Finite Group Databasebeta

← #13137353

Subgroups — #13137353

Order 1728.

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

normal subgroups only

This group has 457 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
1C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C4×#6180C211yesmaximal, characteristic, complemented
3C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#11720317C212yesmaximal
5#11720311C212yesmaximal, complemented
6#10590297—31nomaximal
7#10590261—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2212yes—
13C4×(C3⋊Dic3)⋊C3C411yescharacteristic
14C4×(C3⋊Dic3)⋊C3C411yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C42×C3⋊D3D311yescharacteristic
18C3×C42×D3—31no—
19C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
20C3×C4×C3⋊C8—31no—
21C3×C4×C3⋊C8—31no—
22C3×C4×C3⋊C8—31no—
23C3×C4×C3⋊C8—61no—
24#63572—32no—
25Dic3⋊(C3⋊C8)—32no—
26#63557—32no—
27(C2×D3)⋊(C3⋊C8)—32no—
28C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
33C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
34C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
35C4×(C3⋊D3)⋊C3—22no—
36C4×(C3⋊D3)⋊C3—22no—
37C4×(C3⋊D3)⋊C3C814yescomplemented
38#6180—22no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40#4925—91no—
41#4685—91no—
42#4685—91no—
43C2×C4×C3⋊D3Dic311yescharacteristic
44C2×C4×C3⋊D3Dic311yescharacteristic
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C2×C3×C4×D3—31no—
47C2×C3×C4×D3—31no—
48C2×C3×C4×D3—31no—
49C32×C42—21no—
50C32×C42C2×D311yescharacteristic
51C32×C42C2×D311yescharacteristic
52C4×C3⋊Dic3Dic311yescharacteristic
53C4×C3⋊Dic3Dic311yescharacteristic
54C4×C3⋊Dic3C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×C4×Dic3—31no—
57C3×C4×Dic3—31no—
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—62no—
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3—21no—
66C2×(C3⋊D3)⋊C3C2×C812yes—
67C4×H(3,𝔽3)C2×C812yes—
68C4×H(3,𝔽3)M1612yes—
69C4×H(3,𝔽3)#4112yes—
70(C3⋊Dic3)⋊C3—21no—
71(C3⋊Dic3)⋊C3—21no—
72(C3⋊Dic3)⋊C3C2×C812yes—
73C2×C3×C42—91no—
74C42×D3—31no—
75C42×D3—91no—
76C3×C4×C8—91no—
77C3×C4×C8—91no—
78#830—92no—
79(C2×D3)⋊C8—92no—
80(C2×D3)⋊C8—92no—
81Dic3⋊C8—92no—
82Dic3⋊C8—92no—
83C4⋊(C3⋊C8)—92no—
84C4×C3⋊C8—31no—
85C4×C3⋊C8—31no—
86C4×C3⋊C8—31no—
87C4×C3⋊C8—61no—
88C4×C3⋊C8—91no—
89C22×C3⋊D3C2×Dic311yescharacteristic
90C22×C3×D3—31no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4C4×D311yescharacteristic
95C2×C32×C4C4×D311yescharacteristic
96C2×C32×C4C2×Dic311yescharacteristic
97C2×C32×C4C2×Dic311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3C2×Dic311yescharacteristic
101C2×C3⋊Dic3C2×Dic311yescharacteristic
102C2×C3⋊Dic3C2×Dic311yescharacteristic
103C4×C3⋊D3—22no—
104C4×C3⋊D3—22no—
105C4×C3⋊D3C3⋊C814yescomplemented
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C3×C4×D3—34no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112C3⋊(C3⋊C8)—22no—
113C3×C3⋊C8—64no—
114C3×C3⋊C8—62no—
115C3×C3⋊C8—62no—
116C3×C3⋊C8—62no—
117#431—271noSylow
118C2×H(3,𝔽3)#11711yescharacteristic
119C2×H(3,𝔽3)C22×C811yescharacteristic
120C2×H(3,𝔽3)C2×M1611yescharacteristic
121(C3⋊D3)⋊C3—22no—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—91no—
129C2×C4×D3—91no—
130C2×C4×D3—91no—
131C2×C3×C8—92no—
132C2×C3×C8—92no—
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—61no—
136C3×C42—91no—
137C3×C42—91no—
138C3×C42—91no—
139C3×C42D3211yescharacteristic
140C4×Dic3—31no—
141C4×Dic3—31no—
142C4×Dic3—31no—
143C4×Dic3—91no—
144C4×Dic3—91no—
145C4×Dic3—91no—
146C2×C3⋊C8—32no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—62no—
150C2×C3⋊C8—92no—
151C22×C32—21no—
152C22×C32C2×C4×D311yescharacteristic
153C22×C32C22×Dic311yescharacteristic
154C2×C3⋊D3—21no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3C2×C3⋊C812yes—
