Finite Group Databasebeta

← #13143736

Subgroups — #13143736

Order 1728.

1309 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1309 conjugacy classes of subgroups — too many to draw. Only its 171 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×#88542C212yesmaximal, complemented
3C2×#88542C212yesmaximal, complemented
4C2×#88540C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
6C3×#4672—34nomaximal
7C23×#272C2212yes—
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C2×C4⋊#272—24no—
13C2×C4⋊#272—22no—
14#88542—24no—
15#88542—24no—
16#88540—22no—
17C22×C3×D12—34no—
18C2×C3×(C2×D3)⋊C4—38no—
19C2×C3×(C2×D3)⋊C4—38no—
20C2×C3×Dic3⋊C4—34no—
21C32×#95D314yescomplemented
22C23×H(3,𝔽3)C2311yescharacteristic
23C22×#272—22no—
24C22×#272—22no—
25C22×#272—22no—
26C22×#272—22no—
27C22×#272—22no—
28C22×#272—22no—
29C22×#272D414yes—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)D412yes—
35C2×#1084—21no—
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084—22no—
39C2×#1084D412yescomplemented
40C4⋊#272—28no—
41C3×#394—91no—
42#4672—34no—
43C23×C3×D3—38no—
44C22×C32×C4C2×D318yes—
45C22×C32×C4C2×D314yes—
46C22×C3×Dic3—34no—
47C22×C3×Dic3—34no—
48C2×C3×D12—616no—
49C2×C3×D12—68no—
50C3×(C2×D3)⋊C4—616no—
51C3×(C2×D3)⋊C4—616no—
52C3×Dic3⋊C4—68no—
53C22×H(3,𝔽3)C2×D412yes—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C22×H(3,𝔽3)#4111yescharacteristic
57C22×H(3,𝔽3)#4112yes—
58C2×#272—24no—
59C2×#272—24no—
60C2×#272—24no—
61C2×#272—42no—
62C2×#272—42no—
63C2×#272—42no—
64C2×#272—42no—
65C4×H(3,𝔽3)—22no—
66C4×H(3,𝔽3)—42no—
67#1084—22no—
68#1084—41no—
69C22×C3×D4—91no—
70C22×D12—34no—
71C2×C3×C4⋊C4—91no—
72C2×C3×C22⋊C4—92no—
73C2×C3×C22⋊C4—92no—
74C2×(C2×D3)⋊C4—38no—
75C2×(C2×D3)⋊C4—38no—
76C2×Dic3⋊C4—34no—
77C3×#95—34no—
78C3×#95C3⋊D311yescharacteristic
79C23×C32C22×D314yes—
80C22×C3×D3—316no—
81C22×C3×D3—68no—
82C22×C3×D3—68no—
83C22×C3×D3—68no—
84C22×C3×D3—68no—
85C22×C3×D3—68no—
86C22×C3×D3—68no—
87C2×C32×C4—28no—
88C2×C32×C4—28no—
89C2×C32×C4—28no—
90C2×C32×C4—28no—
91C2×C32×C4D1218yes—
92C2×C3×Dic3—38no—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—68no—
97C3×D12—632no—
98#394—91noSylow
99C2×H(3,𝔽3)#12011yescharacteristic
100C2×H(3,𝔽3)#12112yes—
101C2×H(3,𝔽3)#12111yescharacteristic
102C2×H(3,𝔽3)#12312yes—
103C2×H(3,𝔽3)#12411yescharacteristic
104#272—44no—
105C24×C3—92no—
106C23×D3—38no—
107C2×C3×D4—184no—
108C2×C3×D4—182no—
109C22×C3×C4—34no—
110C22×C3×C4—38no—
111C22×C3×C4—91no—
112C22×C3×C4—91no—
113C22×C3×C4C2×C3⋊D312yes—
114C22×C3×C4C2×C3⋊D311yescharacteristic
115C22×Dic3—34no—
116C22×Dic3—34no—
117C2×D12—68no—
118C2×D12—616no—
119C3×C4⋊C4—182no—
120C3×C22⋊C4—184no—
121C3×C22⋊C4—184no—
122(C2×D3)⋊C4—616no—
123(C2×D3)⋊C4—616no—
124Dic3⋊C4—68no—
125C22×C32C2×D1214yes—
126C22×C32#23518yes—
127C22×C32D3×D418yes—
128C22×C32D3×D414yes—
129C22×C32#23914yes—
130C2×C3×D3—616no—
131C2×C3×D3—616no—
132C2×C3×D3—616no—
133C2×C3×D3—128no—
134C2×C3×D3—128no—
135C2×C3×D3—128no—
136C2×C3×D3—128no—
137C32×C4—28no—
138C32×C4—48no—
139C3×Dic3—68no—
140C3×Dic3—124no—
141C22×D4—91no—
142C2×C4⋊C4—91no—
143C2×C22⋊C4—92no—
144C2×C22⋊C4—92no—
145#95#27211yescharacteristic, complemented
