Finite Group Databasebeta

← #13136677

Subgroups — #13136677

Order 1728.

317 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 21 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#11721163C211yesmaximal, characteristic
2#11720780C211yesmaximal, characteristic, complemented
3#11720325C211yesmaximal, characteristic, complemented
4C22×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic, complemented
5C2×C4⋊#272C2211yescharacteristic
6C2×#6200—21no—
7C22⋊#1088—21no—
8#3752⋊C3—21no—
9#3748⋊C3—21no—
10#88451C411yescharacteristic, complemented
11#63922—61no—
12#63558—61no—
13C22×#272C2×C411yescharacteristic
14C22×#272D411yescharacteristic
15C22×(C3⋊D3)⋊C3—21no—
16C2×((C3⋊D3)⋊C3)⋊C2—41no—
17C2×((C3⋊D3)⋊C3)⋊C2—42no—
18C2×#1088—41no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C4⋊#272—41no—
21C2×(C3⋊Dic3)⋊C3—21no—
22#6200—42no—
23(C3⋊(C3⋊C8))⋊C3—41no—
24#4471—91nomaximal
25C22×D32—61no—
26C2×C3×D12—61no—
27C2×C3×D12—61no—
28C2×#626—61no—
29C2×#625—61no—
30#3752—21no—
31#3748—21no—
32C3×(C2×D3)⋊C4—61no—
33C3×#208—61no—
34(C2×D3)⋊Dic3—61no—
35C22×H(3,𝔽3)C22⋊C411yescharacteristic
36C2×#272—21no—
37C2×#272—21no—
38C2×#272—41no—
39C2×#272—41no—
40C2×(C3⋊D3)⋊C3—22no—
41C2×(C3⋊D3)⋊C3—41no—
42C2×(C3⋊D3)⋊C3—41no—
43((C3⋊D3)⋊C3)⋊C2—42no—
44((C3⋊D3)⋊C3)⋊C2—42no—
45#1088—81no—
46C4×H(3,𝔽3)—21no—
47(C3⋊Dic3)⋊C3—41no—
48#937—91no—
49#882—181no—
50C3×#99—91no—
51#833—91no—
52#823—181no—
53C22×C3⋊D3—21no—
54C22×C3×D3—61no—
55C22×C3×D3—61no—
56C22×C3×D3—61no—
57C22×C3×D3—61no—
58C22×C3×D3—61no—
59C2×D32—122no—
60C2×D32—122no—
61C2×D32—122no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C3⋊Dic3—21no—
65C2×C3×Dic3—61no—
66C3×D12—121no—
67C3×D12—121no—
68C3×D12—122no—
69#626—122no—
70#625—122no—
71C3⋊(C3⋊C8)—41no—
72C3×C3⋊C8—121no—
73AΣL(1,𝔽16)—271noSylow
74C2×H(3,𝔽3)—21no—
75C2×H(3,𝔽3)#9911yescharacteristic
76#272—41no—
77#272—41no—
78(C3⋊D3)⋊C3—42no—
79C23×D3—91no—
80C23×D3—181no—
81C2×C3×D4—91no—
82C2×#67—181no—
83C2×#67—181no—
84C2×D12—61no—
85C2×D12—61no—
86C2×D12—181no—
87C3×M16—181no—
88C3×C22⋊C4—181no—
89C3×C22⋊C4—181no—
90#217—91no—
91(C2×D3)⋊C4—61no—
92(C2×D3)⋊C4—181no—
93(C2×D3)⋊C4—181no—
94#208—61no—
95#208—181no—
96C22×C32—21no—
97C22×C32—21no—
98C2×C3⋊D3—22no—
99C2×C3⋊D3—41no—
100C2×C3⋊D3—41no—
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—121no—
104C2×C3×D3—121no—
105C2×C3×D3—121no—
106C2×C3×D3—121no—
107C2×C3×D3—121no—
108C2×C3×D3—121no—
109C2×C3×D3—121no—
110C2×C3×D3—121no—
111C2×C3×D3—121no—
112C2×C3×D3—241no—
113C2×C3×D3—241no—
114D32—124no—
115D32—124no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—41no—
119C3⋊Dic3—41no—
120C3×Dic3—121no—
121#120—271no—
122#100—271no—
123#99—91no—
124H(3,𝔽3)AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
125C23×C3—91no—
