Finite Group Databasebeta

← #13140128

Subgroups — #13140128

Order 1728.

617 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 125 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×#3750⋊C3C212yesmaximal
3C2×#88449C211yesmaximal, characteristic
4#11720334C212yesmaximal
5#11720334C211yesmaximal, characteristic
6#10591342—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C4⋊#1084—22no—
13#3750⋊C3—24no—
14#88449—22no—
15C2×#3750D312yes—
16C2×C3×C4⋊Dic3—32no—
17C2×C3×C4⋊Dic3—61no—
18C2×C3×Dic3⋊C4—32no—
19C2×Dic3⋊Dic3—32no—
20#63579—32no—
21#63579—32no—
22#63579—32no—
23C23×H(3,𝔽3)C2311yescharacteristic
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)D412yes—
27C2×#1084—22no—
28C2×#1084—21no—
29C2×#1084—22no—
30C2×#1084D412yes—
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—
36C2×(C3⋊Dic3)⋊C3Q814yes—
37#4982—92no—
38#4981—91no—
39C22×C32×C4—21no—
40C22×C32×C4C2×D312yes—
41C22×C3⋊Dic3C2×D312yes—
42C22×C3⋊Dic3C2×D312yes—
43C22×C3×Dic3—32no—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—61no—
48C22×C3×Dic3—61no—
49#3750—24no—
50C3×C4⋊Dic3—64no—
51C3×C4⋊Dic3—64no—
52C3×Dic3⋊C4—64no—
53Dic3⋊Dic3—64no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C22×H(3,𝔽3)C2×Q812yes—
57C22×H(3,𝔽3)#4111yescharacteristic
58C22×H(3,𝔽3)#4112yes—
59C4×H(3,𝔽3)—22no—
60#1084—22no—
61#1084—41no—
62(C3⋊Dic3)⋊C3—24no—
63(C3⋊Dic3)⋊C3—42no—
64C2×C3×C4⋊C4—92no—
65C2×C3×C4⋊C4—91no—
66C2×C4⋊Dic3—32no—
67C2×C4⋊Dic3—61no—
68C2×C4⋊Dic3—91no—
69C2×Dic3⋊C4—32no—
70C2×Dic3⋊C4—92no—
71C2×Dic3⋊C4—92no—
72C3⋊#95—91no—
73C3⋊#95—92no—
74C3⋊#95—92no—
75C3⋊#95—92no—
76C3⋊#95—92no—
77C23×C32—21no—
78C23×C32C22×D312yes—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4#6714yes—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3Dic614yes—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—34no—
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—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—122no—
106C2×C3×Dic3—122no—
107#398—271noSylow
108C2×H(3,𝔽3)#12111yescharacteristic
109C2×H(3,𝔽3)C22⋊Q812yes—
110C2×H(3,𝔽3)#12311yescharacteristic
111C2×H(3,𝔽3)#12512yes—
112C2×H(3,𝔽3)C4⋊Q811yescharacteristic
113C22×C3×C4—32no—
114C22×C3×C4—61no—
115C22×C3×C4—92no—
116C22×C3×C4—91no—
117C22×C3×C4—92no—
118C22×C3×C4—91no—
119C22×C3×C4D3211yescharacteristic
120C22×Dic3—32no—
121C22×Dic3—32no—
122C22×Dic3—32no—
123C22×Dic3—32no—
124C22×Dic3—61no—
125C22×Dic3—61no—
126C22×Dic3—92no—
127C22×Dic3—92no—
128C3×C4⋊C4—182no—
129C3×C4⋊C4—184no—
130C4⋊Dic3—64no—
131C4⋊Dic3—64no—
132C4⋊Dic3—182no—
133Dic3⋊C4—64no—
134Dic3⋊C4—184no—
135Dic3⋊C4—184no—
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32C2×Dic612yes—
142C22×C32#23512yes—
143C22×C32D3×D412yes—
144C22×C32#23712yes—
145C22×C32D3×Q812yes—
146C22×C32#23912yes—
147C22×C32C2×#6712yes—
148C32×C4—24no—
149C32×C4—24no—
150C3⋊Dic3—24no—
151C3⋊Dic3—42no—
152C3×Dic3—64no—
153C3×Dic3—64no—
154C3×Dic3—122no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C2×C4⋊C4—92no—
