Finite Group Databasebeta

← C3×#10589521 #13146896

Subgroups — C3×#10589521 #13146896

Order 1728.

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

normal subgroups only

This group has 331 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic
2C2×C3×#3688C211yesmaximal, characteristic, complemented
3C3×#63421C211yesmaximal, characteristic, complemented
4C3×#5228—31nomaximal
5#10589521C311yesmaximal, characteristic, direct product factor
6C2×C3×Q8⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8C3×#3688—21no—
9C3×#3676—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×C3×#216—31no—
12C2×C3×#216—31no—
13C3×#924—31no—
14C22×Q8⋊C9—21no—
15C22×Q8⋊C9C2×C311yescharacteristic
16C2×#3688C2×C311yescharacteristic, complemented
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20#63421C2×C311yescharacteristic, complemented
21C23×C3×C9—41no—
22C2×C3×Dic9—41no—
23C2×C3×Dic9—41no—
24C3×Q8⋊C9D411yescharacteristic
25C3×#451—91no—
26#5228—31no—
27C2×C32×Q8—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C22×C32×C4—31no—
32C2×C3×Dic6—31no—
33C3×#217—61no—
34C3×#216—61no—
35C3×#216—62no—
36C3×#216—61no—
37C3×C4⋊Dic3—31no—
38C3×Dic3⋊C4—61no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41C2×Q8⋊C9—22no—
42C2×Q8⋊C9—21no—
43C2×Q8⋊C9—21no—
44C2×Q8⋊C9C22×C311yescharacteristic
45#3688—21no—
46#3676—41no—
47C22×C3×C9—41no—
48C22×C3×C9—42no—
49C22×C3×C9—81no—
50C22×C3×C9—81no—
51C3×Dic9—81no—
52C3×Dic9—81no—
53C22×C3×Q8—21no—
54C22×C3×Q8D911yescharacteristic
55C22×C3×Q8C3×D311yescharacteristic
56C2×C3×Q16—91no—
57C2×C3×Q16—91no—
58C3×C22⋊Q8—91no—
59C2×#216—31no—
60C2×#216—31no—
61#924—31no—
62C3×#103—91no—
63C3×#103—91no—
64C3×C22⋊C8—91no—
65#835—31no—
66#835—31no—
67#830—31no—
68C23×C32S411yescharacteristic
69C32×Q8—31no—
70C32×Q8—32no—
71C32×Q8—61no—
72C32×Q8#6711yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C3×Dic6—61no—
80C3×Dic6—61no—
81C23×C9—41no—
82C23×C9—81no—
83C3×C3⋊C8—61no—
84C3×C3⋊C8—61no—
85C2×Dic9—41no—
86C2×Dic9—41no—
87Q8⋊C9—21no—
88Q8⋊C9C3×D411yescharacteristic
89#451—91noSylow
90C2×C3×C9—41no—
91C2×C3×C9—41no—
92C2×C3×C9—41no—
93C2×C3×C9—82no—
94C2×C3×Q8—22no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—31no—
99C2×C3×Q8—31no—
100C2×C3×Q8—61no—
101C2×C3×Q8—61no—
102C2×C3×Q8—61no—
103C2×C3×Q8—62no—
104C2×C3×Q8—91no—
105C2×C3×Q8C2×D911yescharacteristic
106C2×C3×Q8C2×C3×D311yescharacteristic
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—61no—
110C2×Dic6—31no—
111C3×Q16—181no—
112C3×Q16—182no—
113C3×Q16—181no—
114C2×C3×C8—91no—
115C2×C3×C8—91no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—181no—
118C3×C22⋊C4—181no—
119#217—61no—
120#216—61no—
121#216—62no—
122#216—61no—
123C4⋊Dic3—31no—
124Dic3⋊C4—61no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32#22612yes—
130C22×C32C2×S411yescharacteristic
131C32×C4—31no—
132C32×C4—31no—
133C32×C4—61no—
134C3×Dic3—121no—
135C3×Dic3—121no—
136C22×C9—41no—
137C22×C9—42no—
138C22×C9—81no—
139C22×C9—81no—
140C22×C9—82no—
141C22×C9—82no—
142C22×C9—81no—
143C22×C9—82no—
144Dic9—81no—
145Dic9—81no—
146C22×Q8C3×D911yescharacteristic
