Finite Group Databasebeta

← #10592703⋊C3 #13142694

Subgroups — #10592703⋊C3 #13142694

Order 1728.

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

normal subgroups only

This group has 601 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
2H(3,𝔽3)×#117C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#63637⋊C3C214yesmaximal, complemented
5#10592703C311yesmaximal, characteristic, complemented
6C3×#5004—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—
13C4⋊C4×H(3,𝔽3)C2212yes—
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)—21no—
16#3748⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C22×#634C2×C311yescharacteristic
19C22×C3×Dic6—31no—
20C32×#117—21no—
21C32×#117D311yescharacteristic
22C32×#117C2×C311yescharacteristic
23C2×#3748C2×C311yescharacteristic
24C2×C3×#208—31no—
25#63637C2×C314yescomplemented
26C3×#810—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)C2×C412yes—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D412yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—24no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—24no—
36#634⋊C3—22no—
37#634⋊C3C2×C414yes—
38(C3⋊(C3⋊C8))⋊C3—22no—
39C3×#419—91no—
40#5004—31no—
41#5004—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—62no—
54C2×C3×Dic6—61no—
55C32×C4⋊C4—22no—
56C32×C4⋊C4C2×D312yes—
57C32×C4⋊C4C22×C312yes—
58C32×C22⋊C4—22no—
59C32×C22⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C42—22no—
62C32×C42—21no—
63C32×C42—21no—
64#3748—22no—
65C2×C3⋊(C3⋊C8)C22×C312yes—
66C3×#208—62no—
67C2×C3×C3⋊C8—32no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)C22⋊C412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)—42no—
72C4×H(3,𝔽3)C22⋊C412yes—
73C4×H(3,𝔽3)C22×C411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75(C3⋊Dic3)⋊C3—24no—
76C22×C3×Q8—91no—
77C22×Dic6—31no—
78C22×Dic6—91no—
79C2×C3×M16—91no—
80C3×#117—31no—
81C3×#117—31no—
82C3×#117—61no—
83C3×#117C3×D311yescharacteristic
84C2×#208—31no—
85C2×#208—91no—
86C3×#103—94no—
87#810—34no—
88#810—94no—
89C23×C32—21no—
90C23×C32#6711yescharacteristic
91C23×C32C3×D411yescharacteristic
92C2×C32×C4—24no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4C4×D312yes—
100C2×C32×C4D1212yes—
101C2×C32×C4#6711yescharacteristic
102C2×C32×C4C2×C3×C412yes—
103C2×C32×C4C3×D412yes—
104C2×C32×C4C3×D411yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C23×C311yescharacteristic
107C2×C3⋊Dic3—24no—
108C2×C3⋊Dic3—22no—
109#634—24no—
110#634—22no—
111#634C2×C3×C414yes—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—62no—
114C3×Dic6—34no—
115C3×Dic6—64no—
116C3×Dic6—62no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—62no—
119#419—91noSylow
120C2×H(3,𝔽3)—22no—
121C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
122C2×H(3,𝔽3)#13712yes—
123C2×C3×Q8—92no—
124C2×C3×Q8—94no—
125C2×C3×Q8—182no—
126C2×C3×Q8—181no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—61no—
130C22×C3×C4—181no—
131C22×C3×C4C2×C3×D311yescharacteristic
132C22×Dic3—61no—
133C22×Dic3—181no—
134C2×Dic6—34no—
135C2×Dic6—32no—
136C2×Dic6—62no—
137C2×Dic6—61no—
138C2×Dic6—92no—
139C2×Dic6—94no—
140C2×Dic6—182no—
141C2×Dic6—181no—
142C3×M16—182no—
143C2×C3×C8—92no—
144C3×C4⋊C4—32no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4C2×C3×D312yes—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—61no—
150C3×C22⋊C4—61no—
151C3×C22⋊C4—62no—
152C3×C42—21no—
153C3×C42—61no—
154C3×C42—61no—
155C3×C42—62no—
156#208—62no—
157#208—182no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—92no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—42no—
165C22×C32(C2×D3)⋊C412yes—
