Finite Group Databasebeta

← #13139924

Subgroups — #13139924

Order 1728.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721826C211yesmaximal, characteristic
2#63657⋊C3C212yesmaximal
3#11720304C212yesmaximal, complemented
4#11720303C211yesmaximal, characteristic, complemented
5#11720302C211yesmaximal, characteristic, complemented
6#10591056—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8#88552C412yes—
9C4⋊#1084C412yes—
10C4×#1084C2211yescharacteristic
11C4×#1084C2211yescharacteristic
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14C3×#926—32no—
15C3×#926—61no—
16#63657D312yes—
17C3×#830—32no—
18Dic3⋊(C3⋊C8)—32no—
19Dic3⋊(C3⋊C8)—32no—
20#63554—32no—
21Dic3×C3⋊C8—32no—
22C23×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084C2×C412yes—
27C2×#1084C2×C412yes—
28(C3⋊(C3⋊C8))⋊C3—22no—
29(C3⋊(C3⋊C8))⋊C3—22no—
30#5000—91no—
31#4720—92no—
32C22×C32×C4—21no—
33C22×C32×C4C2×D312yes—
34C2×C3⋊(C3⋊C8)C2×D312yes—
35C2×C3⋊(C3⋊C8)C2×D312yes—
36C3×#217—32no—
37C3×#217—32no—
38C3×#217—62no—
39C3×C4⋊Dic3—32no—
40C3×C4⋊Dic3—32no—
41C3×C4⋊Dic3—62no—
42C3×C4×Dic3—32no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C3×C4×Dic3—61no—
46C2×C3×C3⋊C8—32no—
47C2×C3×C3⋊C8—32no—
48C22×H(3,𝔽3)—21no—
49C22×H(3,𝔽3)—22no—
50C22×H(3,𝔽3)C22×C411yescharacteristic
51C4×H(3,𝔽3)—21no—
52C4×H(3,𝔽3)#4112yes—
53#1084—22no—
54#1084—22no—
55C3×#117—91no—
56#926—32no—
57#926—61no—
58C3×C22⋊C8—92no—
59#830—32no—
60#830—92no—
61#815—92no—
62Dic3⋊C8—92no—
63Dic3⋊C8—92no—
64C8×Dic3—92no—
65C4⋊(C3⋊C8)—92no—
66#803—91no—
67C4×C3⋊C8—91no—
68C23×C32—21no—
69C23×C32C4×D312yes—
70C2×C32×C4—24no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4C4×D312yes—
76C2×C32×C4C22×D312yes—
77C2×C3×Dic3—32no—
78C2×C3×Dic3—32no—
79C2×C3×Dic3—32no—
80C2×C3×Dic3—32no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C3⋊(C3⋊C8)—22no—
84C3⋊(C3⋊C8)—22no—
85C3×C3⋊C8—62no—
86C3×C3⋊C8—62no—
87#433—271noSylow
88C2×H(3,𝔽3)—41no—
89C2×H(3,𝔽3)#11711yescharacteristic
90C2×H(3,𝔽3)#13112yes—
91C22×C3×C4—32no—
92C22×C3×C4—61no—
93C22×C3×C4D3211yescharacteristic
94C2×C3×C8—92no—
95C2×C3×C8—92no—
96C3×C4⋊C4—92no—
97C3×C22⋊C4—92no—
98C3×C42—91no—
99C3×C42—91no—
100#217—32no—
101#217—32no—
102#217—62no—
103C4⋊Dic3—32no—
104C4⋊Dic3—32no—
105C4⋊Dic3—62no—
106C4×Dic3—32no—
107C4×Dic3—32no—
108C4×Dic3—61no—
109C4×Dic3—61no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—32no—
112C2×C3⋊C8—92no—
113C2×C3⋊C8—92no—
114C22×C32—22no—
115C22×C32—22no—
116C22×C32—22no—
117C22×C32—22no—
118C22×C32—21no—
119C22×C32—42no—
120C22×C32C2×C4×D312yes—
121C32×C4—22no—
122C32×C4—22no—
123C32×C4—22no—
124C32×C4#23712yes—
125C32×C4#23712yes—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—62no—
129C3×Dic3—62no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132#117—91no—
133C4⋊C8—272no—
134C22⋊C8—92no—
135#97—271no—
136C4×C8—271no—
137H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
138C23×C3—32no—
139C23×C3—61no—
140C23×C3#62411yescharacteristic
141C2×C3×C4—22no—
142C2×C3×C4—32no—
143C2×C3×C4—32no—
144C2×C3×C4—62no—
145C2×C3×C4—62no—
146C2×C3×C4—64no—
147C2×C3×C4—61no—
148C2×C3×C4—61no—
149C2×C3×C4—92no—
150C2×C3×C4—92no—
151C2×C3×C4#62411yescharacteristic
152C2×C3×C4C2×D3211yescharacteristic
153C2×Dic3—32no—
154C2×Dic3—32no—
155C2×Dic3—32no—
156C2×Dic3—32no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C3×C8—182no—
160C3×C8—182no—
161C3⋊C8—62no—
162C3⋊C8—62no—
163C3⋊C8—182no—
164C3⋊C8—182no—
165C2×C32—22no—
166C2×C32—21no—
167C2×C32—42no—
168C2×C32—42no—
169C2×C32#87712yes—
170C2×C32#90112yes—
171C2×C32#90712yes—
172C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
173C2×C8—92no—
174C2×C8—92no—
175C4⋊C4—92no—
176C22⋊C4—92no—
177C42—91no—
178C42—91no—
179C22×C3—22no—
180C22×C3—21no—
181C22×C3—32no—
182C22×C3—62no—
183C22×C3—62no—
184C22×C3—62no—
185C22×C3—62no—
186C22×C3—61no—
187C22×C3—122no—
188C22×C3C2×#62411yescharacteristic
189C3×C4—21no—
190C3×C4—32no—
191C3×C4—32no—
192C3×C4—62no—
193C3×C4—62no—
194C3×C4—62no—
195C3×C4—182no—
196C3×C4—182no—
197C3×C4#380512yes—
198Dic3—62no—
199Dic3—62no—
200Dic3—62no—
201Dic3—62no—
202Dic3—122no—
203Dic3—122no—
204C32—21no—
205C32#472012yescomplemented
206C23#619911yescharacteristic
207C2×C4—22no—
208C2×C4—92no—
209C2×C4—92no—
210C2×C4#619911yescharacteristic
211C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
212C8—182no—
213C8—182no—
214C2×C3—32no—
215C2×C3—32no—
216C2×C3—32no—
217C2×C3—41no—
218C2×C3—61no—
219C2×C3—62no—
220C2×C3—122no—
221C2×C3—122no—
222C2×C3#6381712yes—
223C2×C3#6395711yescharacteristic
224C22—21no—
225C22—22no—
226C22C2×#619911yescharacteristic
227C4—21no—
228C4—182no—
229C4—182no—
230C4#8867212yes—
231C3—32no—
232C3—61no—
233C3#1059106211yescharacteristic
234C2—41no—
235C2#1172106412yes—
236C2#1172129811yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#433
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11720304C2semidirect
4. #11720303C2semidirect
5. #11720302C2semidirect
137. H(3,𝔽3)#433semidirect
205. C32#4720semidirect

Maximal subgroups

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

subgroupindex
#117218262
#63657⋊C32
#117203042
#117203032
#117203022
#105910563