Finite Group Databasebeta

← #10589405⋊C3 #13135606

Subgroups — #10589405⋊C3 #13135606

Order 1728.

316 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 56 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
1H(3,𝔽3)×#123C211yesmaximal, characteristic
2#63661⋊C3C211yesmaximal, characteristic
3#63660⋊C3C211yesmaximal, characteristic
4#10589405C311yesmaximal, characteristic, complemented
5C3×#4539—31nomaximal
6C2×D4×H(3,𝔽3)C2211yescharacteristic
7C22×C4×H(3,𝔽3)C411yescharacteristic, complemented
8C4⋊C4×H(3,𝔽3)—21no—
9C22⋊C4×H(3,𝔽3)—21no—
10C22⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
11#3757⋊C3—21no—
12#3748⋊C3—21no—
13C32×#123—21no—
14C32×#123D311yescharacteristic
15C32×#123C2×C311yescharacteristic
16#63661C2×C311yescharacteristic
17#63660C2×C311yescharacteristic
18C3×#834—31no—
19C3×#833—31no—
20C23×H(3,𝔽3)C2×C411yescharacteristic
21C23×H(3,𝔽3)D411yescharacteristic
22D4×H(3,𝔽3)—41no—
23C2×C4×H(3,𝔽3)—21no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)—41no—
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×(C3⋊Dic3)⋊C3—41no—
28(C3⋊(C3⋊C8))⋊C3—41no—
29C3×#352—91no—
30#4539—31no—
31#4539—91no—
32C2×C32×D4—21no—
33C2×C32×D4C2×D311yescharacteristic
34C2×C32×D4C22×C311yescharacteristic
35C22×C32×C4—21no—
36C22×C32×C4Dic311yescharacteristic, complemented
37C22×C32×C4C3×C411yescharacteristic, complemented
38C32×C4⋊C4—22no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4—21no—
41C32×C22⋊C4—22no—
42C32×C22⋊C4—21no—
43C32×C22⋊C4—21no—
44C32×C22⋊C4—21no—
45C32×C22⋊C4Dic311yescharacteristic, complemented
46C32×C22⋊C4C3×C411yescharacteristic, complemented
47#3757—21no—
48#3748—21no—
49C3×#217—61no—
50C3×#208—61no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)—21no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)C22⋊C411yescharacteristic
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)—41no—
59(C3⋊Dic3)⋊C3—81no—
60C3×#123—31no—
61C3×#123—31no—
62C3×#123—61no—
63C3×#123C3×D311yescharacteristic
64C3×#100—91no—
65C3×#99—91no—
66#834—31no—
67#834—91no—
68#833—31no—
69#833—91no—
70C23×C32—21no—
71C23×C32—21no—
72C23×C32C2×Dic311yescharacteristic
73C23×C32#6711yescharacteristic
74C23×C32C2×C3×C411yescharacteristic
75C23×C32C3×D411yescharacteristic
76C32×D4—41no—
77C32×D4—41no—
78C32×D4—42no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—42no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4C3×D411yescharacteristic
91C2×C3⋊Dic3—41no—
92C2×C3×Dic3—121no—
93C3⋊(C3⋊C8)—41no—
94C3×C3⋊C8—121no—
95#352—91noSylow
96C2×H(3,𝔽3)—21no—
97C2×H(3,𝔽3)—41no—
98C2×H(3,𝔽3)—41no—
99C2×H(3,𝔽3)#9911yescharacteristic
100C2×C3×D4—31no—
101C2×C3×D4—31no—
102C2×C3×D4—61no—
103C2×C3×D4C2×C3×D311yescharacteristic
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—61no—
107C22×C3×C4C3×Dic311yescharacteristic
108C3×M16—181no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—62no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4—31no—
115C3×C22⋊C4—31no—
116C3×C22⋊C4—61no—
117C3×C22⋊C4—61no—
118C3×C22⋊C4—62no—
119C3×C22⋊C4—61no—
120C3×C22⋊C4—181no—
121C3×C22⋊C4C3×Dic311yescharacteristic
122#217—61no—
123#217—181no—
124#208—61no—
125#208—181no—
126C22×C32—22no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—41no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32#21711yescharacteristic
140C22×C32C3×C22⋊C411yescharacteristic
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—41no—
144C32×C4—42no—
145C32×C4—41no—
