Finite Group Databasebeta

← C3×#10590034 #13154884

Subgroups — C3×#10590034 #13154884

Order 1728.

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

normal subgroups only

This group has 332 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×#63865C211yesmaximal, characteristic
2C3×#63781C211yesmaximal, characteristic
3C3×#63557C211yesmaximal, characteristic
4#10590034C311yesmaximal, characteristic, direct product factor
5C22×C3×#157—21no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#627C411yescharacteristic, complemented
8C32×Dic3⋊C4—21no—
9C2×C32×C3⋊C8—21no—
10C3×#3724C411yescharacteristic, complemented
11C3×#3722—21no—
12C3×#896—61no—
13#63865C2×C311yescharacteristic
14#63781C2×C311yescharacteristic
15C3×(C2×D3)⋊C8—61no—
16#63557C2×C311yescharacteristic
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3×#157—22no—
19C2×C3×#157—41no—
20C2×C3×#157—41no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3C2×C411yescharacteristic
23C3×C4×C3⋊D3—41no—
24C2×C32×Dic3—21no—
25C2×C32×Dic3—21no—
26C3×#627—41no—
27C32×C3⋊C8—41no—
28C3×#328—91nomaximal
29C22×#157—21no—
30C2×C4×C3⋊D3C22×C311yescharacteristic
31C2×C3×C4×D3—61no—
32C2×C3×C4×D3—61no—
33C2×C3×Dic6—61no—
34C2×#627C3×C411yescharacteristic, complemented
35C2×C32×C8—61no—
36C32×C4⋊C4—61no—
37C3×(C2×D3)⋊C4—61no—
38C3×(C2×D3)⋊C4—61no—
39C3×Dic3⋊C4—21no—
40C3×Dic3⋊C4—21no—
41C3×Dic3⋊C4—41no—
42C3×Dic3⋊C4—61no—
43C2×C3×C3⋊C8—21no—
44C2×C3×C3⋊C8—21no—
45C2×C3×C3⋊C8—41no—
46#3724C3×C411yescharacteristic, complemented
47#3722—21no—
48C22×C33C22⋊C411yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3SD1611yescharacteristic
52C2×C3×C3⋊D3Q1611yescharacteristic
53C3×#157—42no—
54C33×C4—21no—
55C3×C3⋊Dic3—21no—
56C32×Dic3—41no—
57C32×Dic3—41no—
58C3×C22⋊Q8—91no—
59#896—61no—
60C3×C22⋊C8—91no—
61C3×#95—91no—
62(C2×D3)⋊C8—61no—
63C22×C3⋊D3C3×D411yescharacteristic
64C22×C3×D3—61no—
65C22×C3×D3—61no—
66C2×#157—22no—
67C2×#157—41no—
68C2×#157—41no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4—61no—
74C2×C32×C4—61no—
75C2×C32×C4C3×D411yescharacteristic
76C2×C3⋊Dic3C2×C3×C411yescharacteristic
77C4×C3⋊D3—41no—
78C2×C3×Dic3—21no—
79C2×C3×Dic3—21no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—21no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×C4×D3—122no—
87C3×C4×D3—121no—
88C3×C4×D3—121no—
89C3×Dic6—122no—
90#627—41no—
91C32×C8—121no—
92C3×C3⋊C8—41no—
93C3×C3⋊C8—41no—
94C3×C3⋊C8—42no—
95#328—91noSylow
96C2×C33#9911yescharacteristic
97C2×C33#10311yescharacteristic
98C2×C33#10411yescharacteristic
99C3×C3⋊D3—21no—
100C3×C3⋊D3—21no—
101C2×C3×Q8—91no—
102C22×C3×C4—91no—
103C22×C3×C4—181no—
104C2×C4×D3—61no—
105C2×C4×D3—61no—
106C2×Dic6—61no—
107C2×C3×C8—61no—
108C2×C3×C8—61no—
109C2×C3×C8—121no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—121no—
114C3×C22⋊C4—181no—
115(C2×D3)⋊C4—61no—
116(C2×D3)⋊C4—61no—
117Dic3⋊C4—21no—
118Dic3⋊C4—61no—
119C2×C3⋊C8—21no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32C3×C22⋊C411yescharacteristic
125C2×C3⋊D3—21no—
126C2×C3⋊D3—21no—
127C2×C3⋊D3C3×SD1611yescharacteristic
128C2×C3⋊D3C3×Q1611yescharacteristic
129C2×C3×D3—61no—
130C2×C3×D3—61no—
131C2×C3×D3—61no—
132C2×C3×D3—61no—
133C2×C3×D3—121no—
134C2×C3×D3—121no—
