Finite Group Databasebeta

← C2×#11722372 #13164460

Subgroups — C2×#11722372 #13164460

Order 1728.

563 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 85 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×C9×#41⋊C3C211yesmaximal, characteristic
2#11722372C214yesmaximal, direct product factor
3C2×C9⋊#226C212yesmaximal, complemented
4C2×#64265—31nomaximal
5C2×#63690—32nomaximal
6C2×#63511—31nomaximal
7C2×#6226—41nomaximal
8C9×#41⋊C3C2212yes—
9C2×C9×SL(2,𝔽3)C2211yescharacteristic
10C9⋊#226C2214yescomplemented
11C2×C3×#41⋊C3D311yescharacteristic
12#64265—34no—
13C2×C3⋊#226—32no—
14C2×C9×#41D311yescharacteristic
15C22×Dic18—31no—
16C2×#41⋊C9D312yes—
17#63690—38no—
18C2×#3688—34no—
19#63511—34no—
20#63511—34no—
21C2×#3674—32no—
22C2×#3672—32no—
23C2×#3667—31no—
24C2×C3×C4×C9—41no—
25C2×C3⋊Dic9—42no—
26#6226—44no—
27C9×SL(2,𝔽3)C2311yescharacteristic
28C2×#984—93no—
29C2×#951—91no—
30C2×#634—121no—
31C3×#41⋊C3C2×D312yes—
32C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
33C3⋊#226—34no—
34C9×#41—32no—
35C9×#41C2×D312yes—
36C2×Q8×C9C2×D311yescharacteristic
37C2×D4×C9—31no—
38C22×C4×C9—31no—
39C22×Dic9—61no—
40C2×Dic18—32no—
41C2×Dic18—34no—
42C2×Dic18—62no—
43C2×Dic18—61no—
44C2×Dic18—122no—
45#41⋊C9C2×D314yes—
46C2×Q8⋊C9C2×D312yes—
47#3688—38no—
48#3674—34no—
49#3674—34no—
50#3672—38no—
51#3667—34no—
52C2×C9⋊C8—32no—
53C22×C3×C9—41no—
54C3×C4×C9—42no—
55C3⋊Dic9—44no—
56C2×C3×#41C3⋊D311yescharacteristic
57C22×Dic6—91no—
58C2×#41⋊C3D913yes—
59#984—912no—
60C2×#226—96no—
61#951—94no—
62#951—94no—
63C2×#216—92no—
64C2×#214—92no—
65C2×#208—91no—
66C2×C32×C4—41no—
67C2×C3⋊Dic3—122no—
68#634—124no—
69C3×SL(2,𝔽3)C22×D311yescharacteristic
70C23×C9—31no—
71Q8×C9—31no—
72Q8×C9C22×D311yescharacteristic
73D4×C9—32no—
74D4×C9—61no—
75C2×C4×C9—31no—
76C2×C4×C9—32no—
77C2×C4×C9—32no—
78C2×C4×C9—42no—
79C2×C4×C9S411yescharacteristic
80C2×Dic9—64no—
81C2×Dic9—62no—
82C2×Dic9—124no—
83Dic18—34no—
84Dic18—62no—
85Dic18—64no—
86Dic18—128no—
87Q8⋊C9C22×D312yes—
88C9⋊C8—34no—
89C2×#137—271noSylow
90C2×C3×C9—42no—
91C2×C3×C9—41no—
92C3×#41—32no—
93C3×#41C2×C3⋊D312yes—
94C2×C3×Q8C2×C3⋊D311yescharacteristic
95C2×C3×D4—31no—
96C22×C3×C4—31no—
97C22×Dic3—181no—
98C2×Dic6—92no—
99C2×Dic6—94no—
100C2×Dic6—182no—
101C2×Dic6—181no—
102C2×Dic6—363no—
103#41⋊C3C2×D916yes—
104C2×SL(2,𝔽3)C2×D913yes—
105#226—912no—
106#216—94no—
107#216—94no—
108#214—98no—
109#208—94no—
110C2×C3⋊C8—92no—
111C22×C32—41no—
112C32×C4—42no—
113C3⋊Dic3—124no—
114C22×C9—32no—