157C2×C3×D3—32no—
158C2×C3×D3—61no—
159C2×C3×D3—61no—
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—22no—
163C32×C4D3×C812yes—
164C32×C4ΓL(1,𝔽25)12yes—
165C32×C4C2×C3⋊C812yes—
166C32×C4#20812yes—
167C32×C4#23512yes—
168C32×C4#23512yes—
169C3⋊Dic3—21no—
170C3⋊Dic3—21no—
171C3⋊Dic3C2×C3⋊C812yes—
172C3×Dic3—32no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C2×C42—91no—
176C4⋊C8—272no—
177C22⋊C8—272no—
178C4×C8—91no—
179C4×C8—91no—
180H(3,𝔽3)#43111yescharacteristic, complemented, Sylow
181C23×C3—91no—
182C22×D3—31no—
183C22×D3—91no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—94no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—182no—
198C2×C3×C4—182no—
199C2×C3×C4D3×Dic311yescharacteristic
200C2×C3×C4D3×Dic311yescharacteristic
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—31no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—91no—
206C2×Dic3—91no—
207C2×Dic3—91no—
208C4×D3—34no—
209C4×D3—62no—
210C4×D3—62no—
211C4×D3—94no—
212C4×D3—182no—
213C4×D3—182no—
214C3×C8—182no—
215C3×C8—182no—
216C3⋊C8—62no—
217C3⋊C8—62no—
218C3⋊C8—64no—
219C3⋊C8—62no—
220C3⋊C8—182no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32#87211yescharacteristic
225C2×C32C2×D3×C811yescharacteristic
226C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
227C2×C32C22×C3⋊C811yescharacteristic
228C2×C32C2×#20811yescharacteristic
229C2×C32#92611yescharacteristic
230C3⋊D3—22no—
231C3×D3—62no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4—91no—
235C2×C8—92no—
236C2×C8—92no—
237C42—91no—
238C42—91no—
239C42—91no—
240C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—61no—
244C22×C3—92no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3C2×D3×Dic311yescharacteristic
248C2×D3—32no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—92no—
252C2×D3—181no—
253C2×D3—181no—
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—32no—
258C3×C4—32no—
259C3×C4—32no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4—62no—
263C3×C4—92no—
264C3×C4—181no—
265C3×C4—181no—
266C3×C4D3×C3⋊C812yes—
267C3×C4#371112yes—
268C3×C4#379812yes—
269Dic3—32no—
270Dic3—61no—
271Dic3—61no—
272Dic3—92no—
273Dic3—181no—
274Dic3—181no—
275C32—21no—
276C32#468511yescharacteristic, complemented
277C32#492511yescharacteristic, complemented
278C23—91no—
279C2×C4—91no—
280C2×C4—94no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
286C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
287C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C8—182no—
289C8—182no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—61no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—182no—
300C2×C3C2×D3×C3⋊C811yescharacteristic
301C2×C3C2×#371111yescharacteristic
302C2×C3#6385511yescharacteristic
303D3—62no—
304D3—182no—
305C22—92no—
306C22—181no—
307C22—181no—
308C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
309C4—92no—
310C4—181no—
311C4—181no—
312C4((C3⋊D3)⋊C3)⋊C812yes—
313C4#8843012yes—
314C4#8865012yes—
315C3—31no—
316C3—31no—
317C3—61no—
318C3#1059026111yescharacteristic
319C2—182no—
320C2C2×((C3⋊D3)⋊C3)⋊C811yescharacteristic
321C2C2×#8843011yescharacteristic
322C2#1172110511yescharacteristic
subgroupG/H
center, hypercenterC42((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypocenterH(3,𝔽3)#431
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6180C2semidirect
3. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
5. #11720311C2semidirect
19. C4×C3⋊(C3⋊C8)D3semidirect
37. C4×(C3⋊D3)⋊C3C8semidirect
105. C4×C3⋊D3C3⋊C8semidirect
180. H(3,𝔽3)#431semidirect
276. C32#4685semidirect
277. C32#4925semidirect

Maximal subgroups

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

subgroupindex
C42×(C3⋊D3)⋊C32
C4×#61802
C4×(C3⋊(C3⋊C8))⋊C32
#117203172
#117203112
#105902973
#105902613