Finite Group Databasebeta

← #13140058

Subgroups — #13140058

Order 1728.

743 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 159 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×#3750⋊C3C212yesmaximal
3C2×#88449C211yesmaximal, characteristic
4C2×#88448C212yesmaximal
5C2×#88447C211yesmaximal, characteristic
6#10591244—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12#88540—22no—
13#3750⋊C3—24no—
14#88449—22no—
15#88448—24no—
16#88447C414yescomplemented
17C2×#3750D312yes—
18C2×C3×Dic3⋊C4—32no—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×Dic3⋊C4—61no—
21C2×#3724—32no—
22C2×Dic3⋊Dic3—32no—
23C2×Dic3⋊Dic3—32no—
24C2×Dic32—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—22no—
28C2×#1084—21no—
29C2×#1084—22no—
30C2×#1084C2×C412yes—
31C2×#1084D412yes—
32C2×#1084Q812yes—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3C2×C414yes—
37C2×(C3⋊Dic3)⋊C3D414yes—
38C2×(C3⋊Dic3)⋊C3Q814yes—
39#4975—91no—
40#4975—92no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D312yes—
43C22×C3⋊Dic3C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51#3750—24no—
52C3×Dic3⋊C4—64no—
53C3×Dic3⋊C4—64no—
54C3×Dic3⋊C4—64no—
55#3724—64no—
56Dic3⋊Dic3—64no—
57Dic3⋊Dic3—64no—
58Dic32—38no—
59C22×H(3,𝔽3)C22×C411yescharacteristic
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C22×H(3,𝔽3)C2×D412yes—
62C22×H(3,𝔽3)C2×Q812yes—
63C22×H(3,𝔽3)C2×Q811yescharacteristic
64C4×H(3,𝔽3)—41no—
65#1084—41no—
66#1084C4⋊C414yescomplemented
67(C3⋊Dic3)⋊C3—42no—
68(C3⋊Dic3)⋊C3C4⋊C418yescomplemented
69C2×C3×C4⋊C4—92no—
70C2×C3×C4⋊C4—91no—
71C2×C4⋊Dic3—91no—
72C2×C4⋊Dic3—92no—
73C2×Dic3⋊C4—32no—
74C2×Dic3⋊C4—32no—
75C2×Dic3⋊C4—61no—
76C2×Dic3⋊C4—92no—
77C2×Dic3⋊C4—92no—
78C2×Dic3⋊C4—92no—
79C2×Dic3⋊C4—92no—
80C2×C4×Dic3—91no—
81C2×C4×Dic3—92no—
82C23×C32—21no—
83C23×C32C22×D312yes—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—42no—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3Dic614yes—
93C2×C3⋊Dic3C4×D314yes—
94C2×C3⋊Dic3#6714yes—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—122no—
112#384—271noSylow
113C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
114C2×H(3,𝔽3)C2×C4⋊C412yes—
115C2×H(3,𝔽3)#12711yescharacteristic
116C2×H(3,𝔽3)C4⋊Q811yescharacteristic
117C2×H(3,𝔽3)C4⋊Q812yes—
118C22×C3×C4—32no—
119C22×C3×C4—61no—
120C22×C3×C4—92no—
121C22×C3×C4—91no—
122C22×C3×C4—92no—
123C22×C3×C4—91no—
124C22×C3×C4D3211yescharacteristic
125C22×Dic3—32no—
126C22×Dic3—32no—
127C22×Dic3—32no—
128C22×Dic3—32no—
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3—92no—
132C22×Dic3—92no—
133C3×C4⋊C4—182no—
134C3×C4⋊C4—184no—
135C4⋊Dic3—182no—
136C4⋊Dic3—184no—
137Dic3⋊C4—64no—
138Dic3⋊C4—64no—
139Dic3⋊C4—64no—
140Dic3⋊C4—184no—
141Dic3⋊C4—184no—
142Dic3⋊C4—184no—
143Dic3⋊C4—184no—
144C4×Dic3—94no—
145C4×Dic3—98no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32C2×Dic612yes—
152C22×C32C2×C4×D312yes—
153C22×C32D3×D412yes—
154C22×C32D3×D412yes—
155C22×C32D3×Q812yes—
156C22×C32D3×Q812yes—
157C22×C32C2×#6712yes—
158C32×C4—42no—
159C32×C4—42no—
160C3⋊Dic3—42no—
161C3⋊Dic3Dic3⋊C418yescomplemented
162C3×Dic3—38no—
163C3×Dic3—38no—
164C3×Dic3—64no—
