Finite Group Databasebeta

← C3×#10593753 #13173227

Subgroups — C3×#10593753 #13173227

Order 1728.

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

normal subgroups only

This group has 396 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×#64101C211yesmaximal, characteristic
2C3×#63779C211yesmaximal, characteristic, complemented
3C3×#63778C211yesmaximal, characteristic
4C3×#63778C211yesmaximal, characteristic
5C3×#63777C211yesmaximal, characteristic, complemented
6C3×C4⋊#622C211yesmaximal, characteristic, complemented
7C3×C4×#622C211yesmaximal, characteristic, complemented
8#10593753C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C33×C4⋊C4C411yescharacteristic
15C3×#3752C411yescharacteristic
16C3×#3752C411yescharacteristic
17C3×C4⋊(C3⋊Dic3)C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#64101C2×C311yescharacteristic
21C3×#892—62no—
22#63779C2×C311yescharacteristic, complemented
23#63778C2×C311yescharacteristic
24#63778C2×C311yescharacteristic
25#63777C2×C311yescharacteristic, complemented
26C4⋊#622C2×C311yescharacteristic, complemented
27C4×#622C2×C311yescharacteristic, complemented
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C3×#622—21no—
38C3×#622—21no—
39C3×#622—21no—
40C3×#622—21no—
41C3×#433—91nomaximal
42C2×C4×C3⋊D3C22×C311yescharacteristic
43C2×C3×C4×D3—62no—
44C2×#622C22×C311yescharacteristic
45C2×#622C22×C311yescharacteristic
46C2×#622C22×C311yescharacteristic
47C2×#622C22×C311yescharacteristic
48C32×C4⋊C4—22no—
49C32×C4⋊C4—42no—
50C32×C4⋊C4C3×C411yescharacteristic
51#3752C3×C411yescharacteristic
52#3752C3×C411yescharacteristic
53C4⋊(C3⋊Dic3)C3×C411yescharacteristic
54C4×C3⋊Dic3C22×C311yescharacteristic
55C4×C3⋊Dic3C22×C311yescharacteristic
56C3×(C2×D3)⋊C4—62no—
57C3×(C2×D3)⋊C4—62no—
58C3×C4⋊Dic3—62no—
59C3×C4×Dic3—62no—
60C3×C4×Dic3—62no—
61C22×C33C22×C411yescharacteristic
62C2×C3×C3⋊D3—21no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—21no—
65C33×C4—21no—
66C33×C4—21no—
67C33×C4—21no—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3#4111yescharacteristic
71C3×C3⋊Dic3#4111yescharacteristic
72C3×#117—91no—
73#892—62no—
74C3×C4⋊C8—91no—
75C3×C4⋊C8—91no—
76C3×C22⋊C8—91no—
77C3×C22⋊C8—91no—
78C3×#97—91no—
79C3×C4×C8—91no—
80C22×C3⋊D3C2×C3×C411yescharacteristic
81C22×C3×D3—62no—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4—42no—
87C2×C32×C4—42no—
88C2×C32×C4C2×C3×C411yescharacteristic
89C2×C32×C4C2×C3×C411yescharacteristic
90C2×C32×C4C2×C3×C411yescharacteristic
91C2×C3⋊Dic3C2×C3×C411yescharacteristic
92C2×C3⋊Dic3C2×C3×C411yescharacteristic
93C2×C3⋊Dic3C23×C311yescharacteristic
94C4×C3⋊D3—21no—
95C4×C3⋊D3—21no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C3×C4×D3—64no—
100C3×C4×D3—64no—
101#622—21no—
102#622—21no—
103#622—21no—
104#622—21no—
105#433—91noSylow
106C2×C33#11711yescharacteristic
107C2×C33#13111yescharacteristic
108C2×C33#13111yescharacteristic
109C3×C3⋊D3—41no—
110C22×C3×C4—91no—
111C2×C4×D3—62no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C2×C3×C8—91no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4#15711yescharacteristic
120C3×C22⋊C4—91no—
121C3×C22⋊C4—91no—
122C3×C42—91no—
123C3×C42—91no—
124(C2×D3)⋊C4—62no—
125(C2×D3)⋊C4—62no—
126C4⋊Dic3—62no—
127C4×Dic3—62no—