115C22×C9—42no—
116C22×C9—62no—
117C22×C9C2×S411yescharacteristic
118C4×C9—31no—
119C4×C9—31no—
120C4×C9—44no—
121C4×C9C2×S412yes—
122Dic9—64no—
123Dic9—128no—
124C2×#41C3⋊D911yescharacteristic
125C22×Q8—271no—
126#137—274no—
127#137—274no—
128C2×Q16—272no—
129C2×SD16—272no—
130C2×M16—271no—
131C3×C9—41noSylow
132C23×C3—31no—
133C3×Q8—31no—
134C3×Q8C22×C3⋊D311yescharacteristic
135C3×D4—32no—
136C3×D4—61no—
137C2×C3×C4—31no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—43no—
141C2×C3×C4C3⋊S411yescharacteristic
142C2×Dic3—184no—
143C2×Dic3—182no—
144C2×Dic3—366no—
145Dic6—94no—
146Dic6—182no—
147Dic6—184no—
148Dic6—3612no—
149SL(2,𝔽3)C22×D913yes—
150C3⋊C8—94no—
151C2×C32—42no—
152C2×C32—41no—
153C2×C9—44no—
154C2×C9—42no—
155C2×C9—62no—
156C2×C9#98412yescomplemented
157C2×C9C22×S411yescharacteristic
158#41—32no—
159#41C2×C3⋊D912yes—
160C2×Q8—272no—
161C2×Q8—274no—
162C2×Q8—542no—
163C2×Q8—541no—
164C2×Q8C2×C3⋊D911yescharacteristic
165C2×D4—31no—
166C22×C4—31no—
167C22×C4—541no—
168Q16—274no—
169Q16—274no—
170SD16—278no—
171M16—274no—
172C2×C8—272no—
173C22×C3—32no—
174C22×C3—43no—
175C22×C3—62no—
176C22×C3C2×C3⋊S411yescharacteristic
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—46no—
180C3×C4C2×C3⋊S412yes—
181Dic3—184no—
182Dic3—3612no—
183C32—41no—
184C9—42no—
185C9C2×#98411yescharacteristic, complemented
186C23—31no—
187Q8—31no—
188Q8—274no—
189Q8—542no—
190Q8—544no—
191Q8C22×C3⋊D911yescharacteristic
192D4—32no—
193D4—61no—
194C2×C4—31no—
195C2×C4—32no—
196C2×C4—32no—
197C2×C4—544no—
198C2×C4—542no—
199C2×C4C9⋊S411yescharacteristic
200C8—274no—
201C2×C3—43no—
202C2×C3—46no—
203C2×C3—62no—
204C2×C3#6426512yes—
205C2×C3C22×C3⋊S411yescharacteristic
206C22—32no—
207C22—62no—
208C22C2×C9⋊S411yescharacteristic
209C4—31no—
210C4—31no—
211C4—544no—
212C4C2×C9⋊S412yes—
213C3—43no—
214C3C2×#6426511yescharacteristic
215C2—62no—
216C2#1172237212yesdirect product factor
217C2C22×C9⋊S411yescharacteristic
subgroupG/H
centerC22C2×C9⋊S4
commutator, cosocle, hypocenterC9×SL(2,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC2×C9×#41D3
socleC22×C3C2×C3⋊S4
prime radicalC2×C3×#41⋊C3D3
hypercenterC2×C4C9⋊S4
LCSG ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722372C2direct (N × H)
3. C2×C9⋊#226C2semidirect
10. C9⋊#226C22semidirect
156. C2×C9#984semidirect
185. C9C2×#984semidirect
216. C2#11722372direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C9×#41⋊C32
#117223722
C2×C9⋊#2262
C2×#642653
C2×#636903
C2×#635113
C2×#62264