165C3×Dic3—122no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C2×C4⋊C4—91no—
169C2×C4⋊C4—92no—
170C2×C4⋊C4—271no—
171C2×C4⋊C4—272no—
172C2×C42—271no—
173H(3,𝔽3)#38411yescharacteristic, complemented, Sylow
174C23×C3—32no—
175C23×C3—61no—
176C23×C3C2×D3211yescharacteristic
177C2×C3×C4—22no—
178C2×C3×C4—21no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—94no—
184C2×C3×C4—94no—
185C2×C3×C4—94no—
186C2×C3×C4—92no—
187C2×C3×C4—92no—
188C2×C3×C4—92no—
189C2×C3×C4—122no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—181no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—34no—
200C2×Dic3—34no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—94no—
212C2×Dic3—94no—
213C2×Dic3—94no—
214C2×Dic3—122no—
215C2×Dic3—182no—
216C2×Dic3—182no—
217C2×Dic3—182no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32#88812yes—
224C2×C32#88812yes—
225C2×C32D3×C4⋊C412yes—
226C2×C32D3×C4⋊C412yes—
227C2×C32C2×Dic3⋊C412yes—
228C2×C32#94012yes—
229C2×C32#94412yes—
230C22×C4—91no—
231C22×C4—92no—
232C22×C4—91no—
233C22×C4—92no—
234C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C4⋊C4—182no—
236C4⋊C4—184no—
237C4⋊C4—544no—
238C4⋊C4—542no—
239C42—274no—
240C22×C3—32no—
241C22×C3—32no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—32no—
245C22×C3—32no—
246C22×C3—32no—
247C22×C3—62no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—62no—
251C22×C3—61no—
252C22×C3D3×Dic612yes—
253C22×C3#379711yescharacteristic
254C22×C3C4×D3211yescharacteristic
255C22×C3D3×#6712yes—
256C22×C3#381111yescharacteristic
257C3×C4—41no—
258C3×C4—98no—
259C3×C4—94no—
260C3×C4—122no—
261C3×C4—122no—
262C3×C4—362no—
263C3×C4—361no—
264Dic3—38no—
265Dic3—38no—
266Dic3—64no—
267Dic3—98no—
268Dic3—122no—
269Dic3—122no—
270Dic3—122no—
271Dic3—362no—
272C32—21no—
273C32#497512yescomplemented
274C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—94no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—94no—
281C2×C4—94no—
282C2×C4—92no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—182no—
286C2×C4—181no—
287C2×C4—182no—
288C2×C3—32no—
289C2×C3—32no—
290C2×C3—32no—
291C2×C3—32no—
292C2×C3—32no—
293C2×C3—32no—
294C2×C3—32no—
295C2×C3—61no—
296C2×C3—62no—
297C2×C3—61no—
298C2×C3—62no—
299C2×C3—61no—
300C2×C3#6383912yes—
301C2×C3#6384011yescharacteristic
302C2×C3D3×Dic3⋊C412yes—
303C2×C3#6388311yescharacteristic
304C2×C3#6397911yescharacteristic
305C22((C3⋊D3)⋊C3)⋊Q812yes—
306C22#8865111yescharacteristic
307C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C22#8867612yes—
309C22#8867711yescharacteristic
310C4—41no—
311C4—98no—
312C4—94no—
313C4—361no—
314C4—362no—
315C3—32no—
316C3—61no—
317C3#1059124411yescharacteristic
318C2#1172107912yes—
319C2#1172108011yescharacteristic
320C2#1172111912yes—
321C2#1172114711yescharacteristic
322C2#1172131711yescharacteristic
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)#384
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
16. #88447C4semidirect
66. #1084C4⋊C4semidirect
68. (C3⋊Dic3)⋊C3C4⋊C4semidirect
161. C3⋊Dic3Dic3⋊C4semidirect
173. H(3,𝔽3)#384semidirect
273. C32#4975semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
C2×#3750⋊C32
C2×#884492
C2×#884482
C2×#884472
#105912443