Finite Group Databasebeta

← C3×#10593556 #13167632

Subgroups — C3×#10593556 #13167632

Order 1728.

434 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3777C212yesmaximal
4C3×#63781C212yesmaximal
5C3×#63747C211yesmaximal, characteristic
6#10593556C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C22×C3×#157C2212yes—
10C22×C3×#157C2212yes—
11C2×C3×PSU(3,𝔽2)—24no—
12C2×C3×PSU(3,𝔽2)—22no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C3×C4⋊#157—22no—
15C3×#3777—24no—
16C22×PSU(3,𝔽2)C2×C311yescharacteristic
17C2×C4⋊#157C2×C311yescharacteristic
18C2×#3777C2×C312yes—
19#63781C2×C312yes—
20#63747C2×C311yescharacteristic
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3×#157—22no—
23C2×C3×#157—22no—
24C2×C3×#157—22no—
25C2×C3×#157—22no—
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157—21no—
29C2×C3×#157D414yes—
30C2×C3×#157D412yes—
31C2×C3×#157Q814yescomplemented
32C3×PSU(3,𝔽2)—28no—
33C2×C33×C4Q811yescharacteristic, complemented
34C2×C3×C3⋊Dic3Q811yescharacteristic, complemented
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C3×#395—91nomaximal
38C22×#157C22×C311yescharacteristic
39C22×#157C22×C311yescharacteristic
40C22×#157C22×C312yes—
41C22×#157C22×C312yes—
42C2×PSU(3,𝔽2)—24no—
43C2×PSU(3,𝔽2)—22no—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×C4×D3—121no—
46C4⋊#157—22no—
47#3777—24no—
48C22×C33C2×Q811yescharacteristic
49C2×C3×C3⋊D3C2×D412yes—
50C2×C3×C3⋊D3C2×D411yescharacteristic
51C2×C3×C3⋊D3C2×Q812yes—
52C2×C3×C3⋊D3#4111yescharacteristic
53C3×#157—24no—
54C3×#157—22no—
55C3×#157—24no—
56C3×#157—41no—
57C33×C4—21no—
58C3×C3⋊Dic3—21no—
59C22×C3×Q8—91no—
60C2×C3×C4⋊C4—92no—
61C2×C3×C4⋊C4—91no—
62C3×#95—91no—
63C3×#95—92no—
64C22×C3⋊D3C23×C311yescharacteristic
65C22×C3×D3—121no—
66C2×#157—22no—
67C2×#157—22no—
68C2×#157—22no—
69C2×#157—22no—
70C2×#157—22no—
71C2×#157—22no—
72C2×#157—21no—
73C2×#157C3×D414yes—
74C2×#157C3×D412yes—
75C2×#157C3×Q814yescomplemented
76PSU(3,𝔽2)—28no—
77C2×C32×C4—41no—
78C2×C32×C4—81no—
79C2×C32×C4C3×Q811yescharacteristic, complemented
80C2×C3⋊Dic3C3×Q811yescharacteristic, complemented
81C4×C3⋊D3—21no—
82C4×C3⋊D3—21no—
83C2×C3×Dic3—121no—
84C3×C4×D3—122no—
85C3×C4×D3—122no—
86#395—91noSylow
87C2×C33C22⋊Q811yescharacteristic
88C2×C33C4⋊Q812yes—
89C3×C3⋊D3#12011yescharacteristic
90C3×C3⋊D3C22⋊Q812yes—
91C3×C3⋊D3C4⋊Q811yescharacteristic
92C2×C3×Q8—184no—
93C2×C3×Q8—182no—
94C22×C3×C4—92no—
95C22×C3×C4—91no—
96C22×C3×C4—92no—
97C22×C3×C4—91no—
98C22×C3×C4—91no—
99C2×C4×D3—121no—
100C3×C4⋊C4—184no—
101C3×C4⋊C4—182no—
102C22×C32—41no—
103C22×C32—81no—
104C22×C32C2×C3×Q811yescharacteristic
105C2×C3⋊D3C2×C3×D412yes—
106C2×C3⋊D3C2×C3×D411yescharacteristic
107C2×C3⋊D3C2×C3×Q812yes—
108C2×C3⋊D3C3×#4111yescharacteristic
109C2×C3×D3—121no—
110C2×C3×D3—122no—
111C2×C3×D3—122no—
112C2×C3×D3—121no—
113#157—24no—
114#157—22no—
115#157—24no—
