Finite Group Databasebeta

← #13144297

Subgroups — #13144297

Order 1728.

1023 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 183 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×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88552C212yesmaximal
3C2×C4⋊#1084C211yesmaximal, characteristic, complemented
4C2×#88540C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#4949—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2212yes—
12C22⋊C4×H(3,𝔽3)—22no—
13#88552—24no—
14C4⋊#1084—22no—
15#88540—24no—
16C2×C32×C22⋊C4D314yes—
17C2×C3×#217—38no—
18C2×C3×C4⋊Dic3—34no—
19C2×C3×Dic3⋊C4—38no—
20C3×C3⋊#95—34no—
21C23×H(3,𝔽3)—22no—
22C23×H(3,𝔽3)—21no—
23C23×H(3,𝔽3)—21no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)Q812yes—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084—22no—
34C2×#1084—22no—
35C2×#1084—22no—
36C2×#1084D414yes—
37C3×#399—91no—
38#4949—34no—
39C24×C32C2×D314yes—
40C22×C32×C4C2×D314yes—
41C22×C32×C4C2×D314yes—
42C22×C3×Dic3—38no—
43C22×C3×Dic3—38no—
44C32×C22⋊C4—28no—
45C3×#217—616no—
46C3×C4⋊Dic3—68no—
47C3×Dic3⋊C4—616no—
48C22×H(3,𝔽3)—24no—
49C22×H(3,𝔽3)—22no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)—41no—
53C22×H(3,𝔽3)C2×D412yes—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C22×H(3,𝔽3)C2×Q811yescharacteristic
56C22×H(3,𝔽3)#4111yescharacteristic
57C22×H(3,𝔽3)#4112yes—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)—41no—
60#1084—24no—
61#1084—42no—
62C2×C3×C4⋊C4—92no—
63C2×C3×C4⋊C4—91no—
64C2×C3×C22⋊C4—34no—
65C2×C3×C22⋊C4—92no—
66C2×C3×C22⋊C4C3⋊D311yescharacteristic
67C2×#217—38no—
68C2×C4⋊Dic3—34no—
69C2×Dic3⋊C4—38no—
70C3×#95—91no—
71C3⋊#95—34no—
72C23×C32—24no—
73C23×C32—28no—
74C23×C32—28no—
75C23×C32—24no—
76C23×C32Dic618yes—
77C23×C32C22×D314yes—
78C2×C32×C4—28no—
79C2×C32×C4—28no—
80C2×C32×C4—24no—
81C2×C32×C4#6718yes—
82C2×C3×Dic3—316no—
83C2×C3×Dic3—68no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—68no—
86C2×C3×Dic3—68no—
87C2×C3×Dic3—68no—
88#399—91noSylow
89C2×H(3,𝔽3)—42no—
90C2×H(3,𝔽3)C22⋊Q812yes—
91C2×H(3,𝔽3)C22⋊Q811yescharacteristic
92C2×H(3,𝔽3)#12312yes—
93C2×H(3,𝔽3)#12311yescharacteristic
94C2×H(3,𝔽3)#12711yescharacteristic
95C24×C3—34no—
96C24×C3C2×C3⋊D311yescharacteristic
97C22×C3×C4—34no—
98C22×C3×C4—34no—
99C22×C3×C4—92no—
100C22×C3×C4—92no—
101C22×C3×C4C2×C3⋊D311yescharacteristic
102C22×C3×C4C2×C3⋊D311yescharacteristic
103C22×Dic3—38no—
104C22×Dic3—38no—
105C3×C4⋊C4—184no—
106C3×C4⋊C4—182no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—68no—
109C3×C22⋊C4—184no—
110#217—616no—
111C4⋊Dic3—68no—
112Dic3⋊C4—616no—
113C22×C32—216no—
114C22×C32—28no—
115C22×C32—48no—
116C22×C32—44no—
117C22×C32—44no—
118C22×C32C2×Dic614yes—
119C22×C32#23514yes—
120C22×C32D3×D418yes—
121C22×C32#23718yes—
122C22×C32C2×#6714yes—
123C32×C4—28no—
124C32×C4—44no—
125C3×Dic3—616no—
126C3×Dic3—128no—
127C2×C4⋊C4—92no—
128C2×C4⋊C4—91no—
