Finite Group Databasebeta

← #13144829

Subgroups — #13144829

Order 1728.

1309 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1309 conjugacy classes of subgroups — too many to draw. Only its 195 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×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88552C212yesmaximal, complemented
3C2×#88552C212yesmaximal, complemented
4C2×#88540C211yesmaximal, characteristic, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#5222—34nomaximal
7C24×H(3,𝔽3)C2212yes—
8C2×D4×H(3,𝔽3)—24no—
9C2×D4×H(3,𝔽3)—22no—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×#1084C2212yes—
13C22×#1084C2211yescharacteristic
14#88552—24no—
15#88552—24no—
16#88540—22no—
17C22×C32×D4D314yes—
18C2×C3×#217—38no—
19C2×C3×#217—38no—
20C2×C3×Dic3⋊C4—34no—
21C3×C3⋊#95—34no—
22C23×H(3,𝔽3)—22no—
23C23×H(3,𝔽3)—22no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—22no—
27C23×H(3,𝔽3)—22no—
28C23×H(3,𝔽3)D414yes—
29C23×H(3,𝔽3)C2311yescharacteristic
30D4×H(3,𝔽3)—28no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D412yes—
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084—21no—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×#1084—22no—
39C2×#1084—22no—
40C2×#1084D412yescomplemented
41C3×#394—91no—
42#5222—34no—
43C24×C32C2×D318yes—
44C2×C32×D4—216no—
45C2×C32×D4—28no—
46C22×C32×C4C2×D314yes—
47C22×C3×Dic3—34no—
48C22×C3×Dic3—34no—
49C22×C3×Dic3—38no—
50C3×#217—616no—
51C3×#217—616no—
52C3×Dic3⋊C4—68no—
53C22×H(3,𝔽3)—24no—
54C22×H(3,𝔽3)—24no—
55C22×H(3,𝔽3)—24no—
56C22×H(3,𝔽3)—42no—
57C22×H(3,𝔽3)—42no—
58C22×H(3,𝔽3)—42no—
59C22×H(3,𝔽3)—42no—
60C22×H(3,𝔽3)C2×D412yes—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C22×H(3,𝔽3)#4111yescharacteristic
64C22×H(3,𝔽3)#4112yes—
65C4×H(3,𝔽3)—22no—
66#1084—22no—
67#1084—41no—
68#1084—42no—
69C22×C3×D4—34no—
70C22×C3×D4C3⋊D311yescharacteristic
71C2×C3×C4⋊C4—91no—
72C2×C3×C22⋊C4—92no—
73C2×C3×C22⋊C4—92no—
74C2×#217—38no—
75C2×#217—38no—
76C2×Dic3⋊C4—34no—
77C3×#95—91no—
78C3⋊#95—34no—
79C23×C32—28no—
80C23×C32—28no—
81C23×C32—28no—
82C23×C32—28no—
83C23×C32—28no—
84C23×C32—28no—
85C23×C32#67116yes—
86C23×C32C22×D314yes—
87C32×D4—232no—
88C2×C32×C4—28no—
89C2×C32×C4#6718yes—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—68no—
94C2×C3×Dic3—68no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—68no—
97C2×C3×Dic3—68no—
98#394—91noSylow
99C2×H(3,𝔽3)—44no—
100C2×H(3,𝔽3)#12011yescharacteristic
101C2×H(3,𝔽3)#12111yescharacteristic
102C2×H(3,𝔽3)#12112yes—
103C2×H(3,𝔽3)#12312yes—
104C2×H(3,𝔽3)#12411yescharacteristic
105C24×C3—38no—
106C24×C3C2×C3⋊D312yes—
107C2×C3×D4—24no—
108C2×C3×D4—22no—
109C2×C3×D4—68no—
110C2×C3×D4—616no—
111C22×C3×C4—34no—
112C22×C3×C4—91no—
113C22×C3×C4—92no—
114C22×C3×C4—91no—
115C22×C3×C4C2×C3⋊D311yescharacteristic
116C22×Dic3—34no—
117C22×Dic3—38no—
118C22×Dic3—34no—
119C3×C4⋊C4—182no—
120C3×C22⋊C4—184no—
121C3×C22⋊C4—184no—
122#217—616no—
123#217—616no—
124Dic3⋊C4—68no—
125C22×C32—216no—
126C22×C32—216no—
127C22×C32—216no—
128C22×C32—48no—
129C22×C32—48no—
130C22×C32—48no—
131C22×C32—48no—
132C22×C32D3×D414yes—
133C22×C32#23714yes—
134C22×C32#23718yes—
135C22×C32C2×#6718yes—
136C22×C32C2×#6714yes—
137C32×C4—28no—
138C3×Dic3—68no—
139C3×Dic3—128no—
