Finite Group Databasebeta

← C3×#10593568 #13167644

Subgroups — C3×#10593568 #13167644

Order 1728.

634 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C23×C3×#157C211yesmaximal, characteristic
2C2×C3×C22⋊#157C212yesmaximal
3C2×C3×#3777C212yesmaximal, complemented
4C3×#63747C212yesmaximal, complemented
5#10593568C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×#157—24no—
8C22×C3×#157C414yescomplemented
9C22×C3×#157C2214yes—
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C3×C22⋊#157—24no—
13C3×#3777—24no—
14C23×#157C2×C311yescharacteristic
15C2×C22⋊#157C2×C312yes—
16C2×#3777C2×C312yescomplemented
17#63747C2×C312yescomplemented
18C23×C33Q811yescharacteristic
19C22×C3×C3⋊D3—22no—
20C22×C3×C3⋊D3—22no—
21C22×C3×C3⋊D3C2×C412yes—
22C22×C3×C3⋊D3D412yes—
23C22×C3×C3⋊D3Q811yescharacteristic
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3×#157—22no—
26C2×C3×#157—28no—
27C2×C3×#157—24no—
28C2×C3×#157—24no—
29C2×C3×#157—24no—
30C2×C3×#157C2×C412yes—
31C2×C3×#157C2×C412yes—
32C2×C3×#157D414yes—
33C3×#380—91nomaximal
34C23×C3⋊D3C22×C311yescharacteristic
35C23×C3×D3—121no—
36C22×#157—24no—
37C22×#157C3×C414yescomplemented
38C22×#157C22×C314yes—
39C22×#157C22×C311yescharacteristic
40C22×#157C22×C311yescharacteristic
41C22⋊#157—24no—
42#3777—24no—
43C22×C33—22no—
44C22×C33C4⋊C412yescomplemented
45C22×C33C2×Q811yescharacteristic
46C2×C3×C3⋊D3—22no—
47C2×C3×C3⋊D3—22no—
48C2×C3×C3⋊D3—22no—
49C2×C3×C3⋊D3—24no—
50C2×C3×C3⋊D3C4⋊C412yescomplemented
51C2×C3×C3⋊D3C22×C411yescharacteristic
52C2×C3×C3⋊D3C2×D412yes—
53C2×C3×C3⋊D3C2×D411yescharacteristic
54C2×C3×C3⋊D3#4112yes—
55C3×#157—22no—
56C3×#157—44no—
57C3×#157C22⋊C414yescomplemented
58C23×C3×C4—91no—
59C2×C3×C4⋊C4—92no—
60C2×C3×C22⋊C4—92no—
61C3×#95—92no—
62C23×C32—41no—
63C23×C32—81no—
64C23×C32C3×Q811yescharacteristic
65C22×C3⋊D3—22no—
66C22×C3⋊D3—22no—
67C22×C3⋊D3C2×C3×C412yes—
68C22×C3⋊D3C3×D412yes—
69C22×C3⋊D3C3×Q811yescharacteristic
70C22×C3⋊D3C23×C311yescharacteristic
71C22×C3×D3—122no—
72C22×C3×D3—122no—
73C22×C3×D3—121no—
74C22×C3×D3—121no—
75C22×C3×D3—124no—
76C22×C3×D3—124no—
77C2×#157—22no—
78C2×#157—28no—
79C2×#157—24no—
80C2×#157—24no—
81C2×#157—24no—
82C2×#157C2×C3×C412yes—
83C2×#157C2×C3×C412yes—
84C2×#157C3×D414yes—
85#380—91noSylow
86C2×C33—22no—
87C2×C33C2×C4⋊C411yescharacteristic
88C2×C33C22⋊Q812yes—
89C3×C3⋊D3—22no—
90C3×C3⋊D3C2×C22⋊C411yescharacteristic
91C3×C3⋊D3#11711yescharacteristic
92C3×C3⋊D3#12112yes—
93C24×C3—91no—
94C23×D3—121no—
95C22×C3×C4—91no—
96C22×C3×C4—94no—
97C22×C3×C4—91no—
98C22×C3×C4—94no—
99C22×C3×C4—184no—
100C3×C4⋊C4—184no—
101C3×C22⋊C4—184no—
102C22×C32—22no—
103C22×C32—44no—
104C22×C32—41no—
105C22×C32—42no—
106C22×C32—84no—
107C22×C32—82no—
108C22×C32—81no—
109C22×C32C3×C4⋊C412yescomplemented
110C22×C32C2×C3×Q811yescharacteristic
111C2×C3⋊D3—22no—
112C2×C3⋊D3—22no—
113C2×C3⋊D3—22no—
114C2×C3⋊D3—24no—
115C2×C3⋊D3C3×C4⋊C412yescomplemented
116C2×C3⋊D3C22×C3×C411yescharacteristic
117C2×C3⋊D3C2×C3×D412yes—
118C2×C3⋊D3C2×C3×D411yescharacteristic
119C2×C3⋊D3C3×#4112yes—
120C2×C3×D3—122no—
121C2×C3×D3—121no—
122C2×C3×D3—121no—
123C2×C3×D3—122no—
124C2×C3×D3—122no—
125C2×C3×D3—124no—
126C2×C3×D3—124no—
127C2×C3×D3—128no—
128C2×C3×D3—124no—
129#157—22no—
130#157—44no—
131#157C3×C22⋊C414yescomplemented
132C23×C4—91no—
133C2×C4⋊C4—92no—
134C2×C22⋊C4—92no—
135#95—92no—
