Finite Group Databasebeta

← #10592413⋊C3 #13141525

Subgroups — #10592413⋊C3 #13141525

Order 1728.

390 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#64084⋊C3C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal
5#63643⋊C3C212yesmaximal, complemented
6#10592413C311yesmaximal, characteristic, complemented
7C3×#4714—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3—21no—
10C2×#634⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#3752⋊C3—21no—
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17#64084C2×C311yescharacteristic
18#64078C2×C311yescharacteristic, complemented
19C3×#875—31no—
20C3×#869—31no—
21C32×#97—21no—
22C32×#97D311yescharacteristic, complemented
23C32×#97C2×C311yescharacteristic, complemented
24#63648C2×C312yes—
25#63643C2×C312yescomplemented
26C3×#821—32no—
27C3×#816—32no—
28C22×(C3⋊D3)⋊C3—21no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35#634⋊C3—22no—
36#634⋊C3—42no—
37C8×H(3,𝔽3)—22no—
38C3×#489—91no—
39#4714—31no—
40#4714—91no—
41C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
42C2×#634—21no—
43C2×#634C22×C311yescharacteristic
44C2×C3×D12—31no—
45C2×C3×Dic6—31no—
46C2×C3×Dic6—61no—
47C2×C32×C8—22no—
48C2×C32×C8C2×D312yes—
49C2×C32×C8C22×C312yes—
50C32×C42—21no—
51C32×C42C2×D311yescharacteristic
52C32×C42C22×C311yescharacteristic
53#3752—21no—
54#3752—21no—
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)C22×C312yes—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—61no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×(C3⋊D3)⋊C3—41no—
63C2×(C3⋊D3)⋊C3—41no—
64C2×(C3⋊D3)⋊C3—41no—
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)#4112yes—
68(C3⋊Dic3)⋊C3—41no—
69(C3⋊Dic3)⋊C3—42no—
70C3×C4⋊Q8—91no—
71C3×#124—91no—
72#875—31no—
73#875—91no—
74#869—31no—
75#869—91no—
76C3×#103—92no—
77C3×#102—92no—
78C3×#97—31no—
79C3×#97—31no—
80C3×#97—61no—
81C3×#97C3×D311yescharacteristic
82#821—32no—
83#821—92no—
84#816—32no—
85#816—92no—
86C22×C3⋊D3—21no—
87C22×C3×D3—61no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4D1211yescharacteristic
92C2×C32×C4D1211yescharacteristic
93C2×C32×C4C3×D411yescharacteristic
94C2×C32×C4C3×D411yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C23×C311yescharacteristic
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—21no—
99C4⋊(C3⋊D3)—22no—
100#634—22no—
101#634—42no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—61no—
104C3×D12—62no—
105C3×Dic6—62no—
106C3×Dic6—122no—
107C32×C8—24no—
108C32×C8—22no—
109C32×C8—22no—
110#489—91noSylow
111C2×H(3,𝔽3)#12411yescharacteristic
112C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
113C2×H(3,𝔽3)#13711yescharacteristic
114(C3⋊D3)⋊C3—81no—
115C2×C3×Q8—91no—
116C2×C3×Q8—181no—
117C2×C3×D4—91no—
118C2×D12—31no—
119C2×D12—91no—
120C2×Dic6—31no—
121C2×Dic6—61no—
122C2×Dic6—91no—
123C2×Dic6—181no—
124C2×C3×C8—32no—
125C2×C3×C8—32no—
126C2×C3×C8—62no—
127C2×C3×C8C2×C3×D312yes—
128C3×C4⋊C4—92no—
129C3×C4⋊C4—181no—
130C3×C22⋊C4—181no—
131C3×C22⋊C4—181no—
132C3×C42—31no—
133C3×C42—31no—
134C3×C42—61no—
135C3×C42C2×C3×D311yescharacteristic
136(C2×D3)⋊C4—61no—
137(C2×D3)⋊C4—61no—
138(C2×D3)⋊C4—181no—
139(C2×D3)⋊C4—181no—
140C4⋊Dic3—32no—
141C4⋊Dic3—61no—
142C4⋊Dic3—92no—
143C4⋊Dic3—181no—
144C22×C32—21no—
145C22×C32C2×D1211yescharacteristic
146C22×C32C2×C3×D411yescharacteristic
147C2×C3⋊D3—41no—
148C2×C3⋊D3—41no—
