Finite Group Databasebeta

← #13140209

Subgroups — #13140209

Order 1728.

1139 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1139 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88451C212yesmaximal
3C2×#88447C212yesmaximal, complemented
4#63658⋊C3C212yesmaximal, complemented
5#10591237—32nomaximal
6C23×#272C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C2×C4×#272—24no—
10C2×C4×#272C414yescomplemented
11C22×(C3⋊Dic3)⋊C3C2214yes—
12#88451C418yescomplemented
13#88447—24no—
14C22×C3×C4×D3—32no—
15C22×C3×C4×D3—61no—
16C2×(C2×D3)⋊Dic3—34no—
17C2×Dic32—34no—
18#63658D312yescomplemented
19C3×C3⋊#95—32no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C22×#272—22no—
22C22×#272—22no—
23C22×#272C2×C414yes—
24C22×#272C2×C412yes—
25C2×C4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)C2×C412yes—
27C2×#1084C2×C412yes—
28C2×#1084D414yes—
29C4×#272—28no—
30C2×(C3⋊Dic3)⋊C3—24no—
31C2×(C3⋊Dic3)⋊C3—24no—
32C2×(C3⋊Dic3)⋊C3C2×C418yescomplemented
33C4×#217—91no—
34#4666—92no—
35C23×C3×D3—32no—
36C23×C3×D3—61no—
37C22×C32×C4—21no—
38C22×C32×C4C2×D312yes—
39C22×C3⋊Dic3C2×D314yes—
40C22×C3×Dic3—32no—
41C22×C3×Dic3—34no—
42C22×C3×Dic3—61no—
43C2×C3×C4×D3—38no—
44C2×C3×C4×D3—68no—
45C2×C3×C4×D3—64no—
46C2×C3×C4×D3—68no—
47(C2×D3)⋊Dic3—316no—
48Dic32—68no—
49C22×H(3,𝔽3)C22×C411yescharacteristic
50C22×H(3,𝔽3)C22×C412yes—
51C22×H(3,𝔽3)C2×D412yes—
52C22×H(3,𝔽3)#4112yes—
53C2×#272—22no—
54C2×#272—22no—
55C2×#272—28no—
56C2×#272C4214yescomplemented
57C4×H(3,𝔽3)—22no—
58#1084C22⋊C414yescomplemented
59(C3⋊Dic3)⋊C3—28no—
60C23×C3×C4—91no—
61C22×C4×D3—32no—
62C22×C4×D3—61no—
63C2×#217—92no—
64C2×(C2×D3)⋊C4—94no—
65C2×C4×Dic3—92no—
66C2×C4×Dic3—94no—
67C3×#95—92no—
68C3⋊#95—32no—
69C3⋊#95—92no—
70C23×C32—21no—
71C23×C32C22×D312yes—
72C22×C3×D3—38no—
73C22×C3×D3—34no—
74C22×C3×D3—64no—
75C22×C3×D3—62no—
76C22×C3×D3—64no—
77C22×C3×D3—64no—
78C22×C3×D3—64no—
79C22×C3×D3—62no—
80C22×C3×D3—62no—
81C2×C32×C4—24no—
82C2×C32×C4—24no—
83C2×C32×C4—22no—
84C2×C32×C4C4×D314yes—
85C2×C3⋊Dic3—24no—
86C2×C3⋊Dic3—24no—
87C2×C3⋊Dic3C4×D318yescomplemented
88C2×C3×Dic3—34no—
89C2×C3×Dic3—38no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—64no—
95C3×C4×D3—616no—
96C3×C4×D3—616no—
97C4×C22⋊C4—271noSylow
98C2×H(3,𝔽3)C2×C4211yescharacteristic
99C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
100C2×H(3,𝔽3)#11711yescharacteristic
101C2×H(3,𝔽3)C4×D414yes—
102#272—24no—
103C24×C3—91no—
104C23×D3—32no—
105C23×D3—61no—
106C22×C3×C4—32no—
107C22×C3×C4—61no—
108C22×C3×C4—91no—
109C22×C3×C4—94no—
110C22×C3×C4—94no—
111C22×C3×C4—184no—
112C22×C3×C4D3211yescharacteristic
113C22×Dic3—32no—
114C22×Dic3—34no—
115C22×Dic3—61no—
116C22×Dic3—94no—
117C2×C4×D3—38no—
118C2×C4×D3—64no—
119C2×C4×D3—68no—
120C2×C4×D3—68no—
121#217—98no—
122(C2×D3)⋊C4—916no—
123C4×Dic3—188no—
124C4×Dic3—184no—
125C22×C32—22no—
126C22×C32—22no—
127C22×C32—21no—
128C22×C32—22no—
129C22×C32C2×C4×D314yes—
130C22×C32C2×C4×D312yes—
131C22×C32D3×D414yes—
132C22×C32#23712yes—
133C22×C32#23912yes—
134C2×C3×D3—38no—
135C2×C3×D3—64no—
136C2×C3×D3—64no—
137C2×C3×D3—64no—
138C2×C3×D3—616no—
139C2×C3×D3—64no—
140C2×C3×D3—68no—
141C2×C3×D3—64no—
142C2×C3×D3—68no—
143C32×C4—24no—
144C32×C4—24no—
145C3⋊Dic3—28no—
146C3×Dic3—38no—
147C3×Dic3—64no—
148C3×Dic3—68no—
149C23×C4—91no—
