Finite Group Databasebeta

← #13139966

Subgroups — #13139966

Order 1728.

467 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 131 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×#88538C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720311C214yesmaximal, complemented
5#10591119—31nomaximal
6#10591117—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8#88538—22no—
9C2×#6180C2212yes—
10#3757⋊C3C412yes—
11C4⋊((C3⋊Dic3)⋊C3)C412yes—
12C4×(C3⋊Dic3)⋊C3C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64135D311yescharacteristic
16C22×C3⋊(C3⋊C8)D311yescharacteristic
17C3×#926—31no—
18C2×C3×#208—31no—
19C2×C3×#208—31no—
20C2×C3×#208—61no—
21C22×C3×C3⋊C8—31no—
22#63572—34no—
23Dic3⋊(C3⋊C8)—34no—
24C23×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)D412yes—
27C2×C4×H(3,𝔽3)Q812yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3C2×C414yes—
30#6180—22no—
31(C3⋊(C3⋊C8))⋊C3—22no—
32#5119—91no—
33#5102—91no—
34#4999—91no—
35C22×C32×C4—21no—
36C22×C32×C4C2×D311yescharacteristic
37C22×C32×C4C2×D311yescharacteristic
38#3757Dic312yes—
39C4⋊(C3⋊Dic3)Dic312yes—
40C4×C3⋊Dic3C2×D312yes—
41C2×C3⋊(C3⋊C8)—22no—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×#217—32no—
44C3×C4⋊Dic3—32no—
45C3×C4×Dic3—32no—
46C3×#208—64no—
47C3×#208—62no—
48C3×#208—62no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—32no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—62no—
53C2×C3×C3⋊C8—62no—
54C22×H(3,𝔽3)—22no—
55C22×H(3,𝔽3)C4⋊C412yes—
56C22×H(3,𝔽3)C22×C411yescharacteristic
57C4×H(3,𝔽3)C4⋊C412yes—
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)C2×Q811yescharacteristic
60(C3⋊Dic3)⋊C3—24no—
61C2×C3×M16—91no—
62C22×C3×C8—91no—
63C3×#117—91no—
64#926—31no—
65#926—91no—
66C2×#208—31no—
67C2×#208—31no—
68C2×#208—61no—
69C22×C3⋊C8—31no—
70C22×C3⋊C8—91no—
71Dic3⋊C8—94no—
72Dic3⋊C8—94no—
73C4⋊(C3⋊C8)—94no—
74C23×C32—21no—
75C23×C32C4×D311yescharacteristic
76C23×C32C2×Dic311yescharacteristic
77C2×C32×C4—21no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4Dic612yes—
82C2×C32×C4Dic612yes—
83C2×C32×C4C4×D311yescharacteristic
84C2×C32×C4D1212yes—
85C2×C32×C4C2×Dic311yescharacteristic
86C2×C32×C4#6712yes—
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C22×D311yescharacteristic
89C2×C3⋊Dic3C2×Dic314yes—
90C2×C3×Dic3—34no—
91C3⋊(C3⋊C8)—22no—
92C3×C3⋊C8—64no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95C3×C3⋊C8—62no—
96#424—271noSylow
97C2×H(3,𝔽3)—22no—
98C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
99C2×H(3,𝔽3)#13112yes—
100C22×C3×C4—31no—
101C22×C3×C4—31no—
102C22×C3×C4—61no—
103C22×C3×C4D3211yescharacteristic
104C3×M16—182no—
105C2×C3×C8—92no—
106C2×C3×C8—92no—
107C2×C3×C8—182no—
108C3×C4⋊C4—92no—
109C3×C22⋊C4—92no—
110C3×C42—92no—
111#217—32no—
112#217—92no—
113C4⋊Dic3—32no—
114C4⋊Dic3—92no—
115C4×Dic3—32no—
116C4×Dic3—92no—
117#208—62no—
118#208—62no—
119#208—64no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—32no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—62no—
124C2×C3⋊C8—62no—
125C2×C3⋊C8—92no—
126C2×C3⋊C8—182no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—21no—
