Finite Group Databasebeta

← #10589529⋊C3 #13146921

Subgroups — #10589529⋊C3 #13146921

Order 1728.

318 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C2×#88691C211yesmaximal, characteristic, complemented
2C2×#3689⋊C3C211yesmaximal, characteristic, complemented
3#63421⋊C3C211yesmaximal, characteristic, complemented
4C3×#5238—31nomaximal
5#10589529C311yesmaximal, characteristic, complemented
6#88691—21no—
7C2×#6204C2211yescharacteristic
8#3689⋊C3—21no—
9(C2×AGL(1,ℤ/9))⋊C4—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×C3×#215—31no—
12C2×C3×#213—31no—
13C3×#930—31no—
14C2×#41⋊C9—21no—
15C2×#41⋊C9C2×C311yescharacteristic, complemented
16C2×#3689C2×C311yescharacteristic, complemented
17C3×#835—31no—
18C3×#832—31no—
19C3×#830—31no—
20#63421C2×C311yescharacteristic, complemented
21C22×AGL(1,ℤ/9)—41no—
22C2×C4×3−1+2—41no—
23C2×3−1+2⋊C4—41no—
24#6204D411yescharacteristic, complemented
25C3×#449—91no—
26#5238—31no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×#67—61no—
33C2×C3×D12—31no—
34C3×#215—61no—
35C3×#215—61no—
36C3×#213—62no—
37C3×(C2×D3)⋊C4—61no—
38C3×C4⋊Dic3—31no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41#41⋊C9—22no—
42#41⋊C9—21no—
43C2×Q8⋊C9—21no—
44C2×Q8⋊C9C22×C311yescharacteristic
45#3689—21no—
46(C2×D9)⋊C4—41no—
47C22×3−1+2—41no—
48C2×AGL(1,ℤ/9)—42no—
49C2×AGL(1,ℤ/9)—81no—
50C2×AGL(1,ℤ/9)—81no—
51C4×3−1+2—81no—
523−1+2⋊C4—81no—
53C2×C3×#41—31no—
54C2×C3×#41C3×D311yescharacteristic
55C2×C3×SD16—91no—
56C2×C3×D8—91no—
57C3×#121—91no—
58C2×#215—31no—
59C2×#213—31no—
60#930—31no—
61C3×#103—91no—
62C3×#102—91no—
63C3×C22⋊C8—91no—
64#835—31no—
65#832—31no—
66#830—31no—
67C23×C32—31no—
68C22×C3×D3—61no—
69C32×Q8—31no—
70C32×Q8#6711yescharacteristic
71C32×D4—32no—
72C32×D4—61no—
73C2×C32×C4—31no—
74C2×C32×C4—61no—
75C2×C32×C4—61no—
76C2×C32×C4S411yescharacteristic
77C3×#67—122no—
78C2×C3×Dic3—61no—
79C3×D12—61no—
80C3×D12—61no—
81C22×D9—41no—
82C3×C3⋊C8—61no—
83C3×C3⋊C8—61no—
84C2×C4×C9—41no—
85C2×C4×C9—81no—
86C2×Dic9—41no—
87Q8⋊C9—21no—
88Q8⋊C9C3×D411yescharacteristic, complemented
89#449—91noSylow
90C2×3−1+2—41no—
91C2×3−1+2—41no—
92C2×3−1+2—41no—
93AGL(1,ℤ/9)—82no—
94C3×#41—21no—
95C3×#41—61no—
96C3×#41—62no—
97C3×#41—61no—
98C3×#41—61no—
99C2×C3×Q8—31no—
100C2×C3×Q8C2×C3×D311yescharacteristic
101C2×C3×D4—31no—
102C2×C3×D4—31no—
103C2×C3×D4—61no—
104C2×C3×D4—91no—
105C2×C3×D4—181no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—61no—
109C2×#67—61no—
110C2×D12—31no—
111C3×SD16—181no—
112C3×SD16—181no—
113C3×D8—182no—
114C2×C3×C8—91no—
115C2×C3×C8—91no—
116C3×C4⋊C4—91no—
117C3×C22⋊C4—181no—
118#215—61no—
119#215—61no—
120#213—62no—
121(C2×D3)⋊C4—61no—
122C4⋊Dic3—31no—
123C2×C3⋊C8—31no—
124C2×C3⋊C8—31no—
125C22×C32—61no—
126C22×C32—61no—
127C22×C32—61no—
128C22×C32C2×S411yescharacteristic
129C2×C3×D3—121no—
130C2×C3×D3—121no—
131C2×C3×D3—121no—
132C32×C4—21no—
133C32×C4—31no—
134C32×C4—31no—
135C3×Dic3—121no—
136C22×C9—41no—
137C22×C9—81no—
138C2×D9—42no—
139C2×D9—81no—
140C2×D9—81no—
141C4×C9—81no—
142C4×C9—82no—
143Dic9—81no—
144C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
145C2×SD16—91no—
146C2×D8—91no—
