Finite Group Databasebeta

← #13140129

Subgroups — #13140129

Order 1728.

1055 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 139 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
3C2×#88449C212yesmaximal, complemented
4#63658⋊C3C212yesmaximal, complemented
5#10591295—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—
13#88449—24no—
14C22×C3×C4×D3—32no—
15C22×C3×C4×D3—61no—
16C2×Dic3⋊Dic3—34no—
17C2×(C2×D3)⋊Dic3—34no—
18#63658D312yescomplemented
19C3×C3⋊#95—32no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C22×#272—22no—
22C22×#272—22no—
23C22×#272C2×C412yes—
24C22×#272D412yes—
25C22×#272Q812yes—
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2×C412yes—
28C2×#1084C2×C412yes—
29C2×#1084D414yes—
30C4×#272—28no—
31C2×(C3⋊Dic3)⋊C3—24no—
32C2×(C3⋊Dic3)⋊C3—24no—
33C2×(C3⋊Dic3)⋊C3—24no—
34#5212—91no—
35#4667—92no—
36C23×C3×D3—32no—
37C23×C3×D3—61no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D312yes—
40C22×C3⋊Dic3C2×D314yes—
41C22×C3×Dic3—32no—
42C22×C3×Dic3—34no—
43C22×C3×Dic3—61no—
44C2×C3×C4×D3—38no—
45C2×C3×C4×D3—68no—
46C2×C3×C4×D3—64no—
47C2×C3×C4×D3—68no—
48Dic3⋊Dic3—68no—
49(C2×D3)⋊Dic3—68no—
50C22×H(3,𝔽3)C22×C411yescharacteristic
51C22×H(3,𝔽3)C2×D412yes—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C22×H(3,𝔽3)C2×Q811yescharacteristic
54C22×H(3,𝔽3)#4112yes—
55C2×#272—22no—
56C2×#272—22no—
57C2×#272—24no—
58C2×#272—24no—
59C2×#272C4⋊C414yescomplemented
60C4×H(3,𝔽3)—22no—
61#1084C22⋊C414yescomplemented
62(C3⋊Dic3)⋊C3—44no—
63C23×C3×C4—91no—
64C22×C4×D3—32no—
65C22×C4×D3—61no—
66C2×#217—92no—
67C2×(C2×D3)⋊C4—94no—
68C2×C4⋊Dic3—92no—
69C2×Dic3⋊C4—94no—
70C3×#95—92no—
71C3⋊#95—32no—
72C3⋊#95—92no—
73C23×C32—21no—
74C23×C32C22×D312yes—
75C22×C3×D3—34no—
76C22×C3×D3—34no—
77C22×C3×D3—34no—
78C22×C3×D3—62no—
79C22×C3×D3—62no—
80C22×C3×D3—62no—
81C22×C3×D3—64no—
82C22×C3×D3—64no—
83C22×C3×D3—64no—
84C22×C3×D3—64no—
85C2×C32×C4—24no—
86C2×C32×C4—24no—
87C2×C32×C4—22no—
88C2×C32×C4C4×D314yes—
89C2×C3⋊Dic3—24no—
90C2×C3⋊Dic3—24no—
91C2×C3⋊Dic3—24no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—38no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—64no—
99C3×C4×D3—616no—
100C3×C4×D3—616no—
101#380—271noSylow
102C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
103C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
104C2×H(3,𝔽3)#11711yescharacteristic
105C2×H(3,𝔽3)#12112yes—
106C2×H(3,𝔽3)C22⋊Q812yes—
107#272—24no—
108C24×C3—91no—
109C23×D3—32no—
110C23×D3—61no—
111C22×C3×C4—32no—
112C22×C3×C4—61no—
113C22×C3×C4—91no—
114C22×C3×C4—94no—
115C22×C3×C4—94no—
116C22×C3×C4—184no—
117C22×C3×C4D3211yescharacteristic
118C22×Dic3—32no—
119C22×Dic3—34no—
120C22×Dic3—61no—
121C22×Dic3—94no—
122C2×C4×D3—38no—
123C2×C4×D3—64no—
124C2×C4×D3—68no—
125C2×C4×D3—68no—
126#217—184no—
127(C2×D3)⋊C4—188no—
128C4⋊Dic3—184no—
129Dic3⋊C4—188no—
130C22×C32—21no—
131C22×C32—22no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—22no—
135C22×C32C2×C4×D312yes—
136C22×C32#23514yes—
137C22×C32D3×D412yes—
138C22×C32D3×D414yes—
139C22×C32D3×Q812yes—
140C2×C3×D3—38no—
141C2×C3×D3—64no—
142C2×C3×D3—64no—
143C2×C3×D3—68no—
144C2×C3×D3—64no—
145C2×C3×D3—68no—
146C2×C3×D3—64no—
147C2×C3×D3—64no—
148C2×C3×D3—68no—
149C2×C3×D3—68no—
150C32×C4—24no—
151C32×C4—24no—
152C3⋊Dic3—44no—
153C3×Dic3—38no—
154C3×Dic3—64no—
155C3×Dic3—124no—
156C23×C4—91no—
157C2×C4⋊C4—272no—
