Finite Group Databasebeta

← C2×#11724101 #13180422

Subgroups — C2×#11724101 #13180422

Order 1728.

570 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 89 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×#157⋊D3C211yesmaximal, characteristic
2C22×C3⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
3C2×C3×#3777C211yesmaximal, characteristic, complemented
4#11724101C214yesmaximal, direct product factor
5C2×GO+(4,𝔽2)⋊C4—31nomaximal
6C22×#1113—21no—
7C22×C3×#157—21no—
8C22×C3×#157C2211yescharacteristic
9C2×#157⋊D3—22no—
10C2×#157⋊D3—21no—
11C2×#157⋊D3C414yescomplemented
12C2×#157⋊D3C2212yes—
13C2×C3⋊AGL(1,𝔽9)—22no—
14C2×C3⋊AGL(1,𝔽9)C2212yescomplemented
15C3×#3777—21no—
16C3×#3777C2212yescomplemented
17C22×GO+(4,𝔽2)—31no—
18C22×AGL(1,𝔽9)—31no—
19C2×#3777D311yescharacteristic, complemented
20GO+(4,𝔽2)⋊C4—34no—
21C22×C3×C3⋊D3—41no—
22C22×C3×C3⋊D3D411yescharacteristic
23C2×#1113—24no—
24C2×#1113—22no—
25C2×#1113—42no—
26C2×#1113—41no—
27C2×C3×#157—22no—
28C2×C3×#157—42no—
29C2×C3×#157C2×C412yes—
30C2×C3×#157D411yescharacteristic
31C2×C3×#157D412yes—
32C2×C3×#157C2311yescharacteristic
33#157⋊D3—24no—
34#157⋊D3—22no—
35#157⋊D3C2×C414yescomplemented
36C3⋊AGL(1,𝔽9)—22no—
37C2×#809—91nomaximal
38C22×D32—61no—
39C22×D32—121no—
40C22×#157—21no—
41C22×#157C2×D311yescharacteristic
42C2×GO+(4,𝔽2)—34no—
43C2×GO+(4,𝔽2)—32no—
44C2×GO+(4,𝔽2)—62no—
45C2×GO+(4,𝔽2)—61no—
46C2×AGL(1,𝔽9)—32no—
47C2×AGL(1,𝔽9)—62no—
48#3777—21no—
49#3777C2×D312yescomplemented
50C22×C33SD1611yescharacteristic
51C2×C3×C3⋊D3—44no—
52C2×C3×C3⋊D3—42no—
53C2×C3×C3⋊D3C22⋊C412yes—
54C2×C3×C3⋊D3D812yes—
55C2×C3×C3⋊D3SD1611yescharacteristic
56C2×C3×C3⋊D3C2×D411yescharacteristic
57#1113—24no—
58#1113—44no—
59#1113—42no—
60C3×#157—42no—
61C3×#157C22⋊C412yes—
62C3×#157C22×C411yescharacteristic
63C3×#157C2×D411yescharacteristic
64C22×D12—91no—
65C2×C3×C4⋊C4—91no—
66C22×C3⋊C8—91no—
67#809—94no—
68C22×C3⋊D3—41no—
69C22×C3⋊D3#6711yescharacteristic
70C22×C3×D3—121no—
71C22×C3×D3—121no—
72C22×C3×D3—121no—
73C22×C3×D3—241no—
74C2×D32—64no—
75C2×D32—62no—
76C2×D32—124no—
77C2×D32—122no—
78C2×D32—124no—
79C2×D32—122no—
80C2×D32—122no—
81C2×D32—121no—
82C2×#157—22no—
83C2×#157—42no—
84C2×#157C4×D312yes—
85C2×#157D1212yes—
86C2×#157#6711yescharacteristic
87C2×#157C22×D311yescharacteristic
88GO+(4,𝔽2)—34no—
89GO+(4,𝔽2)—64no—
90GO+(4,𝔽2)—62no—
91AGL(1,𝔽9)—62no—
92C2×#102—271noSylow
93C2×C33#10212yescomplemented
94C2×C33C2×SD1611yescharacteristic
95C3×C3⋊D3—44no—
96C3×C3⋊D3#10212yescomplemented
97C3×C3⋊D3C2×C22⋊C411yescharacteristic
98C3×C3⋊D3C2×D811yescharacteristic
99C23×D3—181no—
100C23×D3—361no—
101C22×C3×C4—91no—
102C22×C3×C4—181no—
103C2×D12—94no—
104C2×D12—92no—
105C2×D12—182no—
106C2×D12—181no—
107C3×C4⋊C4—92no—
108C3×C4⋊C4—181no—
109C2×C3⋊C8—92no—
110C2×C3⋊C8—182no—
111C22×C32—41no—
112C22×C32—81no—
113C22×C32#21511yescharacteristic
114C2×C3⋊D3—42no—
115C2×C3⋊D3—44no—
116C2×C3⋊D3(C2×D3)⋊C412yes—
117C2×C3⋊D3#21312yes—
118C2×C3⋊D3#21511yescharacteristic
119C2×C3⋊D3C2×#6711yescharacteristic
120C2×C3×D3—122no—
121C2×C3×D3—122no—
122C2×C3×D3—121no—
123C2×C3×D3—121no—
124C2×C3×D3—122no—
125C2×C3×D3—124no—
126C2×C3×D3—124no—
127C2×C3×D3—122no—
128C2×C3×D3—244no—
129C2×C3×D3—242no—
130D32—64no—
131D32—124no—
132D32—128no—
133D32—124no—
134D32—124no—
135D32—122no—
136#157—42no—
137#157(C2×D3)⋊C412yes—
138#157C2×C4×D311yescharacteristic
