Finite Group Databasebeta

← C22×#88785 #13176862

Subgroups — C22×#88785 #13176862

Order 1728.

2972 conjugacy classes of subgroups; only the 532 normal subgroups are shown.

normal subgroups only

This group has 2972 conjugacy classes of subgroups — too many to draw. Only its 532 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
1C22×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#88785C2112yesmaximal, direct product factor
5C22×C8×C3⋊D3—31nomaximal
6C22×#3710—34nomaximal
7C23×C3⋊(C3⋊D3)C411yescharacteristic
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C411yescharacteristic
10C2×C4×C3⋊(C3⋊D3)C416yes—
11C2×C4×C3⋊(C3⋊D3)C2216yes—
12C2×C3×C3⋊(C3⋊C8)C2216yescomplemented
13C2×C32×C3⋊C8C2216yescomplemented
14#88785C22116yesdirect product factor
15C22×C4×C3⋊D3—31no—
16C22×C4×C3⋊D3—34no—
17C22×C4×C3⋊D3—64no—
18C22×C32×C8—31no—
19C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
20C2×C8×C3⋊D3—312no—
21C22×C3×C3⋊C8—34no—
22C22×C3×C3⋊C8D314yescomplemented
23C2×#3710—348no—
24C23×C33C2×C411yescharacteristic
25C22×C3⋊(C3⋊D3)C818yescomplemented
26C22×C3⋊(C3⋊D3)C2×C416yes—
27C2×C33×C4C2×C413yes—
28C2×C33×C4C2313yes—
29C2×C3⋊(C3⋊Dic3)C2×C416yes—
30C4×C3⋊(C3⋊D3)C2×C4112yes—
31C4×C3⋊(C3⋊D3)C2314yes—
32C3×C3⋊(C3⋊C8)C2314yescomplemented
33C32×C3⋊C8C2314yescomplemented
34C22×D3×C8—94no—
35C22×D3×C8—91no—
36C23×C3⋊D3—31no—
37C23×C3⋊D3—34no—
38C23×C3⋊D3—64no—
39C22×C32×C4—24no—
40C22×C32×C4C2×D314yes—
41C22×C32×C4C2×D311yescharacteristic
42C22×C3⋊Dic3—31no—
43C22×C3⋊Dic3—34no—
44C22×C3⋊Dic3—64no—
45C2×C4×C3⋊D3—36no—
46C2×C4×C3⋊D3—324no—
47C2×C4×C3⋊D3—36no—
48C2×C4×C3⋊D3—324no—
49C2×C4×C3⋊D3—624no—
50C2×C4×C3⋊D3—624no—
51C2×C32×C8—36no—
52C2×C3⋊(C3⋊C8)C2×D316yescomplemented
53C8×C3⋊D3—316no—
54C2×C3×C3⋊C8—324no—
55C2×C3×C3⋊C8C2×D3124yescomplemented
56#3710—364no—
57C22×C33C2×C814yescomplemented
58C22×C33C22×C413yes—
59C2×C3⋊(C3⋊D3)C2×C8124yescomplemented
60C2×C3⋊(C3⋊D3)C22×C414yes—
61C33×C4C22×C413yes—
62C33×C4C2411yescharacteristic
63C3⋊(C3⋊Dic3)C22×C414yes—
64C22×C4×D3—94no—
65C22×C4×D3—91no—
66C22×C4×D3—184no—
67C22×C3×C8—34no—
68C22×C3×C8—91no—
69C22×C3⋊C8—34no—
70C22×C3⋊C8C3⋊D311yescharacteristic, complemented
71C2×D3×C8—948no—
72C2×D3×C8—912no—
73C23×C32—24no—
74C23×C32C4×D314yes—
75C23×C32C4×D311yescharacteristic
76C22×C3⋊D3—38no—
77C22×C3⋊D3—36no—
78C22×C3⋊D3—332no—
79C22×C3⋊D3—324no—
80C22×C3⋊D3—632no—
81C22×C3⋊D3—624no—
82C2×C32×C4—212no—
83C2×C32×C4—212no—
84C2×C32×C4C4×D3112yes—
85C2×C32×C4C4×D313yes—
86C2×C32×C4C22×D3112yes—