116#157—41no—
117C32×C4—21no—
118C32×C4—42no—
119C32×C4—82no—
120C3⋊Dic3—21no—
121C3×Dic3—122no—
122C22×Q8—91no—
123C2×C4⋊C4—91no—
124C2×C4⋊C4—92no—
125#95—91no—
126#95—92no—
127C33#39511yescharacteristic, complemented, Sylow
128C23×C3—91no—
129C22×D3—121no—
130C3×Q8—188no—
131C2×C3×C4—41no—
132C2×C3×C4—81no—
133C2×C3×C4—94no—
134C2×C3×C4—92no—
135C2×C3×C4—94no—
136C2×C3×C4—91no—
137C2×C3×C4—181no—
138C2×C3×C4—181no—
139C2×C3×C4—182no—
140C2×C3×C4—182no—
141C2×C3×C4—182no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4—182no—
145C2×C3×C4—181no—
146C2×C3×C4PSU(3,𝔽2)11yescharacteristic, complemented
147C2×Dic3—121no—
148C4×D3—122no—
149C4×D3—122no—
150C2×C32—42no—
151C2×C32—41no—
152C2×C32—81no—
153C2×C32—82no—
154C2×C32C3×C22⋊Q811yescharacteristic
155C2×C32C3×C4⋊Q812yes—
156C3⋊D3C3×#12011yescharacteristic
157C3⋊D3C3×C22⋊Q812yes—
158C3⋊D3C3×C4⋊Q811yescharacteristic
159C3×D3—121no—
160C3×D3—122no—
161C3×D3—121no—
162C2×Q8—184no—
163C2×Q8—182no—
164C22×C4—92no—
165C22×C4—91no—
166C22×C4—92no—
167C22×C4—91no—
168C22×C4—91no—
169C4⋊C4—184no—
170C4⋊C4—182no—
171C22×C3—41no—
172C22×C3—81no—
173C22×C3—92no—
174C22×C3—92no—
175C22×C3—91no—
176C22×C3—91no—
177C22×C3C2×PSU(3,𝔽2)11yescharacteristic
178C2×D3—121no—
179C2×D3—121no—
180C2×D3—122no—
181C2×D3—122no—
182C3×C4—21no—
183C3×C4—42no—
184C3×C4—82no—
185C3×C4—184no—
186C3×C4—182no—
187C3×C4—184no—
188C3×C4—181no—
189C3×C4—361no—
190Dic3—122no—
191C32—41no—
192C32—81no—
193C32C3×#39511yescharacteristic, complemented
194C23—91no—
195Q8—188no—
196C2×C4—94no—
197C2×C4—92no—
198C2×C4—94no—
199C2×C4—91no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4—182no—
203C2×C4—182no—
204C2×C4—182no—
205C2×C4—182no—
206C2×C4—182no—
207C2×C4—182no—
208C2×C4—181no—
209C2×C4C3×PSU(3,𝔽2)11yescharacteristic, complemented
210C2×C3—42no—
211C2×C3—41no—
212C2×C3—82no—
213C2×C3—81no—
214C2×C3—91no—
215C2×C3—92no—
216C2×C3—91no—
217C2×C3C4⋊PSU(3,𝔽2)12yes—
218C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
219D3—121no—
220D3—122no—
221D3—121no—
222C22—92no—
223C22—92no—
224C22—91no—
225C22—91no—
226C22C2×C3×PSU(3,𝔽2)11yescharacteristic
227C4—21no—
228C4—184no—
229C4—182no—
230C4—184no—
231C4—181no—
232C4—361no—
233C3—41no—
234C3—81no—
235C3#1059355611yescharacteristic, direct product factor
236C2—91no—
237C2—92no—
238C2—91no—
239C2C3×C4⋊PSU(3,𝔽2)12yes—
240C2C3×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#395
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593556C3direct (N × H)
31. C2×C3×#157Q8semidirect
33. C2×C33×C4Q8semidirect
34. C2×C3×C3⋊Dic3Q8semidirect
75. C2×#157C3×Q8semidirect
79. C2×C32×C4C3×Q8semidirect
80. C2×C3⋊Dic3C3×Q8semidirect
127. C33#395semidirect
146. C2×C3×C4PSU(3,𝔽2)semidirect
193. C32C3×#395semidirect
209. C2×C4C3×PSU(3,𝔽2)semidirect
235. C3#10593556direct (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
#105935563
C3×#3959