Finite Group Databasebeta

← C3×#10593744 #13173215

Subgroups — C3×#10593744 #13173215

Order 1728.

412 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 84 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×#64086C211yesmaximal, characteristic
2C3×C22⋊#622C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4C3×#63779C211yesmaximal, characteristic, complemented
5C3×#63777C211yesmaximal, characteristic, complemented
6C3×C4⋊#622C211yesmaximal, characteristic, complemented
7C3×C4×#622C211yesmaximal, characteristic, complemented
8#10593744C311yesmaximal, characteristic, direct product factor
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C33×C22⋊C4C411yescharacteristic
15C3×#3757C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×#3750C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#64086C2×C311yescharacteristic
21C3×#877—62no—
22C22⋊#622C2×C311yescharacteristic
23C22⋊#622C2×C311yescharacteristic
24#63779C2×C311yescharacteristic, complemented
25#63777C2×C311yescharacteristic, complemented
26C4⋊#622C2×C311yescharacteristic, complemented
27C4×#622C2×C311yescharacteristic, complemented
28C23×C33C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×#622—21no—
38C3×#622—21no—
39C3×#622—21no—
40C3×#622—21no—
41C3×#433—91nomaximal
42C22×C3⋊Dic3C22×C311yescharacteristic
43C22×C3×Dic3—62no—
44C2×#622C22×C311yescharacteristic
45C2×#622C22×C311yescharacteristic
46C2×#622C22×C311yescharacteristic
47C2×#622C22×C311yescharacteristic
48C32×C22⋊C4—22no—
49C32×C22⋊C4—42no—
50C32×C22⋊C4C3×C411yescharacteristic
51#3757C3×C411yescharacteristic
52#3750C3×C411yescharacteristic
53#3750C3×C411yescharacteristic
54C4×C3⋊Dic3C22×C311yescharacteristic
55C4×C3⋊Dic3C22×C311yescharacteristic
56C3×#217—62no—
57C3×Dic3⋊C4—62no—
58C3×Dic3⋊C4—62no—
59C3×C4×Dic3—62no—
60C3×C4×Dic3—62no—
61C22×C33—21no—
62C22×C33—21no—
63C22×C33—21no—
64C22×C33C22×C411yescharacteristic
65C33×C4—21no—
66C33×C4—21no—
67C3×C3⋊Dic3—21no—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3#4111yescharacteristic
71C3×C3⋊Dic3#4111yescharacteristic
72C3×#117—91no—
73#877—62no—
74C3×C4⋊C8—91no—
75C3×C4⋊C8—91no—
76C3×C22⋊C8—91no—
77C3×C22⋊C8—91no—
78C3×#97—91no—
79C3×C4×C8—91no—
80C23×C32—22no—
81C23×C32—42no—
82C23×C32C2×C3×C411yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4—42no—
87C2×C32×C4C2×C3×C411yescharacteristic
88C2×C32×C4C2×C3×C411yescharacteristic
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3C2×C3×C411yescharacteristic
92C2×C3⋊Dic3C2×C3×C411yescharacteristic
93C2×C3⋊Dic3C2×C3×C411yescharacteristic
94C2×C3⋊Dic3C23×C311yescharacteristic
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—64no—
101#622—21no—
102#622—21no—
103#622—21no—
104#622—21no—
105#433—91noSylow
106C2×C33—41no—
107C2×C33#11711yescharacteristic
108C2×C33#13111yescharacteristic
109C2×C33#13111yescharacteristic
110C22×C3×C4—91no—
111C22×Dic3—62no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C2×C3×C8—91no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—91no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—91no—
121C3×C22⋊C4#15711yescharacteristic
122C3×C42—91no—
123C3×C42—91no—
124#217—62no—
125Dic3⋊C4—62no—
126Dic3⋊C4—62no—
127C4×Dic3—62no—
128C4×Dic3—62no—
129C22×C32—21no—
130C22×C32—24no—
131C22×C32—22no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—44no—
135C22×C32—42no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—82no—
139C22×C32—82no—
140C22×C32C22×C3×C411yescharacteristic
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—42no—
144C32×C4—42no—
145C32×C4—82no—
146C32×C4—82no—
147C3⋊Dic3—21no—
148C3⋊Dic3—21no—
149C3⋊Dic3—21no—
150C3⋊Dic3C3×#4111yescharacteristic
151C3⋊Dic3C3×#4111yescharacteristic
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—64no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157#117—91no—
158C4⋊C8—91no—
159C4⋊C8—91no—
160C22⋊C8—91no—
161C22⋊C8—91no—
162#97—91no—
163C4×C8—91no—
164C33#43311yescharacteristic, complemented, Sylow
165C23×C3—22no—
166C23×C3—42no—
167C23×C3C2×#15711yescharacteristic
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4C2×#15711yescharacteristic
179C2×C3×C4C2×#15711yescharacteristic
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—64no—
185C2×Dic3—64no—
186C3×C8—181no—
187C3×C8—181no—
188C3×C8—181no—
189C3×C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—42no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—41no—
197C2×C32—44no—
198C2×C32—84no—
199C2×C32C3×#11711yescharacteristic
200C2×C32C3×#13111yescharacteristic
201C2×C32C3×#13111yescharacteristic
202C22×C4—91no—
203C2×C8—91no—
204C2×C8—91no—
205C2×C8—91no—
206C2×C8—91no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C22⋊C4—91no—
210C22⋊C4C3×#15711yescharacteristic
211C42—91no—
212C42—91no—
213C22×C3—24no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—44no—
221C22×C3—42no—
222C22×C3—82no—
223C22×C3—82no—
224C22×C3C22×#15711yescharacteristic
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—42no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—82no—
231C3×C4—91no—
232C3×C4—91no—
233C3×C4—181no—
234C3×C4—181no—
235C3×C4—181no—
236Dic3—62no—
237Dic3—62no—
238Dic3—64no—
239Dic3—122no—
240Dic3—122no—
241C32—22no—
242C32—42no—
243C32C3×#43311yescharacteristic, complemented
244C23C2×C3×#15711yescharacteristic
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—181no—
250C2×C4—181no—
251C2×C4C2×C3×#15711yescharacteristic
252C2×C4C2×C3×#15711yescharacteristic
253C8—181no—
254C8—181no—
255C8—181no—
256C8—181no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—44no—
261C2×C3—41no—
262C2×C3—42no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3—84no—
266C2×C3#6428611yescharacteristic
267C2×C3#6428711yescharacteristic
268C2×C3#6428711yescharacteristic
269C22—21no—
270C22—21no—
271C22—21no—
272C22C22×C3×#15711yescharacteristic
273C4—21no—
274C4—21no—
275C4—91no—
276C4—91no—
277C4—181no—
278C4—181no—
279C4—181no—
280C3—22no—
281C3—42no—
282C3#1059374411yescharacteristic, direct product factor
283C2—41no—
284C2C3×#6428611yescharacteristic
285C2C3×#6428711yescharacteristic
286C2C3×#6428711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C22⋊C4C4
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C22⋊C4#157
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63779C2semidirect
5. C3×#63777C2semidirect
6. C3×C4⋊#622C2semidirect
7. C3×C4×#622C2semidirect
8. #10593744C3direct (N × H)
24. #63779C2×C3semidirect
25. #63777C2×C3semidirect
26. C4⋊#622C2×C3semidirect
27. C4×#622C2×C3semidirect
164. C33#433semidirect
243. C32C3×#433semidirect
282. C3#10593744direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640862
C3×C22⋊#6222
C3×#637792
C3×#637772
C3×C4⋊#6222
C3×C4×#6222
#105937443
C3×#4339