129C2×C22⋊C4—92no—
130C2×C22⋊C4#27211yescharacteristic
131#95—91no—
132H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
133C23×C3—21no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—21no—
137C23×C3—38no—
138C23×C3—34no—
139C23×C3—68no—
140C23×C3—64no—
141C23×C3—64no—
142C23×C3—68no—
143C23×C3#63412yes—
144C23×C3C22×C3⋊D311yescharacteristic
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—38no—
149C2×C3×C4—68no—
150C2×C3×C4—68no—
151C2×C3×C4—64no—
152C2×C3×C4—94no—
153C2×C3×C4—182no—
154C2×C3×C4—182no—
155C2×C3×C4—182no—
156C2×C3×C4—182no—
157C2×C3×C4—182no—
158C2×C3×C4#63812yes—
159C2×Dic3—316no—
160C2×Dic3—68no—
161C2×Dic3—68no—
162C2×Dic3—68no—
163C2×Dic3—68no—
164C2×Dic3—68no—
165C2×C32—48no—
166C2×C32#87818yes—
167C2×C32#88318yes—
168C2×C32#92414yes—
169C2×C32#93514yes—
170C2×C32#94014yes—
171C24C2×#27211yescharacteristic
172C22×C4—92no—
173C22×C4—92no—
174C22×C4C2×#27211yescharacteristic
175C22×C4C2×#27211yescharacteristic
176C4⋊C4—184no—
177C4⋊C4—182no—
178C22⋊C4—22no—
179C22⋊C4—184no—
180C22×C3—24no—
181C22×C3—22no—
182C22×C3—34no—
183C22×C3—34no—
184C22×C3—38no—
185C22×C3—38no—
186C22×C3—34no—
187C22×C3—42no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3—616no—
191C22×C3—68no—
192C22×C3—128no—
193C22×C3—124no—
194C22×C3—124no—
195C22×C3C2×#63411yescharacteristic
196C22×C3#382811yescharacteristic
197C22×C3D4×C3⋊D312yes—
198C22×C3#383012yes—
199C22×C3C2×#63811yescharacteristic
200C3×C4—22no—
201C3×C4—41no—
202C3×C4—68no—
203C3×C4—124no—
204C3×C4—184no—
205C3×C4—362no—
206Dic3—616no—
207Dic3—128no—
208C32#494914yescomplemented
209C23—21no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23#622912yes—
214C23C22×#27211yescharacteristic
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—94no—
219C2×C4—182no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4—182no—
223C2×C4—182no—
224C2×C4#623512yes—
225C2×C3—38no—
226C2×C3—38no—
227C2×C3—34no—
228C2×C3—34no—
229C2×C3—34no—
230C2×C3—42no—
231C2×C3—128no—
232C2×C3#6408712yes—
233C2×C3#6409212yes—
234C2×C3#6413311yescharacteristic
235C2×C3#6414411yescharacteristic
236C2×C3#6414911yescharacteristic
237C22—24no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22C2×#622911yescharacteristic
243C22#8874011yescharacteristic
244C22D4×#27212yes—
245C22#8874412yes—
246C22C2×#623511yescharacteristic
247C4—22no—
248C4—41no—
249C4—184no—
250C4—362no—
251C3—34no—
252C3#1059264811yescharacteristic
253C2—42no—
254C2#1172173412yes—
255C2#1172173912yes—
256C2#1172182411yescharacteristic
257C2#1172184511yescharacteristic
258C2#1172185011yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#399
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊#1084C2semidirect
4. C2×#88540C2semidirect
5. #11720574C2semidirect
132. H(3,𝔽3)#399semidirect
208. C32#4949semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×C4⋊#10842
C2×#885402
#117205742
C3×#49493