Finite Group Databasebeta

← #10592819⋊C3 #13143208

Subgroups — #10592819⋊C3 #13143208

Order 1728.

593 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 148 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×#634⋊C3C211yesmaximal, characteristic
2C2×M16×H(3,𝔽3)C211yesmaximal, characteristic
3#64135⋊C3C211yesmaximal, characteristic
4#63643⋊C3C214yesmaximal, complemented
5#10592819C311yesmaximal, characteristic, complemented
6C3×#5120—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×(C3⋊Dic3)⋊C3—21no—
9C2×#634⋊C3—21no—
10C2×#634⋊C3—22no—
11C2×#634⋊C3C414yescomplemented
12C2×#634⋊C3C2212yes—
13M16×H(3,𝔽3)—22no—
14C2×C8×H(3,𝔽3)C2212yes—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17C4×(C3⋊Dic3)⋊C3—21no—
18C22×#634C2×C311yescharacteristic
19C22×C3×Dic6—31no—
20C2×C32×M16—21no—
21C2×C32×M16D311yescharacteristic
22C2×C32×M16C2×C311yescharacteristic
23#64135C2×C311yescharacteristic
24C3×#926—31no—
25#63643C2×C314yescomplemented
26C3×#816—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—24no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—24no—
36#634⋊C3—22no—
37#634⋊C3C2×C414yes—
38C8×H(3,𝔽3)—22no—
39C3×#419—91no—
40#5120—31no—
41#5120—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C22×C311yescharacteristic
45C22×C3⋊Dic3—21no—
46C2×#634—21no—
47C2×#634—22no—
48C2×#634C3×C414yescomplemented
49C2×#634C22×C312yes—
50C22×C3×Dic3—61no—
51C2×C3×Dic6—34no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—61no—
54C2×C3×Dic6—62no—
55C32×M16—24no—
56C32×M16—22no—
57C32×M16—22no—
58C2×C32×C8—22no—
59C2×C32×C8C2×D312yes—
60C2×C32×C8C22×C312yes—
61#3757—21no—
62C4⋊(C3⋊Dic3)C22×C312yes—
63C4×C3⋊Dic3—21no—
64C3×#217—61no—
65C3×C4⋊Dic3—32no—
66C3×C4×Dic3—61no—
67C22×H(3,𝔽3)—22no—
68C22×H(3,𝔽3)C22⋊C412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C22⋊C412yes—
71C4×H(3,𝔽3)C22×C411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73(C3⋊Dic3)⋊C3—24no—
74(C3⋊Dic3)⋊C3—42no—
75C22×C3×Q8—91no—
76C22×Dic6—31no—
77C22×Dic6—91no—
78C2×C3×M16—31no—
79C2×C3×M16—31no—
80C2×C3×M16—61no—
81C2×C3×M16C3×D311yescharacteristic
82C3×#117—91no—
83#926—31no—
84#926—91no—
85C3×#103—94no—
86#816—34no—
87#816—94no—
88C23×C32—21no—
89C23×C32D1211yescharacteristic
90C23×C32C3×D411yescharacteristic
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4C4×D312yes—
96C2×C32×C4D1211yescharacteristic
97C2×C32×C4#6712yes—
98C2×C32×C4C2×C3×C412yes—
99C2×C32×C4C3×D411yescharacteristic
100C2×C32×C4C3×D412yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C23×C311yescharacteristic
103C2×C3⋊Dic3—24no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106#634—24no—
107#634—22no—
108#634C2×C3×C414yes—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C3×Dic6—34no—
113C3×Dic6—64no—
114C3×Dic6—62no—
115C32×C8—24no—
116C32×C8—22no—
117C32×C8—22no—
118#419—91noSylow
119C2×H(3,𝔽3)—22no—
120C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
121C2×H(3,𝔽3)#13712yes—
122C2×C3×Q8—92no—
123C2×C3×Q8—94no—
124C2×C3×Q8—181no—
125C2×C3×Q8—182no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—61no—
129C22×C3×C4—181no—
130C22×C3×C4C2×C3×D311yescharacteristic
131C22×Dic3—61no—
132C22×Dic3—181no—
133C2×Dic6—34no—
134C2×Dic6—32no—
135C2×Dic6—61no—
136C2×Dic6—62no—
137C2×Dic6—92no—
138C2×Dic6—94no—
139C2×Dic6—181no—
140C2×Dic6—182no—
141C3×M16—22no—
142C3×M16—62no—
143C3×M16—62no—
144C3×M16—64no—
145C2×C3×C8—32no—
146C2×C3×C8—32no—
147C2×C3×C8—62no—
148C2×C3×C8C2×C3×D312yes—
149C3×C4⋊C4—92no—
150C3×C22⋊C4—181no—
151C3×C42—181no—
152#217—61no—
153#217—181no—
154C4⋊Dic3—32no—
155C4⋊Dic3—92no—
156C4×Dic3—61no—
