Finite Group Databasebeta

← #13150670

Subgroups — #13150670

Order 1728.

813 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 179 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⋊#1084C211yesmaximal, characteristic
2#64135⋊C3C212yesmaximal
3#11721309C214yesmaximal
4#11721116C214yesmaximal, complemented
5#11721093C214yesmaximal, complemented
6#10595051—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2212yes—
9C2×#6196C2214yes—
10C4⋊#1084C2214yes—
11#3757⋊C3C2214yes—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C4×(C3⋊Dic3)⋊C3C2214yes—
15#88449C2214yes—
16#88448C2218yes—
17#64135D312yes—
18C3×#926—32no—
19C2×C3×C4⋊Dic3—32no—
20C2×C3×C4⋊Dic3—61no—
21#63964—38no—
22#63919—34no—
23#63918—34no—
24#63872—34no—
25#63869—34no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)Q814yescomplemented
28C2×C4×H(3,𝔽3)C2312yes—
29C2×#1084—42no—
30C2×#1084C2314yes—
31C2×(C3⋊Dic3)⋊C3C2314yes—
32C2×(C3⋊Dic3)⋊C3C2314yes—
33#6196—44no—
34#5519—92no—
35#5500—91no—
36C22×C32×C4—21no—
37C22×C32×C4C2×D312yes—
38C22×C3×Dic3—34no—
39C22×C3×Dic3—62no—
40C2×#627—38no—
41#3757C2×D314yes—
42C4⋊(C3⋊Dic3)C2×D312yes—
43C4⋊(C3⋊Dic3)C2×D312yes—
44C4×C3⋊Dic3C2×D314yes—
45C3×#217—34no—
46C3×C4⋊Dic3—34no—
47C3×C4⋊Dic3—34no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—64no—
51C3×C4×Dic3—34no—
52#3724—38no—
53Dic3⋊Dic3—38no—
54Dic3⋊Dic3—38no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)—21no—
57C22×H(3,𝔽3)C2×Q812yes—
58C22×H(3,𝔽3)C2411yescharacteristic
59C4×H(3,𝔽3)C2×Q814yes—
60#1084—24no—
61(C3⋊Dic3)⋊C3—24no—
62(C3⋊Dic3)⋊C3—24no—
63C3×#117—92no—
64C2×C3×C4⋊C4—91no—
65#926—32no—
66#926—92no—
67C2×C4⋊Dic3—32no—
68C2×C4⋊Dic3—61no—
69#924—94no—
70#890—94no—
71#890—94no—
72#888—94no—
73#888—94no—
74#878—98no—
75#870—94no—
76#869—94no—
77C23×C32—21no—
78C23×C32C22×D312yes—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4Dic618yescomplemented
82C2×C32×C4C22×D312yes—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3C22×D314yes—
85C2×C3⋊Dic3C22×D314yes—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—38no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—124no—
91C2×C3×Dic3—124no—
92#627—128no—
93#544—271noSylow
94C2×H(3,𝔽3)—22no—
95C2×H(3,𝔽3)C22×Q811yescharacteristic
96C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
97C2×H(3,𝔽3)2−1+411yescharacteristic
98C22×C3×C4—32no—
99C22×C3×C4—61no—
100C22×C3×C4—92no—
101C22×C3×C4D3211yescharacteristic
102C22×Dic3—34no—
103C22×Dic3—62no—
104C2×Dic6—94no—
105C2×Dic6—98no—
106C3×C4⋊C4—92no—
107C3×C4⋊C4—92no—
108C3×C4⋊C4—94no—
109C3×C22⋊C4—94no—
110C3×C42—94no—
111#217—34no—
112#217—94no—
113C4⋊Dic3—34no—
114C4⋊Dic3—34no—
115C4⋊Dic3—32no—
116C4⋊Dic3—32no—
117C4⋊Dic3—64no—
118C4⋊Dic3—92no—
119C4⋊Dic3—92no—
120C4⋊Dic3—94no—
121C4⋊Dic3—98no—
122Dic3⋊C4—98no—
123Dic3⋊C4—98no—
124Dic3⋊C4—98no—
125C4×Dic3—34no—
