Finite Group Databasebeta

← C22×C3⋊D3×#157 #72228456

Subgroups — C22×C3⋊D3×#157 #72228456

Order 2592.

2924 conjugacy classes of subgroups; only the 294 normal subgroups are shown.

normal subgroups only

This group has 2924 conjugacy classes of subgroups — too many to draw. Only its 294 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×C3⋊D32C211yesmaximal, characteristic
2C22×C3⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
3C22×C32×#157C211yesmaximal, characteristic, complemented
4C2×C3⋊D3×#157C2112yesmaximal, direct product factor
5C22×D3×#157—34nomaximal
6C22×C3⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
7C22×C32×C3⋊D3C411yescharacteristic, complemented
8C22×C32×C3⋊D3C2211yescharacteristic
9C2×C3⋊D32C416yescomplemented
10C2×C3⋊D32C2216yes—
11C2×C3⋊(C3⋊#157)C2216yescomplemented
12C2×C32×#157C2216yescomplemented
13C3⋊D3×#157C22116yesdirect product factor
14C22×D3×C3⋊D3—34no—
15C22×D3×C3⋊D3—62no—
16C22×C3⋊#157—34no—
17C22×C3×#157D314yescomplemented
18C2×D3×#157—348no—
19C22×C34C2×C411yescharacteristic, complemented
20C2×C3⋊(C3⋊(C3⋊D3))C2×C416yescomplemented
21C2×C32×C3⋊D3C2×C416yescomplemented
22C2×C32×C3⋊D3C2×C413yescomplemented
23C2×C32×C3⋊D3C2313yes—
24C3⋊D32C2×C4112yescomplemented
25C3⋊D32C2314yes—
26C3⋊(C3⋊#157)C2314yescomplemented
27C32×#157C2314yescomplemented
28C23×#157—91no—
29C22×C4×C3⋊D3—91nomaximal
30C22×C3⋊(C3⋊D3)—34no—
31C22×C3⋊(C3⋊D3)—62no—
32C22×C3⋊(C3⋊D3)—128no—
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3C2×D314yes—
35C22×C32×D3—34no—
36C22×C32×D3—62no—
37C2×D3×C3⋊D3—324no—
38C2×D3×C3⋊D3—324no—
39C2×D3×C3⋊D3—612no—
40C2×D3×C3⋊D3—612no—
41C2×C3⋊#157—324no—
42C2×C3×#157C2×D3124yescomplemented
43D3×#157—364no—
44C2×C34C22×C413yescomplemented
45C3⋊(C3⋊(C3⋊D3))C22×C414yescomplemented
46C32×C3⋊D3C22×C414yescomplemented
47C32×C3⋊D3C22×C413yescomplemented
48C32×C3⋊D3C2411yescharacteristic
49C23×C3⋊D3—91no—
50C23×C3⋊D3—91no—
51C22×D32—188no—
52C22×#157—91no—
53C22×#157—912no—
54C22×#157C3⋊D311yescharacteristic, direct product factor
55C22×C32×C4—91no—
56C22×C3⋊Dic3—91no—
57C2×C4×C3⋊D3—912no—
58C22×C33—22no—
59C22×C33—48no—
60C22×C33C4×D314yescomplemented
61C2×C3⋊(C3⋊D3)—324no—
62C2×C3⋊(C3⋊D3)—612no—
63C2×C3⋊(C3⋊D3)—1248no—
64C2×C3×C3⋊D3—212no—
65C2×C3×C3⋊D3C4×D3112yescomplemented
66C2×C3×C3⋊D3C22×D3112yes—
67C2×C32×D3—324no—
68C2×C32×D3—66no—
69C2×C32×D3—66no—
70D3×C3⋊D3—348no—
71D3×C3⋊D3—316no—
72D3×C3⋊D3—624no—
73D3×C3⋊D3—68no—
74C3⋊#157—316no—
75C3×#157C22×D3116yescomplemented
76C22×C4×D3—274no—
77C34C23×C411yescharacteristic, complemented, Sylow
78C23×C32—91no—
79C23×C32—91no—
80C22×C3⋊D3—91no—
81C22×C3⋊D3—96no—