87C2×C32×C4C22×D313yes—
88C2×C3⋊Dic3—36no—
89C2×C3⋊Dic3—324no—
90C2×C3⋊Dic3—624no—
91C4×C3⋊D3—312no—
92C4×C3⋊D3—34no—
93C4×C3⋊D3—348no—
94C4×C3⋊D3—316no—
95C4×C3⋊D3—648no—
96C4×C3⋊D3—616no—
97C32×C8—34no—
98C3⋊(C3⋊C8)C22×D314yescomplemented
99C3×C3⋊C8—316no—
100C3×C3⋊C8C22×D3116yescomplemented
101C23×C8—271noSylow
102C2×C33C22×C816yescomplemented
103C2×C33C23×C411yescharacteristic
104C3⋊(C3⋊D3)C22×C818yescomplemented
105C23×D3—94no—
106C23×D3—91no—
107C23×D3—184no—
108C22×C3×C4—24no—
109C22×C3×C4D3214yes—
110C22×C3×C4C2×C3⋊D311yescharacteristic
111C22×Dic3—94no—
112C22×Dic3—91no—
113C22×Dic3—184no—
114C2×C4×D3—924no—
115C2×C4×D3—96no—
116C2×C4×D3—924no—
117C2×C4×D3—96no—
118C2×C4×D3—1824no—
119C2×C4×D3—1824no—
120C2×C3×C8—324no—
121C2×C3×C8—96no—
122C2×C3⋊C8—324no—
123C2×C3⋊C8C2×C3⋊D316yescomplemented
124D3×C8—964no—
125D3×C8—916no—
126C22×C32—216no—
127C22×C32—212no—
128C22×C32D3×C8116yescomplemented
129C22×C32D3×C814yescomplemented
130C22×C32C2×C4×D3112yes—
131C22×C32C2×C4×D313yes—
132C2×C3⋊D3—324no—
133C2×C3⋊D3—34no—
134C2×C3⋊D3—396no—
135C2×C3⋊D3—316no—
136C2×C3⋊D3—696no—
137C2×C3⋊D3—616no—
138C32×C4—24no—
139C32×C4—212no—
140C32×C4C2×C4×D3112yes—
141C32×C4C2×C4×D313yes—
142C32×C4C23×D314yes—
143C32×C4C23×D311yescharacteristic
144C3⋊Dic3—34no—
145C3⋊Dic3—316no—
146C3⋊Dic3—616no—
147C23×C4—271no—
148C22×C8—31no—
149C22×C8—91no—
150C22×C8—2712no—
151C33C23×C811yescharacteristic, complemented, Sylow
152C23×C3—24no—
153C23×C3#62414yes—
154C23×C3C4×C3⋊D311yescharacteristic
155C22×D3—932no—
156C22×D3—98no—
157C22×D3—924no—
158C22×D3—96no—
159C22×D3—1824no—
160C22×D3—1832no—
161C2×C3×C4—212no—
162C2×C3×C4—212no—
163C2×C3×C4#624112yes—
164C2×C3×C4C4×C3⋊D313yes—
165C2×C3×C4C2×D32112yes—
166C2×C3×C4C22×C3⋊D313yes—
167C2×Dic3—924no—
168C2×Dic3—96no—
169C2×Dic3—1824no—
170C4×D3—916no—
171C4×D3—94no—
172C4×D3—948no—
173C4×D3—912no—
174C4×D3—1816no—
175C4×D3—1848no—
176C3×C8—316no—
177C3×C8—94no—
178C3⋊C8—316no—
179C3⋊C8C22×C3⋊D314yescomplemented
180C2×C32—24no—
181C2×C32—224no—
182C2×C32C2×D3×C8124yescomplemented
183C2×C32C2×D3×C816yescomplemented
184C2×C32C22×C4×D314yes—
185C2×C32C22×C4×D311yescharacteristic
186C3⋊D3—38no—
187C3⋊D3—332no—
188C3⋊D3—632no—
189C24—271no—
190C22×C4—271no—
191C22×C4—276no—
192C22×C4—276no—
193C22×C4D3×C3⋊D311yescharacteristic
194C2×C8—36no—
195C2×C8—96no—
196C2×C8—2716no—
197C22×C3—212no—
198C22×C3—216no—
199C22×C3#3710116yescomplemented
200C22×C3C8×C3⋊D314yescomplemented
201C22×C3C2×#624112yes—
202C22×C3C2×C4×C3⋊D313yes—
203C2×D3—916no—