128C4×Dic3—62no—
129C22×C32—22no—
130C22×C32—42no—
131C22×C32C22×C3×C411yescharacteristic
132C2×C3⋊D3—21no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3—21no—
135C2×C3×D3—64no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C32×C4—21no—
139C32×C4—24no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—44no—
143C32×C4—42no—
144C32×C4—42no—
145C32×C4—82no—
146C32×C4—82no—
147C3⋊Dic3—21no—
148C3⋊Dic3—21no—
149C3⋊Dic3C3×#4111yescharacteristic
150C3⋊Dic3C3×#4111yescharacteristic
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—122no—
154C3×Dic3—122no—
155#117—91no—
156C4⋊C8—91no—
157C4⋊C8—91no—
158C22⋊C8—91no—
159C22⋊C8—91no—
160#97—91no—
161C4×C8—91no—
162C33#43311yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C22×D3—62no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4C2×#15711yescharacteristic
177C2×C3×C4C2×#15711yescharacteristic
178C2×C3×C4C2×#15711yescharacteristic
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C4×D3—64no—
183C4×D3—64no—
184C3×C8—181no—
185C3×C8—181no—
186C3×C8—181no—
187C3×C8—181no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—22no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32—42no—
194C2×C32C3×#11711yescharacteristic
195C2×C32C3×#13111yescharacteristic
196C2×C32C3×#13111yescharacteristic
197C3⋊D3—41no—
198C3×D3—124no—
199C22×C4—91no—
200C2×C8—91no—
201C2×C8—91no—
202C2×C8—91no—
203C2×C8—91no—
204C4⋊C4—91no—
205C4⋊C4C3×#15711yescharacteristic
206C22⋊C4—91no—
207C22⋊C4—91no—
208C42—91no—
209C42—91no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—181no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3C22×#15711yescharacteristic
216C2×D3—64no—
217C2×D3—122no—
218C2×D3—122no—
219C3×C4—24no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—44no—
226C3×C4—82no—
227C3×C4—82no—
228C3×C4—91no—
229C3×C4—91no—
230C3×C4—181no—
231C3×C4—181no—
232Dic3—62no—
233Dic3—62no—
234Dic3—122no—
235Dic3—122no—
236C32—22no—
237C32—42no—
238C32C3×#43311yescharacteristic, complemented
239C23—91no—
240C2×C4—91no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4C2×C3×#15711yescharacteristic
246C2×C4C2×C3×#15711yescharacteristic
247C2×C4C2×C3×#15711yescharacteristic
248C8—181no—
249C8—181no—
250C8—181no—
251C8—181no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—361no—
259C2×C3#6428611yescharacteristic
260C2×C3#6428711yescharacteristic
261C2×C3#6428911yescharacteristic
262D3—124no—
263C22—181no—
264C22—181no—
265C22—181no—
266C22C22×C3×#15711yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—91no—
271C4—91no—
272C4—181no—
273C4—181no—
274C3—22no—
275C3—42no—
276C3#1059375311yescharacteristic, direct product factor
277C2—361no—
278C2C3×#6428611yescharacteristic
279C2C3×#6428711yescharacteristic
280C2C3×#6428911yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63779C2semidirect
5. C3×#63777C2semidirect
6. C3×C4⋊#622C2semidirect
7. C3×C4×#622C2semidirect
8. #10593753C3direct (N × H)
22. #63779C2×C3semidirect
25. #63777C2×C3semidirect
26. C4⋊#622C2×C3semidirect
27. C4×#622C2×C3semidirect
162. C33#433semidirect
238. C32C3×#433semidirect
276. C3#10593753direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641012
C3×#637792
C3×#637782
C3×#637772
C3×C4⋊#6222
C3×C4×#6222
#105937533
C3×#4339