Finite Group Databasebeta

← C3×#10593516 #13166881

Subgroups — C3×#10593516 #13166881

Order 1728.

800 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C23×C3×#157C211yesmaximal, characteristic
2C2×C3×#3773C212yesmaximal, complemented
3C3×#63974C212yesmaximal
4C3×#63737C212yesmaximal, complemented
5#10593516C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×#157—24no—
8C22×C3×#157C414yescomplemented
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×C3×#624C2214yes—
12C3×#3773—24no—
13C32×#217—22no—
14C3×#3722—22no—
15C23×#157C2×C311yescharacteristic
16C2×#3773C2×C312yescomplemented
17C3×D3×C22⋊C4—62no—
18#63974C2×C312yes—
19#63737C2×C312yescomplemented
20C23×C33D411yescharacteristic
21C22×C3×C3⋊D3—22no—
22C22×C3×C3⋊D3—22no—
23C22×C3×C3⋊D3C2×C412yes—
24C22×C3×C3⋊D3D411yescharacteristic
25C22×C3×C3⋊D3Q812yes—
26C22×C3×C3⋊D3C2311yescharacteristic
27C2×C3×#157—22no—
28C2×C3×#157—28no—
29C2×C3×#157C2×C412yes—
30C2×C3×#157C2×C412yes—
31C2×C3×#157D414yes—
32C2×C32×Dic3—24no—
33C3×#624—24no—
34C3×#624—24no—
35C3×#380—91nomaximal
36C23×C3⋊D3C22×C311yescharacteristic
37C23×C3×D3—62no—
38C22×#157—24no—
39C22×#157C3×C414yescomplemented
40C22×#157C22×C311yescharacteristic
41C22×#157C22×C311yescharacteristic
42C2×C3×C4×D3—64no—
43C2×#624C22×C314yes—
44#3773—24no—
45C32×C22⋊C4—62no—
46C3×#217—22no—
47C3×#217—22no—
48C3×#217—42no—
49C3×(C2×D3)⋊C4—64no—
50#3722—22no—
51C22×C33—22no—
52C22×C33C4⋊C412yescomplemented
53C22×C33C2×D411yescharacteristic
54C2×C3×C3⋊D3—22no—
55C2×C3×C3⋊D3—22no—
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3—24no—
58C2×C3×C3⋊D3C4⋊C412yescomplemented
59C2×C3×C3⋊D3C22×C411yescharacteristic
60C2×C3×C3⋊D3C2×D412yes—
61C2×C3×C3⋊D3C2×Q811yescharacteristic
62C2×C3×C3⋊D3#4112yes—
63C3×#157—22no—
64C3×#157C22⋊C414yescomplemented
65C32×Dic3—44no—
66C23×C3×C4—91no—
67C2×C3×C4⋊C4—92no—
68C2×C3×C22⋊C4—92no—
69D3×C22⋊C4—62no—
70C3×#95—92no—
71C23×C32—22no—
72C23×C32—42no—
73C23×C32C3×D411yescharacteristic
74C22×C3⋊D3—22no—
75C22×C3⋊D3—22no—
76C22×C3⋊D3C2×C3×C412yes—
77C22×C3⋊D3C3×D411yescharacteristic
78C22×C3⋊D3C3×Q812yes—
79C22×C3⋊D3C23×C311yescharacteristic
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C22×C3×D3—64no—
83C22×C3×D3—64no—
84C22×C3×D3—124no—
85C22×C3×D3—124no—
86C2×#157—22no—
87C2×#157—28no—
88C2×#157C2×C3×C412yes—
89C2×#157C2×C3×C412yes—
90C2×#157C3×D414yes—
91C2×C32×C4—64no—
92C2×C3×Dic3—24no—
93C2×C3×Dic3—24no—
94C2×C3×Dic3—44no—
95C3×C4×D3—124no—
96C3×C4×D3—124no—
97#624—24no—
98#624—24no—
99#380—91noSylow
100C2×C33—22no—
101C2×C33C2×C4⋊C411yescharacteristic
102C2×C33#12112yes—
103C3×C3⋊D3—22no—
104C3×C3⋊D3C2×C22⋊C411yescharacteristic
105C3×C3⋊D3#11711yescharacteristic
106C3×C3⋊D3C22⋊Q812yes—
107C24×C3—91no—
108C23×D3—62no—
109C22×C3×C4—94no—
110C22×C3×C4—91no—
111C22×C3×C4—94no—
112C22×C3×C4—91no—
113C22×C3×C4—184no—
114C2×C4×D3—64no—
115C3×C4⋊C4—184no—
116C3×C22⋊C4—62no—
117C3×C22⋊C4—62no—
118C3×C22⋊C4—122no—
119C3×C22⋊C4—182no—
120#217—22no—
121(C2×D3)⋊C4—64no—
122C22×C32—22no—
123C22×C32—24no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32—44no—
127C22×C32—44no—
128C22×C32—84no—
129C22×C32C3×C4⋊C412yescomplemented
130C22×C32C2×C3×D411yescharacteristic
131C2×C3⋊D3—22no—
132C2×C3⋊D3—22no—
133C2×C3⋊D3—22no—
134C2×C3⋊D3—24no—
135C2×C3⋊D3C3×C4⋊C412yescomplemented
136C2×C3⋊D3C22×C3×C411yescharacteristic
137C2×C3⋊D3C2×C3×D412yes—
138C2×C3⋊D3C2×C3×Q811yescharacteristic
139C2×C3⋊D3C3×#4112yes—
140C2×C3×D3—62no—
141C2×C3×D3—64no—
142C2×C3×D3—62no—
143C2×C3×D3—64no—
144C2×C3×D3—64no—
145C2×C3×D3—124no—
146C2×C3×D3—124no—
147C2×C3×D3—128no—
148C2×C3×D3—124no—
