Finite Group Databasebeta

← #13140132

Subgroups — #13140132

Order 1728.

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

normal subgroups only

This group has 647 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×C4×#1084C211yesmaximal, characteristic
2C2×#88449C212yesmaximal
3#63658⋊C3C212yesmaximal
4#11720334C212yesmaximal
5#10591206—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C2214yes—
10C4×#1084C414yescomplemented
11#88449—24no—
12C2×C3×C4×Dic3—32no—
13C2×C3×C4×Dic3—61no—
14C2×Dic3⋊Dic3—34no—
15#63658D312yes—
16C3×C3⋊#95—32no—
17#63579—34no—
18C23×H(3,𝔽3)C2311yescharacteristic
19C2×C4×H(3,𝔽3)—22no—
20C2×C4×H(3,𝔽3)C2×C412yes—
21C2×#1084—22no—
22C2×#1084C2×C412yes—
23C2×#1084C2×C412yes—
24C2×#1084D412yes—
25C2×#1084Q812yes—
26C2×(C3⋊Dic3)⋊C3—24no—
27C2×(C3⋊Dic3)⋊C3—24no—
28C2×(C3⋊Dic3)⋊C3—24no—
29#4936—91no—
30#4650—92no—
31C22×C32×C4—21no—
32C22×C32×C4C2×D312yes—
33C22×C3⋊Dic3C2×D314yes—
34C22×C3×Dic3—32no—
35C22×C3×Dic3—32no—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39C3×C4×Dic3—38no—
40C3×C4×Dic3—64no—
41Dic3⋊Dic3—68no—
42C22×H(3,𝔽3)C22×C411yescharacteristic
43C22×H(3,𝔽3)C2×D411yescharacteristic
44C22×H(3,𝔽3)C2×Q811yescharacteristic
45C22×H(3,𝔽3)#4112yes—
46C22×H(3,𝔽3)#4112yes—
47C4×H(3,𝔽3)—22no—
48#1084—22no—
49#1084C4⋊C414yes—
50(C3⋊Dic3)⋊C3—44no—
51C2×C3×C42—91no—
52C2×C4⋊Dic3—92no—
53C2×Dic3⋊C4—94no—
54C2×C4×Dic3—32no—
55C2×C4×Dic3—61no—
56C3×#95—92no—
57C3⋊#95—32no—
58C3⋊#95—92no—
59C3⋊#95—92no—
60C3⋊#95—94no—
61C23×C32—21no—
62C23×C32C22×D312yes—
63C2×C32×C4—24no—
64C2×C32×C4—24no—
65C2×C32×C4—22no—
66C2×C32×C4C4×D314yes—
67C2×C3⋊Dic3—24no—
68C2×C3⋊Dic3—24no—
69C2×C3⋊Dic3—24no—
70C2×C3×Dic3—34no—
71C2×C3×Dic3—34no—
72C2×C3×Dic3—34no—
73C2×C3×Dic3—34no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—64no—
79C2×C3×Dic3—64no—
80C2×C3×Dic3—64no—
81C2×C3×Dic3—64no—
82C2×C3×Dic3—64no—
83#382—271noSylow
84C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
85C2×H(3,𝔽3)#11711yescharacteristic
86C2×H(3,𝔽3)#11711yescharacteristic
87C2×H(3,𝔽3)#12412yes—
88C2×H(3,𝔽3)#12512yes—
89C22×C3×C4—32no—
90C22×C3×C4—61no—
91C22×C3×C4—91no—
92C22×C3×C4—91no—
93C22×C3×C4—94no—
94C22×C3×C4D3211yescharacteristic
95C22×Dic3—32no—
96C22×Dic3—32no—
97C22×Dic3—34no—
98C22×Dic3—61no—
99C22×Dic3—61no—
100C22×Dic3—94no—
101C3×C42—94no—
102C4⋊Dic3—184no—
103Dic3⋊C4—188no—
104C4×Dic3—38no—
105C4×Dic3—64no—
106C22×C32—21no—
107C22×C32—22no—
108C22×C32—21no—
109C22×C32—22no—
110C22×C32—21no—
111C22×C32C2×C4×D312yes—
112C22×C32#23514yes—
113C22×C32D3×D412yes—
114C22×C32#23714yes—
115C22×C32D3×Q812yes—
116C32×C4—24no—
117C32×C4—24no—
118C3⋊Dic3—44no—
119C3×Dic3—38no—
120C3×Dic3—64no—
121C3×Dic3—64no—
122C3×Dic3—64no—
123C3×Dic3—124no—
124C2×C4⋊C4—272no—
