Finite Group Databasebeta

← C22×#88881 #13180062

Subgroups — C22×#88881 #13180062

Order 1728.

999 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88881C213yesmaximal, direct product factor
2C22×C3×C22⋊A4C311yesmaximal, characteristic
3C22×C22⋊(C22⋊C9)C311yesmaximal, characteristic, complemented
4C24×C22⋊C9C312yesmaximal, complemented
5C22×C9⋊A4—42nomaximal
6#88881C2211yescharacteristic, direct product factor
7C2×C3×C22⋊A4C2×C313yes—
8C2×C22⋊(C22⋊C9)C2×C313yescomplemented
9C23×C22⋊C9—36no—
10C23×C22⋊C9—32no—
11C23×C22⋊C9C2×C316yescomplemented
12C2×C9⋊A4—46no—
13C26×C3C3211yescharacteristic
14C22×C22⋊A4—31no—
15C3×C22⋊A4C22×C311yescharacteristic
16C22×C3×A4—42no—
17C22×C3×A4—121no—
18C24×C9—42no—
19C22⋊(C22⋊C9)C22×C311yescharacteristic, complemented
20C22×C22⋊C9—34no—
21C22×C22⋊C9—36no—
22C22×C22⋊C9—36no—
23C22×C22⋊C9—36no—
24C22×C22⋊C9—42no—
25C22×C22⋊C9—121no—
26C22×C22⋊C9A412yescomplemented
27C22×C22⋊C9C22×C312yescomplemented
28C22×3−1+2—161no—
29C9⋊A4—42no—
30C25×C3—36no—
31C25×C3—32no—
32C25×C3—91no—
33C25×C3—93no—
34C25×C3C2×C3213yes—
35C2×C22⋊A4—33no—
36C2×C3×A4—46no—
37C2×C3×A4—123no—
38C23×C9—46no—
39C23×C9—122no—
40C23×C9—126no—
41C2×C22⋊C9—36no—
42C2×C22⋊C9—32no—
43C2×C22⋊C9—46no—
44C2×C22⋊C9—123no—
45C2×C22⋊C9C2×A416yescomplemented
46C263−1+211yescharacteristic, complemented, Sylow
47C2×3−1+2—163no—
48C24×C3—34no—
49C24×C3—31no—
50C24×C3—31no—
51C24×C3—36no—
52C24×C3—36no—
53C24×C3—36no—
54C24×C3—93no—
55C24×C3—91no—
56C24×C3—96no—
57C24×C3—96no—
58C24×C3—92no—
59C24×C3—93no—
60C24×C3—93no—
61C24×C3—93no—
62C24×C3—93no—
63C24×C3—96no—
64C24×C3—912no—
65C24×C3—912no—
66C24×C3—92no—
67C24×C3—92no—
68C24×C3C3×A412yes—
69C24×C3C22×C3211yescharacteristic
70C22⋊A4—31no—
71C22×A4—123no—
72C22×A4—122no—
73C22×C32—161no—
74C3×A4—42no—
75C3×A4—121no—
76C22×C9—42no—
77C22×C9—42no—
78C22×C9—126no—
79C22×C9—126no—
80C22×C9—126no—
81C22×C9—124no—
82C22×C9—161no—
83C22⋊C9—42no—
84C22⋊C9—121no—
85C22⋊C9C22×A412yescomplemented
86C25—36no—
87C25—32no—
88C25—91no—
89C25—93no—
90C25C2×3−1+213yescomplemented
913−1+2—161noSylow
92C23×C3—36no—
93C23×C3—32no—
94C23×C3—33no—
95C23×C3—33no—
96C23×C3—36no—
97C23×C3—32no—
98C23×C3—91no—
99C23×C3—93no—
100C23×C3—96no—
101C23×C3—93no—
102C23×C3—96no—
103C23×C3—96no—
104C23×C3—912no—
105C23×C3—96no—
106C23×C3—96no—
107C23×C3—94no—
108C23×C3—912no—
109C23×C3—912no—
110C23×C3—93no—
111C23×C3—93no—
112C23×C3—93no—
113C23×C3—912no—
114C23×C3—912no—
115C23×C3—93no—
116C23×C3—93no—
117C23×C3—96no—
118C23×C3—912no—
119C23×C3—912no—
120C23×C3—91no—
121C23×C3C2×C3×A416yes—
122C2×A4—129no—
123C2×A4—126no—
124C2×C32—163no—
125C2×C9—46no—
126C2×C9—122no—
127C2×C9—126no—
128C2×C9—163no—
