Finite Group Databasebeta

← #13140059

Subgroups — #13140059

Order 1728.

631 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 127 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
3C2×#88447C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal
5#11720334C212yesmaximal
6#10591245—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—
14#88447C414yescomplemented
15C2×C3×Dic3⋊C4—32no—
16C2×C3×Dic3⋊C4—61no—
17C2×Dic3⋊Dic3—32no—
18C2×Dic32—32no—
19#63658D312yes—
20C3×C3⋊#95—32no—
21#63579—32no—
22#63579—32no—
23C23×H(3,𝔽3)C2311yescharacteristic
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)—22no—
26C2×#1084—21no—
27C2×#1084—22no—
28C2×#1084C2×C412yes—
29C2×#1084D412yes—
30C2×#1084Q812yes—
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)⋊C3C2×C414yes—
37#4977—91no—
38#4650—92no—
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—
49C3×Dic3⋊C4—64no—
50C3×Dic3⋊C4—64no—
51Dic3⋊Dic3—64no—
52Dic32—38no—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C22×H(3,𝔽3)C2×Q811yescharacteristic
56C22×H(3,𝔽3)#4112yes—
57C22×H(3,𝔽3)#4112yes—
58C4×H(3,𝔽3)—41no—
59#1084—41no—
60#1084C4⋊C414yes—
61(C3⋊Dic3)⋊C3—24no—
62(C3⋊Dic3)⋊C3—42no—
63C2×C3×C4⋊C4—91no—
64C2×C4⋊Dic3—91no—
65C2×Dic3⋊C4—32no—
66C2×Dic3⋊C4—61no—
67C2×Dic3⋊C4—92no—
68C2×C4×Dic3—91no—
69C2×C4×Dic3—92no—
70C3×#95—92no—
71C3⋊#95—32no—
72C3⋊#95—92no—
73C3⋊#95—92no—
74C3⋊#95—92no—
75C3⋊#95—92no—
76C23×C32—21no—
77C23×C32C22×D312yes—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—42no—
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3C4×D314yes—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—34no—
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—
106#382—271noSylow
107C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
108C2×H(3,𝔽3)#11712yes—
109C2×H(3,𝔽3)#12411yescharacteristic
110C2×H(3,𝔽3)#12411yescharacteristic
111C2×H(3,𝔽3)#12512yes—
112C22×C3×C4—32no—
113C22×C3×C4—61no—
114C22×C3×C4—91no—
115C22×C3×C4—92no—
116C22×C3×C4—92no—
117C22×C3×C4—91no—
118C22×C3×C4D3211yescharacteristic
119C22×Dic3—32no—
120C22×Dic3—32no—
121C22×Dic3—32no—
122C22×Dic3—32no—
123C22×Dic3—61no—
124C22×Dic3—61no—
125C22×Dic3—92no—
126C22×Dic3—92no—
127C3×C4⋊C4—182no—
128C4⋊Dic3—182no—
129Dic3⋊C4—64no—
130Dic3⋊C4—64no—
131Dic3⋊C4—184no—
132C4×Dic3—94no—
133C4×Dic3—98no—
134C22×C32—22no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—22no—
138C22×C32—21no—
139C22×C32C2×C4×D312yes—
140C22×C32#23512yes—
141C22×C32#23512yes—
142C22×C32D3×D412yes—
143C22×C32#23712yes—
144C22×C32#23712yes—
145C22×C32D3×Q812yes—
146C32×C4—42no—
147C32×C4—42no—
148C3⋊Dic3—24no—
149C3⋊Dic3—42no—
150C3×Dic3—38no—
151C3×Dic3—64no—
152C3×Dic3—64no—
153C3×Dic3—122no—
154C3×Dic3—122no—
155C3×Dic3—122no—
156C2×C4⋊C4—91no—
157C2×C4⋊C4—271no—
158C2×C42—271no—
159#95—92no—
160#95—272no—
161H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
162C23×C3—32no—
163C23×C3—61no—
164C23×C3C2×D3211yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—92no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—94no—
175C2×C3×C4—122no—
176C2×C3×C4—182no—
177C2×C3×C4—182no—
178C2×C3×C4—182no—
179C2×C3×C4—182no—
180C2×C3×C4—182no—
181C2×C3×C4—182no—
182C2×C3×C4—181no—
183C2×Dic3—34no—
184C2×Dic3—34no—
185C2×Dic3—34no—
186C2×Dic3—34no—
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—62no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—94no—
200C2×Dic3—122no—
201C2×Dic3—182no—
202C2×Dic3—182no—
203C2×Dic3—182no—
204C2×Dic3—182no—
205C2×Dic3—182no—
206C2×C32—22no—
207C2×C32—22no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32#87212yes—
212C2×C32#87712yes—
213C2×C32#88512yes—
214C2×C32#88512yes—
215C2×C32#88912yes—
216C2×C32#88912yes—
217C2×C32D3×C4⋊C412yes—
218C22×C4—91no—
219C22×C4—92no—
220C22×C4—92no—
221C22×C4—91no—
222C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C4⋊C4—182no—
224C4⋊C4—542no—
225C42—274no—
226C22×C3—32no—
227C22×C3—32no—
228C22×C3—32no—
229C22×C3—32no—
230C22×C3—32no—
231C22×C3—32no—
232C22×C3—32no—
233C22×C3—62no—
234C22×C3—61no—
235C22×C3—61no—
236C22×C3—62no—
237C22×C3—61no—
238C22×C3#379512yes—
239C22×C3#379711yescharacteristic
240C22×C3C4×D3211yescharacteristic
241C22×C3#380412yes—
242C22×C3#381111yescharacteristic
243C3×C4—41no—
244C3×C4—94no—
245C3×C4—122no—
246C3×C4—122no—
247C3×C4—184no—
248C3×C4—362no—
249C3×C4—361no—
250Dic3—38no—
251Dic3—64no—
252Dic3—64no—
253Dic3—122no—
254Dic3—122no—
255Dic3—122no—
256Dic3—184no—
257Dic3—362no—
258C32—21no—
259C32#465012yescomplemented
260C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—92no—
264C2×C4—92no—
265C2×C4—92no—
266C2×C4—94no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4—181no—
273C2×C4—182no—
274C2×C3—32no—
275C2×C3—32no—
276C2×C3—32no—
277C2×C3—32no—
278C2×C3—32no—
279C2×C3—32no—
280C2×C3—32no—
281C2×C3—61no—
282C2×C3—62no—
283C2×C3—62no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3#6384712yes—
287C2×C3#6387712yes—
288C2×C3#6388311yescharacteristic
289C2×C3#6391311yescharacteristic
290C2×C3#6395311yescharacteristic
291C22#8864812yes—
292C22#8865111yescharacteristic
293C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
294C22#8867112yes—
295C22#8867711yescharacteristic
296C4—41no—
297C4—94no—
298C4—184no—
299C4—361no—
300C4—362no—
301C3—32no—
302C3—61no—
303C3#1059124711yescharacteristic
304C2#1172108612yes—
305C2#1172112012yes—
306C2#1172114311yescharacteristic
307C2#1172114711yescharacteristic
308C2#1172129411yescharacteristic
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
14. #88447C4semidirect
161. H(3,𝔽3)#382semidirect
259. C32#4650semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
C2×#884492
C2×#884472
#63658⋊C32
#117203342
#105912453