Finite Group Databasebeta

← C3×#10589518 #13146890

Subgroups — C3×#10589518 #13146890

Order 1728.

290 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C3×Q8×C22⋊C9C211yesmaximal, characteristic
2C3×(C22⋊C9)⋊Q8C211yesmaximal, characteristic
3C3×(C22⋊C9)⋊C8C211yesmaximal, characteristic
4C3×#5228—31nomaximal
5#10589518C311yesmaximal, characteristic, direct product factor
6C3×C4×C22⋊C9—21no—
7C3×C4×C22⋊C9C2211yescharacteristic
8C3×(C22⋊C9)⋊C4—21no—
9C3×#3674—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×C3×#216—31no—
12C2×C3×#216—31no—
13C3×#924—31no—
14Q8×C22⋊C9—21no—
15Q8×C22⋊C9C2×C311yescharacteristic
16(C22⋊C9)⋊Q8C2×C311yescharacteristic
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20(C22⋊C9)⋊C8C2×C311yescharacteristic
21C2×C3×C22⋊C9D411yescharacteristic
22C3×Q8×C9—41no—
23C3×Dic18—41no—
24C3×C9⋊C8—41no—
25C3×#451—91no—
26#5228—31no—
27C2×C32×Q8—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C22×C32×C4—21no—
31C22×C32×C4C2×D311yescharacteristic
32C2×C3×Dic6—31no—
33C3×#217—61no—
34C3×#216—62no—
35C3×#216—61no—
36C3×#216—61no—
37C3×C4⋊Dic3—31no—
38C3×Dic3⋊C4—61no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41C4×C22⋊C9—22no—
42C4×C22⋊C9—21no—
43C4×C22⋊C9—21no—
44C4×C22⋊C9C22×C311yescharacteristic
45(C22⋊C9)⋊C4—21no—
46#3674—41no—
47C3×C22⋊C9Q1611yescharacteristic, complemented
48C3×C4×C9—41no—
49C3×C4×C9—81no—
50C3×Dic9—81no—
51C22×C3×Q8—21no—
52C22×C3×Q8D911yescharacteristic
53C22×C3×Q8C3×D311yescharacteristic
54C2×C3×Q16—91no—
55C2×C3×Q16—91no—
56C3×C22⋊Q8—91no—
57C2×#216—31no—
58C2×#216—31no—
59#924—31no—
60C3×#103—91no—
61C3×#103—91no—
62C3×C22⋊C8—91no—
63#835—31no—
64#835—31no—
65#830—31no—
66C23×C32#6711yescharacteristic
67C32×Q8—32no—
68C32×Q8—31no—
69C32×Q8—61no—
70C32×Q8S411yescharacteristic
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—61no—
74C2×C32×C4—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C3×Dic6—61no—
78C3×Dic6—61no—
79C2×C22⋊C9—21no—
80C2×C22⋊C9C3×D411yescharacteristic
81C3×C3⋊C8—61no—
82C3×C3⋊C8—61no—
83Q8×C9—41no—
84Q8×C9—81no—
85Dic18—41no—
86C9⋊C8—41no—
87#451—91noSylow
88C2×C3×C9—41no—
89C2×C3×Q8—31no—
90C2×C3×Q8—31no—
91C2×C3×Q8—31no—
92C2×C3×Q8—31no—
93C2×C3×Q8—61no—
94C2×C3×Q8—61no—
95C2×C3×Q8—61no—
96C2×C3×Q8—61no—
97C2×C3×Q8—62no—
98C2×C3×Q8—91no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D911yescharacteristic
104C22×C3×C4C2×C3×D311yescharacteristic
105C2×Dic6—31no—
106C3×Q16—182no—
107C3×Q16—181no—
108C3×Q16—181no—
109C2×C3×C8—91no—
110C2×C3×C8—91no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—181no—
113C3×C22⋊C4—181no—
114#217—61no—
115#216—62no—
116#216—61no—
117#216—61no—
118C4⋊Dic3—31no—
119Dic3⋊C4—61no—
120C2×C3⋊C8—31no—
121C2×C3⋊C8—31no—
122C22×C32—31no—
123C22×C32—31no—
124C22×C32#21611yescharacteristic
125C32×C4—21no—
126C32×C4—31no—
127C32×C4—61no—
128C32×C4C2×S411yescharacteristic
129C3×Dic3—121no—
130C3×Dic3—121no—
131C22⋊C9—21no—
132C22⋊C9C3×Q1611yescharacteristic, complemented
133C4×C9—41no—
134C4×C9—81no—
135C4×C9—82no—