129C24—34no—
130C24—31no—
131C24—31no—
132C24—36no—
133C24—36no—
134C24—36no—
135C24—93no—
136C24—91no—
137C24—96no—
138C24—96no—
139C24—92no—
140C24—93no—
141C24—93no—
142C24—93no—
143C24—93no—
144C24—96no—
145C24—912no—
146C24—912no—
147C24—92no—
148C24—92no—
149C24C9⋊A412yescomplemented
150C24C22×3−1+211yescharacteristic, complemented
151C22×C3—36no—
152C22×C3—34no—
153C22×C3—36no—
154C22×C3—36no—
155C22×C3—31no—
156C22×C3—31no—
157C22×C3—91no—
158C22×C3—96no—
159C22×C3—93no—
160C22×C3—96no—
161C22×C3—92no—
162C22×C3—96no—
163C22×C3—912no—
164C22×C3—912no—
165C22×C3—93no—
166C22×C3—93no—
167C22×C3—92no—
168C22×C3—92no—
169C22×C3—93no—
170C22×C3—93no—
171C22×C3—481no—
172C22×C3A4211yescharacteristic
173C22×C3C22×C3×A412yes—
174A4—123no—
175A4—122no—
176C32—161no—
177C9—42no—
178C9—161no—
179C23—36no—
180C23—32no—
181C23—33no—
182C23—33no—
183C23—32no—
184C23—36no—
185C23—91no—
186C23—93no—
187C23—96no—
188C23—93no—
189C23—96no—
190C23—96no—
191C23—912no—
192C23—96no—
193C23—96no—
194C23—94no—
195C23—912no—
196C23—912no—
197C23—93no—
198C23—93no—
199C23—93no—
200C23—912no—
201C23—912no—
202C23—93no—
203C23—93no—
204C23—96no—
205C23—912no—
206C23—912no—
207C23—91no—
208C23C2×C9⋊A416yescomplemented
209C2×C3—32no—
210C2×C3—36no—
211C2×C3—91no—
212C2×C3—93no—
213C2×C3—483no—
214C2×C3C2×A4213yes—
215C22—36no—
216C22—34no—
217C22—36no—
218C22—36no—
219C22—31no—
220C22—31no—
221C22—91no—
222C22—96no—
223C22—93no—
224C22—96no—
225C22—92no—
226C22—96no—
227C22—912no—
228C22—912no—
229C22—93no—
230C22—93no—
231C22—92no—
232C22—92no—
233C22—93no—
234C22—93no—
235C22#8888111yescharacteristic, direct product factor
236C22C22×C9⋊A412yescomplemented
237C3—481no—
238C3C22×A4211yescharacteristic
239C2—32no—
240C2—36no—
241C2—91no—
242C2—93no—
243C2C2×#8888113yesdirect product factor
subgroupG/H
center, hypercenterC22×C3A42
commutator, cosocleC24×C3C22×C32
FrattiniC3C22×A42
Fitting, socleC26×C3C32
prime radicalC22×C3×C22⋊A4C3
hypocenterC24C22×3−1+2
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×#88881C2direct (N × H)
3. C22×C22⋊(C22⋊C9)C3semidirect
4. C24×C22⋊C9C3semidirect
6. #88881C22direct (N × H)
8. C2×C22⋊(C22⋊C9)C2×C3semidirect
11. C23×C22⋊C9C2×C3semidirect
19. C22⋊(C22⋊C9)C22×C3semidirect
26. C22×C22⋊C9A4semidirect
27. C22×C22⋊C9C22×C3semidirect
45. C2×C22⋊C9C2×A4semidirect
46. C263−1+2semidirect
85. C22⋊C9C22×A4semidirect
90. C25C2×3−1+2semidirect
149. C24C9⋊A4semidirect
150. C24C22×3−1+2semidirect
208. C23C2×C9⋊A4semidirect
235. C22#88881direct (N × H)
236. C22C22×C9⋊A4semidirect
243. C2C2×#88881direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888812
C22×C3×C22⋊A43
C22×C22⋊(C22⋊C9)3
C24×C22⋊C93
C22×C9⋊A44