139#157C2×D1211yescharacteristic
140C22×D4—271no—
141C22×C8—271no—
142C2×C4⋊C4—91no—
143#102—274no—
144C33C2×#10211yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C23×C3—361no—
147C22×D3—121no—
148C22×D3—121no—
149C22×D3—121no—
150C22×D3—182no—
151C22×D3—184no—
152C22×D3—241no—
153C22×D3—364no—
154C22×D3—362no—
155C22×D3—361no—
156C22×D3—362no—
157C22×D3—364no—
158C22×D3—362no—
159C2×C3×C4—92no—
160C2×C3×C4—92no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—182no—
164C2×C3×C4—362no—
165D12—94no—
166D12—182no—
167D12—184no—
168C3⋊C8—182no—
169C2×C32—41no—
170C2×C32—42no—
171C2×C32—82no—
172C2×C32—81no—
173C2×C32#80912yescomplemented
174C2×C32C2×#21511yescharacteristic
175C3⋊D3—44no—
176C3⋊D3#80912yescomplemented
177C3⋊D3C2×(C2×D3)⋊C411yescharacteristic
178C3⋊D3C2×#21311yescharacteristic
179C3×D3—122no—
180C3×D3—121no—
181C3×D3—121no—
182C3×D3—124no—
183C3×D3—124no—
184C3×D3—244no—
185C24—541no—
186C2×D4—272no—
187C2×D4—274no—
188C2×D4—541no—
189C2×D4—542no—
190C22×C4—91no—
191C22×C4—181no—
192C2×C8—272no—
193C2×C8—542no—
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—364no—
203C22×C3—362no—
204C22×C3AΓL(1,𝔽9)11yescharacteristic
205C2×D3—122no—
206C2×D3—122no—
207C2×D3—121no—
208C2×D3—121no—
209C2×D3—124no—
210C2×D3—122no—
211C2×D3—122no—
212C2×D3—124no—
213C2×D3—184no—
214C2×D3—244no—
215C2×D3—242no—
216C2×D3—364no—
217C2×D3—362no—
218C2×D3—364no—
219C2×D3—368no—
220C2×D3—364no—
221C3×C4—91no—
222C3×C4—91no—
223C3×C4—92no—
224C3×C4—362no—
225C32—41no—
226C32—81no—
227C32C2×#80911yescharacteristic, complemented
228C23—91no—
229C23—361no—
230C23—544no—
231C23—542no—
232C23—1082no—
233C23—1081no—
234D4—274no—
235D4—542no—
236D4—544no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—92no—
240C2×C4—92no—
241C2×C4—182no—
242C2×C4—362no—
243C8—542no—
244C2×C3—42no—
245C2×C3—41no—
246C2×C3—82no—
247C2×C3—81no—
248C2×C3—91no—
249C2×C3—91no—
250C2×C3—92no—
251C2×C3—364no—
252C2×C3GO+(4,𝔽2)⋊C412yescomplemented
253C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
254D3—122no—
255D3—121no—
256D3—121no—
257D3—124no—
258D3—124no—
259D3—244no—
260C22—92no—
261C22—92no—
262C22—91no—
263C22—91no—
264C22—364no—
265C22—362no—
266C22—544no—
267C22—1082no—
268C22—1084no—
269C22#8909211yescharacteristic
270C4—91no—
271C4—91no—
272C4—92no—
273C4—362no—
274C3—41no—
275C3—81no—
276C3C2×GO+(4,𝔽2)⋊C411yescharacteristic, complemented
277C2—91no—
278C2—91no—
279C2—92no—
280C2—364no—
281C2#1172410112yesdirect product factor
282C2C2×#8909211yescharacteristic
subgroupG/H
center, hypercenterC22#89092
commutatorC3×#157C22×C4
FrattiniC2C2×#89092
Fitting, socleC22×C33SD16
prime radicalC22×#157⋊D3C2
cosocleC2×C3×#157C23
hypocenterC33C2×#102
LCSG ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

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. #11724101C2direct (N × H)
11. C2×#157⋊D3C4semidirect
14. C2×C3⋊AGL(1,𝔽9)C22semidirect
16. C3×#3777C22semidirect
19. C2×#3777D3semidirect
35. #157⋊D3C2×C4semidirect
49. #3777C2×D3semidirect
93. C2×C33#102semidirect
96. C3×C3⋊D3#102semidirect
144. C33C2×#102semidirect
173. C2×C32#809semidirect
176. C3⋊D3#809semidirect
227. C32C2×#809semidirect
252. C2×C3GO+(4,𝔽2)⋊C4semidirect
276. C3C2×GO+(4,𝔽2)⋊C4semidirect
281. C2#11724101direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C22×C3⋊AGL(1,𝔽9)2
C2×C3×#37772
#117241012
C2×GO+(4,𝔽2)⋊C43
C2×#8099