166C22×C32C3×C22⋊C412yes—
167C22×C32C2×#6711yescharacteristic
168C22×C32C2×C3×D411yescharacteristic
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4—44no—
173C32×C4—42no—
174C32×C4—42no—
175C32×C4(C2×D3)⋊C412yes—
176C32×C4C3×C22⋊C412yes—
177C32×C4C2×C4×D311yescharacteristic
178C32×C4C2×D1211yescharacteristic
179C32×C4C22×C3×C411yescharacteristic
180C32×C4C2×C3×D411yescharacteristic
181C3⋊Dic3—24no—
182C3×Dic3—64no—
183C22×Q8—91no—
184C2×M16—91no—
185#117(C3⋊D3)⋊C311yescharacteristic
186#103—94no—
187H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—61no—
191C23×C3C3×#6711yescharacteristic
192C3×Q8—94no—
193C3×Q8—182no—
194C3×Q8—184no—
195C2×C3×C4—22no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—64no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—182no—
212C2×C3×C4—184no—
213C2×C3×C4C3×C4×D312yes—
214C2×C3×C4C3×D1212yes—
215C2×C3×C4C3×#6711yescharacteristic
216C2×C3×C4C22×C3×D311yescharacteristic
217C2×Dic3—64no—
218C2×Dic3—62no—
219C2×Dic3—182no—
220C2×Dic3—184no—
221Dic6—34no—
222Dic6—64no—
223Dic6—62no—
224Dic6—94no—
225Dic6—182no—
226Dic6—184no—
227C3×C8—182no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—21no—
233C2×C32—22no—
234C2×C32—42no—
235C2×C32C2×(C2×D3)⋊C411yescharacteristic
236C2×C32#95112yes—
237C2×C32C2×C3×C22⋊C411yescharacteristic
238C2×C32C3×#13712yes—
239C2×Q8—94no—
240C2×Q8—92no—
241C2×Q8—182no—
242C2×Q8—181no—
243C22×C4—181no—
244C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
245M16—182no—
246C2×C8—92no—
247C4⋊C4C2×(C3⋊D3)⋊C312yes—
248C22⋊C4—21no—
249C42—21no—
250C22×C3—22no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—61no—
258C22×C3—62no—
259C22×C3—122no—
260C22×C3C3×(C2×D3)⋊C412yes—
261C22×C3C2×C3×#6711yescharacteristic
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—32no—
267C3×C4—32no—
268C3×C4—42no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—62no—
272C3×C4—122no—
273C3×C4—122no—
274C3×C4—124no—
275C3×C4—184no—
276C3×C4C3×(C2×D3)⋊C412yes—
277C3×C4C2×C3×C4×D311yescharacteristic
278C3×C4C2×C3×D1211yescharacteristic
279Dic3—64no—
280Dic3—184no—
281C32—21no—
282C32#500411yescharacteristic, complemented
283C32C3×#41911yescharacteristic, complemented
284C23#638⋊C311yescharacteristic
285Q8—94no—
286Q8—182no—
287Q8—184no—
288C2×C4—22no—
289C2×C4—182no—
290C2×C4—184no—
291C2×C4C4×(C3⋊D3)⋊C312yes—
292C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
293C2×C4#638⋊C311yescharacteristic
294C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
295C8—182no—
296C2×C3—22no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—32no—
300C2×C3—32no—
301C2×C3—62no—
302C2×C3—62no—
303C2×C3—61no—
304C2×C3—62no—
305C2×C3—122no—
306C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
307C2×C3C3×#95112yes—
308C22—22no—
309C22#3752⋊C312yes—
310C22C2×#638⋊C311yescharacteristic
311C4—42no—
312C4—184no—
313C4#3752⋊C312yes—
314C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
315C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
316C3—31no—
317C3—31no—
318C3—61no—
319C3C3×#500411yescharacteristic
320C2—22no—
321C2C2×#3752⋊C311yescharacteristic
322C2#64160⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C32×C4C23×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#419
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #63637⋊C3C2semidirect
5. #10592703C3semidirect
11. C2×#634⋊C3C4semidirect
25. #63637C2×C3semidirect
48. C2×#634C3×C4semidirect
187. H(3,𝔽3)#419semidirect
282. C32#5004semidirect
283. 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
H(3,𝔽3)×#1172
C2×#3748⋊C32
#63637⋊C32
#105927033
C3×#50043