125C2×C42—91no—
126#95—92no—
127#95—272no—
128H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
129C23×C3—32no—
130C23×C3—61no—
131C23×C3C2×D3211yescharacteristic
132C2×C3×C4—22no—
133C2×C3×C4—34no—
134C2×C3×C4—64no—
135C2×C3×C4—64no—
136C2×C3×C4—62no—
137C2×C3×C4—92no—
138C2×C3×C4—92no—
139C2×C3×C4—92no—
140C2×C3×C4—92no—
141C2×C3×C4—182no—
142C2×C3×C4—184no—
143C2×C3×C4—184no—
144C2×C3×C4—184no—
145C2×C3×C4#62412yes—
146C2×Dic3—34no—
147C2×Dic3—34no—
148C2×Dic3—34no—
149C2×Dic3—34no—
150C2×Dic3—62no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—64no—
155C2×Dic3—64no—
156C2×Dic3—64no—
157C2×Dic3—64no—
158C2×Dic3—64no—
159C2×Dic3—184no—
160C2×Dic3—184no—
161C2×Dic3—184no—
162C2×C32—22no—
163C2×C32—21no—
164C2×C32—21no—
165C2×C32—22no—
166C2×C32—21no—
167C2×C32#87212yes—
168C2×C32#87712yes—
169C2×C32#88514yes—
170C2×C32#88914yes—
171C2×C32D3×C4⋊C412yes—
172C22×C4—91no—
173C22×C4—91no—
174C22×C4—94no—
175C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
176C4⋊C4—544no—
177C42—94no—
178C22×C3—34no—
179C22×C3—32no—
180C22×C3—32no—
181C22×C3—34no—
182C22×C3—32no—
183C22×C3—62no—
184C22×C3—61no—
185C22×C3—61no—
186C22×C3—62no—
187C22×C3—61no—
188C22×C3#379711yescharacteristic
189C22×C3#379812yes—
190C22×C3#380211yescharacteristic
191C22×C3#380512yes—
192C22×C3C2×#62411yescharacteristic
193C3×C4—22no—
194C3×C4—64no—
195C3×C4—64no—
196C3×C4—94no—
197C3×C4—182no—
198C3×C4—364no—
199Dic3—38no—
200Dic3—64no—
201Dic3—64no—
202Dic3—64no—
203Dic3—124no—
204Dic3—364no—
205C32—21no—
206C32#465012yescomplemented
207C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
208C2×C4—22no—
209C2×C4—92no—
210C2×C4—92no—
211C2×C4—92no—
212C2×C4—92no—
213C2×C4—182no—
214C2×C4—184no—
215C2×C4—184no—
216C2×C4—184no—
217C2×C4#619912yes—
218C2×C3—32no—
219C2×C3—32no—
220C2×C3—34no—
221C2×C3—34no—
222C2×C3—32no—
223C2×C3—61no—
224C2×C3—61no—
225C2×C3—62no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3#6387012yes—
229C2×C3#6387411yescharacteristic
230C2×C3#6390011yescharacteristic
231C2×C3#6391512yes—
232C2×C3#6395711yescharacteristic
233C22#8865012yes—
234C22#8865111yescharacteristic
235C22#8865511yescharacteristic
236C22#8867212yes—
237C22C2×#619911yescharacteristic
238C4—22no—
239C4—94no—
240C4—182no—
241C4—364no—
242C3—32no—
243C3—61no—
244C3#1059136011yescharacteristic
245C2#1172109711yescharacteristic
246C2#1172109812yes—
247C2#1172114512yes—
248C2#1172115211yescharacteristic
249C2#1172129811yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#382
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
10. C4×#1084C4semidirect
128. H(3,𝔽3)#382semidirect
206. C32#4650semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×#884492
#63658⋊C32
#117203342
#105912063