Finite Group Databasebeta

← #10592573⋊C3 #13142213

Subgroups — #10592573⋊C3 #13142213

Order 1728.

379 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 108 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
1#64097⋊C3C212yesmaximal
2H(3,𝔽3)×#107C211yesmaximal, characteristic
3#63645⋊C3C211yesmaximal, characteristic
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
5#63634⋊C3C212yesmaximal
6#10592573C311yesmaximal, characteristic, complemented
7C3×#4874—31nomaximal
8C2×#634⋊C3—22no—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)C2212yes—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#64097C2×C312yes—
16C3×#888—32no—
17C32×#107—21no—
18C32×#107D311yescharacteristic
19C32×#107C2×C311yescharacteristic
20#63645C2×C311yescharacteristic
21C8×C3⋊Dic3C2×C311yescharacteristic
22#63634C2×C312yes—
23C3×#818—31no—
24C3×C8×Dic3—31no—
25C3×#807—32no—
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×(C3⋊Dic3)⋊C3—22no—
29C2×(C3⋊Dic3)⋊C3D411yescharacteristic
30C2×(C3⋊Dic3)⋊C3D411yescharacteristic
31#634⋊C3—44no—
32C8×H(3,𝔽3)Q812yescomplemented
33(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
34C3×#500—91no—
35#4874—31no—
36#4874—91no—
37C2×#634—22no—
38C2×C3×Dic6—62no—
39C2×C32×C8—21no—
40C2×C32×C8C2×D311yescharacteristic
41C2×C32×C8C22×C311yescharacteristic
42C32×C4⋊C4—22no—
43C32×C4⋊C4C2×D312yes—
44C32×C4⋊C4C22×C312yes—
45C4⋊(C3⋊Dic3)C22×C312yes—
46#3750—22no—
47C4×C3⋊Dic3C22×C311yescharacteristic
48C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
49C3×C4⋊Dic3—32no—
50C3×Dic3⋊C4—62no—
51C3×C4×Dic3—31no—
52C2×C3×C3⋊C8—31no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C4×H(3,𝔽3)—42no—
55C4×H(3,𝔽3)C2×Q811yescharacteristic
56C4×H(3,𝔽3)C2×Q811yescharacteristic
57(C3⋊Dic3)⋊C3—42no—
58(C3⋊Dic3)⋊C3D812yes—
59(C3⋊Dic3)⋊C3Q1612yes—
60C3×C4⋊Q8—92no—
61#888—32no—
62#888—92no—
63C3×#107—31no—
64C3×#107—31no—
65C3×#107—61no—
66C3×#107—92no—
67C3×#107—91no—
68C3×#107C3×D311yescharacteristic
69C3×C4×C8—91no—
70#818—31no—
71#818—91no—
72C8×Dic3—31no—
73C8×Dic3—91no—
74#807—32no—
75#807—92no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—21no—
79C2×C32×C4—42no—
80C2×C32×C4C22×D311yescharacteristic
81C2×C32×C4C23×C311yescharacteristic
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3C3×D411yescharacteristic
84C2×C3⋊Dic3C3×D411yescharacteristic
85#634—44no—
86C2×C3×Dic3—31no—
87C2×C3×Dic3—31no—
88C2×C3×Dic3—62no—
89C3×Dic6—124no—
90C32×C8—22no—
91C32×C8Dic612yescomplemented
92C32×C8C3×Q812yescomplemented
93C3⋊(C3⋊C8)C3×Q812yescomplemented
94C3×C3⋊C8—32no—
95#500—91noSylow
96C2×H(3,𝔽3)C4⋊Q811yescharacteristic
97C2×H(3,𝔽3)C2×D811yescharacteristic
98C2×H(3,𝔽3)C2×Q1611yescharacteristic
99C2×C3×Q8—182no—
100C2×Dic6—62no—
101C2×Dic6—182no—
102C2×C3×C8—31no—
103C2×C3×C8—31no—
104C2×C3×C8—61no—
105C2×C3×C8—91no—
106C2×C3×C8C2×C3×D311yescharacteristic
107C3×C4⋊C4—32no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—62no—
110C3×C4⋊C4—92no—
111C3×C4⋊C4—182no—
112C3×C4⋊C4C2×C3×D312yes—
113C3×C42—91no—
114C4⋊Dic3—32no—
115C4⋊Dic3—92no—
116Dic3⋊C4—62no—
117Dic3⋊C4—182no—
118C4×Dic3—31no—
119C4×Dic3—91no—
120C2×C3⋊C8—31no—
121C2×C3⋊C8—91no—
122C22×C32—21no—
123C22×C32D3×D411yescharacteristic
124C22×C32C2×C3×D411yescharacteristic
