Finite Group Databasebeta

← #10592548⋊C3 #13142053

Subgroups — #10592548⋊C3 #13142053

Order 1728.

416 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3746⋊C3C212yesmaximal
2#64078⋊C3C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic
5#63643⋊C3C212yesmaximal
6#10592548C311yesmaximal, characteristic, complemented
7C3×#4849—31nomaximal
8C2×#634⋊C3C2212yes—
9C2×#634⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17#3746⋊C3—24no—
18C2×#3746C2×C312yes—
19#64078C2×C311yescharacteristic
20C4×#634C2×C311yescharacteristic
21C2×C3×Dic12—32no—
22C3×#869—31no—
23C3×C4×Dic6—31no—
24C32×C4⋊C8—21no—
25C32×C4⋊C8D311yescharacteristic
26C32×C4⋊C8C2×C311yescharacteristic
27#63643C2×C312yes—
28C3×#816—32no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
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—
35#634⋊C3—21no—
36#634⋊C3—22no—
37#634⋊C3—22no—
38#634⋊C3D412yes—
39C8×H(3,𝔽3)—22no—
40C3×#462—91no—
41#4849—31no—
42#4849—91no—
43C2×#634C22×C312yes—
44C2×#634C22×C311yescharacteristic
45C2×C3×Dic6—31no—
46C2×C3×Dic6—32no—
47C2×C32×C8—22no—
48C2×C32×C8C2×D312yes—
49C2×C32×C8C22×C312yes—
50C32×C42—21no—
51C32×C42C2×D311yescharacteristic
52C32×C42C22×C311yescharacteristic
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)C22×C311yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3—21no—
58#3746—24no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—61no—
62C3×Dic3⋊C4—61no—
63C3×C4×Dic3—61no—
64C3×Dic12—64no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)Q1612yescomplemented
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70(C3⋊Dic3)⋊C3—22no—
71(C3⋊Dic3)⋊C3—42no—
72(C3⋊Dic3)⋊C3—41no—
73C2×C3×Q16—92no—
74C3×C4⋊Q8—91no—
75C3×C4×Q8—91no—
76C2×Dic12—32no—
77C2×Dic12—92no—
78#869—31no—
79#869—91no—
80C4×Dic6—31no—
81C4×Dic6—91no—
82C3×C4⋊C8—31no—
83C3×C4⋊C8—31no—
84C3×C4⋊C8—61no—
85C3×C4⋊C8C3×D311yescharacteristic
86C3×#103—92no—
87#816—32no—
88#816—92no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4D1211yescharacteristic
93C2×C32×C4D1211yescharacteristic
94C2×C32×C4C3×D411yescharacteristic
95C2×C32×C4C3×D411yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C23×C311yescharacteristic
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—22no—
101#634—21no—
102#634—22no—
103#634—22no—
104#634C3×D412yes—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C3×Dic6—32no—
109C3×Dic6—61no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C32×C8—22no—
113C32×C8—22no—
114C32×C8—42no—
115#462—91noSylow
116C2×H(3,𝔽3)#12111yescharacteristic
117C2×H(3,𝔽3)C2×Q1611yescharacteristic
118C2×H(3,𝔽3)#13711yescharacteristic
119C2×C3×Q8—91no—
120C2×C3×Q8—92no—
121C2×Dic6—31no—
122C2×Dic6—32no—
123C2×Dic6—91no—
124C2×Dic6—92no—
125C3×Q16—184no—
126C2×C3×C8—32no—
127C2×C3×C8—32no—
128C2×C3×C8—62no—
129C2×C3×C8C2×C3×D312yes—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—181no—
132C3×C4⋊C4—181no—
133C3×C4⋊C4—181no—
134C3×C42—31no—
135C3×C42—31no—
136C3×C42—61no—
137C3×C42—181no—
138C3×C42C2×C3×D311yescharacteristic
139C4⋊Dic3—31no—
140C4⋊Dic3—61no—
141C4⋊Dic3—61no—
142C4⋊Dic3—91no—
143C4⋊Dic3—181no—
144C4⋊Dic3—181no—
145Dic3⋊C4—61no—
146Dic3⋊C4—181no—
147C4×Dic3—61no—
148C4×Dic3—181no—
149Dic12—64no—
150Dic12—184no—
151C22×C32—21no—
152C22×C32C2×D1211yescharacteristic
153C22×C32C2×C3×D411yescharacteristic
