Finite Group Databasebeta

← #10593622⋊C3 #13146821

Subgroups — #10593622⋊C3 #13146821

Order 1728.

277 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 43 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⋊#226)⋊C3C211yesmaximal, characteristic
2C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
3(C3⋊#859)⋊C3C211yesmaximal, characteristic
4#10593622C311yesmaximal, characteristic, complemented
5C3×#5030—31nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7(C3⋊#226)⋊C3C412yescomplemented
8#3750⋊C3—41nomaximal
9C2×C3⋊#226C2×C311yescharacteristic
10C32×C4×Q8D311yescharacteristic
11C2×C3×#216—31no—
12C3×C4×Dic6—31no—
13C3×C4×SL(2,𝔽3)—21no—
14C3×C4×SL(2,𝔽3)C2×C311yescharacteristic
15C3⋊#859C2×C311yescharacteristic
16C3×#835—31no—
17C3×#810—31no—
18C3×#808—31no—
19C3×#803—31no—
20C2×C4×H(3,𝔽3)—41no—
21C2×(C3⋊Dic3)⋊C3—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23Q8⋊H(3,𝔽3)C2×C411yescharacteristic
24#5388—91no—
25C3×#441—91no—
26#5030—31no—
27C2×C32×Q8C2×D311yescharacteristic
28C2×C3×Dic6—31no—
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
31C3⋊#226C3×C412yescomplemented
32C32×C4⋊C4—31no—
33C32×C42—31no—
34#3750—41no—
35C3×#216—32no—
36C3×#216—32no—
37C3×C4⋊Dic3—31no—
38C3×Dic3⋊C4—61no—
39C3×C4×Dic3—61no—
40C2×C3×C3⋊C8—31no—
41C2×C3×C3⋊C8—31no—
42C22×H(3,𝔽3)—41no—
43C4×H(3,𝔽3)—81no—
44(C3⋊Dic3)⋊C3—42no—
45(C3⋊Dic3)⋊C3—81no—
46C2×#226—91no—
47C2×C3×Q16—91no—
48C3×C4×Q8—31no—
49C3×C4×Q8—91no—
50C3×C4×Q8C3×D311yescharacteristic
51C2×#216—31no—
52C4×Dic6—31no—
53C4×SL(2,𝔽3)—31no—
54C4×SL(2,𝔽3)—61no—
55#859—91no—
56C3×#106—91no—
57C3×#103—91no—
58C3×#103—91no—
59C3×#97—91no—
60#835—31no—
61#810—31no—
62#808—31no—
63#803—31no—
64C32×Q8—31no—
65C32×Q8C4×D311yescharacteristic
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—41no—
69C2×C32×C4—81no—
70C2×C32×C4S411yescharacteristic
71C2×C3⋊Dic3—41no—
72C2×C3⋊Dic3—41no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—61no—
75C3×Dic6—32no—
76C3×Dic6—61no—
77C3×SL(2,𝔽3)—21no—
78C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
79C3×C3⋊C8—32no—
80C3×C3⋊C8—61no—
81#441—91noSylow
82C2×H(3,𝔽3)—41no—
83C2×H(3,𝔽3)—41no—
84C2×H(3,𝔽3)—41no—
85C2×C3×Q8—31no—
86C2×C3×Q8—91no—
87C2×C3×Q8C2×C3×D311yescharacteristic
88C2×Dic6—31no—
89C2×SL(2,𝔽3)—31no—
90C2×SL(2,𝔽3)—61no—
91#226—92no—
92C3×Q16—92no—
93C3×Q16—92no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C3×C4⋊C4—31no—
97C3×C4⋊C4—31no—
98C3×C4⋊C4—61no—
99C3×C4⋊C4—91no—
100C3×C4⋊C4—181no—
101C3×C42—31no—
102C3×C42—31no—
103C3×C42—61no—
104C3×C42—181no—
105#216—32no—
106#216—32no—
107C4⋊Dic3—31no—
108Dic3⋊C4—61no—
109Dic3⋊C4—361no—
110C4×Dic3—61no—
111C2×C3⋊C8—31no—
112C2×C3⋊C8—31no—
113C22×C32—41no—
114C22×C32—81no—
115C22×C32C2×S411yescharacteristic
116C32×C4—21no—
117C32×C4—31no—
118C32×C4—31no—
119C32×C4—61no—
120C32×C4—81no—
