Finite Group Databasebeta

← C3×#10593555 #13167631

Subgroups — C3×#10593555 #13167631

Order 1728.

442 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 118 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×PSU(3,𝔽2)C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic
3C2×C3×#3777C211yesmaximal, characteristic
4C3×#63781C212yesmaximal
5C3×#63747C212yesmaximal
6#10593555C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C22×C3×#157C2212yes—
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×C3×PSU(3,𝔽2)—22no—
12C2×C3×PSU(3,𝔽2)—22no—
13C2×C3×PSU(3,𝔽2)C414yescomplemented
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C3×C4×#157—22no—
16C3×#3777—22no—
17C22×PSU(3,𝔽2)C2×C311yescharacteristic
18C2×C4×#157C2×C311yescharacteristic
19C2×#3777C2×C311yescharacteristic
20#63781C2×C312yes—
21#63747C2×C312yes—
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×C3×#157—22no—
24C2×C3×#157—22no—
25C2×C3×#157—21no—
26C2×C3×#157—21no—
27C2×C3×#157—22no—
28C2×C3×#157—22no—
29C2×C3×#157—22no—
30C2×C3×#157C2×C414yes—
31C2×C3×#157C2×C412yes—
32C2×C3×#157D412yes—
33C2×C3×#157D412yes—
34C2×C3×#157Q812yescomplemented
35C3×PSU(3,𝔽2)—28no—
36C2×C33×C4Q811yescharacteristic, complemented
37C2×C3×C3⋊Dic3Q811yescharacteristic, complemented
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3—21no—
40C3×#391—91nomaximal
41C22×#157C22×C312yes—
42C22×#157C22×C312yes—
43C22×#157C22×C311yescharacteristic
44C22×#157C22×C311yescharacteristic
45C2×PSU(3,𝔽2)—22no—
46C2×PSU(3,𝔽2)—22no—
47C2×PSU(3,𝔽2)C3×C414yescomplemented
48C2×C4×C3⋊D3C22×C311yescharacteristic
49C2×C3×C4×D3—121no—
50C4×#157—22no—
51#3777—22no—
52C22×C33C2×Q811yescharacteristic
53C2×C3×C3⋊D3C22×C411yescharacteristic
54C2×C3×C3⋊D3C2×D411yescharacteristic
55C2×C3×C3⋊D3C2×D411yescharacteristic
56C2×C3×C3⋊D3C2×Q811yescharacteristic
57C2×C3×C3⋊D3#4111yescharacteristic
58C2×C3×C3⋊D3#4111yescharacteristic
59C3×#157—24no—
60C3×#157—22no—
61C3×#157—42no—
62C3×#157C22⋊C414yes—
63C33×C4—21no—
64C3×C3⋊Dic3—21no—
65C22×C3×Q8—91no—
66C2×C3×C4⋊C4—91no—
67C2×C3×C42—91no—
68C3×#95—92no—
69C3×#95—92no—
70C22×C3⋊D3C23×C311yescharacteristic
71C22×C3×D3—121no—
72C2×#157—22no—
73C2×#157—22no—
74C2×#157—21no—
75C2×#157—21no—
76C2×#157—22no—
77C2×#157—22no—
78C2×#157—22no—
79C2×#157C2×C3×C414yes—
80C2×#157C2×C3×C412yes—
81C2×#157C3×D412yes—
82C2×#157C3×D412yes—
83C2×#157C3×Q812yescomplemented
84PSU(3,𝔽2)—28no—
85C2×C32×C4—41no—
86C2×C32×C4—81no—
87C2×C32×C4C3×Q811yescharacteristic, complemented
88C2×C3⋊Dic3C3×Q811yescharacteristic, complemented
89C4×C3⋊D3—21no—
90C4×C3⋊D3—21no—
91C2×C3×Dic3—121no—
92C3×C4×D3—122no—
93C3×C4×D3—122no—
94#391—91noSylow
95C2×C33C4×Q811yescharacteristic
96C2×C33C22⋊Q811yescharacteristic
97C2×C33C4⋊Q811yescharacteristic
98C3×C3⋊D3C2×C22⋊C411yescharacteristic
99C3×C3⋊D3C4×Q811yescharacteristic
100C3×C3⋊D3C22⋊Q811yescharacteristic
101C3×C3⋊D3#12411yescharacteristic
102C2×C3×Q8—94no—
103C2×C3×Q8—182no—
104C2×C3×Q8—182no—
105C22×C3×C4—91no—
106C22×C3×C4—91no—
107C22×C3×C4—92no—
108C22×C3×C4—91no—
109C22×C3×C4—92no—
110C2×C4×D3—121no—
111C3×C4⋊C4—182no—
112C3×C42—182no—
113C22×C32—41no—
114C22×C32—81no—
115C22×C32C2×C3×Q811yescharacteristic
116C2×C3⋊D3C22×C3×C411yescharacteristic
117C2×C3⋊D3C2×C3×D411yescharacteristic
118C2×C3⋊D3C2×C3×D411yescharacteristic
119C2×C3⋊D3C2×C3×Q811yescharacteristic
120C2×C3⋊D3C3×#4111yescharacteristic
121C2×C3⋊D3C3×#4111yescharacteristic
122C2×C3×D3—121no—
123C2×C3×D3—121no—
124C2×C3×D3—121no—
125C2×C3×D3—121no—
126C2×C3×D3—121no—
127C2×C3×D3—121no—
128#157—24no—
129#157—22no—
