Finite Group Databasebeta

← #10592720⋊C3 #13142791

Subgroups — #10592720⋊C3 #13142791

Order 1728.

406 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3#63662⋊C3C212yesmaximal
4#63634⋊C3C212yesmaximal
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
6#10592720C311yesmaximal, characteristic, complemented
7C3×#5021—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×#634⋊C3—21no—
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×C4×Q8—21no—
18C32×C4×Q8D311yescharacteristic
19C32×C4×Q8C2×C311yescharacteristic
20#64078C2×C311yescharacteristic
21C3×#869—31no—
22#63662C2×C312yes—
23#63634C2×C312yes—
24C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
25C3×#835—32no—
26C3×#807—32no—
27C3×C4⋊(C3⋊C8)—31no—
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)Q812yescomplemented
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3—22no—
36#634⋊C3—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38C3×#477—91no—
39#5021—31no—
40#5021—91no—
41C2×C32×Q8—21no—
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×Q8C22×C311yescharacteristic
44C2×#634—21no—
45C2×C3×Dic6—61no—
46C32×C4⋊C4—22no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4C2×D311yescharacteristic
51C32×C4⋊C4C22×C311yescharacteristic
52C32×C42—22no—
53C32×C42—21no—
54C32×C42—21no—
55C32×C42—21no—
56C32×C42C2×D311yescharacteristic
57C32×C42C22×C311yescharacteristic
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C312yes—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—61no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C4×H(3,𝔽3)—22no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)Q1612yes—
69C4×H(3,𝔽3)C2×Q811yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71(C3⋊Dic3)⋊C3—42no—
72C3×C4⋊Q8—91no—
73C3×C4×Q8—31no—
74C3×C4×Q8—31no—
75C3×C4×Q8—61no—
76C3×C4×Q8C3×D311yescharacteristic
77#869—31no—
78#869—91no—
79C3×#107—92no—
80C3×C4⋊C8—91no—
81C3×#103—92no—
82#835—32no—
83#835—92no—
84#807—32no—
85#807—92no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—91no—
88C32×Q8—22no—
89C32×Q8—21no—
90C32×Q8—21no—
91C32×Q8—22no—
92C32×Q8Dic612yescomplemented
93C32×Q8C3×Q812yescomplemented
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C3×D411yescharacteristic
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C23×C311yescharacteristic
109C2×C3⋊Dic3—22no—
110#634—42no—
111C2×C3×Dic3—62no—
112C3×Dic6—122no—
113C3⋊(C3⋊C8)—22no—
114C3×C3⋊C8—62no—
115#477—91noSylow
116C2×H(3,𝔽3)C22⋊Q811yescharacteristic
117C2×H(3,𝔽3)C2×Q1611yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
119C2×C3×Q8—31no—
120C2×C3×Q8—31no—
121C2×C3×Q8—61no—
122C2×C3×Q8—181no—
123C2×C3×Q8C2×C3×D311yescharacteristic
124C2×Dic6—61no—
125C2×Dic6—181no—
126C2×C3×C8—92no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—62no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—92no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4C2×C3×D311yescharacteristic
137C3×C42—21no—
138C3×C42—31no—
139C3×C42—31no—
140C3×C42—61no—
141C3×C42—61no—
142C3×C42—62no—
143C3×C42—61no—
144C3×C42C2×C3×D311yescharacteristic
145C4⋊Dic3—32no—
146C4⋊Dic3—61no—
147C4⋊Dic3—92no—
148C4⋊Dic3—181no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—92no—
151C22×C32—21no—
152C22×C32C2×#6711yescharacteristic
153C22×C32C2×C3×D411yescharacteristic
154C32×C4—24no—
155C32×C4—22no—
156C32×C4—21no—
157C32×C4—22no—
158C32×C4—21no—
159C32×C4—22no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—41no—
163C32×C4—42no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4#21612yes—
167C32×C4C3×Q1612yes—
168C32×C4C2×Dic611yescharacteristic
169C32×C4#23511yescharacteristic
170C32×C4C2×C3×Q811yescharacteristic
171C32×C4C3×#4111yescharacteristic
172C3⋊Dic3—42no—
173C3×Dic3—122no—
174C4⋊Q8—91no—
175C4×Q8(C3⋊D3)⋊C311yescharacteristic
176#107—92no—
177C4⋊C8—91no—
178#103—92no—
179H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
180C3×Q8—21no—
181C3×Q8—32no—
182C3×Q8—32no—
183C3×Q8—61no—
184C3×Q8—61no—
185C3×Q8—62no—
186C3×Q8—62no—
187C3×Q8—362no—
188C3×Q8C3×Dic612yes—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—62no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—182no—
207C2×C3×C4C3×#6711yescharacteristic
208C2×C3×C4C3×#6711yescharacteristic
209C2×C3×C4C22×C3×D311yescharacteristic
210C2×Dic3—62no—
211C2×Dic3—182no—
212Dic6—122no—
213Dic6—362no—
214C3×C8—182no—
215C3⋊C8—62no—
216C3⋊C8—182no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32#92411yescharacteristic
221C2×C32C2×#21611yescharacteristic
222C2×C32#94911yescharacteristic
223C2×C32C3×C22⋊Q811yescharacteristic
224C2×C32C2×C3×Q1611yescharacteristic
225C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
226C2×Q8—181no—
227C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
228C2×C8—92no—
229C4⋊C4—21no—
230C4⋊C4—92no—
231C4⋊C4—181no—
232C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
233C42—21no—
234C42C2×(C3⋊D3)⋊C311yescharacteristic
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—61no—
238C22×C3C2×C3×#6711yescharacteristic
239C3×C4—22no—
240C3×C4—21no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—32no—
246C3×C4—32no—
247C3×C4—41no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—62no—
251C3×C4—62no—
252C3×C4—64no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—62no—
256C3×C4—121no—
257C3×C4—121no—
258C3×C4—121no—
259C3×C4—122no—
260C3×C4—362no—
261C3×C4C3×#21612yes—
262C3×C4C2×C3×Dic611yescharacteristic
263C3×C4C3×#23511yescharacteristic
264Dic3—122no—
265Dic3—362no—
266C32—21no—
267C32#502111yescharacteristic, complemented
268C32C3×#47711yescharacteristic, complemented
269Q8—21no—
270Q8—362no—
271Q8#634⋊C312yescomplemented
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—182no—
275C2×C4#638⋊C311yescharacteristic
276C2×C4#638⋊C311yescharacteristic
277C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
278C8—182no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—61no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3C3×#92411yescharacteristic
289C2×C3C2×C3×#21611yescharacteristic
290C2×C3C3×#94911yescharacteristic
291C22C2×#638⋊C311yescharacteristic
292C4—22no—
293C4—21no—
294C4—41no—
295C4—362no—
296C4#3756⋊C312yes—
297C4C2×#634⋊C311yescharacteristic
298C4#3828⋊C311yescharacteristic
299C3—31no—
300C3—31no—
301C3—61no—
302C3C3×#502111yescharacteristic
303C2#64133⋊C311yescharacteristic
304C2C2×#3756⋊C311yescharacteristic
305C2#64158⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#477
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
6. #10592720C3semidirect
29. Q8×H(3,𝔽3)Q8semidirect
92. C32×Q8Dic6semidirect
93. C32×Q8C3×Q8semidirect
179. H(3,𝔽3)#477semidirect
267. C32#5021semidirect
268. C32C3×#477semidirect
271. Q8#634⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#64078⋊C32
#63662⋊C32
#63634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927203
C3×#50213