Finite Group Databasebeta

← 3−1+2×#833 #72219976

Subgroups — 3−1+2×#833 #72219976

Order 2592.

665 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 665 conjugacy classes of subgroups — too many to draw. Only its 147 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×C3×D4×3−1+2C211yesmaximal, characteristic
23−1+2×#208C212yesmaximal, complemented
3C32×#833C311yesmaximal, characteristic
43−1+2×#100—31nomaximal
5C9×#833C313yesmaximal, complemented
6C23×C3×3−1+2C412yes—
7C3×D4×3−1+2—22no—
8C2×C3×C4×3−1+2C2211yescharacteristic
9C3⋊C8×3−1+2—22no—
10C2×C33×D4C2×C311yescharacteristic
11C32×#208C2×C312yes—
12C2×D4×3−1+2—21no—
13C2×D4×3−1+2—23no—
14C2×D4×3−1+2D311yescharacteristic
15C2×C3×D4×C9C2×C313yes—
16M16×3−1+2—32no—
17C9×#208C2×C316yescomplemented
18C22×C3×3−1+2—22no—
19C22×C3×3−1+2—42no—
20C22×C3×3−1+2C2×C411yescharacteristic
21C3×C4×3−1+2D412yes—
22C32×#100—31no—
23C3×#833—31no—
24C3×#833C3211yescharacteristic
25C9×#100—33no—
26C23×C33C3×C412yes—
27C33×D4—22no—
28C2×C33×C4C22×C311yescharacteristic
29C23×3−1+2—22no—
30C23×3−1+2—26no—
31C23×3−1+2Dic312yes—
32C23×C3×C9C3×C416yes—
33C32×C3⋊C8—22no—
34D4×3−1+2—24no—
35D4×3−1+2—212no—
36D4×3−1+2—22no—
37C3×D4×C9—26no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—23no—
40C2×C4×3−1+2C2×D311yescharacteristic
41C2×C3×C4×C9C22×C313yes—
42C8×3−1+2—62no—
43C9×C3⋊C8—26no—
44C2×C3×3−1+2—21no—
45C2×C3×3−1+2—42no—
46C2×C3×3−1+2C22⋊C411yescharacteristic
47C2×C32×D4—21no—
48C2×C32×D4—31no—
49C2×C32×D4—61no—
50C2×C32×D4C3×D311yescharacteristic
51C2×C32×D4C2×C3211yescharacteristic
52C32×M16—32no—
53C3×#208—32no—
54C3×#208C2×C3212yes—
55C2×D4×C9—23no—
56C2×D4×C9C3×D313yes—
57C9×M16—36no—
58C22×C33—22no—
59C22×C33—42no—
60C22×C33C2×C3×C411yescharacteristic
61C33×C4C3×D412yes—
62C22×3−1+2—21no—
63C22×3−1+2—23no—
64C22×3−1+2—24no—
65C22×3−1+2—212no—
66C22×3−1+2—22no—
67C22×3−1+2—412no—
68C22×3−1+2—44no—
69C22×3−1+2—42no—
70C22×3−1+2C2×Dic311yescharacteristic
71C22×C3×C9—26no—
72C22×C3×C9—46no—
73C22×C3×C9C2×C3×C413yes—
74C4×3−1+2—22no—
75C4×3−1+2—26no—
76C4×3−1+2#6712yes—
77C3×C4×C9C3×D416yes—
78C3×#100—31no—
79C3×#100—91no—
80#8333−1+211yescharacteristic, direct product factor
81C3×3−1+2#10011yescharacteristic, complemented, Sylow
82C23×C32—22no—
83C23×C32—32no—
84C23×C32—62no—
85C23×C32C3×Dic312yes—
86C23×C32C32×C412yes—
87C32×D4—22no—
88C32×D4—24no—
89C32×D4—22no—
90C32×D4—62no—
91C32×D4—64no—
92C2×C32×C4—21no—
93C2×C32×C4—31no—
94C2×C32×C4—61no—
95C2×C32×C4C2×C3×D311yescharacteristic
96C2×C32×C4C22×C3211yescharacteristic
97C23×C9—26no—
98C23×C9C3×Dic316yes—
99C32×C8—62no—
100C3×C3⋊C8—22no—
101C3×C3⋊C8—62no—
102D4×C9—212no—
103D4×C9—26no—
104C2×C4×C9—23no—
105C2×C4×C9C2×C3×D313yes—
106C8×C9—66no—
107C2×C33—21no—
108C2×C33—42no—
109C2×C33C3×C22⋊C411yescharacteristic
110C2×3−1+2—22no—
111C2×3−1+2—26no—
112C2×3−1+2—21no—
113C2×3−1+2—21no—
114C2×3−1+2—23no—
115C2×3−1+2—44no—
116C2×3−1+2—412no—
117C2×3−1+2—42no—
118C2×3−1+2#21711yescharacteristic
119C2×C3×C9—23no—
120C2×C3×C9—46no—
121C2×C3×C9C3×C22⋊C413yes—
122C2×C3×D4—21no—
123C2×C3×D4—31no—
124C2×C3×D4—61no—
125C2×C3×D4C2×3−1+211yescharacteristic
126C2×C3×D4C32×D311yescharacteristic
127C3×M16—32no—
128C3×M16—92no—
129#208C2×3−1+212yescomplemented
130C22×C32—22no—
131C22×C32—24no—
132C22×C32—22no—
133C22×C32—21no—
134C22×C32—31no—
135C22×C32—42no—