157C4×Dic3—181no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—42no—
163C22×C32(C2×D3)⋊C412yes—
164C22×C32C3×C22⋊C412yes—
165C22×C32C2×D1211yescharacteristic
166C22×C32C2×C3×D411yescharacteristic
167C32×C4—22no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4(C2×D3)⋊C412yes—
171C32×C4C3×C22⋊C412yes—
172C32×C4C2×C4×D311yescharacteristic
173C32×C4C2×#6711yescharacteristic
174C32×C4C22×C3×C411yescharacteristic
175C32×C4C2×C3×D411yescharacteristic
176C3⋊Dic3—24no—
177C3⋊Dic3—42no—
178C3×Dic3—64no—
179C3×Dic3—122no—
180C22×Q8—91no—
181C2×M16(C3⋊D3)⋊C311yescharacteristic
182#117—91no—
183#103—94no—
184H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3C3×D1211yescharacteristic
189C3×Q8—94no—
190C3×Q8—182no—
191C3×Q8—184no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—182no—
205C2×C3×C4—184no—
206C2×C3×C4—182no—
207C2×C3×C4C3×C4×D312yes—
208C2×C3×C4C3×D1211yescharacteristic
209C2×C3×C4C3×#6712yes—
210C2×C3×C4C22×C3×D311yescharacteristic
211C2×Dic3—64no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—182no—
215C2×Dic3—184no—
216C2×Dic3—182no—
217Dic6—34no—
218Dic6—64no—
219Dic6—62no—
220Dic6—94no—
221Dic6—182no—
222Dic6—184no—
223C3×C8—22no—
224C3×C8—62no—
225C3×C8—62no—
226C3×C8—64no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—42no—
232C2×C32#90912yes—
233C2×C32C2×(C2×D3)⋊C411yescharacteristic
234C2×C32C2×C3×C22⋊C411yescharacteristic
235C2×C32C3×#13712yes—
236C2×Q8—94no—
237C2×Q8—92no—
238C2×Q8—181no—
239C2×Q8—182no—
240C22×C4—181no—
241C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
242M16—22no—
243C2×C8C2×(C3⋊D3)⋊C312yes—
244C4⋊C4—92no—
245C22⋊C4—181no—
246C42—181no—
247C22×C3—22no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—62no—
253C22×C3—62no—
254C22×C3—61no—
255C22×C3—62no—
256C22×C3—122no—
257C22×C3C3×(C2×D3)⋊C412yes—
258C22×C3C2×C3×D1211yescharacteristic
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—32no—
264C3×C4—32no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—184no—
269C3×C4—362no—
270C3×C4C3×(C2×D3)⋊C412yes—
271C3×C4C2×C3×C4×D311yescharacteristic
272C3×C4C2×C3×#6711yescharacteristic
273Dic3—64no—
274Dic3—122no—
275Dic3—184no—
276Dic3—362no—
277C32—21no—
278C32#512011yescharacteristic, complemented
279C32C3×#41911yescharacteristic, complemented
280C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
281Q8—94no—
282Q8—182no—
283Q8—184no—
284C2×C4—182no—
285C2×C4—184no—
286C2×C4—182no—
287C2×C4C4×(C3⋊D3)⋊C312yes—
288C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
289C2×C4#638⋊C312yes—
290C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
291C8—22no—
292C2×C3—22no—
293C2×C3—32no—
294C2×C3—32no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—62no—
298C2×C3—62no—
299C2×C3—62no—
300C2×C3—61no—
301C2×C3—122no—
302C2×C3C3×#90912yes—
303C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
304C22—22no—
305C22#3752⋊C312yes—
306C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
307C4—184no—
308C4—362no—
309C4#3752⋊C312yes—
310C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
311C4C2×#638⋊C311yescharacteristic
312C3—31no—
313C3—31no—
314C3—61no—
315C3C3×#512011yescharacteristic
316C2—22no—
317C2#64118⋊C312yes—
318C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#419
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #63643⋊C3C2semidirect
5. #10592819C3semidirect
11. C2×#634⋊C3C4semidirect
25. #63643C2×C3semidirect
48. C2×#634C3×C4semidirect
184. H(3,𝔽3)#419semidirect
278. C32#5120semidirect
279. C32C3×#419semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×M16×H(3,𝔽3)2
#64135⋊C32
#63643⋊C32
#105928193
C3×#51203