140C3×Dic3—124no—
141C22×D4#27211yescharacteristic
142C2×C4⋊C4—91no—
143C2×C22⋊C4—92no—
144C2×C22⋊C4—92no—
145#95—91no—
146H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
147C23×C3—22no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—316no—
154C23×C3—34no—
155C23×C3—68no—
156C23×C3—68no—
157C23×C3—68no—
158C23×C3—68no—
159C23×C3—68no—
160C23×C3—68no—
161C23×C3#63814yes—
162C23×C3C22×C3⋊D311yescharacteristic
163C3×D4—28no—
164C3×D4—632no—
165C2×C3×C4—22no—
166C2×C3×C4—38no—
167C2×C3×C4—68no—
168C2×C3×C4—92no—
169C2×C3×C4—181no—
170C2×C3×C4—181no—
171C2×C3×C4—182no—
172C2×C3×C4—182no—
173C2×C3×C4—182no—
174C2×C3×C4—182no—
175C2×C3×C4—181no—
176C2×C3×C4#63812yes—
177C2×Dic3—38no—
178C2×Dic3—64no—
179C2×Dic3—64no—
180C2×Dic3—64no—
181C2×Dic3—68no—
182C2×Dic3—68no—
183C2×Dic3—68no—
184C2×Dic3—68no—
185C2×C32—416no—
186C2×C32#93518yes—
187C2×C32#93614yes—
188C2×C32#93814yes—
189C2×C32#93918yes—
190C2×C32#95314yes—
191C24C2×#27212yes—
192C2×D4—24no—
193C2×D4—22no—
194C22×C4—91no—
195C22×C4—92no—
196C22×C4—91no—
197C22×C4C2×#27211yescharacteristic
198C4⋊C4—182no—
199C22⋊C4—184no—
200C22⋊C4—184no—
201C22×C3—24no—
202C22×C3—24no—
203C22×C3—24no—
204C22×C3—34no—
205C22×C3—34no—
206C22×C3—38no—
207C22×C3—34no—
208C22×C3—38no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—616no—
214C22×C3—616no—
215C22×C3—616no—
216C22×C3—128no—
217C22×C3—128no—
218C22×C3—128no—
219C22×C3—128no—
220C22×C3D4×C3⋊D311yescharacteristic
221C22×C3#383011yescharacteristic
222C22×C3#383012yes—
223C22×C3C2×#63812yes—
224C22×C3C2×#63811yescharacteristic
225C3×C4—22no—
226C3×C4—68no—
227C3×C4—182no—
228C3×C4—362no—
229C3×C4—361no—
230Dic3—68no—
231Dic3—128no—
232Dic3—124no—
233C32#522214yescomplemented
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23#623514yes—
241C23C22×#27211yescharacteristic
242D4—28no—
243C2×C4—22no—
244C2×C4—92no—
245C2×C4—181no—
246C2×C4—181no—
247C2×C4—182no—
248C2×C4—182no—
249C2×C4—182no—
250C2×C4—182no—
251C2×C4—181no—
252C2×C4#623512yes—
253C2×C3—38no—
254C2×C3—34no—
255C2×C3—34no—
256C2×C3—38no—
257C2×C3—34no—
258C2×C3—44no—
259C2×C3—1216no—
260C2×C3#6414412yes—
261C2×C3#6414511yescharacteristic
262C2×C3#6414711yescharacteristic
263C2×C3#6414812yes—
264C2×C3#6416211yescharacteristic
265C22—24no—
266C22—24no—
267C22—24no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22D4×#27211yescharacteristic
273C22#8874411yescharacteristic
274C22#8874412yes—
275C22C2×#623512yes—
276C22C2×#623511yescharacteristic
277C4—22no—
278C4—182no—
279C4—362no—
280C4—361no—
281C3—34no—
282C3#1059292111yescharacteristic
283C2—44no—
284C2#1172184512yes—
285C2#1172184611yescharacteristic
286C2#1172184811yescharacteristic
287C2#1172184912yes—
288C2#1172187611yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
Fitting, prime radicalC22×D4×H(3,𝔽3)C2
hypercenterC22×C3×D4C3⋊D3
hypocenterH(3,𝔽3)#394
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88552C2semidirect
3. C2×#88552C2semidirect
4. C2×#88540C2semidirect
5. #11720574C2semidirect
40. C2×#1084D4semidirect
146. H(3,𝔽3)#394semidirect
233. C32#5222semidirect

Maximal subgroups

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

subgroupindex
C22×D4×H(3,𝔽3)2
C2×#885522
C2×#885402
#117205742
C3×#52223