Finite Group Databasebeta

← C3×#10589519 #13146892

Subgroups — C3×#10589519 #13146892

Order 1728.

314 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C3×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
2C3×C4⋊((C22⋊C9)⋊C2)C211yesmaximal, characteristic
3C3×(C22⋊C9)⋊C8C211yesmaximal, characteristic, complemented
4C3×#5227—31nomaximal
5#10589519C311yesmaximal, characteristic, direct product factor
6C2×C3×(C22⋊C9)⋊C2—21no—
7C3×C4×C22⋊C9—21no—
8C3×C4×C22⋊C9C2211yescharacteristic
9C3×#3675—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×C3×#215—31no—
12C2×C3×#215—31no—
13C3×#930—31no—
14Q8×C22⋊C9—21no—
15Q8×C22⋊C9C2×C311yescharacteristic, complemented
16C4⋊((C22⋊C9)⋊C2)C2×C311yescharacteristic
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20(C22⋊C9)⋊C8C2×C311yescharacteristic, complemented
21C2×C3×C22⋊C9D411yescharacteristic
22C3×(C22⋊C9)⋊C2—41no—
23C3×Q8×C9—41no—
24C3×D36—41no—
25C3×C9⋊C8—41no—
26C3×#448—91no—
27#5227—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C22×C32×C4—21no—
32C22×C32×C4C2×D311yescharacteristic
33C2×C3×#67—61no—
34C2×C3×D12—31no—
35C2×(C22⋊C9)⋊C2—21no—
36C3×#215—62no—
37C3×#215—61no—
38C3×#215—61no—
39C3×(C2×D3)⋊C4—61no—
40C3×C4⋊Dic3—31no—
41C2×C3×C3⋊C8—31no—
42C2×C3×C3⋊C8—31no—
43C4×C22⋊C9—22no—
44C4×C22⋊C9—21no—
45C4×C22⋊C9—21no—
46C4×C22⋊C9C22×C311yescharacteristic
47#3675—41no—
48C2×C3×D9—81no—
49C3×C22⋊C9SD1611yescharacteristic, complemented
50C3×C4×C9—41no—
51C3×C4×C9—81no—
52C22×C3×Q8—21no—
53C22×C3×Q8D911yescharacteristic, complemented
54C22×C3×Q8C3×D311yescharacteristic
55C2×C3×SD16—91no—
56C2×C3×SD16—91no—
57C3×#121—91no—
58C2×#215—31no—
59C2×#215—31no—
60#930—31no—
61C3×#103—91no—
62C3×#103—91no—
63C3×C22⋊C8—91no—
64#835—31no—
65#835—31no—
66#830—31no—
67C23×C32#6711yescharacteristic
68C22×C3×D3—61no—
69C32×Q8—32no—
70C32×Q8—31no—
71C32×Q8—61no—
72C32×Q8S411yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C3×#67—121no—
78C3×#67—121no—
79C2×C3×Dic3—61no—
80C3×D12—61no—
81C3×D12—61no—
82C2×C22⋊C9—21no—
83C2×C22⋊C9C3×D411yescharacteristic
84(C22⋊C9)⋊C2—41no—
85C3×C3⋊C8—61no—
86C3×C3⋊C8—61no—
87Q8×C9—41no—
88Q8×C9—81no—
89D36—41no—
90C9⋊C8—41no—
91#448—91noSylow
92C2×C3×C9—41no—
93C3×D9—161no—
94C2×C3×Q8—31no—
95C2×C3×Q8—31no—
96C2×C3×Q8—31no—
97C2×C3×Q8—31no—
98C2×C3×Q8—61no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—61no—
102C2×C3×Q8—62no—
103C2×C3×D4—91no—
104C2×C3×D4—181no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D911yescharacteristic
110C22×C3×C4C2×C3×D311yescharacteristic
111C2×#67—61no—
112C2×D12—31no—
113C3×SD16—182no—
114C3×SD16—181no—
115C3×SD16—181no—
116C2×C3×C8—91no—
117C2×C3×C8—91no—
118C3×C4⋊C4—91no—
119C3×C22⋊C4—181no—
120#215—62no—
121#215—61no—
122#215—61no—
123(C2×D3)⋊C4—61no—
124C4⋊Dic3—31no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C22×C32—31no—
128C22×C32—31no—
129C22×C32#21511yescharacteristic
130C2×C3×D3—121no—
131C2×C3×D3—121no—
132C2×C3×D3—121no—
133C32×C4—21no—
134C32×C4—31no—
135C32×C4—61no—
136C32×C4C2×S411yescharacteristic
137C3×Dic3—121no—
138C2×D9—81no—
139C22⋊C9—21no—
140C22⋊C9C3×SD1611yescharacteristic, complemented
141C4×C9—41no—
142C4×C9—81no—
143C4×C9—82no—
144C4×C9—81no—
145C22×Q8C3×D911yescharacteristic, complemented
146C2×SD16—91no—
147C2×SD16—91no—
148#121—91no—