146H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
147C23×C3—34no—
148C23×C3—94no—
149C23×C3—182no—
150C23×C3—182no—
151C23×C3—182no—
152C23×C3—182no—
153C23×C3—182no—
154C23×C3—182no—
155C23×C3C22×C3⋊D311yescharacteristic
156C22×D3—316no—
157C22×D3—68no—
158C22×D3—68no—
159C22×D3—68no—
160C22×D3—68no—
161C22×D3—68no—
162C22×D3—68no—
163C3×D4—188no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—38no—
169C2×C3×C4—68no—
170C2×C3×C4—68no—
171C2×C3×C4—68no—
172C2×C3×C4—68no—
173C2×C3×C4—92no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4—182no—
178C2×C3×C4C4⋊(C3⋊D3)12yes—
179C2×Dic3—38no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C2×Dic3—64no—
183C2×Dic3—68no—
184D12—632no—
185C2×C32#87514yes—
186C2×C32#88214yes—
187C2×C32#88418yes—
188C2×C32#89418yes—
189C2×C32#89514yes—
190C3×D3—1216no—
191C24—92no—
192C2×D4—184no—
193C2×D4—182no—
194C22×C4—91no—
195C22×C4—91no—
196C22×C4C2×#27212yes—
197C22×C4C2×#27211yescharacteristic
198C4⋊C4—182no—
199C22⋊C4—184no—
200C22⋊C4—184no—
201C22×C3—34no—
202C22×C3—38no—
203C22×C3—34no—
204C22×C3—38no—
205C22×C3—34no—
206C22×C3—184no—
207C22×C3—184no—
208C22×C3—184no—
209C22×C3—362no—
210C22×C3—362no—
211C22×C3—362no—
212C22×C3—362no—
213C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
214C22×C3#382812yes—
215C22×C3D4×C3⋊D312yes—
216C22×C3D4×C3⋊D311yescharacteristic
217C22×C3#383211yescharacteristic
218C2×D3—616no—
219C2×D3—616no—
220C2×D3—616no—
221C2×D3—128no—
222C2×D3—128no—
223C2×D3—128no—
224C2×D3—128no—
225C3×C4—22no—
226C3×C4—42no—
227C3×C4—68no—
228C3×C4—128no—
229C3×C4—182no—
230C3×C4—361no—
231Dic3—68no—
232Dic3—124no—
233C32#467214yescomplemented
234C23—94no—
235C23—182no—
236C23—182no—
237C23—182no—
238C23—182no—
239C23—182no—
240C23—182no—
241C23C22×#27211yescharacteristic
242D4—188no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—92no—
248C2×C4—181no—
249C2×C4—181no—
250C2×C4—181no—
251C2×C4—182no—
252C2×C4C4⋊#27212yes—
253C2×C3—34no—
254C2×C3—38no—
255C2×C3—38no—
256C2×C3—34no—
257C2×C3—34no—
258C2×C3—364no—
259C2×C3#6408411yescharacteristic
260C2×C3#6409111yescharacteristic
261C2×C3#6409312yes—
262C2×C3#6410312yes—
263C2×C3#6410411yescharacteristic
264D3—1216no—
265C22—184no—
266C22—184no—
267C22—184no—
268C22—362no—
269C22—362no—
270C22—362no—
271C22—362no—
272C22C2×C4⋊#27211yescharacteristic
273C22#8874012yes—
274C22D4×#27212yes—
275C22D4×#27211yescharacteristic
276C22#8874811yescharacteristic
277C4—22no—
278C4—42no—
279C4—182no—
280C4—361no—
281C3—34no—
282C3#1059237111yescharacteristic
283C2—364no—
284C2#1172172111yescharacteristic
285C2#1172173811yescharacteristic
286C2#1172174012yes—
287C2#1172176212yes—
288C2#1172176311yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
prime radicalC22×C4⋊#272C2
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#394
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
2. C2×#88542C2semidirect
3. C2×#88542C2semidirect
4. C2×#88540C2semidirect
5. H(3,𝔽3)×#95C2semidirect
21. C32×#95D3semidirect
39. C2×#1084D4semidirect
145. #95#272semidirect
146. H(3,𝔽3)#394semidirect
233. C32#4672semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×#885422
C2×#885402
H(3,𝔽3)×#952
C3×#46723