Finite Group Databasebeta

← #10589338⋊C3 #13135340

Subgroups — #10589338⋊C3 #13135340

Order 1728.

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

normal subgroups only

This group has 290 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#64144⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#99C211yesmaximal, characteristic
3#63660⋊C3C211yesmaximal, characteristic
4#10589338C311yesmaximal, characteristic, complemented
5C3×#4472—31nomaximal
6C2×D4×H(3,𝔽3)C2211yescharacteristic
7C22×(C3⋊Dic3)⋊C3C411yescharacteristic, complemented
8C22⋊C4×H(3,𝔽3)—21no—
9#3757⋊C3—21no—
10#3757⋊C3C411yescharacteristic, complemented
11#3750⋊C3—21no—
12#3748⋊C3—21no—
13#64144C2×C311yescharacteristic
14C3×#935—31no—
15C32×#99—21no—
16C32×#99D311yescharacteristic
17C32×#99C2×C311yescharacteristic
18#63660C2×C311yescharacteristic
19C3×#833—31no—
20C23×H(3,𝔽3)C2×C411yescharacteristic
21C23×H(3,𝔽3)D411yescharacteristic
22D4×H(3,𝔽3)—41no—
23C2×C4×H(3,𝔽3)—41no—
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×(C3⋊Dic3)⋊C3—21no—
26C2×(C3⋊Dic3)⋊C3—21no—
27C2×(C3⋊Dic3)⋊C3—41no—
28(C3⋊(C3⋊C8))⋊C3—41no—
29C3×#352—91no—
30#4472—31no—
31#4472—91no—
32C2×C32×D4—21no—
33C2×C32×D4C2×D311yescharacteristic
34C2×C32×D4C22×C311yescharacteristic
35C22×C3⋊Dic3C3×C411yescharacteristic, complemented
36C22×C3×Dic3—31no—
37C32×C22⋊C4—22no—
38C32×C22⋊C4—21no—
39C32×C22⋊C4—21no—
40#3757—21no—
41#3757C3×C411yescharacteristic, complemented
42#3750—21no—
43#3748—21no—
44C3×#217—31no—
45C3×#217—61no—
46C3×Dic3⋊C4—61no—
47C3×#208—61no—
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—
57C3×#123—91no—
58#935—31no—
59#935—91no—
60C3×#100—91no—
61C3×#99—31no—
62C3×#99—31no—
63C3×#99—61no—
64C3×#99C3×D311yescharacteristic
65#833—31no—
66#833—91no—
67C23×C32—21no—
68C23×C32—21no—
69C23×C32C4×D311yescharacteristic
70C23×C32#6711yescharacteristic
71C23×C32C2×C3×C411yescharacteristic
72C23×C32C3×D411yescharacteristic
73C32×D4—41no—
74C32×D4—41no—
75C32×D4—42no—
76C2×C32×C4—21no—
77C2×C32×C4—42no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4D1211yescharacteristic
81C2×C32×C4C3×D411yescharacteristic
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3—41no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—121no—
88C3⋊(C3⋊C8)—41no—
89C3×C3⋊C8—121no—
90#352—91noSylow
91C2×H(3,𝔽3)—21no—
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)#9911yescharacteristic
95C2×C3×D4—31no—
96C2×C3×D4—31no—
97C2×C3×D4—61no—
98C2×C3×D4C2×C3×D311yescharacteristic
99C22×C3×C4—91no—
100C22×Dic3—31no—
101C22×Dic3—91no—
102C3×M16—181no—
103C3×C4⋊C4—181no—
104C3×C22⋊C4—21no—
105C3×C22⋊C4—61no—
106C3×C22⋊C4—61no—
107C3×C22⋊C4—62no—
108C3×C22⋊C4—91no—
109C3×C22⋊C4—181no—
110#217—31no—
111#217—61no—
112#217—91no—
113#217—181no—
114Dic3⋊C4—61no—
115Dic3⋊C4—181no—
116#208—61no—
117#208—181no—
118C22×C32—22no—
119C22×C32—21no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—41no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32—41no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32(C2×D3)⋊C411yescharacteristic
132C22×C32C3×C22⋊C411yescharacteristic
133C32×C4—21no—
134C32×C4—21no—
135C32×C4—41no—
136C32×C4—82no—