135C2×C3×D3—121no—
136C2×C3×D3—121no—
137#157—42no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—41no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—81no—
145C32×C4—121no—
146C32×C4—121no—
147C3⋊Dic3—21no—
148C3×Dic3—41no—
149C3×Dic3—41no—
150C3×Dic3—42no—
151C3×Dic3—41no—
152C3×Dic3—41no—
153C3×Dic3—62no—
154C3×Dic3—81no—
155C3×Dic3—121no—
156C22⋊Q8—91no—
157C22⋊C8—91no—
158#95—91no—
159C33#32811yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—61no—
162C22×D3—61no—
163C3×Q8—361no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—61no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—91no—
173C2×C3×C4—121no—
174C2×C3×C4—121no—
175C2×C3×C4—182no—
176C2×C3×C4—361no—
177C2×C3×C4—361no—
178C2×C3×C4—361no—
179C2×C3×C4GO+(4,𝔽2)11yescharacteristic
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C4×D3—122no—
185C4×D3—121no—
186C4×D3—121no—
187Dic6—122no—
188C3×C8—121no—
189C3×C8—121no—
190C3×C8—122no—
191C3⋊C8—41no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32C3×#9911yescharacteristic
205C2×C32C3×#10311yescharacteristic
206C2×C32C3×#10411yescharacteristic
207C3⋊D3—21no—
208C3⋊D3—21no—
209C3×D3—121no—
210C3×D3—121no—
211C3×D3—121no—
212C3×D3—121no—
213C2×Q8—91no—
214C22×C4—91no—
215C22×C4—181no—
216C2×C8—61no—
217C4⋊C4—61no—
218C4⋊C4—91no—
219C22⋊C4—181no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—91no—
225C22×C3—91no—
226C22×C3—181no—
227C22×C3—181no—
228C22×C3D32⋊C411yescharacteristic
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—121no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—81no—
244C3×C4—121no—
245C3×C4—121no—
246C3×C4—122no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—181no—
250C3×C4—241no—
251C3×C4—362no—
252Dic3—41no—
253Dic3—41no—
254Dic3—62no—
255Dic3—121no—
256C32—21no—
257C32—21no—
258C32—41no—
259C32—41no—
260C32C3×#32811yescharacteristic, complemented
261C23—91no—
262Q8—361no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—91no—
266C2×C4—182no—
267C2×C4—361no—
268C2×C4—361no—
269C2×C4—361no—
270C2×C4C3×GO+(4,𝔽2)11yescharacteristic
271C8—121no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—181no—
285C2×C3—181no—
286C2×C3#6373011yescharacteristic
287C2×C3#6373111yescharacteristic
288C2×C3#6373811yescharacteristic
289D3—121no—
290D3—121no—
291D3—121no—
292D3—121no—
293C22—91no—
294C22—91no—
295C22—181no—
296C22—181no—
297C22C3×D32⋊C411yescharacteristic
298C4—21no—
299C4—121no—
300C4—121no—
301C4—181no—
302C4—362no—
303C3—21no—
304C3—21no—
305C3—41no—
306C3—41no—
307C3#1059003411yescharacteristic, direct product factor
308C2—181no—
309C2—181no—
310C2C3×#6373011yescharacteristic
311C2C3×#6373111yescharacteristic
312C2C3×#6373811yescharacteristic
subgroupG/H
centerC22×C3D32⋊C4
commutatorC2×C3⋊Dic3C2×C3×C4
FrattiniC2×C4C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#328
LCSG ▷ C2×C3⋊Dic3 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10590034C3direct (N × H)
7. C2×C3×#627C4semidirect
10. C3×#3724C4semidirect
34. C2×#627C3×C4semidirect
46. #3724C3×C4semidirect
159. C33#328semidirect
260. C32C3×#328semidirect
307. C3#10590034direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638652
C3×#637812
C3×#635572
#105900343
C3×#3289