Finite Group Databasebeta

← #13140219

Subgroups — #13140219

Order 1728.

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

normal subgroups only

This group has 999 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88451C212yesmaximal
3#63658⋊C3C212yesmaximal, complemented
4#11720334C212yesmaximal, complemented
5#10591294—32nomaximal
6C23×#272C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C2×C4×#272—24no—
10C2×C4×#272C414yescomplemented
11C22×(C3⋊Dic3)⋊C3C2214yes—
12#88451—24no—
13C22×C3×C4×D3—32no—
14C22×C3×C4×D3—61no—
15C2×(C2×D3)⋊Dic3—34no—
16#63658D312yescomplemented
17C3×C3⋊#95—32no—
18#63579—34no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C22×#272—22no—
21C22×#272—22no—
22C22×#272C2×C412yes—
23C22×#272D414yes—
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)C2×C412yes—
26C2×#1084—22no—
27C2×#1084C2×C412yes—
28C4×#272—28no—
29C2×(C3⋊Dic3)⋊C3—24no—
30C2×(C3⋊Dic3)⋊C3—24no—
31C2×(C3⋊Dic3)⋊C3—24no—
32#5211—91no—
33#4664—92no—
34C23×C3×D3—32no—
35C23×C3×D3—61no—
36C22×C32×C4—21no—
37C22×C32×C4C2×D312yes—
38C22×C3⋊Dic3C2×D314yes—
39C22×C3×Dic3—32no—
40C22×C3×Dic3—34no—
41C22×C3×Dic3—61no—
42C2×C3×C4×D3—38no—
43C2×C3×C4×D3—68no—
44C2×C3×C4×D3—64no—
45C2×C3×C4×D3—68no—
46(C2×D3)⋊Dic3—68no—
47C22×H(3,𝔽3)C22×C411yescharacteristic
48C22×H(3,𝔽3)C2×D412yes—
49C22×H(3,𝔽3)#4112yes—
50C22×H(3,𝔽3)#4112yes—
51C2×#272—22no—
52C2×#272—22no—
53C2×#272—28no—
54C2×#272C22⋊C414yes—
55C4×H(3,𝔽3)—22no—
56#1084—22no—
57(C3⋊Dic3)⋊C3—44no—
58C23×C3×C4—91no—
59C22×C4×D3—32no—
60C22×C4×D3—61no—
61C2×#217—92no—
62C2×(C2×D3)⋊C4—94no—
63C3×#95—92no—
64C3⋊#95—32no—
65C3⋊#95—92no—
66C3⋊#95—92no—
67C3⋊#95—94no—
68C23×C32—21no—
69C23×C32C22×D312yes—
70C22×C3×D3—34no—
71C22×C3×D3—38no—
72C22×C3×D3—62no—
73C22×C3×D3—64no—
74C22×C3×D3—64no—
75C22×C3×D3—64no—
76C22×C3×D3—64no—
77C22×C3×D3—62no—
78C22×C3×D3—62no—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4—22no—
82C2×C32×C4C4×D314yes—
83C2×C3⋊Dic3—24no—
84C2×C3⋊Dic3—24no—
85C2×C3⋊Dic3—24no—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—64no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—64no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—64no—
93C3×C4×D3—616no—
94C3×C4×D3—616no—
95#381—271noSylow
96C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
97C2×H(3,𝔽3)#11711yescharacteristic
98C2×H(3,𝔽3)#11711yescharacteristic
99C2×H(3,𝔽3)#12314yes—
100#272—24no—
101C24×C3—91no—
102C23×D3—32no—
103C23×D3—61no—
104C22×C3×C4—32no—
105C22×C3×C4—61no—
106C22×C3×C4—91no—
107C22×C3×C4—94no—
108C22×C3×C4—94no—
109C22×C3×C4—184no—
110C22×C3×C4D3211yescharacteristic
111C22×Dic3—32no—
112C22×Dic3—34no—
113C22×Dic3—61no—
114C22×Dic3—94no—
115C2×C4×D3—38no—
116C2×C4×D3—64no—
117C2×C4×D3—68no—
118C2×C4×D3—68no—
119#217—184no—
120(C2×D3)⋊C4—188no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—21no—
124C22×C32—22no—
125C22×C32C2×C4×D312yes—
126C22×C32#23514yes—
127C22×C32D3×D414yes—
128C22×C32#23712yes—
129C22×C32#23912yes—
130C2×C3×D3—38no—
131C2×C3×D3—64no—
132C2×C3×D3—64no—
133C2×C3×D3—616no—
134C2×C3×D3—64no—
135C2×C3×D3—64no—
136C2×C3×D3—68no—
137C2×C3×D3—64no—
138C2×C3×D3—68no—
139C32×C4—24no—
140C32×C4—24no—
141C3⋊Dic3—44no—
142C3×Dic3—64no—