149#157—22no—
150#157C3×C22⋊C414yescomplemented
151C32×C4—124no—
152C3×Dic3—44no—
153C3×Dic3—44no—
154C3×Dic3—84no—
155C23×C4—91no—
156C2×C4⋊C4—92no—
157C2×C22⋊C4—92no—
158#95—92no—
159C33#38011yescharacteristic, complemented, Sylow
160C23×C3—22no—
161C23×C3—42no—
162C23×C3—91no—
163C23×C3—91no—
164C23×C3—92no—
165C23×C3—92no—
166C23×C3—182no—
167C23×C3—182no—
168C23×C3GO+(4,𝔽2)11yescharacteristic
169C22×D3—62no—
170C22×D3—64no—
171C22×D3—62no—
172C22×D3—64no—
173C22×D3—124no—
174C22×D3—124no—
175C2×C3×C4—64no—
176C2×C3×C4—64no—
177C2×C3×C4—92no—
178C2×C3×C4—94no—
179C2×C3×C4—92no—
180C2×C3×C4—124no—
181C2×C3×C4—188no—
182C2×C3×C4—184no—
183C2×C3×C4—184no—
184C2×C3×C4—182no—
185C2×Dic3—24no—
186C4×D3—124no—
187C4×D3—124no—
188C2×C32—22no—
189C2×C32—24no—
190C2×C32—22no—
191C2×C32—42no—
192C2×C32—44no—
193C2×C32—44no—
194C2×C32—84no—
195C2×C32C2×C3×C4⋊C411yescharacteristic
196C2×C32C3×#12112yes—
197C3⋊D3—22no—
198C3⋊D3C2×C3×C22⋊C411yescharacteristic
199C3⋊D3C3×#11711yescharacteristic
200C3⋊D3C3×C22⋊Q812yes—
201C3×D3—62no—
202C3×D3—62no—
203C3×D3—64no—
204C3×D3—124no—
205C24—91no—
206C22×C4—94no—
207C22×C4—91no—
208C22×C4—94no—
209C22×C4—91no—
210C22×C4—184no—
211C4⋊C4—184no—
212C22⋊C4—62no—
213C22⋊C4—182no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—24no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—44no—
220C22×C3—84no—
221C22×C3—91no—
222C22×C3—91no—
223C22×C3—92no—
224C22×C3—92no—
225C22×C3—92no—
226C22×C3—182no—
227C22×C3—182no—
228C22×C3—184no—
229C22×C3—182no—
230C22×C3#377312yescomplemented
231C22×C3C2×GO+(4,𝔽2)11yescharacteristic
232C2×D3—64no—
233C2×D3—64no—
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—64no—
237C2×D3—124no—
238C2×D3—124no—
239C2×D3—128no—
240C2×D3—124no—
241C3×C4—94no—
242C3×C4—124no—
243C3×C4—124no—
244C3×C4—182no—
245C3×C4—244no—
246Dic3—44no—
247C32—22no—
248C32—42no—
249C32C3×#38011yescharacteristic, complemented
250C23—91no—
251C23—91no—
252C23—92no—
253C23—92no—
254C23—182no—
255C23—182no—
256C23C3×GO+(4,𝔽2)11yescharacteristic
257C2×C4—64no—
258C2×C4—94no—
259C2×C4—92no—
260C2×C4—92no—
261C2×C4—188no—
262C2×C4—184no—
263C2×C4—184no—
264C2×C4—182no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—24no—
268C2×C3—44no—
269C2×C3—42no—
270C2×C3—44no—
271C2×C3—84no—
272C2×C3—91no—
273C2×C3—91no—
274C2×C3—92no—
275C2×C3—182no—
276C2×C3C2×#377311yescharacteristic
277C2×C3#6424212yes—
278D3—62no—
279D3—62no—
280D3—64no—
281D3—124no—
282C22—22no—
283C22—91no—
284C22—91no—
285C22—92no—
286C22—92no—
287C22—92no—
288C22—182no—
289C22—182no—
290C22—184no—
291C22—182no—
292C22C3×#377312yescomplemented
293C22C2×C3×GO+(4,𝔽2)11yescharacteristic
294C4—94no—
295C4—124no—
296C4—182no—
297C3—22no—
298C3—42no—
299C3#1059351611yescharacteristic, direct product factor
300C2—22no—
301C2—91no—
302C2—91no—
303C2—92no—
304C2—182no—
305C2C2×C3×#377311yescharacteristic
306C2C3×#6424212yes—
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#380
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3773C2semidirect
4. C3×#63737C2semidirect
5. #10593516C3direct (N × H)
8. C22×C3×#157C4semidirect
16. C2×#3773C2×C3semidirect
19. #63737C2×C3semidirect
39. C22×#157C3×C4semidirect
52. C22×C33C4⋊C4semidirect
58. C2×C3×C3⋊D3C4⋊C4semidirect
64. C3×#157C22⋊C4semidirect
129. C22×C32C3×C4⋊C4semidirect
135. C2×C3⋊D3C3×C4⋊C4semidirect
150. #157C3×C22⋊C4semidirect
159. C33#380semidirect
230. C22×C3#3773semidirect
249. C32C3×#380semidirect
292. C22C3×#3773semidirect
299. C3#10593516direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×C3×#37732
C3×#639742
C3×#637372
#105935163
C3×#3809