131C22×C32—42no—
132C22×C32Dic3⋊C412yes—
133C22×C32C4⋊Dic312yes—
134C22×C32C2×C4×D311yescharacteristic
135C22×C32C22×Dic311yescharacteristic
136C32×C4—22no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4Dic3⋊C412yes—
140C32×C4C4⋊Dic312yes—
141C32×C4C2×Dic611yescharacteristic
142C32×C4C2×Dic611yescharacteristic
143C32×C4C2×D1211yescharacteristic
144C32×C4C2×#6711yescharacteristic
145C3⋊Dic3—24no—
146C3×Dic3—64no—
147C2×M16—91no—
148C22×C8—91no—
149#117—91no—
150C4⋊C8—274no—
151H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—61no—
155C23×C3D3×Dic311yescharacteristic
156C2×C3×C4—32no—
157C2×C3×C4—32no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—62no—
165C2×C3×C4—61no—
166C2×C3×C4—62no—
167C2×C3×C4—61no—
168C2×C3×C4—94no—
169C2×C3×C4D3×Dic311yescharacteristic
170C2×C3×C4#62612yes—
171C2×C3×C4#62712yes—
172C2×C3×C4C2×D3211yescharacteristic
173C2×Dic3—34no—
174C2×Dic3—94no—
175C3×C8—182no—
176C3×C8—182no—
177C3⋊C8—62no—
178C3⋊C8—62no—
179C3⋊C8—64no—
180C3⋊C8—62no—
181C3⋊C8—182no—
182C2×C32—22no—
183C2×C32—22no—
184C2×C32—22no—
185C2×C32—21no—
186C2×C32—42no—
187C2×C32#90112yes—
188C2×C32C2×Dic3⋊C411yescharacteristic
189C2×C32C2×C4⋊Dic311yescharacteristic
190C2×C32#94812yes—
191C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
192M16—182no—
193C2×C8—92no—
194C2×C8—92no—
195C2×C8—182no—
196C4⋊C4—92no—
197C22⋊C4—92no—
198C42—92no—
199C22×C3—22no—
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—32no—
203C22×C3—32no—
204C22×C3—62no—
205C22×C3—62no—
206C22×C3—61no—
207C22×C3—62no—
208C22×C3—122no—
209C22×C3Dic3⋊Dic312yes—
210C22×C3C2×D3×Dic311yescharacteristic
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—32no—
216C3×C4—32no—
217C3×C4—61no—
218C3×C4—61no—
219C3×C4—62no—
220C3×C4—184no—
221C3×C4Dic3⋊Dic312yes—
222C3×C4C2×#62611yescharacteristic
223C3×C4C2×#62711yescharacteristic
224Dic3—64no—
225Dic3—184no—
226C32—21no—
227C32#499911yescharacteristic, complemented
228C32#510211yescharacteristic, complemented
229C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
230C2×C4—94no—
231C2×C4#619612yes—
232C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
233C2×C4#620012yes—
234C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C8—182no—
236C8—182no—
237C2×C3—22no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—62no—
243C2×C3—62no—
244C2×C3—61no—
245C2×C3—62no—
246C2×C3—122no—
247C2×C3#6381412yes—
248C2×C3C2×Dic3⋊Dic311yescharacteristic
249C22—22no—
250C22#8844812yes—
251C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
252C4—184no—
253C4#8844812yes—
254C4C2×#619611yescharacteristic
255C4C2×#620011yescharacteristic
256C3—31no—
257C3—31no—
258C3—61no—
259C3#1059111611yescharacteristic
260C2—22no—
261C2#1172105412yes—
262C2C2×#8844811yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720311C2semidirect
151. H(3,𝔽3)#424semidirect
227. C32#4999semidirect
228. C32#5102semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64135⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#117203112
#105911193
#105911173