143C3×Dic3—64no—
144C3×Dic3—124no—
145C23×C4—91no—
146C2×C22⋊C4—272no—
147#95—92no—
148#95—272no—
149H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
150C23×C3—32no—
151C23×C3—61no—
152C23×C3—92no—
153C23×C3—94no—
154C23×C3—182no—
155C23×C3—182no—
156C23×C3C2×D3211yescharacteristic
157C22×D3—38no—
158C22×D3—34no—
159C22×D3—62no—
160C22×D3—64no—
161C22×D3—62no—
162C22×D3—62no—
163C22×D3—64no—
164C22×D3—64no—
165C22×D3—64no—
166C2×C3×C4—22no—
167C2×C3×C4—34no—
168C2×C3×C4—64no—
169C2×C3×C4—64no—
170C2×C3×C4—62no—
171C2×C3×C4—92no—
172C2×C3×C4—188no—
173C2×C3×C4—182no—
174C2×C3×C4—184no—
175C2×C3×C4—184no—
176C2×C3×C4—184no—
177C2×C3×C4#62412yes—
178C2×Dic3—34no—
179C2×Dic3—62no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C2×Dic3—64no—
183C2×Dic3—64no—
184C2×Dic3—64no—
185C2×Dic3—184no—
186C2×Dic3—184no—
187C2×Dic3—184no—
188C4×D3—616no—
189C4×D3—616no—
190C2×C32—21no—
191C2×C32—24no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32#87212yes—
195C2×C32D3×C22⋊C412yes—
196C2×C32#88314yes—
197C2×C32#89212yes—
198C2×C32#89414yes—
199C3×D3—68no—
200C3×D3—68no—
201C24—91no—
202C22×C4—91no—
203C22×C4—94no—
204C22×C4—94no—
205C22×C4—184no—
206C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C22⋊C4—544no—
208C22×C3—32no—
209C22×C3—32no—
210C22×C3—34no—
211C22×C3—32no—
212C22×C3—34no—
213C22×C3—62no—
214C22×C3—62no—
215C22×C3—61no—
216C22×C3—62no—
217C22×C3—94no—
218C22×C3—182no—
219C22×C3—188no—
220C22×C3—182no—
221C22×C3#379612yes—
222C22×C3#379812yes—
223C22×C3C2×#62411yescharacteristic
224C22×C3#381112yes—
225C2×D3—38no—
226C2×D3—64no—
227C2×D3—64no—
228C2×D3—64no—
229C2×D3—68no—
230C2×D3—68no—
231C2×D3—64no—
232C2×D3—64no—
233C2×D3—616no—
234C3×C4—22no—
235C3×C4—64no—
236C3×C4—64no—
237C3×C4—182no—
238C3×C4—364no—
239Dic3—64no—
240Dic3—64no—
241Dic3—124no—
242Dic3—364no—
243C32—21no—
244C32#466412yescomplemented
245C23—92no—
246C23—94no—
247C23—182no—
248C23—182no—
249C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
250C2×C4—22no—
251C2×C4—92no—
252C2×C4—188no—
253C2×C4—182no—
254C2×C4—184no—
255C2×C4—184no—
256C2×C4—184no—
257C2×C4#619912yes—
258C2×C3—32no—
259C2×C3—32no—
260C2×C3—32no—
261C2×C3—34no—
262C2×C3—34no—
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—64no—
267C2×C3—184no—
268C2×C3#6384911yescharacteristic
269C2×C3#6385314yes—
270C2×C3#6387411yescharacteristic
271C2×C3#6397411yescharacteristic
272D3—68no—
273D3—68no—
274C22—94no—
275C22—182no—
276C22—182no—
277C22—188no—
278C22#8864912yes—
279C22#8865012yes—
280C22C2×#619911yescharacteristic
281C22#8867712yes—
282C4—22no—
283C4—182no—
284C4—364no—
285C3—32no—
286C3—61no—
287C3#1059121911yescharacteristic
288C2—184no—
289C2#1172108911yescharacteristic
290C2#1172109711yescharacteristic
291C2#1172110414yes—
292C2#1172132011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
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)#381
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
3. #63658⋊C3C2semidirect
4. #11720334C2semidirect
10. C2×C4×#272C4semidirect
16. #63658D3semidirect
149. H(3,𝔽3)#381semidirect
244. C32#4664semidirect

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
#63658⋊C32
#117203342
#105912943