ/8)11yescharacteristic
105C2×C33#13711yescharacteristic
106C3⋊(C3⋊D3)—22no—
107C2×C3×D4C2×C3⋊D3113yes—
108C2×#67—1813no—
109C22×Dic3—1813no—
110#237—952no—
111#237—1826no—
112C2×C4×D3—913no—
113C2×C4×D3—1813no—
114C2×Dic6—913no—
115C2×C3×C8C2×C3⋊D3113yes—
116C3×C4⋊C4C2×C3⋊D3113yes—
117C4⋊Dic3—913no—
118Dic3⋊C4—1813no—
119C2×C3⋊C8—913no—
120ΓL(1,𝔽25)—1826no—
121C22×C32—226no—
122C22×C32—413no—
123C22×C32—413no—
124C22×C32D3×D4113yes—
125C2×C3⋊D3—326no—
126C2×C3⋊D3—613no—
127C2×C3⋊D3—613no—
128C32×C4—413no—
129C32×C4C2×C4×D3113yes—
130C32×C4D3×D4113yes—
131C3⋊Dic3—326no—
132C3⋊Dic3—626no—
133C3⋊Dic3—1213no—
134C2×#41—271no—
135C2×M16—271no—
136C2×C4⋊C4—271no—
137#103—271no—
138#103—271no—
139#102—271no—
140#102C3⋊(C3⋊D3)11yescharacteristic, complemented
141C33#41711yescharacteristic, complemented, Sylow
142C23×C3—213no—
143C22×D3—913no—
144C3×D4—213no—
145C3×D4C4×C3⋊D3126yescomplemented
146C2×C3×C4—213no—
147C2×C3×C4C22×C3⋊D3113yes—
148#67—1852no—
149C2×Dic3—913no—
150C2×Dic3—1852no—
151C2×Dic3—1813no—
152C2×Dic3—1813no—
153C4×D3—926no—
154C4×D3—926no—
155C4×D3—3626no—
156Dic6—926no—
157Dic6—1813no—
158C3×C8—213no—
159C3⋊C8—1813no—
160C2×C32—426no—
161C2×C32D3×C22⋊C4113yes—
162C2×C32#911113yes—
163C2×C32#915113yes—
164C3⋊D3—626no—
165#41—274no—
166#41—542no—
167C2×Q8—271no—
168C2×D4—541no—
169C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
170C22×C4—271no—
171C22×C4—541no—
172C22×C4—541no—
173M16—542no—
174C2×C8—271no—
175C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
176C4⋊C4—271no—
177C4⋊C4—541no—
178C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
179C22×C3—226no—
180C22×C3—413no—
181C22×C3—413no—
182C22×C3D4×C3⋊D3113yes—
183C2×D3—926no—
184C2×D3—1813no—
185C2×D3—1813no—
186C3×C4—413no—
187C3×C4C2×C4×C3⋊D3113yes—
188C3×C4D4×C3⋊D3113yes—
189Dic3—926no—
190Dic3—1826no—
191Dic3—3613no—
192C32#4771113yescomplemented
193C23—21no—
194C23—271no—
195Q8—272no—
196Q8—541no—
197D4—21no—
198D4—544no—
199D4C4×C3⋊(C3⋊D3)12yescomplemented
200C2×C4—21no—
201C2×C4—271no—
202C2×C4—272no—
203C2×C4—272no—
204C2×C4—544no—
205C2×C4—541no—
206C2×C4—541no—
207C2×C4—1082no—
208C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
209C8—21no—
210C8—541no—
211C2×C3—426no—
212C2×C3C22⋊C4×C3⋊D3113yes—
213C2×C3#64120113yes—
214C2×C3#64124113yes—
215D3—1826no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—272no—
220C22—541no—
221C22—541no—
222C22D4×C3⋊(C3⋊D3)11yescharacteristic
223C4—41no—
224C4—272no—
225C4—542no—
226C4—1081no—
227C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
228C4D4×C3⋊(C3⋊D3)11yescharacteristic
229C3#10592470113yescomplemented
230C2—42no—
231C2—542no—
232C2C22⋊C4×C3⋊(C3⋊D3)11yescharacteristic
233C2#1172395211yescharacteristic
234C2#1172395611yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×#89078C2
cosocleC2×C33×C4C23
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#88850C2semidirect
3. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
4. C33×#102C2semidirect
5. #11722202C2semidirect
6. #11722186C2semidirect
7. #11722180C2semidirect
13. #89078C4semidirect
26. C32×#102D3semidirect
33. C33×D4C2×C4semidirect
45. #6308C2×C4semidirect
79. C3×#102C3⋊D3semidirect
86. C32×D4C4×D3semidirect
140. #102C3⋊(C3⋊D3)semidirect
141. C33#417semidirect
145. C3×D4C4×C3⋊D3semidirect
192. C32#4771semidirect
199. D4C4×C3⋊(C3⋊D3)semidirect
229. C3#10592470semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×#888502
C4⋊C4×C3⋊(C3⋊D3)2
C33×#1022
#117222022
#117221862
#117221802
#105924703