Finite Group Databasebeta

← #10592592⋊C3 #13142267

Subgroups — #10592592⋊C3 #13142267

Order 1728.

387 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 114 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
1M16×(C3⋊D3)⋊C3C212yesmaximal
2C2×#3743⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#108C211yesmaximal, characteristic, complemented
4#63649⋊C3C212yesmaximal, complemented
5#63646⋊C3C211yesmaximal, characteristic, complemented
6#10592592C311yesmaximal, characteristic, complemented
7C3×#4893—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9M16×H(3,𝔽3)C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#3748⋊C3C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
13#3743⋊C3—22no—
14#3743⋊C3C414yes—
15C8×(C3⋊D3)⋊C3—22no—
16M16×C3⋊D3C2×C312yes—
17C2×#3743C2×C311yescharacteristic
18C3×D3×M16—32no—
19C2×C3×ΓL(1,𝔽25)—31no—
20C32×#108—21no—
21C32×#108D311yescharacteristic, complemented
22C32×#108C2×C311yescharacteristic, complemented
23#63649C2×C312yescomplemented
24#63646C2×C311yescharacteristic, complemented
25C3×#822—32no—
26C3×#819—31no—
27C22×(C3⋊D3)⋊C3Q811yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3Q811yescharacteristic
30C4×(C3⋊D3)⋊C3C2×C412yes—
31C4×(C3⋊D3)⋊C3D412yes—
32C8×H(3,𝔽3)—22no—
33C8×H(3,𝔽3)C2×C412yes—
34(C3⋊(C3⋊C8))⋊C3—22no—
35(C3⋊(C3⋊C8))⋊C3C2×C412yes—
36C3×#430—91no—
37#4893—31no—
38#4893—91no—
39C2×C4×C3⋊D3C22×C311yescharacteristic
40C2×C3×C4×D3—31no—
41C32×M16—22no—
42C32×M16C2×D312yes—
43C32×M16C22×C312yes—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C22×C311yescharacteristic
47#3748C22×C312yes—
48C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
49#3743—22no—
50#3743C3×C414yes—
51C8×C3⋊D3—22no—
52C3×#208—32no—
53C2×C3×C3⋊C8—31no—
54C3×ΓL(1,𝔽25)—34no—
55C3×ΓL(1,𝔽25)—62no—
56C3×D3×C8—62no—
57C22×H(3,𝔽3)C2×Q811yescharacteristic
58C2×(C3⋊D3)⋊C3—41no—
59C2×(C3⋊D3)⋊C3C4⋊C412yes—
60C4×H(3,𝔽3)C22×C411yescharacteristic
61C4×H(3,𝔽3)C2×D411yescharacteristic
62(C3⋊Dic3)⋊C3C4⋊C412yes—
63C2×C3×M16—91no—
64C2×C3×M16—92no—
65D3×M16—32no—
66D3×M16—92no—
67C2×ΓL(1,𝔽25)—31no—
68C2×ΓL(1,𝔽25)—91no—
69C3×#108—31no—
70C3×#108—31no—
71C3×#108—61no—
72C3×#108—91no—
73C3×#108—92no—
74C3×#108C3×D311yescharacteristic
75#822—32no—
76#822—92no—
77#819—31no—
78#819—91no—
79C22×C3⋊D3C3×Q811yescharacteristic
80C22×C3×D3—31no—
81C2×C32×C4—21no—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C23×C311yescharacteristic
84C2×C3⋊Dic3C3×Q811yescharacteristic
85C4×C3⋊D3C2×C3×C412yes—
86C4×C3⋊D3C3×D412yes—
87C2×C3×Dic3—31no—
88C3×C4×D3—32no—
89C3×C4×D3—32no—
90C32×C8—22no—
91C32×C8—22no—
92C32×C8—22no—
93C32×C8—42no—
94C32×C8C4×D312yes—
95C32×C8C2×C3×C412yes—
96C3⋊(C3⋊C8)—22no—
97C3⋊(C3⋊C8)C2×C3×C412yes—
98C3×C3⋊C8—32no—
99C3×C3⋊C8—62no—
100#430—91noSylow
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
103(C3⋊D3)⋊C3—22no—
104C22×C3×C4—91no—
105C2×C4×D3—31no—
106C2×C4×D3—91no—
107C3×M16—32no—
108C3×M16—32no—
109C3×M16—62no—
110C3×M16—94no—
111C3×M16—92no—
112C3×M16—182no—
113C3×M16C2×C3×D312yes—
114C2×C3×C8—31no—
115C2×C3×C8—31no—
116C2×C3×C8—61no—
117C2×C3×C8—91no—
118C2×C3×C8—182no—
119C2×C3×C8C2×C3×D311yescharacteristic
120#208—32no—
121#208—92no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—91no—
124ΓL(1,𝔽25)—34no—
125ΓL(1,𝔽25)—62no—
126ΓL(1,𝔽25)—94no—
127ΓL(1,𝔽25)—182no—
128D3×C8—62no—