146C32×C4—41no—
147C32×C4—42no—
148C32×C4—41no—
149C32×C4—41no—
150C3⋊Dic3—81no—
151C3×Dic3—241no—
152#123(C3⋊D3)⋊C311yescharacteristic
153#100—91no—
154#99—91no—
155H(3,𝔽3)#35211yescharacteristic, complemented, Sylow
156C23×C3—31no—
157C23×C3—31no—
158C23×C3—31no—
159C23×C3—31no—
160C23×C3—61no—
161C23×C3—61no—
162C23×C3C2×C3×Dic311yescharacteristic
163C23×C3C3×#6711yescharacteristic
164C3×D4—41no—
165C3×D4—121no—
166C3×D4—121no—
167C3×D4—122no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—41no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—61no—
180C2×C3×C4—122no—
181C2×C3×C4—121no—
182C2×C3×C4—121no—
183C2×C3×C4—361no—
184C2×C3×C4C3×#6711yescharacteristic
185C2×Dic3—121no—
186C2×Dic3—361no—
187C3×C8—361no—
188C3⋊C8—121no—
189C3⋊C8—361no—
190C2×C32—22no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—42no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—81no—
200C2×C32#83411yescharacteristic
201C2×C32C3×#9911yescharacteristic
202C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
203C22×C4(C3⋊Dic3)⋊C311yescharacteristic, complemented
204M16—181no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—181no—
208C22⋊C4(C3⋊Dic3)⋊C311yescharacteristic, complemented
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—31no—
212C22×C3—31no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—61no—
216C22×C3—61no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—62no—
220C22×C3—61no—
221C22×C3—121no—
222C22×C3—121no—
223C22×C3—121no—
224C22×C3—121no—
225C22×C3—121no—
226C22×C3—122no—
227C22×C3—122no—
228C22×C3C3×#21711yescharacteristic
229C3×C4—21no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—121no—
235C3×C4—121no—
236C3×C4—121no—
237C3×C4—122no—
238C3×C4—121no—
239C3×C4—121no—
240C3×C4—122no—
241C3×C4—721no—
242Dic3—241no—
243Dic3—721no—
244C32—21no—
245C32#453911yescharacteristic, complemented
246C32C3×#35211yescharacteristic, complemented
247C23C2×(C3⋊Dic3)⋊C311yescharacteristic
248C23#638⋊C311yescharacteristic
249D4—41no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—361no—
254C2×C4#638⋊C311yescharacteristic
255C8—361no—
256C2×C3—21no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3—41no—
260C2×C3—41no—
261C2×C3—61no—
262C2×C3—61no—
263C2×C3—62no—
264C2×C3—61no—
265C2×C3—121no—
266C2×C3—121no—
267C2×C3—121no—
268C2×C3—121no—
269C2×C3—122no—
270C2×C3—241no—
271C2×C3C3×#83411yescharacteristic
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22#3757⋊C311yescharacteristic
277C4—21no—
278C4—41no—
279C4—41no—
280C4—721no—
281C3—31no—
282C3—31no—
283C3—61no—
284C3C3×#453911yescharacteristic
285C2—21no—
286C2—41no—
287C2—41no—
288C2#63661⋊C311yescharacteristic
subgroupG/H
centerC2#63661⋊C3
commutatorC23×C32C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
FittingH(3,𝔽3)×#123C2
socleC2×C3C3×#834
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C32×D4C22×C3
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#352
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #10589405C3semidirect
7. C22×C4×H(3,𝔽3)C4semidirect
10. C22⋊C4×H(3,𝔽3)C4semidirect
36. C22×C32×C4Dic3semidirect
37. C22×C32×C4C3×C4semidirect
45. C32×C22⋊C4Dic3semidirect
46. C32×C22⋊C4C3×C4semidirect
155. H(3,𝔽3)#352semidirect
203. C22×C4(C3⋊Dic3)⋊C3semidirect
208. C22⋊C4(C3⋊Dic3)⋊C3semidirect
245. C32#4539semidirect
246. C32C3×#352semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1232
#63661⋊C32
#63660⋊C32
#105894053
C3×#45393