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—42no—
128C32×C4—42no—
129C32×C4—82no—
130C32×C4C2×Dic611yescharacteristic
131C32×C4D3×Q811yescharacteristic
132C32×C4C2×C3×Q811yescharacteristic
133C32×C4C2×C3×Q811yescharacteristic
134C3⋊Dic3—42no—
135C3⋊Dic3C3×D812yes—
136C3⋊Dic3C3×Q1612yes—
137C3×Dic3—32no—
138C3×Dic3—32no—
139C3×Dic3—122no—
140C4⋊Q8—92no—
141#107—91no—
142#107—92no—
143#107(C3⋊D3)⋊C311yescharacteristic
144C4×C8—91no—
145H(3,𝔽3)#50011yescharacteristic, complemented, Sylow
146C3×Q8—364no—
147C2×C3×C4—22no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—61no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—122no—
156C2×C3×C4—182no—
157C2×C3×C4C22×C3×D311yescharacteristic
158C2×Dic3—31no—
159C2×Dic3—31no—
160C2×Dic3—62no—
161C2×Dic3—91no—
162C2×Dic3—91no—
163C2×Dic3—182no—
164Dic6—124no—
165Dic6—364no—
166C3×C8—32no—
167C3×C8—32no—
168C3×C8—62no—
169C3×C8—92no—
170C3×C8C3×Dic612yes—
171C3⋊C8—32no—
172C3⋊C8—92no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32#88811yescharacteristic
177C2×C32D3×D811yescharacteristic
178C2×C32D3×Q1611yescharacteristic
179C2×C32C3×C4⋊Q811yescharacteristic
180C2×C32C2×C3×D811yescharacteristic
181C2×C32C2×C3×Q1611yescharacteristic
182C2×Q8—182no—
183C2×C8—91no—
184C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
185C4⋊C4—92no—
186C4⋊C4—182no—
187C4⋊C4C2×(C3⋊D3)⋊C312yes—
188C42—91no—
189C22×C3—31no—
190C22×C3—31no—
191C22×C3—61no—
192C22×C3C3×D3×D411yescharacteristic
193C3×C4—31no—
194C3×C4—31no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—42no—
198C3×C4—61no—
199C3×C4—61no—
200C3×C4—92no—
201C3×C4—92no—
202C3×C4—122no—
203C3×C4—122no—
204C3×C4—242no—
205C3×C4—362no—
206C3×C4C2×C3×Dic611yescharacteristic
207C3×C4C3×D3×Q811yescharacteristic
208Dic3—32no—
209Dic3—32no—
210Dic3—92no—
211Dic3—92no—
212Dic3—122no—
213Dic3—362no—
214C32—21no—
215C32#487411yescharacteristic, complemented
216C32C3×#50011yescharacteristic, complemented
217Q8—364no—
218C2×C4—22no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—182no—
222C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
223C8—92no—
224C8#634⋊C312yescomplemented
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—31no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—31no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3C3×#88811yescharacteristic
235C2×C3C3×D3×D811yescharacteristic
236C2×C3C3×D3×Q1611yescharacteristic
237C22D4×(C3⋊D3)⋊C311yescharacteristic
238C4—42no—
239C4—92no—
240C4—92no—
241C4—362no—
242C4C2×#634⋊C311yescharacteristic
243C4Q8×(C3⋊D3)⋊C311yescharacteristic
244C3—31no—
245C3—31no—
246C3—61no—
247C3C3×#487411yescharacteristic
248C2#64097⋊C311yescharacteristic
249C2D8×(C3⋊D3)⋊C311yescharacteristic
250C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#500
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
6. #10592573C3semidirect
32. C8×H(3,𝔽3)Q8semidirect
33. (C3⋊(C3⋊C8))⋊C3Q8semidirect
91. C32×C8Dic6semidirect
92. C32×C8C3×Q8semidirect
93. C3⋊(C3⋊C8)C3×Q8semidirect
145. H(3,𝔽3)#500semidirect
215. C32#4874semidirect
216. C32C3×#500semidirect
224. C8#634⋊C3semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
H(3,𝔽3)×#1072
#63645⋊C32
C8×(C3⋊Dic3)⋊C32
#63634⋊C32
#105925733
C3×#48743