Finite Group Databasebeta

← #13164549

Subgroups — #13164549

Order 1728.

318 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 26 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
1#11722587C211yesmaximal, characteristic
2C3×#5930—33nomaximal
3C3×#5228—31nomaximal
4#88613—41nomaximal
5#88610—42nomaximal
6C22×C32×Q8D311yescharacteristic
7C3×C22⋊SL(2,𝔽3)D313yes—
8C2×C3×#216—32no—
9C3×#924—31no—
10C3×#835—32no—
11C3×#830—31no—
12C2×C22⋊H(3,𝔽3)—41no—
13Q8⋊H(3,𝔽3)—42no—
14Q8⋊H(3,𝔽3)—81no—
15#5930—33no—
16C3×#451—91no—
17#5228—31no—
18C2×C32×Q8—32no—
19C2×C32×Q8—61no—
20C22×C32×C4—31no—
21C2×C3×Dic6—31no—
22C3×SL(2,ℤ/4)—123no—
23C3×#226—126no—
24C3×#217—61no—
25C3×#216—62no—
26C3×#216—62no—
27C3×C4⋊Dic3—31no—
28C3×Dic3⋊C4—61no—
29C2×C3×C3⋊C8—32no—
30C22⋊H(3,𝔽3)—41no—
31#1084—161no—
32C22×C3×Q8—31no—
33C22×C3×Q8C3⋊D311yescharacteristic
34C22⋊SL(2,𝔽3)—33no—
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
44C2×C3×A4—43no—
45C32×Q8—21no—
46C32×Q8—32no—
47C32×Q8—61no—
48C32×Q8S412yes—
49C2×C32×C4—31no—
50C2×C32×C4—31no—
51C2×C32×C4—62no—
52C2×C3×Dic3—61no—
53C2×C3×Dic3—61no—
54C3×Dic6—62no—
55C3×SL(2,𝔽3)—46no—
56C3×SL(2,𝔽3)—83no—
57C3×C3⋊C8—62no—
58#451—91noSylow
59C2×H(3,𝔽3)—161no—
60C2×C3×Q8—32no—
61C2×C3×Q8—32no—
62C2×C3×Q8—62no—
63C2×C3×Q8—61no—
64C2×C3×Q8—61no—
65C2×C3×Q8—62no—
66C2×C3×Q8—91no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—61no—
70C2×Dic6—31no—
71SL(2,ℤ/4)—123no—
72#226—126no—
73C3×Q16—182no—
74C3×Q16—182no—
75C2×C3×C8—92no—
76C3×C4⋊C4—91no—
77C3×C4⋊C4—181no—
78C3×C22⋊C4—181no—
79#217—61no—
80#216—62no—
81#216—62no—
82C4⋊Dic3—31no—
83Dic3⋊C4—61no—
84C2×C3⋊C8—32no—
85C22×C32—31no—
86C22×C32—31no—
87C22×C32#22611yescharacteristic, complemented
88C3×A4—43no—
89C32×C4—32no—
90C32×C4—61no—
91C3×Dic3—121no—
92C3×Dic3—121no—
93C3×Dic3—483no—
94C22×Q8#27211yescharacteristic
95C2×Q16—92no—
96C22⋊Q8—91no—
97#103—92no—
98C22⋊C8—91no—
99H(3,𝔽3)—161noSylow
100C23×C3—31no—
101C23×C3C3⋊S411yescharacteristic
102C2×A4—123no—
103C3×Q8—21no—
104C3×Q8—32no—
105C3×Q8—32no—
106C3×Q8—32no—
107C3×Q8—61no—
108C3×Q8—62no—
109C3×Q8—61no—
110C3×Q8—62no—
111C3×Q8—61no—
112C3×Q8—182no—
113C3×Q8C3⋊S412yes—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4—62no—
121C2×C3×C4—62no—
122C2×C3×C4—62no—
123C2×C3×C4—62no—
124C2×C3×C4—181no—
125C2×C3×C4—181no—
126C2×Dic3—61no—
127C2×Dic3—61no—
128Dic6—62no—
129SL(2,𝔽3)—126no—
130SL(2,𝔽3)—243no—
131C3×C8—182no—
132C3⋊C8—62no—
133C2×C32—31no—
134C2×C32—31no—
135C2×C32—163no—
136C2×C32#102011yescharacteristic
137C2×Q8—32no—
138C2×Q8—61no—
139C2×Q8—91no—
140C22×C4—31no—
141Q16—182no—
142Q16—182no—
143C2×C8—92no—
144C4⋊C4—91no—
145C4⋊C4—181no—
146C22⋊C4—181no—
147C22×C3—31no—
148C22×C3—31no—
149C22×C3—31no—
150C22×C3—31no—
151C22×C3—31no—
152C22×C3—61no—
153C22×C3—61no—
154C22×C3C3⋊#22611yescharacteristic
155A4—123no—
156C3×C4—32no—
157C3×C4—32no—
158C3×C4—62no—
159C3×C4—61no—
160C3×C4—62no—
161C3×C4—61no—
162C3×C4—361no—
163C3×C4—361no—
164Dic3—121no—
165Dic3—121no—
166Dic3—483no—
167C32—163no—
168C32#593011yescharacteristic, complemented
169C23#625811yescharacteristic
170Q8—21no—
171Q8—32no—
172Q8—61no—
173Q8—182no—
174Q8#625812yes—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—62no—
178C2×C4—181no—
179C2×C4—181no—
180C8—182no—
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—31no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3—61no—
187C2×C3—61no—
188C2×C3—483no—
189C2×C3C3⋊#102011yescharacteristic
190C22—31no—
191C22—31no—
192C22#8861011yescharacteristic, complemented
193C4—32no—
194C4—61no—
195C4—361no—
196C4—361no—
197C3—31no—
198C3—483no—
199C3C3⋊#593011yescharacteristic
200C2—31no—
201C2—31no—
202C2#1172405511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C3C3⋊#1020
commutator, prime radical, cosocle, hypocenter#11722587C2
FittingC22×C32×Q8D3
socleC23×C3C3⋊S4
LCSG ▷ #11722587
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
87. C22×C32#226semidirect
168. C32#5930semidirect
192. C22#88610semidirect

Maximal subgroups

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

subgroupindex
#117225872
C3×#59303
C3×#52283
#886134
#886104