Finite Group Databasebeta

← #13140141

Subgroups — #13140141

Order 1728.

561 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88540C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic
3#63658⋊C3C212yesmaximal
4#11720334C212yesmaximal
5#11720334C211yesmaximal, characteristic
6#10591341—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12#88540—22no—
13#88449—22no—
14C2×C3×Dic3⋊C4—32no—
15C2×C3×Dic3⋊C4—61no—
16C2×Dic3⋊Dic3—32no—
17#63658D312yes—
18C3×C3⋊#95—32no—
19#63579—32no—
20#63579—32no—
21#63579—32no—
22C23×H(3,𝔽3)C2311yescharacteristic
23C2×C4×H(3,𝔽3)—21no—
24C2×C4×H(3,𝔽3)—22no—
25C2×#1084—21no—
26C2×#1084—21no—
27C2×#1084—22no—
28C2×#1084D412yes—
29C2×#1084Q812yes—
30C2×(C3⋊Dic3)⋊C3—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36#4986—91no—
37#4658—92no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D312yes—
40C22×C3⋊Dic3C2×D312yes—
41C22×C3⋊Dic3C2×D312yes—
42C22×C3×Dic3—32no—
43C22×C3×Dic3—32no—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C3×Dic3⋊C4—64no—
49C3×Dic3⋊C4—64no—
50Dic3⋊Dic3—64no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C22×H(3,𝔽3)C2×Q811yescharacteristic
53C22×H(3,𝔽3)#4111yescharacteristic
54C22×H(3,𝔽3)#4112yes—
55C22×H(3,𝔽3)#4112yes—
56C4×H(3,𝔽3)—41no—
57#1084—22no—
58#1084—22no—
59(C3⋊Dic3)⋊C3—42no—
60(C3⋊Dic3)⋊C3—42no—
61C2×C3×C4⋊C4—91no—
62C2×C4⋊Dic3—91no—
63C2×Dic3⋊C4—32no—
64C2×Dic3⋊C4—61no—
65C2×Dic3⋊C4—92no—
66C3×#95—92no—
67C3⋊#95—32no—
68C3⋊#95—92no—
69C3⋊#95—91no—
70C3⋊#95—92no—
71C3⋊#95—92no—
72C3⋊#95—92no—
73C3⋊#95—92no—
74C23×C32—21no—
75C23×C32C22×D312yes—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—42no—
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—22no—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104#400—271noSylow
105C2×H(3,𝔽3)C22⋊Q811yescharacteristic
106C2×H(3,𝔽3)#12311yescharacteristic
107C2×H(3,𝔽3)#12411yescharacteristic
108C2×H(3,𝔽3)#12512yes—
109C2×H(3,𝔽3)#12612yes—
110C22×C3×C4—32no—
111C22×C3×C4—61no—
112C22×C3×C4—91no—
113C22×C3×C4—91no—
114C22×C3×C4—92no—
115C22×C3×C4—92no—
116C22×C3×C4D3211yescharacteristic
117C22×Dic3—32no—
118C22×Dic3—32no—
119C22×Dic3—32no—
120C22×Dic3—32no—
121C22×Dic3—61no—
122C22×Dic3—61no—
123C22×Dic3—92no—
124C22×Dic3—92no—
125C3×C4⋊C4—182no—
126C4⋊Dic3—182no—
127Dic3⋊C4—64no—
128Dic3⋊C4—64no—
129Dic3⋊C4—184no—
130C22×C32—22no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—22no—
134C22×C32—21no—
135C22×C32#23512yes—
136C22×C32#23512yes—
137C22×C32#23512yes—
138C22×C32D3×D412yes—
139C22×C32#23712yes—
140C22×C32#23712yes—
141C22×C32D3×Q812yes—
142C32×C4—42no—
143C32×C4—42no—
144C3⋊Dic3—42no—
145C3⋊Dic3—42no—
146C3×Dic3—64no—
147C3×Dic3—64no—
148C3×Dic3—64no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—122no—
152C2×C4⋊C4—91no—
153C2×C4⋊C4—271no—
154#95—92no—
155#95—272no—
156#95—271no—
157H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
158C23×C3—32no—
159C23×C3—61no—
160C23×C3C2×D3211yescharacteristic
161C2×C3×C4—22no—
162C2×C3×C4—21no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—62no—
166C2×C3×C4—62no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4—122no—
170C2×C3×C4—181no—
171C2×C3×C4—182no—
172C2×C3×C4—181no—
173C2×C3×C4—182no—
174C2×C3×C4—182no—
175C2×C3×C4—182no—
176C2×C3×C4—182no—
177C2×C3×C4—182no—
178C2×C3×C4—182no—
179C2×Dic3—34no—
180C2×Dic3—34no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—122no—
196C2×Dic3—182no—
197C2×Dic3—182no—
198C2×Dic3—182no—
199C2×Dic3—182no—
200C2×Dic3—182no—
201C2×Dic3—182no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32#87612yes—
208C2×C32#87912yes—
209C2×C32#88312yes—
210C2×C32#88512yes—
211C2×C32#88912yes—
212C2×C32#88912yes—
213C2×C32#89612yes—
214C22×C4—91no—
215C22×C4—91no—
216C22×C4—92no—
217C22×C4—92no—
218C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C4⋊C4—182no—
220C4⋊C4—542no—
221C22×C3—32no—
222C22×C3—32no—
223C22×C3—32no—
224C22×C3—32no—
225C22×C3—32no—
226C22×C3—32no—
227C22×C3—32no—
228C22×C3—62no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—62no—
232C22×C3—61no—
233C22×C3#379512yes—
234C22×C3#379711yescharacteristic
235C22×C3#379811yescharacteristic
236C22×C3#380412yes—
237C22×C3#381111yescharacteristic
238C3×C4—41no—
239C3×C4—122no—
240C3×C4—122no—
241C3×C4—182no—
242C3×C4—182no—
243C3×C4—362no—
244C3×C4—362no—
245Dic3—64no—
246Dic3—64no—
247Dic3—64no—
248Dic3—122no—
249Dic3—122no—
250Dic3—122no—
251Dic3—362no—
252Dic3—362no—
253C32—21no—
254C32#465812yescomplemented
255C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—92no—
259C2×C4—92no—
260C2×C4—181no—
261C2×C4—182no—
262C2×C4—181no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4—182no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C3—32no—
270C2×C3—32no—
271C2×C3—32no—
272C2×C3—32no—
273C2×C3—32no—
274C2×C3—32no—
275C2×C3—32no—
276C2×C3—61no—
277C2×C3—62no—
278C2×C3—61no—
279C2×C3—62no—
280C2×C3—61no—
281C2×C3#6384712yes—
282C2×C3#6385912yes—
283C2×C3#6386511yescharacteristic
284C2×C3#6391311yescharacteristic
285C2×C3#6397511yescharacteristic
286C22#8864812yes—
287C22#8865011yescharacteristic
288C22#8865111yescharacteristic
289C22#8867112yes—
290C22#8867711yescharacteristic
291C4—41no—
292C4—182no—
293C4—182no—
294C4—362no—
295C4—362no—
296C3—32no—
297C3—61no—
298C3#1059121711yescharacteristic
299C2#1172108612yes—
300C2#1172109611yescharacteristic
301C2#1172110912yes—
302C2#1172114311yescharacteristic
303C2#1172131911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
157. H(3,𝔽3)#400semidirect
254. C32#4658semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
C2×#884492
#63658⋊C32
#117203342
#105913413