Finite Group Databasebeta

← C2×C3×GO+(4,𝔽2)⋊C4 #13180388

Subgroups — C2×C3×GO+(4,𝔽2)⋊C4 #13180388

Order 1728.

570 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×C3×GO+(4,𝔽2)C211yesmaximal, characteristic
2C22×C3×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
3C2×C3×#3777C211yesmaximal, characteristic, complemented
4C3×GO+(4,𝔽2)⋊C4C214yesmaximal, direct product factor
5C2×GO+(4,𝔽2)⋊C4C311yesmaximal, characteristic, direct product factor
6C22×C3×D32—21no—
7C22×C3×#157—21no—
8C22×C3×#157C2211yescharacteristic
9C2×C3×GO+(4,𝔽2)—21no—
10C2×C3×GO+(4,𝔽2)—22no—
11C2×C3×GO+(4,𝔽2)C414yescomplemented
12C2×C3×GO+(4,𝔽2)C2212yes—
13C2×C3×AGL(1,𝔽9)—22no—
14C2×C3×AGL(1,𝔽9)C2212yescomplemented
15C3×#3777—21no—
16C3×#3777C2212yescomplemented
17C22×GO+(4,𝔽2)C2×C311yescharacteristic
18C22×AGL(1,𝔽9)C2×C311yescharacteristic, complemented
19C2×#3777C2×C311yescharacteristic, complemented
20GO+(4,𝔽2)⋊C4C2×C314yesdirect product factor
21C22×C3×C3⋊D3D411yescharacteristic
22C22×C32×D3—41no—
23C2×C3×D32—24no—
24C2×C3×D32—22no—
25C2×C3×D32—42no—
26C2×C3×D32—41no—
27C2×C3×#157—22no—
28C2×C3×#157—42no—
29C2×C3×#157C2×C412yes—
30C2×C3×#157D411yescharacteristic
31C2×C3×#157D412yes—
32C2×C3×#157C2311yescharacteristic
33C3×GO+(4,𝔽2)—24no—
34C3×GO+(4,𝔽2)—22no—
35C3×GO+(4,𝔽2)C2×C414yescomplemented
36C3×AGL(1,𝔽9)—22no—
37C2×C3×#102—91nomaximal
38C23×C3×D3—121no—
39C22×D32—21no—
40C22×#157—21no—
41C22×#157C22×C311yescharacteristic
42C2×GO+(4,𝔽2)—21no—
43C2×GO+(4,𝔽2)—22no—
44C2×GO+(4,𝔽2)C3×C414yescomplemented
45C2×GO+(4,𝔽2)C22×C312yes—
46C2×AGL(1,𝔽9)—22no—
47C2×AGL(1,𝔽9)C22×C312yescomplemented
48#3777—21no—
49#3777C22×C312yescomplemented
50C22×C33SD1611yescharacteristic
51C2×C3×C3⋊D3C22⋊C412yes—
52C2×C3×C3⋊D3D812yes—
53C2×C3×C3⋊D3SD1611yescharacteristic
54C2×C3×C3⋊D3C2×D411yescharacteristic
55C2×C32×D3—44no—
56C2×C32×D3—42no—
57C3×D32—24no—
58C3×D32—44no—
59C3×D32—42no—
60C3×#157—42no—
61C3×#157C22⋊C412yes—
62C3×#157C22×C411yescharacteristic
63C3×#157C2×D411yescharacteristic
64C22×C3×D4—91no—
65C22×C3×C8—91no—
66C2×C3×C4⋊C4—91no—
67C3×#102—94no—
68C23×C32—121no—
69C22×C3⋊D3C3×D411yescharacteristic
70C22×C3×D3—41no—
71C22×C3×D3—41no—
72C22×C3×D3—81no—
73C22×C3×D3—121no—
74C22×C3×D3—124no—
75C22×C3×D3—122no—
76C22×C3×D3—124no—
77C22×C3×D3—122no—
78C2×D32—24no—
79C2×D32—22no—
80C2×D32—42no—
81C2×D32—41no—
82C2×#157—22no—
83C2×#157—42no—
84C2×#157C2×C3×C412yes—
85C2×#157C3×D411yescharacteristic
86C2×#157C3×D412yes—
