Finite Group Databasebeta

← #10589339⋊C3 #13135341

Subgroups — #10589339⋊C3 #13135341

Order 1728.

297 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 50 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#64145⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#99C211yesmaximal, characteristic
3#63661⋊C3C211yesmaximal, characteristic
4#10589339C311yesmaximal, characteristic, complemented
5C3×#4473—31nomaximal
6C2×D4×H(3,𝔽3)C2211yescharacteristic
7C2×#634⋊C3C411yescharacteristic, complemented
8C22⋊C4×H(3,𝔽3)—21no—
9#3757⋊C3—21no—
10#3757⋊C3—21no—
11#3757⋊C3—21no—
12C4×(C3⋊Dic3)⋊C3C411yescharacteristic, complemented
13#64145C2×C311yescharacteristic
14C3×#936—31no—
15C32×#99—21no—
16C32×#99D311yescharacteristic
17C32×#99C2×C311yescharacteristic
18#63661C2×C311yescharacteristic
19C3×#834—31no—
20C23×H(3,𝔽3)D411yescharacteristic
21C23×H(3,𝔽3)D411yescharacteristic
22D4×H(3,𝔽3)—41no—
23C2×C4×H(3,𝔽3)—41no—
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×(C3⋊Dic3)⋊C3—21no—
26C2×(C3⋊Dic3)⋊C3—21no—
27C2×(C3⋊Dic3)⋊C3—41no—
28#634⋊C3—41no—
29C3×#354—91no—
30#4473—31no—
31#4473—91no—
32C2×C32×D4—21no—
33C2×C32×D4C2×D311yescharacteristic
34C2×C32×D4C22×C311yescharacteristic
35C2×#634C3×C411yescharacteristic, complemented
36C2×C3×Dic6—31no—
37C32×C22⋊C4—22no—
38C32×C22⋊C4—21no—
39C32×C22⋊C4—21no—
40#3757—21no—
41#3757—21no—
42#3757—21no—
43C4×C3⋊Dic3C3×C411yescharacteristic, complemented
44C3×#217—61no—
45C3×#217—61no—
46C3×#217—61no—
47C3×C4×Dic3—31no—
48C22×H(3,𝔽3)—21no—
49C22×H(3,𝔽3)—21no—
50C22×H(3,𝔽3)—41no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)C22⋊C411yescharacteristic
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)—81no—
55(C3⋊Dic3)⋊C3—41no—
56(C3⋊Dic3)⋊C3—41no—
57(C3⋊Dic3)⋊C3—81no—
58C3×#124—91no—
59#936—31no—
60#936—91no—
61C3×#99—31no—
62C3×#99—31no—
63C3×#99—61no—
64C3×#99—91no—
65C3×#99C3×D311yescharacteristic
66#834—31no—
67#834—91no—
68C23×C32—21no—
69C23×C32—21no—
70C23×C32D1211yescharacteristic
71C23×C32#6711yescharacteristic
72C23×C32C3×D411yescharacteristic
73C23×C32C3×D411yescharacteristic
74C32×D4—41no—
75C32×D4—41no—
76C32×D4—42no—
77C2×C32×C4—21no—
78C2×C32×C4—42no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4C4×D311yescharacteristic
82C2×C32×C4C2×C3×C411yescharacteristic
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3—21no—
85C2×C3⋊Dic3—41no—
86#634—41no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—121no—
90C3×Dic6—121no—
91#354—91noSylow
92C2×H(3,𝔽3)—21no—
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)—41no—
95C2×H(3,𝔽3)#9911yescharacteristic
96C2×C3×Q8—91no—
97C2×C3×D4—31no—
98C2×C3×D4—31no—
99C2×C3×D4—61no—
100C2×C3×D4C2×C3×D311yescharacteristic
101C2×Dic6—31no—
102C2×Dic6—91no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—61no—
105C3×C22⋊C4—61no—
106C3×C22⋊C4—62no—
107C3×C22⋊C4—181no—
108C3×C22⋊C4—181no—
109C3×C22⋊C4—181no—
110C3×C42—91no—
111#217—61no—
112#217—61no—
113#217—61no—
114#217—181no—
115#217—181no—
116#217—181no—
117C4×Dic3—31no—
118C4×Dic3—91no—
119C22×C32—22no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32—81no—
131C22×C32—81no—
132C22×C32(C2×D3)⋊C411yescharacteristic
133C22×C32C3×C22⋊C411yescharacteristic
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—41no—
137C32×C4—82no—
138C32×C4—81no—
139C32×C4—81no—