136C33#38011yescharacteristic, complemented, Sylow
137C23×C3—41no—
138C23×C3—81no—
139C23×C3—91no—
140C23×C3—91no—
141C23×C3—92no—
142C23×C3—92no—
143C23×C3—182no—
144C23×C3—182no—
145C23×C3PSU(3,𝔽2)11yescharacteristic
146C22×D3—122no—
147C22×D3—122no—
148C22×D3—121no—
149C22×D3—121no—
150C22×D3—124no—
151C22×D3—124no—
152C2×C3×C4—92no—
153C2×C3×C4—94no—
154C2×C3×C4—92no—
155C2×C3×C4—188no—
156C2×C3×C4—182no—
157C2×C3×C4—184no—
158C2×C3×C4—184no—
159C2×C3×C4—184no—
160C2×C32—22no—
161C2×C32—44no—
162C2×C32—41no—
163C2×C32—42no—
164C2×C32—84no—
165C2×C32—81no—
166C2×C32—82no—
167C2×C32C2×C3×C4⋊C411yescharacteristic
168C2×C32C3×C22⋊Q812yes—
169C3⋊D3—22no—
170C3⋊D3C2×C3×C22⋊C411yescharacteristic
171C3⋊D3C3×#11711yescharacteristic
172C3⋊D3C3×#12112yes—
173C3×D3—121no—
174C3×D3—121no—
175C3×D3—122no—
176C3×D3—124no—
177C24—91no—
178C22×C4—91no—
179C22×C4—94no—
180C22×C4—91no—
181C22×C4—94no—
182C22×C4—184no—
183C4⋊C4—184no—
184C22⋊C4—184no—
185C22×C3—22no—
186C22×C3—44no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3—84no—
190C22×C3—81no—
191C22×C3—82no—
192C22×C3—91no—
193C22×C3—91no—
194C22×C3—92no—
195C22×C3—92no—
196C22×C3—92no—
197C22×C3—182no—
198C22×C3—184no—
199C22×C3—182no—
200C22×C3—182no—
201C22×C3#377712yescomplemented
202C22×C3C2×PSU(3,𝔽2)11yescharacteristic
203C2×D3—122no—
204C2×D3—121no—
205C2×D3—121no—
206C2×D3—122no—
207C2×D3—122no—
208C2×D3—124no—
209C2×D3—124no—
210C2×D3—128no—
211C2×D3—124no—
212C3×C4—94no—
213C3×C4—182no—
214C3×C4—364no—
215C32—41no—
216C32—81no—
217C32C3×#38011yescharacteristic, complemented
218C23—91no—
219C23—91no—
220C23—92no—
221C23—92no—
222C23—182no—
223C23—182no—
224C23C3×PSU(3,𝔽2)11yescharacteristic
225C2×C4—94no—
226C2×C4—92no—
227C2×C4—92no—
228C2×C4—188no—
229C2×C4—182no—
230C2×C4—184no—
231C2×C4—184no—
232C2×C4—184no—
233C2×C3—22no—
234C2×C3—44no—
235C2×C3—41no—
236C2×C3—42no—
237C2×C3—84no—
238C2×C3—81no—
239C2×C3—82no—
240C2×C3—91no—
241C2×C3—92no—
242C2×C3—91no—
243C2×C3—182no—
244C2×C3C2×#377711yescharacteristic
245C2×C3C22⋊PSU(3,𝔽2)12yes—
246D3—121no—
247D3—121no—
248D3—122no—
249D3—124no—
250C22—22no—
251C22—91no—
252C22—91no—
253C22—92no—
254C22—92no—
255C22—92no—
256C22—182no—
257C22—184no—
258C22—182no—
259C22—182no—
260C22C3×#377712yescomplemented
261C22C2×C3×PSU(3,𝔽2)11yescharacteristic
262C4—94no—
263C4—182no—
264C4—364no—
265C3—41no—
266C3—81no—
267C3#1059356811yescharacteristic, direct product factor
268C2—22no—
269C2—91no—
270C2—92no—
271C2—91no—
272C2—182no—
273C2C2×C3×#377711yescharacteristic
274C2C3×C22⋊PSU(3,𝔽2)12yes—
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC23×C33Q8
socleC22×C33C2×Q8
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3PSU(3,𝔽2)
hypocenterC32C3×#380
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3777C2semidirect
4. C3×#63747C2semidirect
5. #10593568C3direct (N × H)
8. C22×C3×#157C4semidirect
16. C2×#3777C2×C3semidirect
17. #63747C2×C3semidirect
37. C22×#157C3×C4semidirect
44. C22×C33C4⋊C4semidirect
50. C2×C3×C3⋊D3C4⋊C4semidirect
57. C3×#157C22⋊C4semidirect
109. C22×C32C3×C4⋊C4semidirect
115. C2×C3⋊D3C3×C4⋊C4semidirect
131. #157C3×C22⋊C4semidirect
136. C33#380semidirect
201. C22×C3#3777semidirect
217. C32C3×#380semidirect
260. C22C3×#3777semidirect
267. C3#10593568direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×C3×C22⋊#1572
C2×C3×#37772
C3×#637472
#105935683
C3×#3809