Finite Group Databasebeta

← C3×#10593554 #13167629

Subgroups — C3×#10593554 #13167629

Order 1728.

410 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×C3×C4⋊#157C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3C2×C3×C4×#157C211yesmaximal, characteristic
4C2×C3×#3777C212yesmaximal
5C3×#63747C212yesmaximal
6#10593554C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C22×C3×#157C2212yes—
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C3×C4⋊#157—22no—
13C3×C4⋊#157C414yescomplemented
14C3×C4×#157—22no—
15C3×#3777—24no—
16C2×C4⋊#157C2×C311yescharacteristic
17C2×C4⋊#157C2×C311yescharacteristic
18C2×C4×#157C2×C311yescharacteristic
19C2×#3777C2×C312yes—
20#63747C2×C312yes—
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3×#157—22no—
23C2×C3×#157—21no—
24C2×C3×#157—21no—
25C2×C3×#157—22no—
26C2×C3×#157—21no—
27C2×C3×#157—21no—
28C2×C3×#157—22no—
29C2×C3×#157—22no—
30C2×C3×#157—22no—
31C2×C3×#157C2×C414yes—
32C2×C3×#157D412yes—
33C2×C3×#157Q812yes—
34C2×C33×C4Q811yescharacteristic
35C2×C3×C3⋊Dic3Q811yescharacteristic
36C3×C4×C3⋊D3C2×C412yes—
37C3×C4×C3⋊D3D412yes—
38C3×#389—91nomaximal
39C22×#157C22×C312yes—
40C22×#157C22×C312yes—
41C22×#157C22×C311yescharacteristic
42C22×#157C22×C311yescharacteristic
43C2×C4×C3⋊D3C22×C311yescharacteristic
44C2×C3×C4×D3—121no—
45C4⋊#157—22no—
46C4⋊#157C3×C414yescomplemented
47C4×#157—22no—
48#3777—24no—
49C22×C33C2×Q811yescharacteristic
50C2×C3×C3⋊D3C22×C411yescharacteristic
51C2×C3×C3⋊D3C2×D411yescharacteristic
52C2×C3×C3⋊D3C2×D411yescharacteristic
53C2×C3×C3⋊D3C2×Q811yescharacteristic
54C2×C3×C3⋊D3#4111yescharacteristic
55C2×C3×C3⋊D3#4111yescharacteristic
56C3×#157—24no—
57C3×#157—22no—
58C3×#157—22no—
59C3×#157—42no—
60C33×C4C4⋊C412yescomplemented
61C3×C3⋊Dic3C4⋊C412yescomplemented
62C2×C3×C4⋊C4—91no—
63C2×C3×C4⋊C4—92no—
64C2×C3×C4⋊C4—91no—
65C2×C3×C42—91no—
66C3×#95—92no—
67C22×C3⋊D3C23×C311yescharacteristic
68C22×C3×D3—121no—
69C2×#157—22no—
70C2×#157—21no—
71C2×#157—21no—
72C2×#157—22no—
73C2×#157—21no—
74C2×#157—21no—
75C2×#157—22no—
76C2×#157—22no—
77C2×#157—22no—
78C2×#157C2×C3×C414yes—
79C2×#157C3×D412yes—
80C2×#157C3×Q812yes—
81C2×C32×C4—41no—
82C2×C32×C4—81no—
83C2×C32×C4C3×Q811yescharacteristic
84C2×C3⋊Dic3C3×Q811yescharacteristic
85C4×C3⋊D3C2×C3×C412yes—
86C4×C3⋊D3C3×D412yes—
87C2×C3×Dic3—121no—
88C3×C4×D3—122no—
89C3×C4×D3—122no—
90#389—91noSylow
91C2×C33C2×C4⋊C411yescharacteristic
92C2×C33#12511yescharacteristic
93C2×C33C4⋊Q811yescharacteristic
94C3×C3⋊D3C4×D411yescharacteristic
95C3×C3⋊D3C4×Q811yescharacteristic
96C3×C3⋊D3#12111yescharacteristic
97C3×C3⋊D3C22⋊Q811yescharacteristic
98C22×C3×C4—91no—
99C22×C3×C4—91no—
100C22×C3×C4—92no—
101C22×C3×C4—91no—
102C22×C3×C4—92no—
103C2×C4×D3—121no—
104C3×C4⋊C4—94no—
105C3×C4⋊C4—184no—
106C3×C4⋊C4—182no—
107C3×C42—182no—
108C22×C32—41no—
109C22×C32—81no—
110C22×C32C2×C3×Q811yescharacteristic
111C2×C3⋊D3C22×C3×C411yescharacteristic
112C2×C3⋊D3C2×C3×D411yescharacteristic
113C2×C3⋊D3C2×C3×D411yescharacteristic
114C2×C3⋊D3C2×C3×Q811yescharacteristic
115C2×C3⋊D3C3×#4111yescharacteristic
116C2×C3⋊D3C3×#4111yescharacteristic
117C2×C3×D3—121no—
118C2×C3×D3—121no—
119C2×C3×D3—121no—
120C2×C3×D3—121no—
121C2×C3×D3—121no—
122C2×C3×D3—121no—
123#157—24no—
124#157—22no—
125#157—22no—
126#157—42no—
127C32×C4—42no—
128C32×C4—82no—