140C3⋊Dic3—41no—
141C3⋊Dic3—41no—
142C3⋊Dic3—81no—
143C3×Dic3—121no—
144C3×Dic3—121no—
145C3×Dic3—241no—
146#124—91no—
147#99—91no—
148#99(C3⋊D3)⋊C311yescharacteristic
149H(3,𝔽3)#35411yescharacteristic, complemented, Sylow
150C23×C3—31no—
151C23×C3—31no—
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—61no—
155C23×C3—61no—
156C23×C3C3×D1211yescharacteristic
157C23×C3C3×#6711yescharacteristic
158C3×Q8—361no—
159C3×D4—41no—
160C3×D4—121no—
161C3×D4—121no—
162C3×D4—122no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—41no—
166C2×C3×C4—61no—
167C2×C3×C4—121no—
168C2×C3×C4—121no—
169C2×C3×C4—122no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C2×C3×C4—361no—
173C2×C3×C4C3×C4×D311yescharacteristic
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—121no—
177C2×Dic3—181no—
178C2×Dic3—181no—
179C2×Dic3—361no—
180Dic6—121no—
181Dic6—361no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—41no—
186C2×C32—42no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—81no—
192C2×C32#80611yescharacteristic
193C2×C32C3×#9911yescharacteristic
194C2×Q8—91no—
195C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—181no—
198C22⋊C4—181no—
199C22⋊C4—181no—
200C42—91no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—61no—
208C22×C3—61no—
209C22×C3—61no—
210C22×C3—61no—
211C22×C3—62no—
212C22×C3—61no—
213C22×C3—121no—
214C22×C3—121no—
215C22×C3—121no—
216C22×C3—121no—
217C22×C3—121no—
218C22×C3—241no—
219C22×C3—241no—
220C22×C3C3×(C2×D3)⋊C411yescharacteristic
221C3×C4—21no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—81no—
225C3×C4—121no—
226C3×C4—241no—
227C3×C4—241no—
228C3×C4—242no—
229C3×C4—361no—
230C3×C4—361no—
231C3×C4—721no—
232Dic3—121no—
233Dic3—121no—
234Dic3—241no—
235Dic3—361no—
236Dic3—361no—
237Dic3—721no—
238C32—21no—
239C32#447311yescharacteristic, complemented
240C32C3×#35411yescharacteristic, complemented
241C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
242C23#638⋊C311yescharacteristic
243Q8—361no—
244D4—41no—
245C2×C4—41no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4—361no—
249C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
250C2×C3—21no—
251C2×C3—31no—
252C2×C3—31no—
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—61no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3—121no—
259C2×C3—121no—
260C2×C3—121no—
261C2×C3—121no—
262C2×C3—121no—
263C2×C3—122no—
264C2×C3—241no—
265C2×C3C3×#80611yescharacteristic
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22#3752⋊C311yescharacteristic
271C4—21no—
272C4—81no—
273C4—361no—
274C4—361no—
275C4—721no—
276C3—31no—
277C3—31no—
278C3—61no—
279C3C3×#447311yescharacteristic
280C2—21no—
281C2—41no—
282C2—41no—
283C2#63633⋊C311yescharacteristic
subgroupG/H
centerC2#63633⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
FittingH(3,𝔽3)×#99C2
socleC2×C3C3×#806
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C32×D4C22×C3
hypercenter#99(C3⋊D3)⋊C3
hypocenterC32C3×#354
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
4. #10589339C3semidirect
7. C2×#634⋊C3C4semidirect
12. C4×(C3⋊Dic3)⋊C3C4semidirect
35. C2×#634C3×C4semidirect
43. C4×C3⋊Dic3C3×C4semidirect
149. H(3,𝔽3)#354semidirect
239. C32#4473semidirect
240. C32C3×#354semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
H(3,𝔽3)×#992
#63661⋊C32
#105893393
C3×#44733