87C2×#157C23×C311yescharacteristic
88GO+(4,𝔽2)—24no—
89GO+(4,𝔽2)—22no—
90GO+(4,𝔽2)C2×C3×C414yescomplemented
91AGL(1,𝔽9)—22no—
92C2×#102—91noSylow
93C2×C33#10212yescomplemented
94C2×C33C2×SD1611yescharacteristic
95C3×C3⋊D3#10212yescomplemented
96C3×C3⋊D3C2×C22⋊C411yescharacteristic
97C3×C3⋊D3C2×D811yescharacteristic
98C32×D3—44no—
99C24×C3—181no—
100C23×D3—121no—
101C2×C3×D4—94no—
102C2×C3×D4—92no—
103C2×C3×D4—181no—
104C2×C3×D4—182no—
105C22×C3×C4—91no—
106C22×C3×C4—181no—
107C2×C3×C8—92no—
108C2×C3×C8—182no—
109C3×C4⋊C4—92no—
110C3×C4⋊C4—181no—
111C22×C32—41no—
112C22×C32—81no—
113C22×C32—122no—
114C22×C32—124no—
115C22×C32C3×SD1611yescharacteristic
116C2×C3⋊D3C3×C22⋊C412yes—
117C2×C3⋊D3C3×D812yes—
118C2×C3⋊D3C3×SD1611yescharacteristic
119C2×C3⋊D3C2×C3×D411yescharacteristic
120C2×C3×D3—42no—
121C2×C3×D3—44no—
122C2×C3×D3—44no—
123C2×C3×D3—42no—
124C2×C3×D3—84no—
125C2×C3×D3—82no—
126C2×C3×D3—122no—
127C2×C3×D3—122no—
128C2×C3×D3—121no—
129C2×C3×D3—121no—
130C2×C3×D3—124no—
131C2×C3×D3—128no—
132C2×C3×D3—124no—
133D32—24no—
134D32—44no—
135D32—42no—
136#157—42no—
137#157C3×C22⋊C412yes—
138#157C22×C3×C411yescharacteristic
139#157C2×C3×D411yescharacteristic
140C22×D4—91no—
141C22×C8—91no—
142C2×C4⋊C4—91no—
143#102—94no—
144C33C2×#10211yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C23×C3—121no—
147C23×C3—121no—
148C23×C3—182no—
149C23×C3—184no—
150C23×C3—241no—
151C23×C3—361no—
152C23×C3—362no—
153C22×D3—41no—
154C22×D3—121no—
155C22×D3—124no—
156C22×D3—122no—
157C22×D3—124no—
158C22×D3—122no—
159C3×D4—94no—
160C3×D4—182no—
161C3×D4—184no—
162C2×C3×C4—92no—
163C2×C3×C4—91no—
164C2×C3×C4—92no—
165C2×C3×C4—91no—
166C2×C3×C4—182no—
167C2×C3×C4—362no—
168C3×C8—182no—
169C2×C32—41no—
170C2×C32—42no—
171C2×C32—82no—
172C2×C32—81no—
173C2×C32—124no—
174C2×C32C3×#10212yescomplemented
175C2×C32C2×C3×SD1611yescharacteristic
176C3⋊D3C3×#10212yescomplemented
177C3⋊D3C2×C3×C22⋊C411yescharacteristic
178C3⋊D3C2×C3×D811yescharacteristic
179C3×D3—44no—
180C3×D3—44no—
181C3×D3—84no—
182C3×D3—122no—
183C3×D3—121no—
184C3×D3—121no—
185C24—181no—
186C2×D4—94no—
187C2×D4—92no—
188C2×D4—181no—
189C2×D4—182no—
190C22×C4—91no—
191C22×C4—181no—
192C2×C8—92no—
193C2×C8—182no—
194C4⋊C4—92no—
195C4⋊C4—181no—
196C22×C3—41no—
197C22×C3—81no—
198C22×C3—91no—
199C22×C3—91no—
200C22×C3—92no—
201C22×C3—92no—
202C22×C3—124no—
203C22×C3—122no—
204C22×C3—122no—
205C22×C3—124no—
206C22×C3—184no—
207C22×C3—244no—