204C2×D3—996no—
205C2×D3—94no—
206C2×D3—924no—
207C2×D3—1816no—
208C2×D3—1896no—
209C3×C4—24no—
210C3×C4—212no—
211C3×C4C2×#624112yes—
212C3×C4C2×C4×C3⋊D313yes—
213C3×C4C22×D3214yes—
214C3×C4C23×C3⋊D311yescharacteristic
215Dic3—916no—
216Dic3—94no—
217Dic3—1816no—
218C32—24no—
219C32C22×D3×C814yescomplemented
220C32C22×D3×C811yescharacteristic, complemented
221C23—276no—
222C23—278no—
223C23#628911yescharacteristic
224C2×C4—276no—
225C2×C4—274no—
226C2×C4—2712no—
227C2×C4#628913yes—
228C2×C4C2×D3×C3⋊D313yes—
229C8—34no—
230C8—94no—
231C2×C3—24no—
232C2×C3—224no—
233C2×C3C2×#3710124yescomplemented
234C2×C3C2×C8×C3⋊D316yescomplemented
235C2×C3C22×#62414yes—
236C2×C3C22×C4×C3⋊D311yescharacteristic
237D3—98no—
238D3—932no—
239D3—1832no—
240C22—274no—
241C22—2724no—
242C22#8878514yesdirect product factor
243C22C2×#628913yes—
244C4—274no—
245C4C2×#628913yes—
246C4C22×D3×C3⋊D311yescharacteristic
247C3—24no—
248C3C22×#371014yescomplemented
249C3C22×C8×C3⋊D311yescharacteristic, complemented
250C2—278no—
251C2C2×#8878516yesdirect product factor
252C2C22×#628911yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×C3⋊D3
commutator, hypocenterC33C23×C8
FrattiniC4C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C2×C4
prime radicalC23×C3⋊(C3⋊D3)C4
cosocleC33×C4C24
LCSG ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C2×#88785C2direct (N × H)
12. C2×C3×C3⋊(C3⋊C8)C22semidirect
13. C2×C32×C3⋊C8C22semidirect
14. #88785C22direct (N × H)
19. C22×C3⋊(C3⋊C8)D3semidirect
22. C22×C3×C3⋊C8D3semidirect
25. C22×C3⋊(C3⋊D3)C8semidirect
32. C3×C3⋊(C3⋊C8)C23semidirect
33. C32×C3⋊C8C23semidirect
52. C2×C3⋊(C3⋊C8)C2×D3semidirect
55. C2×C3×C3⋊C8C2×D3semidirect
57. C22×C33C2×C8semidirect
59. C2×C3⋊(C3⋊D3)C2×C8semidirect
70. C22×C3⋊C8C3⋊D3semidirect
98. C3⋊(C3⋊C8)C22×D3semidirect
100. C3×C3⋊C8C22×D3semidirect
102. C2×C33C22×C8semidirect
104. C3⋊(C3⋊D3)C22×C8semidirect
123. C2×C3⋊C8C2×C3⋊D3semidirect
128. C22×C32D3×C8semidirect
129. C22×C32D3×C8semidirect
151. C33C23×C8semidirect
179. C3⋊C8C22×C3⋊D3semidirect
182. C2×C32C2×D3×C8semidirect
183. C2×C32C2×D3×C8semidirect
199. C22×C3#3710semidirect
200. C22×C3C8×C3⋊D3semidirect
219. C32C22×D3×C8semidirect
220. C32C22×D3×C8semidirect
233. C2×C3C2×#3710semidirect
234. C2×C3C2×C8×C3⋊D3semidirect
242. C22#88785direct (N × H)
248. C3C22×#3710semidirect
249. C3C22×C8×C3⋊D3semidirect
251. C2C2×#88785direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊(C3⋊D3)2
C22×C3×C3⋊(C3⋊C8)2
C22×C32×C3⋊C82
C2×#887852
C22×C8×C3⋊D33
C22×#37103