Finite Group Databasebeta

← #10589523⋊C3 #13146901

Subgroups — #10589523⋊C3 #13146901

Order 1728.

350 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C22×#6204C211yesmaximal, characteristic, complemented
2C2×#3689⋊C3C211yesmaximal, characteristic, complemented
3#63421⋊C3C211yesmaximal, characteristic, complemented
4C3×#5227—31nomaximal
5#10589523C311yesmaximal, characteristic, complemented
6C2×#611⋊C3—41nomaximal
7C2×#6204—21no—
8C2×#6204C2211yescharacteristic
9#3689⋊C3—21no—
10C22×C32×Q8D311yescharacteristic
11C2×C3×#215—31no—
12C2×C3×#215—31no—
13C3×#930—31no—
14C22×Q8⋊C9—21no—
15C22×Q8⋊C9C2×C311yescharacteristic, complemented
16C2×#3689C2×C311yescharacteristic, complemented
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20#63421C2×C311yescharacteristic, complemented
21C23×3−1+2—41no—
22C22×AGL(1,ℤ/9)—41no—
23#611⋊C3—44no—
24C2×3−1+2⋊C4—41no—
25#6204D411yescharacteristic, complemented
26C3×#448—91no—
27#5227—31no—
28C2×C32×Q8—21no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×Q8C2×D311yescharacteristic
32C22×C32×C4—31no—
33C2×C3×#67—61no—
34C2×C3×D12—31no—
35C3×#215—61no—
36C3×#215—62no—
37C3×#215—61no—
38C3×(C2×D3)⋊C4—61no—
39C3×C4⋊Dic3—31no—
40C2×C3×C3⋊C8—31no—
41C2×C3×C3⋊C8—31no—
42C2×#611—41no—
43C2×Q8⋊C9—22no—
44C2×Q8⋊C9—21no—
45C2×Q8⋊C9—21no—
46C2×Q8⋊C9C22×C311yescharacteristic
47#3689—21no—
48C22×3−1+2—41no—
49C22×3−1+2—42no—
50C22×3−1+2—81no—
51C22×3−1+2—81no—
52C2×AGL(1,ℤ/9)—42no—
53C2×AGL(1,ℤ/9)—81no—
54C2×AGL(1,ℤ/9)—81no—
553−1+2⋊C4—42no—
56C22×C3×Q8—31no—
57C22×C3×Q8C3×D311yescharacteristic
58C2×C3×SD16—91no—
59C2×C3×SD16—91no—
60C3×#121—91no—
61C2×#215—31no—
62C2×#215—31no—
63#930—31no—
64C3×#103—91no—
65C3×#103—91no—
66C3×C22⋊C8—91no—
67#835—31no—
68#835—31no—
69#830—31no—
70C23×C32S411yescharacteristic
71C22×C3×D3—61no—
72C32×Q8—31no—
73C32×Q8—32no—
74C32×Q8—61no—
75C32×Q8#6711yescharacteristic
76C2×C32×C4—31no—
77C2×C32×C4—31no—
78C2×C32×C4—61no—
79C2×C32×C4—61no—
80C3×#67—122no—
81C2×C3×Dic3—61no—
82C3×D12—61no—
83C3×D12—61no—
84C23×C9—41no—
85C23×C9—81no—
86C22×D9—41no—
87C3×C3⋊C8—61no—
88C3×C3⋊C8—61no—
89#611—44no—
90C2×Dic9—41no—
91Q8⋊C9—21no—
92Q8⋊C9C3×D411yescharacteristic, complemented
93#448—91noSylow
94C2×3−1+2—41no—
95C2×3−1+2—41no—
96C2×3−1+2—41no—
97C2×3−1+2—82no—
98AGL(1,ℤ/9)—82no—
99C2×C3×Q8—21no—
100C2×C3×Q8—31no—
101C2×C3×Q8—31no—
102C2×C3×Q8—31no—
103C2×C3×Q8—61no—
104C2×C3×Q8—61no—
105C2×C3×Q8—62no—
106C2×C3×Q8—61no—
107C2×C3×Q8—61no—
108C2×C3×Q8C2×C3×D311yescharacteristic
109C2×C3×D4—91no—
110C2×C3×D4—181no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—61no—
114C2×#67—61no—
115C2×D12—31no—
116C3×SD16—181no—
117C3×SD16—182no—
118C3×SD16—181no—
119C2×C3×C8—91no—
120C2×C3×C8—91no—
121C3×C4⋊C4—91no—
122C3×C22⋊C4—181no—
123#215—61no—
124#215—62no—
125#215—61no—
126(C2×D3)⋊C4—61no—
127C4⋊Dic3—31no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—31no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32GL(2,𝔽3)12yes—
133C22×C32C2×S411yescharacteristic
134C2×C3×D3—121no—
135C2×C3×D3—121no—
136C2×C3×D3—121no—
137C32×C4—31no—
138C32×C4—31no—
139C32×C4—61no—
140C3×Dic3—121no—
141C22×C9—41no—
142C22×C9—42no—
143C22×C9—81no—
144C22×C9—81no—
145C22×C9—82no—
146C22×C9—82no—
147C22×C9—81no—
148C22×C9—82no—