149C2×C3⋊D3—41no—
150C2×C3×D3—121no—
151C2×C3×D3—121no—
152C2×C3×D3—121no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—22no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4#23512yes—
161C32×C4C3×#4112yes—
162C3⋊Dic3—41no—
163C3⋊Dic3—42no—
164C3×Dic3—121no—
165C3×Dic3—122no—
166C4⋊Q8—91no—
167#124—91no—
168#103—92no—
169#102—92no—
170#97(C3⋊D3)⋊C311yescharacteristic, complemented
171H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
172C23×C3—181no—
173C22×D3—61no—
174C22×D3—181no—
175C3×Q8—182no—
176C3×Q8—362no—
177C3×D4—182no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—181no—
188C2×C3×C4—182no—
189C2×C3×C4C3×D1211yescharacteristic
190C2×C3×C4C3×D1211yescharacteristic
191C2×C3×C4C22×C3×D311yescharacteristic
192C2×Dic3—62no—
193C2×Dic3—61no—
194C2×Dic3—182no—
195C2×Dic3—181no—
196D12—62no—
197D12—182no—
198Dic6—62no—
199Dic6—122no—
200Dic6—182no—
201Dic6—362no—
202C3×C8—22no—
203C3×C8—62no—
204C3×C8—62no—
205C3×C8—64no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32#87511yescharacteristic
210C2×C32#90811yescharacteristic
211C2×C32#90911yescharacteristic
212C2×C32C3×#12411yescharacteristic
213C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
214C2×C32C3×#13711yescharacteristic
215C3⋊D3—81no—
216C3×D3—241no—
217C2×Q8—91no—
218C2×Q8—181no—
219C2×D4—91no—
220C2×C8C2×(C3⋊D3)⋊C312yes—
221C4⋊C4—92no—
222C4⋊C4—181no—
223C22⋊C4—181no—
224C22⋊C4—181no—
225C42C2×(C3⋊D3)⋊C311yescharacteristic
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—61no—
229C22×C3—361no—
230C22×C3—361no—
231C22×C3—361no—
232C22×C3C2×C3×D1211yescharacteristic
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—361no—
237C2×D3—361no—
238C2×D3—361no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—32no—
242C3×C4—32no—
243C3×C4—61no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—62no—
248C3×C4—121no—
249C3×C4—121no—
250C3×C4—361no—
251C3×C4—362no—
252C3×C4C3×#23512yes—
253Dic3—122no—
254Dic3—121no—
255Dic3—361no—
256Dic3—362no—
257C32—21no—
258C32#471411yescharacteristic, complemented
259C32C3×#48911yescharacteristic, complemented
260C23—181no—
261Q8—182no—
262Q8—362no—
263D4—182no—
264C2×C4—181no—
265C2×C4—182no—
266C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
267C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
268C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
269C8—22no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—61no—
279C2×C3—721no—
280C2×C3C3×#87511yescharacteristic
281C2×C3C3×#90811yescharacteristic
282C2×C3C3×#90911yescharacteristic
283D3—241no—
284D3—721no—
285C22—361no—
286C22—361no—
287C22—361no—
288C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
289C4—21no—
290C4—21no—
291C4—362no—
292C4—361no—
293C4#3828⋊C312yes—
294C3—31no—
295C3—31no—
296C3—61no—
297C3C3×#471411yescharacteristic
298C2—721no—
299C2#64084⋊C311yescharacteristic
300C2#64117⋊C311yescharacteristic
301C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. H(3,𝔽3)×#97C2semidirect
5. #63643⋊C3C2semidirect
6. #10592413C3semidirect
18. #64078C2×C3semidirect
22. C32×#97D3semidirect
23. C32×#97C2×C3semidirect
25. #63643C2×C3semidirect
170. #97(C3⋊D3)⋊C3semidirect
171. H(3,𝔽3)#489semidirect
258. C32#4714semidirect
259. C32C3×#489semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
#64078⋊C32
H(3,𝔽3)×#972
#63648⋊C32
#63643⋊C32
#105924133
C3×#47143