208C22×C3—242no—
209C22×C3—362no—
210C22×C3—364no—
211C22×C3AΓL(1,𝔽9)11yescharacteristic
212C2×D3—44no—
213C2×D3—42no—
214C2×D3—122no—
215C2×D3—122no—
216C2×D3—121no—
217C2×D3—121no—
218C2×D3—124no—
219C2×D3—128no—
220C2×D3—124no—
221C3×C4—91no—
222C3×C4—91no—
223C3×C4—92no—
224C3×C4—362no—
225C32—41no—
226C32—81no—
227C32C2×C3×#10211yescharacteristic, complemented
228C23—91no—
229C23—121no—
230C23—184no—
231C23—182no—
232C23—362no—
233C23—361no—
234D4—94no—
235D4—182no—
236D4—184no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—92no—
240C2×C4—92no—
241C2×C4—182no—
242C2×C4—362no—
243C8—182no—
244C2×C3—42no—
245C2×C3—41no—
246C2×C3—82no—
247C2×C3—81no—
248C2×C3—91no—
249C2×C3—91no—
250C2×C3—92no—
251C2×C3—124no—
252C2×C3—124no—
253C2×C3—244no—
254C2×C3GO+(4,𝔽2)⋊C412yesdirect product factor
255C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
256D3—44no—
257D3—122no—
258D3—121no—
259D3—121no—
260C22—92no—
261C22—92no—
262C22—91no—
263C22—91no—
264C22—124no—
265C22—122no—
266C22—184no—
267C22—362no—
268C22—364no—
269C22C3×AΓL(1,𝔽9)11yescharacteristic
270C4—91no—
271C4—91no—
272C4—92no—
273C4—362no—
274C3—41no—
275C3—81no—
276C3C2×GO+(4,𝔽2)⋊C411yescharacteristic, direct product factor
277C2—91no—
278C2—91no—
279C2—92no—
280C2—124no—
281C2C3×GO+(4,𝔽2)⋊C412yesdirect product factor
282C2C2×C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3AΓL(1,𝔽9)
commutator#157C22×C3×C4
FrattiniC2C2×C3×AΓL(1,𝔽9)
Fitting, socleC22×C33SD16
prime radicalC22×C3×GO+(4,𝔽2)C2
cosocleC2×#157C23×C3
hypocenterC32C2×C3×#102
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×AGL(1,𝔽9)C2semidirect
3. C2×C3×#3777C2semidirect
4. C3×GO+(4,𝔽2)⋊C4C2direct (N × H)
5. C2×GO+(4,𝔽2)⋊C4C3direct (N × H)
11. C2×C3×GO+(4,𝔽2)C4semidirect
14. C2×C3×AGL(1,𝔽9)C22semidirect
16. C3×#3777C22semidirect
18. C22×AGL(1,𝔽9)C2×C3semidirect
19. C2×#3777C2×C3semidirect
20. GO+(4,𝔽2)⋊C4C2×C3direct (N × H)
35. C3×GO+(4,𝔽2)C2×C4semidirect
44. C2×GO+(4,𝔽2)C3×C4semidirect
47. C2×AGL(1,𝔽9)C22×C3semidirect
49. #3777C22×C3semidirect
90. GO+(4,𝔽2)C2×C3×C4semidirect
93. C2×C33#102semidirect
95. C3×C3⋊D3#102semidirect
144. C33C2×#102semidirect
174. C2×C32C3×#102semidirect
176. C3⋊D3C3×#102semidirect
227. C32C2×C3×#102semidirect
254. C2×C3GO+(4,𝔽2)⋊C4direct (N × H)
276. C3C2×GO+(4,𝔽2)⋊C4direct (N × H)
281. C2C3×GO+(4,𝔽2)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C22×C3×AGL(1,𝔽9)2
C2×C3×#37772
C3×GO+(4,𝔽2)⋊C42
C2×GO+(4,𝔽2)⋊C43
C2×C3×#1029