136C22×C32—42no—
137C22×C32—44no—
138C22×C32—61no—
139C22×C32—62no—
140C22×C32—64no—
141C22×C32—122no—
142C22×C32—124no—
143C22×C32C2×C3×Dic311yescharacteristic
144C22×C32C2×C32×C411yescharacteristic
145C32×C4—22no—
146C32×C4—32no—
147C32×C4—62no—
148C32×C4C3×#6712yes—
149C32×C4C32×D412yes—
150C22×C9—212no—
151C22×C9—26no—
152C22×C9—23no—
153C22×C9—46no—
154C22×C9—412no—
155C22×C9C2×C3×Dic313yes—
156C4×C9—26no—
157C4×C9C3×#6716yes—
158#100—31noSylow
159C33C3×#10011yescharacteristic
1603−1+2—21no—
1613−1+2—23no—
1623−1+2#83311yescharacteristic, direct product factor
163C3×C9C3×#10013yescomplemented
164C23×C3—22no—
165C23×C3—32no—
166C23×C3—62no—
167C23×C3C4×3−1+212yes—
168C23×C3C32×Dic312yes—
169C3×D4—24no—
170C3×D4—22no—
171C3×D4—22no—
172C3×D4—62no—
173C3×D4—64no—
174C2×C3×C4—21no—
175C2×C3×C4—31no—
176C2×C3×C4—61no—
177C2×C3×C4C22×3−1+211yescharacteristic
178C2×C3×C4C2×C32×D311yescharacteristic
179C3×C8—62no—
180C3×C8—182no—
181C3⋊C8—22no—
182C2×C32—21no—
183C2×C32—22no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—31no—
187C2×C32—42no—
188C2×C32—44no—
189C2×C32—42no—
190C2×C32—61no—
191C2×C32—61no—
192C2×C32—62no—
193C2×C32—122no—
194C2×C32—124no—
195C2×C32C3×#21711yescharacteristic
196C2×C32C32×C22⋊C411yescharacteristic
197C2×C9—26no—
198C2×C9—23no—
199C2×C9—23no—
200C2×C9—412no—
201C2×C9—46no—
202C2×C9C3×#21713yes—
203C2×D4D3×3−1+211yescharacteristic
204M16—32no—
205C22×C3—24no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—21no—
209C22×C3—31no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—44no—
213C22×C3—62no—
214C22×C3—64no—
215C22×C3—61no—
216C22×C3—122no—
217C22×C3—124no—
218C22×C3C2×C4×3−1+211yescharacteristic
219C22×C3C2×C32×Dic311yescharacteristic
220C3×C4—22no—
221C3×C4—32no—
222C3×C4—62no—
223C3×C4D4×3−1+212yes—
224C3×C4C32×#6712yes—
225C32—21no—
226C32—31no—
227C32—61no—
228C32C3×#83311yescharacteristic
229C32C32×#10011yescharacteristic
230C9—23no—
231C9C3×#83313yescomplemented
232C23Dic3×3−1+212yes—
233D4—22no—
234C2×C4C2×D3×3−1+211yescharacteristic
235C8—62no—
236C2×C3—22no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—31no—
241C2×C3—44no—
242C2×C3—42no—
243C2×C3—42no—
244C2×C3—61no—
245C2×C3—62no—
246C2×C3—61no—
247C2×C3—122no—
248C2×C3—124no—
249C2×C3C22⋊C4×3−1+211yescharacteristic
250C2×C3C32×#21711yescharacteristic
251C22—22no—
252C22—42no—
253C22C2×Dic3×3−1+211yescharacteristic
254C4#67×3−1+212yes—
255C3—21no—
256C3—31no—
257C3—61no—
258C33−1+2×#10011yescharacteristic, complemented
259C3C32×#83311yescharacteristic
260C2—21no—
261C2—42no—
262C23−1+2×#21711yescharacteristic
subgroupG/H
centerC2×C3C32×#217
commutatorC22×C32C2×C32×C4
FrattiniC2×C3×C4C2×C32×D3
FittingC2×C3×D4×3−1+2C2
socleC2×C32C32×C22⋊C4
prime radicalC2×C33×D4C2×C3
cosocleC2×C32×C4C22×C32
hypercenterC2×D4×3−1+2D3
hypocenterC33−1+2×#100
LCSG ▷ C22×C32 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C4×3−1+2 ◁ C2×D4×3−1+2

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#208C2semidirect
5. C9×#833C3semidirect
17. C9×#208C2×C3semidirect
80. #8333−1+2direct (N × H)
81. C3×3−1+2#100semidirect
129. #208C2×3−1+2semidirect
162. 3−1+2#833direct (N × H)
163. C3×C9C3×#100semidirect
231. C9C3×#833semidirect
258. C33−1+2×#100semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×3−1+22
3−1+2×#2082
C32×#8333
3−1+2×#1003
C9×#8333