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—22no—
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—41no—
160C32×C4Dic1212yescomplemented
161C32×C4C3×Q1612yescomplemented
162C32×C4D3×D411yescharacteristic
163C32×C4#23911yescharacteristic
164C32×C4C2×C3×D411yescharacteristic
165C32×C4C3×#4111yescharacteristic
166C3⋊Dic3—22no—
167C3⋊Dic3—42no—
168C3⋊Dic3—41no—
169C3×Dic3—62no—
170C3×Dic3—122no—
171C3×Dic3—121no—
172C2×Q16—92no—
173C4⋊Q8—91no—
174C4×Q8—91no—
175C4⋊C8(C3⋊D3)⋊C311yescharacteristic
176#103—92no—
177H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
178C3×Q8—92no—
179C3×Q8—181no—
180C3×Q8—182no—
181C3×Q8—182no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4—182no—
194C2×C3×C4C3×D1211yescharacteristic
195C2×C3×C4C3×D1211yescharacteristic
196C2×C3×C4C22×C3×D311yescharacteristic
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—62no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—182no—
203Dic6—32no—
204Dic6—61no—
205Dic6—62no—
206Dic6—62no—
207Dic6—92no—
208Dic6—181no—
209Dic6—182no—
210Dic6—182no—
211C3×C8—22no—
212C3×C8—62no—
213C3×C8—62no—
214C3×C8—122no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32#89511yescharacteristic
219C2×C32C2×Dic1211yescharacteristic
220C2×C32#90911yescharacteristic
221C2×C32C3×#12111yescharacteristic
222C2×C32C2×C3×Q1611yescharacteristic
223C2×C32C3×#13711yescharacteristic
224C2×Q8—91no—
225C2×Q8—92no—
226Q16—184no—
227C2×C8C2×(C3⋊D3)⋊C312yes—
228C4⋊C4—91no—
229C4⋊C4—181no—
230C4⋊C4—181no—
231C4⋊C4—181no—
232C42—181no—
233C42C2×(C3⋊D3)⋊C311yescharacteristic
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—61no—
237C22×C3C2×C3×D1211yescharacteristic
238C3×C4—21no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—32no—
244C3×C4—32no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4—121no—
251C3×C4—182no—
252C3×C4—362no—
253C3×C4—361no—
254C3×C4C3×Dic1212yes—
255C3×C4C3×D3×D411yescharacteristic
256C3×C4C3×#23911yescharacteristic
257Dic3—62no—
258Dic3—122no—
259Dic3—121no—
260Dic3—182no—
261Dic3—362no—
262Dic3—361no—
263C32—21no—
264C32#484911yescharacteristic, complemented
265C32C3×#46211yescharacteristic, complemented
266Q8—92no—
267Q8—181no—
268Q8—182no—
269Q8—182no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—182no—
273C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
274C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
275C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
276C8—22no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3C3×#89511yescharacteristic
287C2×C3C2×C3×Dic1211yescharacteristic
288C2×C3C3×#90911yescharacteristic
289C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
290C4—21no—
291C4—182no—
292C4—361no—
293C4—362no—
294C4#3746⋊C312yescomplemented
295C4D4×(C3⋊D3)⋊C311yescharacteristic
296C4#8872211yescharacteristic
297C3—31no—
298C3—31no—
299C3—61no—
300C3C3×#484911yescharacteristic
301C2#64104⋊C311yescharacteristic
302C2C2×#3746⋊C311yescharacteristic
303C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
6. #10592548C3semidirect
67. C4×H(3,𝔽3)Q16semidirect
160. C32×C4Dic12semidirect
161. C32×C4C3×Q16semidirect
177. H(3,𝔽3)#462semidirect
264. C32#4849semidirect
265. C32C3×#462semidirect
294. C4#3746⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
#64078⋊C32
C4×#634⋊C32
H(3,𝔽3)×C4⋊C82
#63643⋊C32
#105925483
C3×#48493