147C2×Q16—91no—
148C2×Q16—91no—
149C22⋊Q8—91no—
150#103—91no—
151#103—91no—
152C22⋊C8—91no—
153C3×C9—41noSylow
154C23×C3—21no—
155C23×C3(C22⋊C9)⋊C211yescharacteristic
156C23×C3C3×S411yescharacteristic
157C3×Q8—21no—
158C3×Q8—31no—
159C3×Q8—31no—
160C3×Q8—32no—
161C3×Q8—32no—
162C3×Q8—61no—
163C3×Q8—62no—
164C3×Q8—61no—
165C3×Q8—61no—
166C3×Q8—62no—
167C3×Q8—181no—
168C3×Q8—181no—
169C3×Q8#61111yescharacteristic
170C3×Q8C3×#6711yescharacteristic
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187Dic6—61no—
188Dic6—61no—
189C3×C8—181no—
190C3×C8—181no—
191C3⋊C8—61no—
192C3⋊C8—61no—
193C2×C32—22no—
194C2×C32C2×#22611yescharacteristic
195C2×C32#98311yescharacteristic
196C2×C32GL(2,ℤ/4)11yescharacteristic
197C2×C9—41no—
198C2×C9—41no—
199C2×C9—41no—
200C2×C9—82no—
201C2×C9—84no—
202C2×C9—81no—
203C2×C9—81no—
204C2×C9—81no—
205C2×Q8—21no—
206C2×Q8—31no—
207C2×Q8—61no—
208C2×Q8—91no—
209C2×Q8C2×C3×D911yescharacteristic
210C22×C4—31no—
211Q16—181no—
212Q16—182no—
213Q16—181no—
214C2×C8—91no—
215C2×C8—91no—
216C4⋊C4—91no—
217C4⋊C4—181no—
218C22⋊C4—181no—
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—22no—
226C22×C3—21no—
227C22×C3#368812yescomplemented
228C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
229C22×C3C3×#22612yes—
230C22×C3C2×C3×S411yescharacteristic
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—61no—
236C3×C4—61no—
237C3×C4—62no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—361no—
241C3×C4—361no—
242Dic3—121no—
243Dic3—121no—
244C32#541911yescharacteristic
245C9—41no—
246C9—81no—
247C23C3×(C22⋊C9)⋊C211yescharacteristic
248Q8—31no—
249Q8—32no—
250Q8—61no—
251Q8—181no—
252Q8—181no—
253Q8C3×#61111yescharacteristic
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—181no—
259C2×C4—181no—
260C8—181no—
261C8—181no—
262C2×C3—24no—
263C2×C3—22no—
264C2×C3—22no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3C2×#368811yescharacteristic
269C2×C3#6368911yescharacteristic
270C2×C3#6369411yescharacteristic
271C2×C3C2×C3×#22611yescharacteristic
272C2×C3C3×#98311yescharacteristic
273C2×C3C3×GL(2,ℤ/4)11yescharacteristic
274C22—21no—
275C22—21no—
276C22C3×#368812yescomplemented
277C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
278C4—31no—
279C4—31no—
280C4—61no—
281C4—361no—
282C4—361no—
283C3—21no—
284C3#1058952111yescharacteristic, direct product factor
285C3C3×#541911yescharacteristic
286C2—22no—
287C2C2×C3×#368811yescharacteristic
288C2C3×#6368911yescharacteristic
289C2C3×#6369411yescharacteristic
subgroupG/H
centerC22×C3C2×(C22⋊C9)⋊C2
commutator, cosocleC2×Q8⋊C9C22×C3
FrattiniC22×C3C2×C3×S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC23×C32S4
hypercenterC23×C3(C22⋊C9)⋊C2
hypocenterQ8⋊C9C3×D4
LCSG ▷ C2×Q8⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3688C2semidirect
3. C3×#63421C2semidirect
5. #10589521C3direct (N × H)
16. C2×#3688C2×C3semidirect
20. #63421C2×C3semidirect
227. C22×C3#3688semidirect
276. C22C3×#3688semidirect
284. C3#10589521direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
C2×C3×#36882
C3×#634212
C3×#52283
#105895213
C3×#36764