158C2×C22⋊C4—272no—
159#95—92no—
160H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
161C23×C3—32no—
162C23×C3—61no—
163C23×C3—92no—
164C23×C3—92no—
165C23×C3—92no—
166C23×C3—182no—
167C23×C3—182no—
168C23×C3C2×D3211yescharacteristic
169C22×D3—34no—
170C22×D3—34no—
171C22×D3—34no—
172C22×D3—62no—
173C22×D3—62no—
174C22×D3—62no—
175C22×D3—64no—
176C22×D3—64no—
177C22×D3—64no—
178C22×D3—64no—
179C2×C3×C4—22no—
180C2×C3×C4—34no—
181C2×C3×C4—64no—
182C2×C3×C4—64no—
183C2×C3×C4—62no—
184C2×C3×C4—92no—
185C2×C3×C4—94no—
186C2×C3×C4—188no—
187C2×C3×C4—184no—
188C2×C3×C4—184no—
189C2×C3×C4—184no—
190C2×C3×C4#62412yes—
191C2×Dic3—38no—
192C2×Dic3—34no—
193C2×Dic3—62no—
194C2×Dic3—64no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C2×Dic3—64no—
198C2×Dic3—184no—
199C2×Dic3—184no—
200C2×Dic3—184no—
201C4×D3—616no—
202C4×D3—616no—
203C2×C32—22no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32#87212yes—
209C2×C32D3×C22⋊C412yes—
210C2×C32#88414yes—
211C2×C32D3×C4⋊C412yes—
212C2×C32#89614yes—
213C3×D3—68no—
214C3×D3—68no—
215C24—91no—
216C22×C4—91no—
217C22×C4—94no—
218C22×C4—94no—
219C22×C4—184no—
220C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
221C4⋊C4—544no—
222C22⋊C4—544no—
223C22×C3—34no—
224C22×C3—32no—
225C22×C3—32no—
226C22×C3—32no—
227C22×C3—34no—
228C22×C3—62no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—62no—
233C22×C3—94no—
234C22×C3—184no—
235C22×C3—182no—
236C22×C3—184no—
237C22×C3—182no—
238C22×C3#379711yescharacteristic
239C22×C3#379812yes—
240C22×C3#380211yescharacteristic
241C22×C3C2×#62411yescharacteristic
242C22×C3#381112yes—
243C2×D3—38no—
244C2×D3—64no—
245C2×D3—64no—
246C2×D3—68no—
247C2×D3—64no—
248C2×D3—68no—
249C2×D3—64no—
250C2×D3—68no—
251C2×D3—64no—
252C2×D3—68no—
253C3×C4—22no—
254C3×C4—64no—
255C3×C4—64no—
256C3×C4—94no—
257C3×C4—364no—
258Dic3—38no—
259Dic3—64no—
260Dic3—124no—
261Dic3—364no—
262C32—21no—
263C32#466712yescomplemented
264C23—92no—
265C23—92no—
266C23—92no—
267C23—182no—
268C23—182no—
269C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C2×C4—22no—
271C2×C4—92no—
272C2×C4—94no—
273C2×C4—188no—
274C2×C4—184no—
275C2×C4—184no—
276C2×C4—184no—
277C2×C4#619912yes—
278C2×C3—32no—
279C2×C3—34no—
280C2×C3—32no—
281C2×C3—34no—
282C2×C3—32no—
283C2×C3—61no—
284C2×C3—62no—
285C2×C3—61no—
286C2×C3—62no—
287C2×C3—61no—
288C2×C3—184no—
289C2×C3#6386512yes—
290C2×C3#6387411yescharacteristic
291C2×C3#6390011yescharacteristic
292C2×C3#6391212yes—
293C2×C3#6397411yescharacteristic
294D3—68no—
295D3—68no—
296C22—94no—
297C22—182no—
298C22—184no—
299C22—182no—
300C22—184no—
301C22#8865012yes—
302C22#8865111yescharacteristic
303C22#8865511yescharacteristic
304C22C2×#619911yescharacteristic
305C22#8867712yes—
306C4—22no—
307C4—94no—
308C4—364no—
309C3—32no—
310C3—61no—
311C3#1059135511yescharacteristic
312C2—184no—
313C2#1172109612yes—
314C2#1172109711yescharacteristic
315C2#1172114212yes—
316C2#1172115211yescharacteristic
317C2#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)#380
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. C2×#88449C2semidirect
4. #63658⋊C3C2semidirect
10. C2×C4×#272C4semidirect
18. #63658D3semidirect
59. C2×#272C4⋊C4semidirect
61. #1084C22⋊C4semidirect
160. H(3,𝔽3)#380semidirect
263. C32#4667semidirect

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
C2×#884492
#63658⋊C32
#105912953