82C22×C3⋊D3—96no—
83C22×C3⋊D3—91no—
84C22×C3⋊D3—96no—
85C22×C3⋊D3—96no—
86C22×C3⋊D3—188no—
87C22×C3⋊D3—368no—
88C22×C3⋊D3—368no—
89C22×C3⋊D3—3612no—
90C22×C3⋊D3#15711yescharacteristic, direct product factor
91C22×C3⋊D3C2×C3⋊D311yescharacteristic
92C22×C3×D3—68no—
93C22×C3×D3—68no—
94C2×D32—1848no—
95C2×D32—1848no—
96C2×#157—96no—
97C2×#157—916no—
98C2×#157C2×C3⋊D316yesdirect product factor
99C2×C32×C4—96no—
100C2×C3⋊Dic3—96no—
101C4×C3⋊D3—916no—
102C2×C33—26no—
103C2×C33—424no—
104C2×C33C2×C4×D3112yescomplemented
105C3⋊(C3⋊D3)—316no—
106C3⋊(C3⋊D3)—68no—
107C3⋊(C3⋊D3)—1232no—
108C3×C3⋊D3—28no—
109C3×C3⋊D3C2×C4×D3112yescomplemented
110C3×C3⋊D3C23×D314yes—
111C32×D3—316no—
112C32×D3—66no—
113C32×D3—62no—
114C23×D3—274no—
115C23×D3—542no—
116C22×C3×C4—94no—
117C22×Dic3—274no—
118C2×C4×D3—2748no—
119C22×C32—28no—
120C22×C32—48no—
121C22×C32—412no—
122C22×C32—48no—
123C22×C32—93no—
124C22×C32—93no—
125C22×C32—96no—
126C22×C32C4×C3⋊D311yescharacteristic, complemented
127C22×C32C2×#15711yescharacteristic, complemented
128C2×C3⋊D3—96no—
129C2×C3⋊D3—94no—
130C2×C3⋊D3—912no—
131C2×C3⋊D3—96no—
132C2×C3⋊D3—94no—
133C2×C3⋊D3—912no—
134C2×C3⋊D3—1848no—
135C2×C3⋊D3—3648no—
136C2×C3⋊D3—3648no—
137C2×C3⋊D3—3672no—
138C2×C3⋊D3C4×C3⋊D313yescomplemented
139C2×C3⋊D3C2×#15716yesdirect product factor
140C2×C3⋊D3C22×C3⋊D313yes—
141C2×C3×D3—648no—
142C2×C3×D3—624no—
143C2×C3×D3—624no—
144D32—1832no—
145D32—1896no—
146#157—94no—
147#157C22×C3⋊D314yesdirect product factor
148C32×C4—94no—
149C3⋊Dic3—94no—
150C23×C4—811noSylow
151C33—22no—
152C33—48no—
153C33C22×C4×D314yescomplemented
154C23×C3—94no—
155C23×C3—182no—
156C22×D3—34no—
157C22×D3—62no—
158C22×D3—2724no—
159C22×D3—2724no—
160C22×D3—274no—
161C22×D3—5412no—
162C22×D3—5412no—
163C22×D3—542no—
164C22×D3—1088no—
165C2×C3×C4—924no—
166C2×Dic3—2724no—
167C4×D3—2764no—
168C2×C32—224no—
169C2×C32—424no—
170C2×C32—436no—
171C2×C32—424no—
172C2×C32—91no—
173C2×C32—93no—
174C2×C32—94no—
175C2×C32C2×C4×C3⋊D313yescomplemented
176C2×C32C22×#15713yescomplemented
177C3⋊D3—94no—
178C3⋊D3—94no—
179C3⋊D3—1832no—
180C3⋊D3—3632no—
181C3⋊D3—3632no—
182C3⋊D3—3648no—
183C3⋊D3C2×C4×C3⋊D313yescomplemented
184C3⋊D3C22×#15714yesdirect product factor
185C3⋊D3C23×C3⋊D311yescharacteristic
186C3×D3—632no—
187C3×D3—624no—
188C3×D3—68no—
189C24—811no—
190C22×C4—91no—
191C22×C4—8112no—
192C22×C4—811no—
193C22×C3—22no—
194C22×C3—48no—
195C22×C3—912no—
196C22×C3—912no—
197C22×C3—1812no—
198C22×C3D3×#15714yescomplemented
199C2×D3—324no—
200C2×D3—66no—
201C2×D3—66no—
202C2×D3—2716no—
203C2×D3—2748no—
204C2×D3—2724no—
205C2×D3—548no—
206C2×D3—5424no—
207C2×D3—5412no—
208C2×D3—10848no—