149C2×D9—42no—
150C2×D9—81no—
151C2×D9—81no—
152Dic9—42no—
153C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
154C2×SD16—91no—
155C2×SD16—91no—
156#121—91no—
157#103—91no—
158#103—91no—
159C22⋊C8—91no—
1603−1+2—41noSylow
161C23×C3—31no—
162C23×C3—181no—
163C23×C3C3×S411yescharacteristic
164C22×D3—61no—
165C3×Q8—31no—
166C3×Q8—32no—
167C3×Q8—31no—
168C3×Q8—32no—
169C3×Q8—31no—
170C3×Q8—61no—
171C3×Q8—62no—
172C3×Q8—61no—
173C3×Q8—61no—
174C3×Q8—62no—
175C3×Q8C3×#6711yescharacteristic
176C3×D4—181no—
177C3×D4—181no—
178C3×D4—362no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—181no—
192#67—122no—
193C2×Dic3—61no—
194D12—61no—
195D12—61no—
196C3×C8—181no—
197C3×C8—181no—
198C3⋊C8—61no—
199C3⋊C8—61no—
200C2×C32—22no—
201C2×C32C2×GL(2,𝔽3)11yescharacteristic
202C2×C32#98311yescharacteristic
203C2×C32GL(2,ℤ/4)11yescharacteristic
204C3×D3—241no—
205C2×C9—41no—
206C2×C9—41no—
207C2×C9—41no—
208C2×C9—82no—
209C2×C9—84no—
210C2×C9—81no—
211C2×C9—81no—
212C2×C9—81no—
213D9—82no—
214C2×Q8—21no—
215C2×Q8—31no—
216C2×Q8—61no—
217C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
218C2×D4—91no—
219C2×D4—181no—
220C22×C4—31no—
221SD16—181no—
222SD16—182no—
223SD16—181no—
224C2×C8—91no—
225C2×C8—91no—
226C4⋊C4—91no—
227C22⋊C4—181no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—32no—
231C22×C3—31no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—361no—
235C22×C3—361no—
236C22×C3—361no—
237C22×C3C3×GL(2,𝔽3)12yes—
238C22×C3C2×C3×S411yescharacteristic
239C2×D3—121no—
240C2×D3—121no—
241C2×D3—121no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4—61no—
251C3×C4—361no—
252Dic3—121no—
253C32#542111yescharacteristic
254C9—41no—
255C9—81no—
256C23—181no—
257C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
258Q8—31no—
259Q8—32no—
260Q8—61no—
261Q8#611⋊C311yescharacteristic, complemented
262D4—181no—
263D4—181no—
264D4—362no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—181no—
270C8—181no—
271C8—181no—
272C2×C3—22no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—62no—
277C2×C3—721no—
278C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
279C2×C3C3×#98311yescharacteristic
280C2×C3C3×GL(2,ℤ/4)11yescharacteristic
281D3—241no—
282C22—21no—
283C22—21no—
284C22—361no—
285C22—361no—
286C22—361no—
287C22#3689⋊C312yescomplemented
288C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
289C4—31no—
290C4—31no—
291C4—61no—
292C4—361no—
293C3—31no—
294C3C3×#542111yescharacteristic
295C2—22no—
296C2—721no—
297C2C2×#3689⋊C311yescharacteristic
298C2#63689⋊C311yescharacteristic
299C2#63694⋊C311yescharacteristic
subgroupG/H
centerC22C2×((C22⋊C9)⋊C2)⋊C3
commutator, cosocleC2×Q8⋊C9C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC22×C32×Q8D3
hypercenterC23((C22⋊C9)⋊C2)⋊C3
hypocenterQ8⋊C9C3×D4
LCSG ▷ C2×Q8⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6204C2semidirect
2. C2×#3689⋊C3C2semidirect
3. #63421⋊C3C2semidirect
5. #10589523C3semidirect
15. C22×Q8⋊C9C2×C3semidirect
16. C2×#3689C2×C3semidirect
20. #63421C2×C3semidirect
25. #6204D4semidirect
92. Q8⋊C9C3×D4semidirect
153. C22×Q8AGL(1,ℤ/9)semidirect
261. Q8#611⋊C3semidirect
287. C22#3689⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#62042
C2×#3689⋊C32
#63421⋊C32
C3×#52273
#105895233
C2×#611⋊C34