129D3×C8—182no—
130C22×C32—21no—
131C22×C32D3×Q811yescharacteristic
132C22×C32C2×C3×Q811yescharacteristic
133C2×C3⋊D3—41no—
134C2×C3⋊D3C3×C4⋊C412yes—
135C2×C3×D3—32no—
136C2×C3×D3—121no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4C2×C4×D311yescharacteristic
140C32×C4D3×D411yescharacteristic
141C32×C4C22×C3×C411yescharacteristic
142C32×C4C2×C3×D411yescharacteristic
143C3⋊Dic3C3×C4⋊C412yes—
144C3×Dic3—32no—
145C2×M16—91no—
146C2×M16—92no—
147#108—92no—
148#108—91no—
149#108(C3⋊D3)⋊C311yescharacteristic, complemented
150H(3,𝔽3)#43011yescharacteristic, complemented, Sylow
151C23×C3—91no—
152C22×D3—31no—
153C22×D3—91no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—61no—
157C2×C3×C4—92no—
158C2×C3×C4—92no—
159C2×C3×C4—91no—
160C2×C3×C4C22×C3×D311yescharacteristic
161C2×Dic3—31no—
162C2×Dic3—91no—
163C4×D3—32no—
164C4×D3—32no—
165C4×D3—92no—
166C4×D3—92no—
167C3×C8—22no—
168C3×C8—32no—
169C3×C8—32no—
170C3×C8—62no—
171C3×C8—62no—
172C3×C8—62no—
173C3×C8—92no—
174C3×C8—122no—
175C3×C8—182no—
176C3×C8C3×C4×D312yes—
177C3⋊C8—32no—
178C3⋊C8—62no—
179C3⋊C8—92no—
180C3⋊C8—182no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—41no—
185C2×C32D3×C4⋊C411yescharacteristic
186C2×C32C2×C3×C4⋊C411yescharacteristic
187C3⋊D3—22no—
188C3×D3—62no—
189C22×C4—91no—
190M16—94no—
191M16—92no—
192M16—182no—
193M16C2×(C3⋊D3)⋊C312yes—
194C2×C8—91no—
195C2×C8—182no—
196C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
197C22×C3—31no—
198C22×C3—31no—
199C22×C3—61no—
200C22×C3—92no—
201C22×C3—361no—
202C22×C3C3×D3×Q811yescharacteristic
203C2×D3—32no—
204C2×D3—92no—
205C2×D3—121no—
206C2×D3—361no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—92no—
214C3×C4C2×C3×C4×D311yescharacteristic
215C3×C4C3×D3×D411yescharacteristic
216Dic3—32no—
217Dic3—92no—
218C32—21no—
219C32#489311yescharacteristic, complemented
220C32C3×#43011yescharacteristic, complemented
221C23—91no—
222C2×C4—91no—
223C2×C4—92no—
224C2×C4—92no—
225C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
226C8—22no—
227C8—92no—
228C8—182no—
229C8C4×(C3⋊D3)⋊C312yes—
230C2×C3—21no—
231C2×C3—31no—
232C2×C3—31no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—121no—
237C2×C3—182no—
238C2×C3C3×D3×C4⋊C411yescharacteristic
239D3—62no—
240D3—182no—
241C22—92no—
242C22—361no—
243C22Q8×(C3⋊D3)⋊C311yescharacteristic
244C4—92no—
245C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
246C4D4×(C3⋊D3)⋊C311yescharacteristic
247C3—31no—
248C3—31no—
249C3—61no—
250C3C3×#489311yescharacteristic
251C2—21no—
252C2—182no—
253C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC4D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#108C2
socleC2×C3C3×D3×C4⋊C4
prime radicalC22×(C3⋊D3)⋊C3Q8
cosocleC2×C32×C4C23×C3
hypercenter#108(C3⋊D3)⋊C3
hypocenterC32C3×#430
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#108C2semidirect
4. #63649⋊C3C2semidirect
5. #63646⋊C3C2semidirect
6. #10592592C3semidirect
21. C32×#108D3semidirect
22. C32×#108C2×C3semidirect
23. #63649C2×C3semidirect
24. #63646C2×C3semidirect
149. #108(C3⋊D3)⋊C3semidirect
150. H(3,𝔽3)#430semidirect
219. C32#4893semidirect
220. C32C3×#430semidirect

Maximal subgroups

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

subgroupindex
M16×(C3⋊D3)⋊C32
C2×#3743⋊C32
H(3,𝔽3)×#1082
#63649⋊C32
#63646⋊C32
#105925923
C3×#48933