Finite Group Databasebeta

← #10592670⋊C3 #13142602

Subgroups — #10592670⋊C3 #13142602

Order 1728.

456 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 102 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×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#63637⋊C3C212yesmaximal, complemented
5#63635⋊C3C212yesmaximal, complemented
6#10592670C311yesmaximal, characteristic, complemented
7C3×#4971—31nomaximal
8C22×C4×H(3,𝔽3)—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#634⋊C3C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2212yes—
13#3757⋊C3—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×C32×C4⋊C4—21no—
18C2×C32×C4⋊C4D311yescharacteristic
19C2×C32×C4⋊C4C2×C311yescharacteristic
20#64133C2×C311yescharacteristic
21C3×#924—31no—
22#63657C2×C311yescharacteristic
23#63637C2×C312yescomplemented
24#63635C2×C312yescomplemented
25C3×#830—31no—
26C3×#810—32no—
27C3×#808—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—42no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—21no—
36#634⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38C3×#483—91no—
39#4971—31no—
40#4971—91no—
41C22×C32×C4—21no—
42C22×C32×C4—22no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C22×C311yescharacteristic
47C2×#634C22×C311yescharacteristic
48C2×C3×Dic6—31no—
49C32×C4⋊C4—22no—
50C32×C4⋊C4—22no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×D312yes—
54C32×C4⋊C4C22×C312yes—
55#3757—21no—
56C4⋊(C3⋊Dic3)C22×C311yescharacteristic
57#3750—21no—
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C3×#217—61no—
60C3×C4⋊Dic3—31no—
61C3×Dic3⋊C4—61no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)SD1612yes—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—42no—
69C4×H(3,𝔽3)#4112yes—
70(C3⋊Dic3)⋊C3—41no—
71(C3⋊Dic3)⋊C3—41no—
72C3×C22⋊Q8—91no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—61no—
76C2×C3×C4⋊C4C3×D311yescharacteristic
77#924—31no—
78#924—91no—
79C3×#106—92no—
80C3×#103—92no—
81C3×C22⋊C8—91no—
82#830—31no—
83#830—91no—
84#810—32no—
85#810—92no—
86#808—32no—
87#808—92no—
88C23×C32—21no—
89C23×C32#6711yescharacteristic
90C23×C32C3×D411yescharacteristic
91C2×C32×C4—24no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—24no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—44no—
100C2×C32×C4—42no—
101C2×C32×C4—42no—
102C2×C32×C4#6711yescharacteristic
103C2×C32×C4C3×D411yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C23×C311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108#634—22no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×Dic6—62no—
112C3⋊(C3⋊C8)—22no—
113C3×C3⋊C8—62no—
114#483—91noSylow
115C2×H(3,𝔽3)—22no—
116C2×H(3,𝔽3)#12311yescharacteristic
117C2×H(3,𝔽3)C2×SD1611yescharacteristic
118C2×H(3,𝔽3)#13711yescharacteristic
119C2×C3×Q8—91no—
120C22×C3×C4—21no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C22×C3×C4—61no—
124C22×C3×C4—61no—
125C22×C3×C4—62no—
126C22×C3×C4—61no—
127C22×C3×C4C2×C3×D311yescharacteristic
128C2×Dic6—31no—
129C2×Dic6—91no—
130C2×C3×C8—92no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—32no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—62no—
137C3×C4⋊C4—62no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4—181no—
140C3×C4⋊C4C2×C3×D312yes—
141C3×C22⋊C4—181no—
142#217—61no—
143#217—181no—
144C4⋊Dic3—31no—
145C4⋊Dic3—91no—
146Dic3⋊C4—61no—
147Dic3⋊C4—181no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—92no—
150C22×C32—22no—
151C22×C32—22no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32#21412yes—
159C22×C32C3×SD1612yes—
160C22×C32C2×#6711yescharacteristic
161C22×C32C2×C3×D411yescharacteristic
162C32×C4—22no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4—22no—
166C32×C4—44no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4#23512yes—
170C32×C4C3×#4112yes—
171C3⋊Dic3—41no—
172C3⋊Dic3—41no—
173C3×Dic3—121no—
174C3×Dic3—121no—
175C22⋊Q8—91no—
176C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
177#106—92no—
178#103—92no—
179C22⋊C8—91no—
180H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C23×C3C3×#6711yescharacteristic
185C3×Q8—182no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—42no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—64no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—64no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—122no—
202C2×C3×C4—122no—
203C2×C3×C4—124no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4C3×#6711yescharacteristic
207C2×C3×C4C22×C3×D311yescharacteristic
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212Dic6—62no—
213Dic6—182no—
214C3×C8—182no—
215C3⋊C8—62no—
216C3⋊C8—182no—
217C2×C32—24no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32#92911yescharacteristic
224C2×C32C2×#21411yescharacteristic
225C2×C32#94211yescharacteristic
226C2×C32C3×#12311yescharacteristic
227C2×C32C2×C3×SD1611yescharacteristic
228C2×C32C3×#13711yescharacteristic
229C2×Q8—91no—
230C22×C4—21no—
231C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
232C2×C8—92no—
233C4⋊C4—21no—
234C4⋊C4—91no—
235C4⋊C4—181no—
236C4⋊C4C2×(C3⋊D3)⋊C312yes—
237C22⋊C4—181no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—61no—
251C22×C3—62no—
252C22×C3C3×#21412yes—
253C22×C3C2×C3×#6711yescharacteristic
254C3×C4—21no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—42no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4—122no—
263C3×C4—122no—
264C3×C4—124no—
265C3×C4—361no—
266C3×C4—361no—
267C3×C4C3×#23512yes—
268Dic3—121no—
269Dic3—121no—
270Dic3—361no—
271Dic3—361no—
272C32—21no—
273C32#497111yescharacteristic, complemented
274C32C3×#48311yescharacteristic, complemented
275C23#638⋊C311yescharacteristic
276Q8—182no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4#638⋊C311yescharacteristic
283C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
284C8—182no—
285C2×C3—22no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—62no—
293C2×C3—62no—
294C2×C3—64no—
295C2×C3—61no—
296C2×C3—61no—
297C2×C3—61no—
298C2×C3C3×#92911yescharacteristic
299C2×C3C2×C3×#21411yescharacteristic
300C2×C3C3×#94211yescharacteristic
301C22—21no—
302C22—21no—
303C22#3754⋊C312yes—
304C22C2×#638⋊C311yescharacteristic
305C4—21no—
306C4—42no—
307C4—361no—
308C4—361no—
309C4#3828⋊C312yes—
310C3—31no—
311C3—31no—
312C3—61no—
313C3C3×#497111yescharacteristic
314C2—22no—
315C2#64138⋊C311yescharacteristic
316C2C2×#3754⋊C311yescharacteristic
317C2#64151⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#483
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #63637⋊C3C2semidirect
5. #63635⋊C3C2semidirect
6. #10592670C3semidirect
23. #63637C2×C3semidirect
24. #63635C2×C3semidirect
180. H(3,𝔽3)#483semidirect
273. C32#4971semidirect
274. C32C3×#483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#64133⋊C32
#63657⋊C32
#63637⋊C32
#63635⋊C32
#105926703
C3×#49713