150C2×C22⋊C4—272no—
151C2×C42—272no—
152#95—92no—
153H(3,𝔽3)C4×C22⋊C411yescharacteristic, complemented, Sylow
154C23×C3—32no—
155C23×C3—61no—
156C23×C3—94no—
157C23×C3—92no—
158C23×C3—182no—
159C23×C3—182no—
160C23×C3C2×D3211yescharacteristic
161C22×D3—38no—
162C22×D3—34no—
163C22×D3—62no—
164C22×D3—64no—
165C22×D3—64no—
166C22×D3—62no—
167C22×D3—62no—
168C22×D3—64no—
169C22×D3—64no—
170C2×C3×C4—22no—
171C2×C3×C4—34no—
172C2×C3×C4—64no—
173C2×C3×C4—64no—
174C2×C3×C4—62no—
175C2×C3×C4—92no—
176C2×C3×C4—94no—
177C2×C3×C4—98no—
178C2×C3×C4—188no—
179C2×C3×C4—184no—
180C2×C3×C4—184no—
181C2×C3×C4#62412yes—
182C2×Dic3—38no—
183C2×Dic3—34no—
184C2×Dic3—38no—
185C2×Dic3—62no—
186C2×Dic3—64no—
187C2×Dic3—64no—
188C2×Dic3—64no—
189C2×Dic3—98no—
190C2×Dic3—184no—
191C2×Dic3—184no—
192C4×D3—616no—
193C4×D3—616no—
194C2×C32—21no—
195C2×C32—24no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32C42×D312yes—
199C2×C32D3×C22⋊C412yes—
200C2×C32#67⋊C414yes—
201C2×C32#89212yes—
202C2×C32#89314yes—
203C3×D3—68no—
204C3×D3—68no—
205C24—91no—
206C22×C4—91no—
207C22×C4—94no—
208C22×C4—94no—
209C22×C4—184no—
210C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
211C22⋊C4—278no—
212C42—544no—
213C22×C3—32no—
214C22×C3—32no—
215C22×C3—34no—
216C22×C3—32no—
217C22×C3—34no—
218C22×C3—62no—
219C22×C3—62no—
220C22×C3—61no—
221C22×C3—62no—
222C22×C3—94no—
223C22×C3—182no—
224C22×C3—188no—
225C22×C3—182no—
226C22×C3#379612yes—
227C22×C3C4×D3212yes—
228C22×C3C2×#62411yescharacteristic
229C22×C3#381112yes—
230C2×D3—38no—
231C2×D3—64no—
232C2×D3—64no—
233C2×D3—64no—
234C2×D3—68no—
235C2×D3—68no—
236C2×D3—64no—
237C2×D3—64no—
238C2×D3—616no—
239C3×C4—22no—
240C3×C4—64no—
241C3×C4—64no—
242C3×C4—94no—
243C3×C4—188no—
244Dic3—38no—
245Dic3—64no—
246Dic3—68no—
247Dic3—188no—
248C32—21no—
249C32#466612yescomplemented
250C23—94no—
251C23—92no—
252C23—182no—
253C23—182no—
254C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C2×C4—22no—
256C2×C4—92no—
257C2×C4—94no—
258C2×C4—98no—
259C2×C4—188no—
260C2×C4—184no—
261C2×C4—184no—
262C2×C4#619912yes—
263C2×C3—32no—
264C2×C3—32no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—32no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—64no—
271C2×C3—61no—
272C2×C3—184no—
273C2×C3#6384911yescharacteristic
274C2×C3#6388114yes—
275C2×C3C4×#62411yescharacteristic
276C2×C3#6397411yescharacteristic
277D3—68no—
278D3—68no—
279C22—94no—
280C22—182no—
281C22—182no—
282C22—188no—
283C22#8864912yes—
284C22C4×((C3⋊D3)⋊C3)⋊C212yes—
285C22C2×#619911yescharacteristic
286C22#8867712yes—
287C4—22no—
288C4—94no—
289C4—188no—
290C3—32no—
291C3—61no—
292C3#1059126011yescharacteristic
293C2—184no—
294C2#1172108911yescharacteristic
295C2#1172112414yes—
296C2C4×#619911yescharacteristic
297C2#1172132011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×C22⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88447C2semidirect
4. #63658⋊C3C2semidirect
10. C2×C4×#272C4semidirect
12. #88451C4semidirect
18. #63658D3semidirect
32. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
56. C2×#272C42semidirect
58. #1084C22⋊C4semidirect
87. C2×C3⋊Dic3C4×D3semidirect
153. H(3,𝔽3)C4×C22⋊C4semidirect
249. C32#4666semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#884512
C2×#884472
#63658⋊C32
#105912373