130#157—42no—
131#157C3×C22⋊C414yes—
132C32×C4—21no—
133C32×C4—42no—
134C32×C4—82no—
135C3⋊Dic3—21no—
136C3×Dic3—122no—
137C22×Q8—91no—
138C2×C4⋊C4—91no—
139C2×C42—91no—
140#95—92no—
141#95—92no—
142C33#39111yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C22×D3—121no—
145C3×Q8—188no—
146C2×C3×C4—41no—
147C2×C3×C4—81no—
148C2×C3×C4—92no—
149C2×C3×C4—92no—
150C2×C3×C4—92no—
151C2×C3×C4—94no—
152C2×C3×C4—92no—
153C2×C3×C4—91no—
154C2×C3×C4—181no—
155C2×C3×C4—181no—
156C2×C3×C4—181no—
157C2×C3×C4—181no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4—182no—
161C2×C3×C4—182no—
162C2×C3×C4—182no—
163C2×C3×C4PSU(3,𝔽2)11yescharacteristic, complemented
164C2×Dic3—121no—
165C4×D3—122no—
166C4×D3—122no—
167C2×C32—41no—
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—81no—
171C2×C32—81no—
172C2×C32—81no—
173C2×C32C3×C4×Q811yescharacteristic
174C2×C32C3×C22⋊Q811yescharacteristic
175C2×C32C3×C4⋊Q811yescharacteristic
176C3⋊D3C2×C3×C22⋊C411yescharacteristic
177C3⋊D3C3×C4×Q811yescharacteristic
178C3⋊D3C3×C22⋊Q811yescharacteristic
179C3⋊D3C3×#12411yescharacteristic
180C3×D3—121no—
181C3×D3—121no—
182C3×D3—121no—
183C3×D3—121no—
184C2×Q8—94no—
185C2×Q8—182no—
186C2×Q8—182no—
187C22×C4—91no—
188C22×C4—91no—
189C22×C4—92no—
190C22×C4—91no—
191C22×C4—92no—
192C4⋊C4—182no—
193C42—182no—
194C22×C3—41no—
195C22×C3—81no—
196C22×C3—91no—
197C22×C3—91no—
198C22×C3—91no—
199C22×C3—91no—
200C22×C3—91no—
201C22×C3—91no—
202C22×C3C2×PSU(3,𝔽2)11yescharacteristic
203C2×D3—121no—
204C2×D3—121no—
205C2×D3—121no—
206C2×D3—121no—
207C2×D3—121no—
208C2×D3—121no—
209C3×C4—21no—
210C3×C4—42no—
211C3×C4—82no—
212C3×C4—94no—
213C3×C4—184no—
214C3×C4—182no—
215C3×C4—181no—
216C3×C4—362no—
217Dic3—122no—
218C32—41no—
219C32—81no—
220C32C3×#39111yescharacteristic, complemented
221C23—91no—
222Q8—188no—
223C2×C4—92no—
224C2×C4—92no—
225C2×C4—92no—
226C2×C4—94no—
227C2×C4—92no—
228C2×C4—91no—
229C2×C4—181no—
230C2×C4—181no—
231C2×C4—181no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4—181no—
235C2×C4—182no—
236C2×C4—182no—
237C2×C4—182no—
238C2×C4C3×PSU(3,𝔽2)11yescharacteristic, complemented
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—81no—
243C2×C3—81no—
244C2×C3—81no—
245C2×C3—91no—
246C2×C3—91no—
247C2×C3—91no—
248C2×C3—91no—
249C2×C3C4×PSU(3,𝔽2)11yescharacteristic
250C2×C3C4⋊PSU(3,𝔽2)11yescharacteristic
251C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
252D3—121no—
253D3—121no—
254D3—121no—
255D3—121no—
256C22—91no—
257C22—91no—
258C22—91no—
259C22—91no—
260C22—91no—
261C22—91no—
262C22C2×C3×PSU(3,𝔽2)11yescharacteristic
263C4—21no—
264C4—94no—
265C4—184no—
266C4—182no—
267C4—181no—
268C4—362no—
269C3—41no—
270C3—81no—
271C3#1059355511yescharacteristic, direct product factor
272C2—91no—
273C2—91no—
274C2—91no—
275C2—91no—
276C2C3×C4×PSU(3,𝔽2)11yescharacteristic
277C2C3×C4⋊PSU(3,𝔽2)11yescharacteristic
278C2C3×C22⋊PSU(3,𝔽2)11yescharacteristic
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×#391
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. #10593555C3direct (N × H)
13. C2×C3×PSU(3,𝔽2)C4semidirect
34. C2×C3×#157Q8semidirect
36. C2×C33×C4Q8semidirect
37. C2×C3×C3⋊Dic3Q8semidirect
47. C2×PSU(3,𝔽2)C3×C4semidirect
83. C2×#157C3×Q8semidirect
87. C2×C32×C4C3×Q8semidirect
88. C2×C3⋊Dic3C3×Q8semidirect
142. C33#391semidirect
163. C2×C3×C4PSU(3,𝔽2)semidirect
220. C32C3×#391semidirect
238. C2×C4C3×PSU(3,𝔽2)semidirect
271. C3#10593555direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×PSU(3,𝔽2)2
C2×C3×C4×#1572
C2×C3×#37772
C3×#637812
C3×#637472
#105935553
C3×#3919