126C23×C3—91no—
127C23×C3—181no—
128C22×D3—61no—
129C22×D3—61no—
130C22×D3—61no—
131C22×D3—61no—
132C22×D3—61no—
133C22×D3—181no—
134C22×D3—362no—
135C22×D3—362no—
136C22×D3—362no—
137C22×D3—362no—
138C22×D3—361no—
139C3×D4—361no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—181no—
143C2×C3×C4—361no—
144C2×C3×C4GO+(4,𝔽2)11yescharacteristic
145#67—362no—
146#67—362no—
147C2×Dic3—61no—
148C2×Dic3—181no—
149D12—121no—
150D12—121no—
151D12—122no—
152D12—362no—
153C3×C8—361no—
154C3⋊C8—121no—
155C3⋊C8—361no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32—41no—
161C3⋊D3—42no—
162C3×D3—122no—
163C3×D3—122no—
164C3×D3—241no—
165C3×D3—241no—
166C3×D3—241no—
167C24—271no—
168C2×D4—91no—
169C2×D4—541no—
170M16—181no—
171C22⋊C4—181no—
172C22⋊C4—181no—
173C22⋊C4—271no—
174C22×C3—61no—
175C22×C3—61no—
176C22×C3—182no—
177C22×C3—181no—
178C22×C3—181no—
179C22×C3—361no—
180C22×C3—361no—
181C22×C3—361no—
182C22×C3—361no—
183C22×C3D32⋊C411yescharacteristic
184C2×D3—62no—
185C2×D3—62no—
186C2×D3—121no—
187C2×D3—121no—
188C2×D3—121no—
189C2×D3—121no—
190C2×D3—121no—
191C2×D3—121no—
192C2×D3—121no—
193C2×D3—121no—
194C2×D3—121no—
195C2×D3—182no—
196C2×D3—241no—
197C2×D3—241no—
198C2×D3—361no—
199C2×D3—361no—
200C2×D3—362no—
201C2×D3—364no—
202C2×D3—364no—
203C2×D3—362no—
204C3×C4—21no—
205C3×C4—61no—
206C3×C4—61no—
207C3×C4—121no—
208C3×C4—361no—
209C3×C4—721no—
210Dic3—121no—
211Dic3—361no—
212C32—21no—
213C32—21no—
214C23—91no—
215C23—91no—
216C23—181no—
217C23—1082no—
218C23—1081no—
219D4—361no—
220D4—1082no—
221C2×C4—181no—
222C2×C4—361no—
223C2×C4#1088⋊C211yescharacteristic
224C8—361no—
225C2×C3—21no—
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3—61no—
230C2×C3—121no—
231C2×C3—362no—
232C2×C3—361no—
233C2×C3—361no—
234C2×C3#6373811yescharacteristic
235D3—122no—
236D3—122no—
237D3—241no—
238D3—241no—
239D3—241no—
240D3—362no—
241C22—182no—
242C22—181no—
243C22—181no—
244C22—361no—
245C22—361no—
246C22—361no—
247C22—361no—
248C22—1082no—
249C22—1082no—
250C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
251C4—21no—
252C4—361no—
253C4—721no—
254C3—61no—
255C3—61no—
256C3#1059005711yescharacteristic
257C2—21no—
258C2—361no—
259C2—362no—
260C2—361no—
261C2#1172070111yescharacteristic
subgroupG/H
centerC2#11720701
commutatorC22×#272C2×C4
FrattiniC2×C3×C4GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC2×C3#63738
prime radical#11721163C2
cosocleC2×C4⋊#272C22
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)AΣL(1,𝔽16)
LCSG ▷ C22×#272 ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11720780C2semidirect
3. #11720325C2semidirect
4. C22×((C3⋊D3)⋊C3)⋊C2C4semidirect
10. #88451C4semidirect
124. H(3,𝔽3)AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
#117211632
#117207802
#117203252
#44719