Finite Group Databasebeta

← #10592421⋊C3 #13141605

Subgroups — #10592421⋊C3 #13141605

Order 1728.

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

normal subgroups only

This group has 414 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⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
4#63645⋊C3C212yesmaximal, complemented
5#63643⋊C3C212yesmaximal, complemented
6#10592421C311yesmaximal, characteristic, complemented
7C3×#4722—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10C2×#634⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2212yes—
12#3757⋊C3—21no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3750⋊C3—21no—
17C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
18#64133C2×C311yescharacteristic
19C2×C3×C4⋊Dic3—31no—
20C3×#924—31no—
21C32×C22⋊C8—21no—
22C32×C22⋊C8D311yescharacteristic
23C32×C22⋊C8C2×C311yescharacteristic
24#63645C2×C312yescomplemented
25#63643C2×C312yescomplemented
26C3×#818—32no—
27C3×#816—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—42no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)—22no—
38C3×#484—91no—
39#4722—31no—
40#4722—91no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C22×C311yescharacteristic
44C22×C3⋊Dic3—21no—
45C2×#634C22×C311yescharacteristic
46C22×C3×Dic3—61no—
47C2×C3×Dic6—31no—
48C2×C32×C8—22no—
49C2×C32×C8C2×D312yes—
50C2×C32×C8C22×C312yes—
51#3757—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)C22×C311yescharacteristic
54C4⋊(C3⋊Dic3)C22×C312yes—
55#3750—21no—
56C3×#217—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—61no—
60C3×Dic3⋊C4—61no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)Q1612yes—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)#4112yes—
67(C3⋊Dic3)⋊C3—41no—
68(C3⋊Dic3)⋊C3—41no—
69(C3⋊Dic3)⋊C3—42no—
70C3×C22⋊Q8—91no—
71C2×C3×C4⋊C4—91no—
72C2×C4⋊Dic3—31no—
73C2×C4⋊Dic3—91no—
74#924—31no—
75#924—91no—
76C3×#107—92no—
77C3×#103—92no—
78C3×C22⋊C8—31no—
79C3×C22⋊C8—31no—
80C3×C22⋊C8—61no—
81C3×C22⋊C8C3×D311yescharacteristic
82#818—32no—
83#818—92no—
84#816—32no—
85#816—92no—
86C23×C32—21no—
87C23×C32D1211yescharacteristic
88C23×C32C3×D411yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—42no—
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4C3×D411yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C23×C311yescharacteristic
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—42no—
102#634—22no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—122no—
107C3×Dic6—62no—
108C32×C8—22no—
109C32×C8—22no—
110C32×C8—42no—
111#484—91noSylow
112C2×H(3,𝔽3)—22no—
113C2×H(3,𝔽3)#12311yescharacteristic
114C2×H(3,𝔽3)C2×Q1611yescharacteristic
115C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
116C2×C3×Q8—91no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—61no—
120C22×C3×C4—181no—
121C22×C3×C4C2×C3×D311yescharacteristic
122C22×Dic3—61no—
123C22×Dic3—181no—
124C2×Dic6—31no—
125C2×Dic6—91no—
126C2×C3×C8—32no—
127C2×C3×C8—32no—
128C2×C3×C8—62no—
129C2×C3×C8C2×C3×D312yes—
130C3×C4⋊C4—92no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—181no—
133C3×C4⋊C4—181no—
134C3×C22⋊C4—181no—
135#217—61no—
136#217—181no—
137C4⋊Dic3—32no—
138C4⋊Dic3—31no—
139C4⋊Dic3—61no—
140C4⋊Dic3—92no—
141C4⋊Dic3—91no—
142C4⋊Dic3—181no—
143Dic3⋊C4—61no—
144Dic3⋊C4—181no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32Dic1212yes—
154C22×C32C3×Q1612yes—
155C22×C32C2×D1211yescharacteristic
156C22×C32C2×C3×D411yescharacteristic
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—22no—
160C32×C4—41no—
161C32×C4#23712yes—
162C32×C4C3×#4112yes—
163C3⋊Dic3—41no—
164C3⋊Dic3—41no—
165C3⋊Dic3—42no—
166C3×Dic3—121no—
167C3×Dic3—121no—
168C3×Dic3—122no—
169C22⋊Q8—91no—
170C2×C4⋊C4—91no—
171#107—92no—
172#103—92no—
173C22⋊C8(C3⋊D3)⋊C311yescharacteristic
174H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—61no—
178C23×C3C3×D1211yescharacteristic
179C3×Q8—182no—
180C2×C3×C4—22no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—122no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—182no—
193C2×C3×C4—362no—
194C2×C3×C4C3×D1211yescharacteristic
195C2×C3×C4C22×C3×D311yescharacteristic
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—62no—
199C2×Dic3—122no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—182no—
203C2×Dic3—362no—
204Dic6—62no—
205Dic6—182no—
206C3×C8—22no—
207C3×C8—62no—
208C3×C8—62no—
209C3×C8—122no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—42no—
216C2×C32#88611yescharacteristic
217C2×C32C2×Dic1211yescharacteristic
218C2×C32#90811yescharacteristic
219C2×C32C3×#12311yescharacteristic
220C2×C32C2×C3×Q1611yescharacteristic
221C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
222C2×Q8—91no—
223C22×C4—181no—
224C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
225C2×C8C2×(C3⋊D3)⋊C312yes—
226C4⋊C4—92no—
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C22⋊C4—181no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—62no—
243C22×C3—121no—
244C22×C3—121no—
245C22×C3C3×Dic1212yes—
246C22×C3C2×C3×D1211yescharacteristic
247C3×C4—21no—
248C3×C4—32no—
249C3×C4—32no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—62no—
253C3×C4—121no—
254C3×C4—362no—
255C3×C4—361no—
256C3×C4—361no—
257C3×C4C3×#23712yes—
258Dic3—121no—
259Dic3—122no—
260Dic3—121no—
261Dic3—361no—
262Dic3—362no—
263Dic3—361no—
264C32—21no—
265C32#472211yescharacteristic, complemented
266C32C3×#48411yescharacteristic, complemented
267C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
268Q8—182no—
269C2×C4—22no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—182no—
273C2×C4—362no—
274C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
275C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
276C8—22no—
277C2×C3—22no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—62no—
285C2×C3—62no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—122no—
290C2×C3C3×#88611yescharacteristic
291C2×C3C2×C3×Dic1211yescharacteristic
292C2×C3C3×#90811yescharacteristic
293C22—21no—
294C22—21no—
295C22#3746⋊C312yes—
296C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
297C4—21no—
298C4—362no—
299C4—361no—
300C4—361no—
301C4#3830⋊C312yes—
302C3—31no—
303C3—31no—
304C3—61no—
305C3C3×#472211yescharacteristic
306C2—22no—
307C2#64095⋊C311yescharacteristic
308C2C2×#3746⋊C311yescharacteristic
309C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC23×H(3,𝔽3)D4
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#484
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #63645⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #10592421C3semidirect
24. #63645C2×C3semidirect
25. #63643C2×C3semidirect
174. H(3,𝔽3)#484semidirect
265. C32#4722semidirect
266. C32C3×#484semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊Dic3)⋊C3)2
#64133⋊C32
H(3,𝔽3)×C22⋊C82
#63645⋊C32
#63643⋊C32
#105924213
C3×#47223