129C32×C4C3×C4⋊C412yescomplemented
130C3⋊Dic3C3×C4⋊C412yescomplemented
131C3×Dic3—122no—
132C2×C4⋊C4—91no—
133C2×C4⋊C4—91no—
134C2×C4⋊C4—92no—
135C2×C42—91no—
136#95—92no—
137C33#38911yescharacteristic, complemented, Sylow
138C23×C3—91no—
139C22×D3—121no—
140C2×C3×C4—41no—
141C2×C3×C4—81no—
142C2×C3×C4—91no—
143C2×C3×C4—92no—
144C2×C3×C4—92no—
145C2×C3×C4—94no—
146C2×C3×C4—92no—
147C2×C3×C4—92no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4—182no—
151C2×C3×C4—181no—
152C2×C3×C4—182no—
153C2×C3×C4—181no—
154C2×C3×C4—182no—
155C2×C3×C4—182no—
156C2×C3×C4—182no—
157C2×C3×C4PSU(3,𝔽2)11yescharacteristic
158C2×Dic3—121no—
159C4×D3—122no—
160C4×D3—122no—
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—81no—
165C2×C32—81no—
166C2×C32—81no—
167C2×C32C2×C3×C4⋊C411yescharacteristic
168C2×C32C3×#12511yescharacteristic
169C2×C32C3×C4⋊Q811yescharacteristic
170C3⋊D3C3×C4×D411yescharacteristic
171C3⋊D3C3×C4×Q811yescharacteristic
172C3⋊D3C3×#12111yescharacteristic
173C3⋊D3C3×C22⋊Q811yescharacteristic
174C3×D3—121no—
175C3×D3—121no—
176C3×D3—121no—
177C3×D3—121no—
178C22×C4—91no—
179C22×C4—91no—
180C22×C4—92no—
181C22×C4—91no—
182C22×C4—92no—
183C4⋊C4—94no—
184C4⋊C4—184no—
185C4⋊C4—182no—
186C42—182no—
187C22×C3—41no—
188C22×C3—81no—
189C22×C3—91no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3—91no—
193C22×C3—91no—
194C22×C3—91no—
195C22×C3C2×PSU(3,𝔽2)11yescharacteristic
196C2×D3—121no—
197C2×D3—121no—
198C2×D3—121no—
199C2×D3—121no—
200C2×D3—121no—
201C2×D3—121no—
202C3×C4—42no—
203C3×C4—82no—
204C3×C4—92no—
205C3×C4—184no—
206C3×C4—182no—
207C3×C4—182no—
208C3×C4—362no—
209C3×C4#377712yescomplemented
210Dic3—122no—
211C32—41no—
212C32—81no—
213C32C3×#38911yescharacteristic, complemented
214C23—91no—
215C2×C4—91no—
216C2×C4—92no—
217C2×C4—92no—
218C2×C4—94no—
219C2×C4—92no—
220C2×C4—92no—
221C2×C4—181no—
222C2×C4—181no—
223C2×C4—182no—
224C2×C4—181no—
225C2×C4—182no—
226C2×C4—181no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4—182no—
230C2×C4C3×PSU(3,𝔽2)11yescharacteristic
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—81no—
235C2×C3—81no—
236C2×C3—81no—
237C2×C3—91no—
238C2×C3—91no—
239C2×C3—91no—
240C2×C3—91no—
241C2×C3#6424311yescharacteristic
242C2×C3C4⋊PSU(3,𝔽2)11yescharacteristic
243C2×C3C2×#377711yescharacteristic
244D3—121no—
245D3—121no—
246D3—121no—
247D3—121no—
248C22—91no—
249C22—91no—
250C22—91no—
251C22—91no—
252C22—91no—
253C22—91no—
254C22C2×C3×PSU(3,𝔽2)11yescharacteristic
255C4—92no—
256C4—184no—
257C4—182no—
258C4—182no—
259C4—362no—
260C4C3×#377712yescomplemented
261C3—41no—
262C3—81no—
263C3#1059355411yescharacteristic, direct product factor
264C2—91no—
265C2—91no—
266C2—91no—
267C2—91no—
268C2C3×#6424311yescharacteristic
269C2C3×C4⋊PSU(3,𝔽2)11yescharacteristic
270C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#389
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593554C3direct (N × H)
13. C3×C4⋊#157C4semidirect
46. C4⋊#157C3×C4semidirect
60. C33×C4C4⋊C4semidirect
61. C3×C3⋊Dic3C4⋊C4semidirect
129. C32×C4C3×C4⋊C4semidirect
130. C3⋊Dic3C3×C4⋊C4semidirect
137. C33#389semidirect
209. C3×C4#3777semidirect
213. C32C3×#389semidirect
260. C4C3×#3777semidirect
263. C3#10593554direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C2×C3×#37772
C3×#637472
#105935543
C3×#3899