Finite Group Databasebeta

← #10589394⋊C3 #13135566

Subgroups — #10589394⋊C3 #13135566

Order 1728.

365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 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×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2#63657⋊C3C212yesmaximal
3#10589394C311yesmaximal, characteristic, complemented
4C3×#4528—31nomaximal
5C22×C4×H(3,𝔽3)C411yescharacteristic
6C22×C4×H(3,𝔽3)C411yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C4⋊C4×H(3,𝔽3)—22no—
9C2×(C3⋊(C3⋊C8))⋊C3—22no—
10C2×C32×C4⋊C4—21no—
11C2×C32×C4⋊C4D311yescharacteristic
12C2×C32×C4⋊C4C2×C311yescharacteristic
13#63657C2×C312yes—
14C3×#830—32no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)—21no—
17C2×C4×H(3,𝔽3)—21no—
18C2×C4×H(3,𝔽3)—21no—
19C2×C4×H(3,𝔽3)—41no—
20C2×C4×H(3,𝔽3)—41no—
21C2×C4×H(3,𝔽3)C812yescomplemented
22C2×C4×H(3,𝔽3)D412yes—
23(C3⋊(C3⋊C8))⋊C3—42no—
24C3×#324—91no—
25#4528—31no—
26#4528—91no—
27C22×C32×C4—21no—
28C22×C32×C4—21no—
29C22×C32×C4—21no—
30C22×C32×C4Dic311yescharacteristic
31C22×C32×C4Dic311yescharacteristic
32C22×C32×C4C3×C411yescharacteristic
33C22×C32×C4C3×C411yescharacteristic
34C22×C32×C4C2×D311yescharacteristic
35C22×C32×C4C22×C311yescharacteristic
36C32×C4⋊C4—24no—
37C32×C4⋊C4—22no—
38C32×C4⋊C4—22no—
39C2×C3⋊(C3⋊C8)—22no—
40C2×C3×C3⋊C8—62no—
41C22×H(3,𝔽3)—21no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)C22⋊C411yescharacteristic
44C22×H(3,𝔽3)C2×C811yescharacteristic
45C22×H(3,𝔽3)M1611yescharacteristic
46C4×H(3,𝔽3)—22no—
47C4×H(3,𝔽3)—22no—
48C4×H(3,𝔽3)—41no—
49C2×C3×C4⋊C4—31no—
50C2×C3×C4⋊C4—31no—
51C2×C3×C4⋊C4—61no—
52C2×C3×C4⋊C4C3×D311yescharacteristic
53C3×C22⋊C8—92no—
54#830—32no—
55#830—92no—
56C23×C32—21no—
57C23×C32C2×Dic311yescharacteristic
58C23×C32C2×C3×C411yescharacteristic
59C2×C32×C4—22no—
60C2×C32×C4—22no—
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4—22no—
66C2×C32×C4—21no—
67C2×C32×C4—21no—
68C2×C32×C4—22no—
69C2×C32×C4—22no—
70C2×C32×C4—41no—
71C2×C32×C4—41no—
72C2×C32×C4—42no—
73C2×C32×C4—42no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4C3⋊C812yescomplemented
77C2×C32×C4C3×C812yescomplemented
78C2×C32×C4#6712yes—
79C2×C32×C4C3×D412yes—
80C3⋊(C3⋊C8)—42no—
81C3×C3⋊C8—122no—
82#324—91noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)C22⋊C811yescharacteristic
86C2×H(3,𝔽3)#9911yescharacteristic
87C2×H(3,𝔽3)#10111yescharacteristic
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—61no—
95C22×C3×C4—61no—
96C22×C3×C4—61no—
97C22×C3×C4C3×Dic311yescharacteristic
98C22×C3×C4C3×Dic311yescharacteristic
99C22×C3×C4C2×C3×D311yescharacteristic
100C2×C3×C8—182no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—62no—
103C3×C4⋊C4—62no—
104C3×C4⋊C4—64no—
105C2×C3⋊C8—62no—
106C2×C3⋊C8—182no—
107C22×C32—22no—
108C22×C32—22no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32C2×C3⋊C811yescharacteristic
117C22×C32#20811yescharacteristic
118C22×C32#21711yescharacteristic
119C22×C32C3×C22⋊C411yescharacteristic
120C22×C32C2×C3×C811yescharacteristic
121C22×C32C3×M1611yescharacteristic
122C32×C4—24no—
123C32×C4—22no—
124C32×C4—22no—
125C32×C4—22no—
126C32×C4—22no—
127C32×C4—42no—
128C32×C4—42no—
129C32×C4—41no—
130C32×C4—41no—
