Finite Group Databasebeta

← C3×#10593105 #13164519

Subgroups — C3×#10593105 #13164519

Order 1728.

271 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 32 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×C22⋊(Q8⋊C9)C211yesmaximal, characteristic
2#10593105C311yesmaximal, characteristic, direct product factor
3C3×#5228—31nomaximal
4C3×(C22⋊C9)⋊C4—41nomaximal
5C3×#3688—42nomaximal
6C22×C32×Q8D311yescharacteristic
7C2×C3×#216—32no—
8C3×#924—31no—
9C22⋊(Q8⋊C9)—21no—
10C22⋊(Q8⋊C9)C2×C311yescharacteristic
11C3×#835—32no—
12C3×#830—31no—
13C2×C3×C22⋊C9—41no—
14C3×Q8⋊C9—42no—
15C3×Q8⋊C9—81no—
16C3×#451—91no—
17#5228—31no—
18C2×C32×Q8—32no—
19C2×C32×Q8—61no—
20C22×C32×C4—31no—
21C2×C3×Dic6—31no—
22C3×#217—61no—
23C3×#216—62no—
24C3×#216—62no—
25C3×C4⋊Dic3—31no—
26C3×Dic3⋊C4—61no—
27C2×C3×C3⋊C8—32no—
28(C22⋊C9)⋊C4—41no—
29#3688—42no—
30C3×C22⋊C9—41no—
31C3×Dic9—161no—
32C22×C3×Q8—21no—
33C22×C3×Q8D911yescharacteristic
34C22×C3×Q8C3×D311yescharacteristic
35C2×C3×Q16—92no—
36C3×C22⋊Q8—91no—
37C2×#216—32no—
38#924—31no—
39C3×#103—92no—
40C3×C22⋊C8—91no—
41#835—32no—
42#830—31no—
43C23×C32S411yescharacteristic
44C32×Q8—21no—
45C32×Q8—32no—
46C32×Q8—61no—
47C32×Q8S412yes—
48C2×C32×C4—31no—
49C2×C32×C4—31no—
50C2×C32×C4—62no—
51C2×C3×Dic3—61no—
52C2×C3×Dic3—61no—
53C3×Dic6—62no—
54C2×C22⋊C9—41no—
55C2×C22⋊C9—81no—
56C3×C3⋊C8—62no—
57Q8⋊C9—42no—
58Q8⋊C9—81no—
59Q8⋊C9—82no—
60Q8⋊C9—82no—
61#451—91noSylow
62C2×C3×C9—161no—
63C2×C3×Q8—32no—
64C2×C3×Q8—32no—
65C2×C3×Q8—62no—
66C2×C3×Q8—61no—
67C2×C3×Q8—61no—
68C2×C3×Q8—62no—
69C2×C3×Q8—91no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×C3×C4—61no—
73C2×Dic6—31no—
74C3×Q16—182no—
75C3×Q16—182no—
76C2×C3×C8—92no—
77C3×C4⋊C4—91no—
78C3×C4⋊C4—181no—
79C3×C22⋊C4—181no—
80#217—61no—
81#216—62no—
82#216—62no—
83C4⋊Dic3—31no—
84Dic3⋊C4—61no—
85C2×C3⋊C8—32no—
86C22×C32—31no—
87C22×C32—31no—
88C22×C32#22611yescharacteristic
89C32×C4—32no—
90C32×C4—61no—
91C3×Dic3—121no—
92C3×Dic3—121no—
93C22⋊C9—41no—
94C22⋊C9—81no—
95Dic9—161no—
96C22×Q8C3×D911yescharacteristic
97C2×Q16—92no—
98C22⋊Q8—91no—
99#103—92no—
100C22⋊C8—91no—
101C3×C9—161noSylow
102C23×C3—21no—
103C23×C3(C22⋊C9)⋊C211yescharacteristic
104C23×C3C3×S411yescharacteristic
105C3×Q8—21no—
106C3×Q8—22no—
107C3×Q8—21no—
108C3×Q8—22no—
109C3×Q8—32no—
110C3×Q8—32no—
111C3×Q8—62no—
112C3×Q8—61no—
113C3×Q8—61no—
114C3×Q8—62no—
115C3×Q8—182no—
116C3×Q8(C22⋊C9)⋊C212yes—
117C3×Q8C3×S412yes—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—62no—
125C2×C3×C4—62no—
126C2×C3×C4—64no—
127C2×C3×C4—181no—
128C2×C3×C4—181no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131Dic6—62no—
132C3×C8—182no—
133C3⋊C8—62no—
134C2×C32—31no—
135C2×C32—31no—
136C2×C32#102011yescharacteristic
137C2×C9—161no—
138C2×C9—321no—
139C2×Q8—32no—
140C2×Q8—61no—
141C2×Q8—91no—
142C22×C4—31no—
143Q16—182no—
144Q16—182no—
145C2×C8—92no—
146C4⋊C4—91no—
147C4⋊C4—181no—
148C22⋊C4—181no—
149C22×C3—21no—
150C22×C3—31no—
151C22×C3—31no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—61no—
155C22×C3—61no—
156C22×C3#368811yescharacteristic, complemented
157C22×C3C3×#22611yescharacteristic
158C3×C4—32no—
159C3×C4—32no—
160C3×C4—62no—
161C3×C4—62no—
162C3×C4—61no—
163C3×C4—61no—
164C3×C4—361no—
165C3×C4—361no—
166Dic3—121no—
167Dic3—121no—
168C32#593011yescharacteristic
169C9—161no—
170C9—321no—
171C23C3×(C22⋊C9)⋊C211yescharacteristic
172Q8—21no—
173Q8—32no—
174Q8—61no—
175Q8—182no—
176Q8C3×(C22⋊C9)⋊C212yes—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—62no—
180C2×C4—181no—
181C2×C4—181no—
182C8—182no—
183C2×C3—21no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3—31no—
187C2×C3—31no—
188C2×C3—61no—
189C2×C3—61no—
190C2×C3#6418811yescharacteristic
191C2×C3C3×#102011yescharacteristic
192C22—31no—
193C22—31no—
194C22C3×#368811yescharacteristic, complemented
195C4—32no—
196C4—61no—
197C4—361no—
198C4—361no—
199C3—21no—
200C3#1059310511yescharacteristic, direct product factor
201C3C3×#593011yescharacteristic
202C2—31no—
203C2—31no—
204C2C3×#6418811yescharacteristic
subgroupG/H
center, hypercenterC2×C3#64188
commutator, cosocle, hypocenterC22⋊(Q8⋊C9)C2×C3
FrattiniC2×C3C3×#1020
FittingC22×C32×Q8D3
socle, prime radicalC23×C32S4
LCSG ▷ C22⋊(Q8⋊C9)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. #10593105C3direct (N × H)
156. C22×C3#3688semidirect
194. C22C3×#3688semidirect
200. C3#10593105direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C22⋊(Q8⋊C9)2
#105931053
C3×#52283
C3×(C22⋊C9)⋊C44
C3×#36884