137C32×C4—81no—
138C32×C4—81no—
139C3⋊Dic3—41no—
140C3⋊Dic3—41no—
141C3×Dic3—121no—
142C3×Dic3—121no—
143#123—91no—
144#100—91no—
145#99(C3⋊D3)⋊C311yescharacteristic
146H(3,𝔽3)#35211yescharacteristic, complemented, Sylow
147C23×C3—31no—
148C23×C3—31no—
149C23×C3—31no—
150C23×C3—31no—
151C23×C3—61no—
152C23×C3—61no—
153C23×C3C3×C4×D311yescharacteristic
154C23×C3C3×#6711yescharacteristic
155C3×D4—41no—
156C3×D4—121no—
157C3×D4—121no—
158C3×D4—122no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—41no—
162C2×C3×C4—61no—
163C2×C3×C4—121no—
164C2×C3×C4—121no—
165C2×C3×C4—122no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4—361no—
169C2×C3×C4C3×D1211yescharacteristic
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—121no—
173C2×Dic3—181no—
174C2×Dic3—181no—
175C2×Dic3—361no—
176C3×C8—361no—
177C3⋊C8—121no—
178C3⋊C8—361no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—41no—
183C2×C32—42no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—81no—
189C2×C32#80611yescharacteristic
190C2×C32C3×#9911yescharacteristic
191C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
192C22×C4—91no—
193M16—181no—
194C4⋊C4—181no—
195C22⋊C4—21no—
196C22⋊C4—91no—
197C22⋊C4—181no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—61no—
205C22×C3—61no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3—62no—
209C22×C3—61no—
210C22×C3—121no—
211C22×C3—121no—
212C22×C3—121no—
213C22×C3—121no—
214C22×C3—121no—
215C22×C3—241no—
216C22×C3—241no—
217C22×C3C3×(C2×D3)⋊C411yescharacteristic
218C3×C4—21no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—81no—
222C3×C4—121no—
223C3×C4—241no—
224C3×C4—241no—
225C3×C4—242no—
226C3×C4—361no—
227C3×C4—361no—
228Dic3—121no—
229Dic3—121no—
230Dic3—361no—
231Dic3—361no—
232C32—21no—
233C32#447211yescharacteristic, complemented
234C32C3×#35211yescharacteristic, complemented
235C23C4×(C3⋊D3)⋊C311yescharacteristic
236C23#638⋊C311yescharacteristic
237D4—41no—
238C2×C4—41no—
239C2×C4—181no—
240C2×C4—181no—
241C2×C4—361no—
242C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
243C8—361no—
244C2×C3—21no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—61no—
250C2×C3—61no—
251C2×C3—61no—
252C2×C3—121no—
253C2×C3—121no—
254C2×C3—121no—
255C2×C3—121no—
256C2×C3—121no—
257C2×C3—122no—
258C2×C3—241no—
259C2×C3C3×#80611yescharacteristic
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22#3752⋊C311yescharacteristic
265C4—21no—
266C4—81no—
267C4—361no—
268C4—361no—
269C3—31no—
270C3—31no—
271C3—61no—
272C3C3×#447211yescharacteristic
273C2—21no—
274C2—41no—
275C2—41no—
276C2#63633⋊C311yescharacteristic
subgroupG/H
centerC2#63633⋊C3
commutatorC23×C32C2×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×#352
LCSG ▷ C23×C32 ▷ 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. #10589338C3semidirect
7. C22×(C3⋊Dic3)⋊C3C4semidirect
10. #3757⋊C3C4semidirect
35. C22×C3⋊Dic3C3×C4semidirect
41. #3757C3×C4semidirect
146. H(3,𝔽3)#352semidirect
233. C32#4472semidirect
234. C32C3×#352semidirect

Maximal subgroups

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

subgroupindex
#64144⋊C32
H(3,𝔽3)×#992
#63660⋊C32
#105893383
C3×#44723