149#103—91no—
150#103—91no—
151C22⋊C8—91no—
152C3×C9—41noSylow
153C23×C3—21no—
154C23×C3—181no—
155C23×C3#61111yescharacteristic
156C23×C3C3×#6711yescharacteristic
157C22×D3—61no—
158C3×Q8—21no—
159C3×Q8—32no—
160C3×Q8—32no—
161C3×Q8—31no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8—62no—
165C3×Q8—61no—
166C3×Q8—61no—
167C3×Q8—62no—
168C3×Q8(C22⋊C9)⋊C211yescharacteristic, complemented
169C3×Q8C3×S411yescharacteristic
170C3×D4—181no—
171C3×D4—181no—
172C3×D4—361no—
173C3×D4—361no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—62no—
186C2×C3×C4—181no—
187#67—121no—
188#67—121no—
189C2×Dic3—61no—
190D12—61no—
191D12—61no—
192C3×C8—181no—
193C3×C8—181no—
194C3⋊C8—61no—
195C3⋊C8—61no—
196C2×C32—31no—
197C2×C32—31no—
198C2×C32GL(2,ℤ/4)11yescharacteristic
199C3×D3—241no—
200C2×C9—41no—
201C2×C9—81no—
202D9—161no—
203C2×Q8—31no—
204C2×Q8—31no—
205C2×Q8—61no—
206C2×D4—91no—
207C2×D4—181no—
208C22×C4—21no—
209C22×C4C2×C3×D911yescharacteristic
210SD16—182no—
211SD16—181no—
212SD16—181no—
213C2×C8—91no—
214C2×C8—91no—
215C4⋊C4—91no—
216C22⋊C4—181no—
217C22×C3—21no—
218C22×C3—31no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—361no—
225C22×C3—361no—
226C22×C3—361no—
227C22×C3#367511yescharacteristic, complemented
228C22×C3C3×#21511yescharacteristic
229C2×D3—121no—
230C2×D3—121no—
231C2×D3—121no—
232C3×C4—22no—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—61no—
239C3×C4—62no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—361no—
243C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
244C3×C4C2×C3×S411yescharacteristic
245Dic3—121no—
246C32#541711yescharacteristic
247C9—41no—
248C9—81no—
249C23—181no—
250C23C3×#61111yescharacteristic
251Q8—32no—
252Q8—31no—
253Q8—61no—
254Q8C3×(C22⋊C9)⋊C211yescharacteristic, complemented
255D4—181no—
256D4—181no—
257D4—361no—
258D4—361no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—181no—
264C8—181no—
265C8—181no—
266C2×C3—21no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—721no—
274C2×C3#6369411yescharacteristic
275C2×C3C3×GL(2,ℤ/4)11yescharacteristic
276D3—241no—
277C22—31no—
278C22—31no—
279C22—361no—
280C22—361no—
281C22—361no—
282C22C3×#367511yescharacteristic, complemented
283C4—21no—
284C4—31no—
285C4—61no—
286C4—361no—
287C4C2×C3×(C22⋊C9)⋊C211yescharacteristic
288C3—21no—
289C3#1058951911yescharacteristic, direct product factor
290C3C3×#541711yescharacteristic
291C2—31no—
292C2—31no—
293C2—721no—
294C2C3×#6369411yescharacteristic
subgroupG/H
centerC2×C3#63694
commutator, cosocleC4×C22⋊C9C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×Q8D3
socleC23×C32#67
prime radicalC3×C4⋊((C22⋊C9)⋊C2)C2
hypercenterC3×Q8(C22⋊C9)⋊C2
hypocenterC22⋊C9C3×SD16
LCSG ▷ C4×C22⋊C9 ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C22⋊C9C2semidirect
3. C3×(C22⋊C9)⋊C8C2semidirect
5. #10589519C3direct (N × H)
15. Q8×C22⋊C9C2×C3semidirect
20. (C22⋊C9)⋊C8C2×C3semidirect
49. C3×C22⋊C9SD16semidirect
53. C22×C3×Q8D9semidirect
140. C22⋊C9C3×SD16semidirect
145. C22×Q8C3×D9semidirect
168. C3×Q8(C22⋊C9)⋊C2semidirect
227. C22×C3#3675semidirect
254. Q8C3×(C22⋊C9)⋊C2semidirect
282. C22C3×#3675semidirect
289. C3#10589519direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
C3×C4⋊((C22⋊C9)⋊C2)2
C3×(C22⋊C9)⋊C82
C3×#52273
#105895193
C3×#36754