126C4×Dic3—94no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—21no—
132C22×C32—22no—
133C22×C32C2×Dic614yes—
134C22×C32C23×D312yes—
135C32×C4—24no—
136C32×C4C2×Dic614yes—
137C32×C4C2×Dic614yes—
138C3⋊Dic3—24no—
139C3⋊Dic3—24no—
140C3×Dic3—64no—
141C3×Dic3—68no—
142C3×Dic3—64no—
143C3×Dic3—124no—
144C4⋊Q8—274no—
145#125—274no—
146C22⋊Q8—274no—
147#117—92no—
148C2×C4⋊C4—91no—
149H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
150C23×C3—32no—
151C23×C3—61no—
152C23×C3C2×D3211yescharacteristic
153C2×C3×C4—38no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—64no—
157C2×C3×C4—62no—
158C2×C3×C4—94no—
159C2×C3×C4—94no—
160C2×C3×C4—94no—
161C2×C3×C4—362no—
162C2×C3×C4#62714yes—
163C2×C3×C4C2×D3212yes—
164C2×Dic3—38no—
165C2×Dic3—34no—
166C2×Dic3—34no—
167C2×Dic3—64no—
168C2×Dic3—94no—
169C2×Dic3—94no—
170C2×Dic3—124no—
171C2×Dic3—124no—
172Dic6—364no—
173Dic6—368no—
174C2×C32—24no—
175C2×C32—24no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32—21no—
179C2×C32C22×Dic612yes—
180C2×C32#100912yes—
181C2×C32#101012yes—
182C2×Q8—274no—
183C22×C4—92no—
184C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C4⋊C4—94no—
186C4⋊C4—92no—
187C4⋊C4—92no—
188C4⋊C4—274no—
189C4⋊C4—278no—
190C22⋊C4—94no—
191C42—94no—
192C22×C3—21no—
193C22×C3—21no—
194C22×C3—32no—
195C22×C3—34no—
196C22×C3—62no—
197C22×C3—62no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—61no—
201C22×C3—62no—
202C22×C3C2×#62712yes—
203C22×C3C22×D3211yescharacteristic
204C3×C4—34no—
205C3×C4—34no—
206C3×C4—64no—
207C3×C4—184no—
208C3×C4—184no—
209C3×C4—184no—
210C3×C4C2×#62714yes—
211Dic3—68no—
212Dic3—64no—
213Dic3—64no—
214Dic3—124no—
215Dic3—184no—
216Dic3—184no—
217C32—21no—
218C32#551912yescomplemented
219C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
220Q8—1084no—
221C2×C4—94no—
222C2×C4—94no—
223C2×C4—94no—
224C2×C4—362no—
225C2×C4#619614yescomplemented
226C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
227C2×C3—22no—
228C2×C3—32no—
229C2×C3—32no—
230C2×C3—32no—
231C2×C3—64no—
232C2×C3—64no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3#6429811yescharacteristic
237C2×C3#6430811yescharacteristic
238C2×C3C22×#62711yescharacteristic
239C22—21no—
240C22—21no—
241C22C2×#619612yes—
242C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C4—184no—
244C4—184no—
245C4—184no—
246C4C2×#619614yes—
247C3—32no—
248C3—61no—
249C3#1059505211yescharacteristic
250C2—22no—
251C2#1172248311yescharacteristic
252C2#1172249311yescharacteristic
253C2C22×#619611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11721116C2semidirect
5. #11721093C2semidirect
27. C2×C4×H(3,𝔽3)Q8semidirect
81. C2×C32×C4Dic6semidirect
149. H(3,𝔽3)#544semidirect
218. C32#5519semidirect
225. C2×C4#6196semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#64135⋊C32
#117213092
#117211162
#117210932
#105950513