209C3×C4—916no—
210Dic3—2716no—
211C32—28no—
212C32—48no—
213C32—412no—
214C32—48no—
215C32C22×C4×C3⋊D311yescharacteristic, complemented
216C32C23×#15711yescharacteristic, complemented
217C23—91no—
218C23—91no—
219C23—816no—
220C23—816no—
221C23—811no—
222C2×C4—96no—
223C2×C4—8116no—
224C2×C4—816no—
225C2×C3—26no—
226C2×C3—424no—
227C2×C3—94no—
228C2×C3—912no—
229C2×C3—188no—
230C2×C3C2×D3×#157112yescomplemented
231D3—316no—
232D3—62no—
233D3—66no—
234D3—2716no—
235D3—548no—
236D3—10832no—
237C22—93no—
238C22—96no—
239C22—93no—
240C22—814no—
241C22—8112no—
242C22—816no—
243C22C3⋊D3×#15711yescharacteristic, direct product factor
244C4—94no—
245C4—814no—
246C3—22no—
247C3—48no—
248C3C22×D3×#15714yescomplemented
249C2—91no—
250C2—94no—
251C2—93no—
252C2—814no—
253C2C2×C3⋊D3×#15713yesdirect product factor
subgroupG/H
center, hypercenterC22C3⋊D3×#157
commutator, hypocenterC34C23×C4
FrattiniC1G
Fitting, socleC22×C34C2×C4
prime radicalC22×C3⋊D32C2
cosocleC32×C3⋊D3C24
LCSG ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊(C3⋊#157)C2semidirect
3. C22×C32×#157C2semidirect
4. C2×C3⋊D3×#157C2direct (N × H)
6. C22×C3⋊(C3⋊(C3⋊D3))C4semidirect
7. C22×C32×C3⋊D3C4semidirect
9. C2×C3⋊D32C4semidirect
11. C2×C3⋊(C3⋊#157)C22semidirect
12. C2×C32×#157C22semidirect
13. C3⋊D3×#157C22direct (N × H)
17. C22×C3×#157D3semidirect
19. C22×C34C2×C4semidirect
20. C2×C3⋊(C3⋊(C3⋊D3))C2×C4semidirect
21. C2×C32×C3⋊D3C2×C4semidirect
22. C2×C32×C3⋊D3C2×C4semidirect
24. C3⋊D32C2×C4semidirect
26. C3⋊(C3⋊#157)C23semidirect
27. C32×#157C23semidirect
42. C2×C3×#157C2×D3semidirect
44. C2×C34C22×C4semidirect
45. C3⋊(C3⋊(C3⋊D3))C22×C4semidirect
46. C32×C3⋊D3C22×C4semidirect
47. C32×C3⋊D3C22×C4semidirect
54. C22×#157C3⋊D3direct (N × H)
60. C22×C33C4×D3semidirect
65. C2×C3×C3⋊D3C4×D3semidirect
75. C3×#157C22×D3semidirect
77. C34C23×C4semidirect
90. C22×C3⋊D3#157direct (N × H)
98. C2×#157C2×C3⋊D3direct (N × H)
104. C2×C33C2×C4×D3semidirect
109. C3×C3⋊D3C2×C4×D3semidirect
126. C22×C32C4×C3⋊D3semidirect
127. C22×C32C2×#157semidirect
138. C2×C3⋊D3C4×C3⋊D3semidirect
139. C2×C3⋊D3C2×#157direct (N × H)
147. #157C22×C3⋊D3direct (N × H)
153. C33C22×C4×D3semidirect
175. C2×C32C2×C4×C3⋊D3semidirect
176. C2×C32C22×#157semidirect
183. C3⋊D3C2×C4×C3⋊D3semidirect
184. C3⋊D3C22×#157direct (N × H)
198. C22×C3D3×#157semidirect
215. C32C22×C4×C3⋊D3semidirect
216. C32C23×#157semidirect
230. C2×C3C2×D3×#157semidirect
243. C22C3⋊D3×#157direct (N × H)
248. C3C22×D3×#157semidirect
253. C2C2×C3⋊D3×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊D322
C22×C3⋊(C3⋊#157)2
C22×C32×#1572
C2×C3⋊D3×#1572
C22×D3×#1573
C22×C4×C3⋊D39