159C2×C4⋊C4—91no—
160C2×C4⋊C4—271no—
161#95—271no—
162#95—272no—
163H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
164C23×C3—32no—
165C23×C3—61no—
166C23×C3C2×D3211yescharacteristic
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—34no—
170C2×C3×C4—62no—
171C2×C3×C4—62no—
172C2×C3×C4—62no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—94no—
176C2×C3×C4—92no—
177C2×C3×C4—182no—
178C2×C3×C4—182no—
179C2×C3×C4—182no—
180C2×C3×C4—182no—
181C2×C3×C4—182no—
182C2×C3×C4—182no—
183C2×C3×C4—182no—
184C2×C3×C4—181no—
185C2×C3×C4#62512yes—
186C2×Dic3—34no—
187C2×Dic3—34no—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—94no—
201C2×Dic3—122no—
202C2×Dic3—122no—
203C2×Dic3—182no—
204C2×Dic3—182no—
205C2×Dic3—182no—
206C2×Dic3—182no—
207C2×Dic3—182no—
208C2×C32—21no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32#88812yes—
214C2×C32#88912yes—
215C2×C32#89012yes—
216C2×C32#89412yes—
217C2×C32#92412yes—
218C2×C32#93912yes—
219C2×C32#94612yes—
220C22×C4—92no—
221C22×C4—91no—
222C22×C4—91no—
223C22×C4—92no—
224C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C4⋊C4—182no—
226C4⋊C4—184no—
227C4⋊C4—542no—
228C22×C3—32no—
229C22×C3—32no—
230C22×C3—32no—
231C22×C3—32no—
232C22×C3—32no—
233C22×C3—32no—
234C22×C3—32no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3D3×Dic612yes—
241C22×C3#379912yes—
242C22×C3#380511yescharacteristic
243C22×C3C2×#62511yescharacteristic
244C22×C3#381111yescharacteristic
245C3×C4—22no—
246C3×C4—64no—
247C3×C4—64no—
248C3×C4—184no—
249C3×C4—182no—
250C3×C4—362no—
251C3×C4—361no—
252Dic3—64no—
253Dic3—64no—
254Dic3—122no—
255Dic3—122no—
256Dic3—122no—
257Dic3—122no—
258Dic3—184no—
259Dic3—362no—
260C32—21no—
261C32#498212yescomplemented
262C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—94no—
266C2×C4—92no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4—181no—
274C2×C4—182no—
275C2×C4#619812yes—
276C2×C3—32no—
277C2×C3—32no—
278C2×C3—32no—
279C2×C3—32no—
280C2×C3—32no—
281C2×C3—32no—
282C2×C3—32no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—62no—
286C2×C3—62no—
287C2×C3—61no—
288C2×C3#6384011yescharacteristic
289C2×C3#6384512yes—
290C2×C3#6385612yes—
291C2×C3#6389711yescharacteristic
292C2×C3#6398011yescharacteristic
293C22((C3⋊D3)⋊C3)⋊Q812yes—
294C22#8865212yes—
295C22#8867211yescharacteristic
296C22C2×#619811yescharacteristic
297C22#8867711yescharacteristic
298C4—22no—
299C4—184no—
300C4—182no—
301C4—362no—
302C4—361no—
303C3—32no—
304C3—61no—
305C3#1059121511yescharacteristic
306C2#1172108011yescharacteristic
307C2#1172108512yes—
308C2#1172110612yes—
309C2#1172115111yescharacteristic
310C2#1172131811yescharacteristic
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)#398
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
163. H(3,𝔽3)#398semidirect
261. C32#4982semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#3750⋊C32
C2×#884492
#117203342
#105913423