121C32×C4—82no—
122C3⋊Dic3—42no—
123C3⋊Dic3—81no—
124C3×Dic3—62no—
125C3×Dic3—121no—
126C2×Q16—91no—
127C4×Q8—91no—
128C4×Q8(C3⋊D3)⋊C311yescharacteristic
129#106—91no—
130#103—91no—
131#103—91no—
132#97—91no—
133H(3,𝔽3)—41noSylow
134C3×Q8—31no—
135C3×Q8—31no—
136C3×Q8—31no—
137C3×Q8—61no—
138C3×Q8—92no—
139C3×Q8—181no—
140C3×Q8C3×C4×D311yescharacteristic
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—61no—
147C2×C3×C4—61no—
148C2×C3×C4—121no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4—241no—
152C2×C3×C4C3×S411yescharacteristic
153C2×Dic3—61no—
154C2×Dic3—61no—
155C2×Dic3—361no—
156C2×Dic3—361no—
157Dic6—32no—
158Dic6—61no—
159SL(2,𝔽3)—31no—
160SL(2,𝔽3)—61no—
161C3×C8—92no—
162C3×C8—181no—
163C3⋊C8—32no—
164C3⋊C8—61no—
165C2×C32—41no—
166C2×C32—41no—
167C2×C32—41no—
168C2×C32—81no—
169C2×C32—81no—
170C2×C32—81no—
171C2×C32C4×S411yescharacteristic
172C2×C32#98311yescharacteristic
173C2×C32#98411yescharacteristic
174C2×Q8—91no—
175C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
176Q16—92no—
177Q16—92no—
178C2×C8—91no—
179C2×C8—91no—
180C4⋊C4—31no—
181C4⋊C4—91no—
182C4⋊C4—181no—
183C42—31no—
184C42—181no—
185C22×C3—31no—
186C22×C3—121no—
187C22×C3—241no—
188C22×C3C2×C3×S411yescharacteristic
189C3×C4—21no—
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—61no—
195C3×C4—61no—
196C3×C4—61no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—121no—
200C3×C4—182no—
201C3×C4—241no—
202C3×C4—242no—
203C3×C4—361no—
204Dic3—62no—
205Dic3—121no—
206Dic3—362no—
207Dic3—721no—
208C32—41no—
209C32—81no—
210C32#538811yescharacteristic, complemented
211Q8—31no—
212Q8—92no—
213Q8—181no—
214Q8C4×(C3⋊D3)⋊C311yescharacteristic
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—181no—
218C2×C4—181no—
219C2×C4(C3⋊S4)⋊C311yescharacteristic
220C8—92no—
221C8—181no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—31no—
225C2×C3—121no—
226C2×C3—121no—
227C2×C3—121no—
228C2×C3—241no—
229C2×C3—241no—
230C2×C3—241no—
231C2×C3C3×C4×S411yescharacteristic
232C2×C3C3×#98311yescharacteristic
233C2×C3C3×#98411yescharacteristic
234C22C2×(C3⋊S4)⋊C311yescharacteristic
235C4—21no—
236C4—31no—
237C4—31no—
238C4—61no—
239C4—182no—
240C4—361no—
241C3—31no—
242C3—121no—
243C3—241no—
244C3C3×#538811yescharacteristic
245C2C4×(C3⋊S4)⋊C311yescharacteristic
246C2#64264⋊C311yescharacteristic
247C2#64265⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
Frattini, socleC22×C3C2×C3×S4
FittingC32×C4×Q8D3
prime radicalC2×Q8⋊H(3,𝔽3)C22
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC2×C4(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #10593622C3semidirect
7. (C3⋊#226)⋊C3C4semidirect
31. C3⋊#226C3×C4semidirect
210. C32#5388semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊#226)⋊C32
C4×Q8⋊H(3,𝔽3)2
(C3⋊#859)⋊C32
#105936223
C3×#50303
#3750⋊C34