131C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
132C22⋊C8—92no—
133H(3,𝔽3)#32411yescharacteristic, complemented, Sylow
134C23×C3—31no—
135C23×C3—31no—
136C23×C3—61no—
137C23×C3C2×C3×Dic311yescharacteristic
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—41no—
146C2×C3×C4—41no—
147C2×C3×C4—61no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—62no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—121no—
159C2×C3×C4—121no—
160C2×C3×C4—122no—
161C2×C3×C4—122no—
162C2×C3×C4—121no—
163C2×C3×C4—121no—
164C2×C3×C4C3×C3⋊C812yes—
165C2×C3×C4C3×#6712yes—
166C3×C8—362no—
167C3⋊C8—122no—
168C3⋊C8—362no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32#83011yescharacteristic
179C2×C32#83411yescharacteristic
180C2×C32#83611yescharacteristic
181C2×C32C3×C22⋊C811yescharacteristic
182C2×C32C3×#9911yescharacteristic
183C2×C32C3×#10111yescharacteristic
184C22×C4(C3⋊Dic3)⋊C311yescharacteristic
185C22×C4(C3⋊Dic3)⋊C311yescharacteristic
186C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
187C2×C8—182no—
188C4⋊C4—22no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—31no—
192C22×C3—31no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—61no—
198C22×C3—61no—
199C22×C3—61no—
200C22×C3—61no—
201C22×C3—62no—
202C22×C3—62no—
203C22×C3—61no—
204C22×C3—61no—
205C22×C3—61no—
206C22×C3C2×C3×C3⋊C811yescharacteristic
207C22×C3C3×#20811yescharacteristic
208C22×C3C3×#21711yescharacteristic
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—41no—
212C3×C4—62no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—62no—
216C3×C4—64no—
217C3×C4—122no—
218C3×C4—121no—
219C3×C4—121no—
220C3×C4—122no—
221C32—21no—
222C32#452811yescharacteristic, complemented
223C32C3×#32411yescharacteristic, complemented
224C23C2×(C3⋊Dic3)⋊C311yescharacteristic
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4(C3⋊(C3⋊C8))⋊C312yescomplemented
231C2×C4#638⋊C312yes—
232C8—362no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—31no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—62no—
246C2×C3—62no—
247C2×C3—61no—
248C2×C3—61no—
249C2×C3—61no—
250C2×C3C3×#83011yescharacteristic
251C2×C3C3×#83411yescharacteristic
252C2×C3C3×#83611yescharacteristic
253C22—21no—
254C22—21no—
255C22C2×(C3⋊(C3⋊C8))⋊C311yescharacteristic
256C22#3748⋊C311yescharacteristic
257C22#3757⋊C311yescharacteristic
258C4—22no—
259C4—22no—
260C4—41no—
261C3—31no—
262C3—31no—
263C3—61no—
264C3C3×#452811yescharacteristic
265C2—21no—
266C2—21no—
267C2#63657⋊C311yescharacteristic
268C2#63661⋊C311yescharacteristic
269C2#63663⋊C311yescharacteristic
subgroupG/H
centerC22#3757⋊C3
commutatorC22×C32C2×C3×C8
FrattiniC22×C3×C4C2×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC22×C3C3×#217
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC22×C32×C4C22×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#324
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10589394C3semidirect
21. C2×C4×H(3,𝔽3)C8semidirect
76. C2×C32×C4C3⋊C8semidirect
77. C2×C32×C4C3×C8semidirect
133. H(3,𝔽3)#324semidirect
222. C32#4528semidirect
223. C32C3×#324semidirect
230. C2×C4(C3⋊(C3⋊C8))⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#63657⋊C32
#105893943
C3×#45283