147#121—91no—
148#103—91no—
149#102—91no—
150C22⋊C8—91no—
1513−1+2—41noSylow
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—61no—
155C23×C3—181no—
156C22×D3—61no—
157C3×Q8—31no—
158C3×Q8—31no—
159C3×Q8—31no—
160C3×Q8—61no—
161C3×Q8C3×#6711yescharacteristic
162C3×D4—32no—
163C3×D4—32no—
164C3×D4—62no—
165C3×D4—61no—
166C3×D4—61no—
167C3×D4—62no—
168C3×D4—181no—
169C3×D4—181no—
170C3×D4—362no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—181no—
182C2×C3×C4C3×S411yescharacteristic
183#67—122no—
184C2×Dic3—61no—
185D12—61no—
186D12—61no—
187C3×C8—181no—
188C3×C8—181no—
189C3⋊C8—61no—
190C3⋊C8—61no—
191C2×C32—121no—
192C2×C32#98511yescharacteristic
193C2×C32#98611yescharacteristic
194C2×C32GL(2,ℤ/4)11yescharacteristic
195C3×D3—241no—
196C2×C9—41no—
197C2×C9—41no—
198C2×C9—41no—
199C2×C9—81no—
200C2×C9—81no—
201C2×C9—81no—
202D9—82no—
203#41—21no—
204#41—61no—
205C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
206C2×D4—31no—
207C2×D4—91no—
208C2×D4—181no—
209C22×C4—31no—
210SD16—181no—
211SD16—181no—
212D8—182no—
213C2×C8—91no—
214C2×C8—91no—
215C4⋊C4—91no—
216C22⋊C4—181no—
217C22×C3—31no—
218C22×C3—62no—
219C22×C3—61no—
220C22×C3—61no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—61no—
225C22×C3—121no—
226C22×C3—121no—
227C22×C3—361no—
228C22×C3—361no—
229C22×C3—361no—
230C22×C3C2×C3×S411yescharacteristic
231C2×D3—121no—
232C2×D3—121no—
233C2×D3—121no—
234C3×C4—21no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—31no—
239C3×C4—61no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—361no—
243Dic3—121no—
244C32#542711yescharacteristic
245C9—41no—
246C9—81no—
247C23—31no—
248C23—181no—
249Q8—31no—
250Q8#611⋊C311yescharacteristic
251D4—32no—
252D4—61no—
253D4—181no—
254D4—181no—
255D4—362no—
256C2×C4—31no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—181no—
260C2×C4((C22⋊C9)⋊C2)⋊C311yescharacteristic
261C8—181no—
262C8—181no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—121no—
267C2×C3—122no—
268C2×C3—121no—
269C2×C3—721no—
270C2×C3C3×#98511yescharacteristic
271C2×C3C3×#98611yescharacteristic
272C2×C3C3×GL(2,ℤ/4)11yescharacteristic
273D3—241no—
274C22—61no—
275C22—61no—
276C22—61no—
277C22—361no—
278C22—361no—
279C22—361no—
280C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
281C4—21no—
282C4—31no—
283C4—31no—
284C4—361no—
285C3—31no—
286C3C3×#542711yescharacteristic
287C2—121no—
288C2—721no—
289C2#63691⋊C311yescharacteristic
290C2#63692⋊C311yescharacteristic
291C2#63694⋊C311yescharacteristic
subgroupG/H
centerC22C2×((C22⋊C9)⋊C2)⋊C3
commutator, cosocleC2×Q8⋊C9C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC2×C32×#41D3
hypercenterC2×C4((C22⋊C9)⋊C2)⋊C3
hypocenterQ8⋊C9C3×D4
LCSG ▷ C2×Q8⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88691C2semidirect
2. C2×#3689⋊C3C2semidirect
3. #63421⋊C3C2semidirect
5. #10589529C3semidirect
15. C2×#41⋊C9C2×C3semidirect
16. C2×#3689C2×C3semidirect
20. #63421C2×C3semidirect
24. #6204D4semidirect
88. Q8⋊C9C3×D4semidirect
144. C2×#41AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×#886912
C2×#3689⋊C32
#63421⋊C32
C3×#52383
#105895293
(C2×AGL(1,ℤ/9))⋊C44