136C4×C9—81no—
137Dic9—81no—
138C22×Q8C3×D911yescharacteristic
139C2×Q16—91no—
140C2×Q16—91no—
141C22⋊Q8—91no—
142#103—91no—
143#103—91no—
144C22⋊C8—91no—
145C3×C9—41noSylow
146C23×C3—21no—
147C23×C3#61111yescharacteristic
148C23×C3C3×#6711yescharacteristic
149C3×Q8—21no—
150C3×Q8—32no—
151C3×Q8—32no—
152C3×Q8—31no—
153C3×Q8—31no—
154C3×Q8—61no—
155C3×Q8—62no—
156C3×Q8—61no—
157C3×Q8—61no—
158C3×Q8—62no—
159C3×Q8—181no—
160C3×Q8—181no—
161C3×Q8(C22⋊C9)⋊C211yescharacteristic
162C3×Q8C3×S411yescharacteristic
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—61no—
168C2×C3×C4—61no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—62no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—62no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179Dic6—61no—
180Dic6—61no—
181C3×C8—181no—
182C3×C8—181no—
183C3⋊C8—61no—
184C3⋊C8—61no—
185C2×C32—31no—
186C2×C32—31no—
187C2×C32GL(2,ℤ/4)11yescharacteristic
188C2×C9—41no—
189C2×C9—81no—
190C2×Q8—31no—
191C2×Q8—31no—
192C2×Q8—61no—
193C2×Q8—91no—
194C22×C4—21no—
195C22×C4C2×C3×D911yescharacteristic
196Q16—182no—
197Q16—181no—
198Q16—181no—
199C2×C8—91no—
200C2×C8—91no—
201C4⋊C4—91no—
202C4⋊C4—181no—
203C22⋊C4—181no—
204C22×C3—21no—
205C22×C3—31no—
206C22×C3—31no—
207C22×C3—31no—
208C22×C3—31no—
209C22×C3—61no—
210C22×C3—61no—
211C22×C3#367411yescharacteristic, complemented
212C22×C3C3×#21611yescharacteristic
213C3×C4—22no—
214C3×C4—21no—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—61no—
220C3×C4—62no—
221C3×C4—61no—
222C3×C4—61no—
223C3×C4—361no—
224C3×C4—361no—
225C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
226C3×C4C2×C3×S411yescharacteristic
227Dic3—121no—
228Dic3—121no—
229C32#541611yescharacteristic
230C9—41no—
231C9—81no—
232C23C3×#61111yescharacteristic
233Q8—32no—
234Q8—31no—
235Q8—61no—
236Q8—181no—
237Q8—181no—
238Q8C3×(C22⋊C9)⋊C211yescharacteristic
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—181no—
244C2×C4—181no—
245C8—181no—
246C8—181no—
247C2×C3—21no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—61no—
253C2×C3—61no—
254C2×C3#6369411yescharacteristic
255C2×C3C3×GL(2,ℤ/4)11yescharacteristic
256C22—31no—
257C22—31no—
258C22C3×#367411yescharacteristic, complemented
259C4—21no—
260C4—31no—
261C4—61no—
262C4—361no—
263C4—361no—
264C4C2×C3×(C22⋊C9)⋊C211yescharacteristic
265C3—21no—
266C3#1058951811yescharacteristic, direct product factor
267C3C3×#541611yescharacteristic
268C2—31no—
269C2—31no—
270C2C3×#6369411yescharacteristic
subgroupG/H
centerC2×C3#63694
commutator, cosocleC4×C22⋊C9C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×Q8D3
socle, prime radicalC23×C32#67
hypercenterC3×Q8(C22⋊C9)⋊C2
hypocenterC22⋊C9C3×Q16
LCSG ▷ C4×C22⋊C9 ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
5. #10589518C3direct (N × H)
47. C3×C22⋊C9Q16semidirect
132. C22⋊C9C3×Q16semidirect
211. C22×C3#3674semidirect
258. C22C3×#3674semidirect
266. C3#10589518direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
C3×(C22⋊C9)⋊Q82
C3×(C22⋊C9)⋊C82
C3×#52283
#105895183
C3×#36744