Finite Group Databasebeta

← C3×#10593754 #13173225

Subgroups — C3×#10593754 #13173225

Order 1728.

380 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 92 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×#64096C211yesmaximal, characteristic
2C3×#63779C211yesmaximal, characteristic
3C3×#63779C211yesmaximal, characteristic
4C3×#63777C211yesmaximal, characteristic
5C3×#63777C211yesmaximal, characteristic
6C3×C4⋊#622C211yesmaximal, characteristic
7C3×C4×#622C211yesmaximal, characteristic
8#10593754C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C411yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C33×C4⋊C4C411yescharacteristic
15C3×#3750C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#64096C2×C311yescharacteristic
21C3×#887—62no—
22#63779C2×C311yescharacteristic
23#63779C2×C311yescharacteristic
24#63777C2×C311yescharacteristic
25#63777C2×C311yescharacteristic
26C4⋊#622C2×C311yescharacteristic
27C4×#622C2×C311yescharacteristic
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×#634—22no—
36C3×#622—21no—
37C3×#622—21no—
38C3×#622—21no—
39C3×#622Q812yescomplemented
40C3×#446—91nomaximal
41C2×#634C3×C411yescharacteristic
42C2×C3×Dic6—62no—
43C2×#622C22×C311yescharacteristic
44C2×#622C22×C311yescharacteristic
45C2×#622C22×C311yescharacteristic
46C2×#622C22×C311yescharacteristic
47C32×C4⋊C4—22no—
48C32×C4⋊C4—42no—
49C32×C4⋊C4C3×C411yescharacteristic
50#3750C3×C411yescharacteristic
51#3750C3×C411yescharacteristic
52C4×C3⋊Dic3C22×C311yescharacteristic
53C4×C3⋊Dic3C22×C311yescharacteristic
54C4×C3⋊Dic3C22×C311yescharacteristic
55C3×Dic3⋊C4—62no—
56C3×Dic3⋊C4—62no—
57C3×C4×Dic3—62no—
58C3×C4×Dic3—62no—
59C3×C4×Dic3—62no—
60C22×C33C22×C411yescharacteristic
61C33×C4—21no—
62C33×C4—21no—
63C33×C4—21no—
64C3×C3⋊Dic3—21no—
65C3×C3⋊Dic3—21no—
66C3×C3⋊Dic3M1612yes—
67C3×C3⋊Dic3C2×Q811yescharacteristic
68C3×C3⋊Dic3#4111yescharacteristic
69C3×C4×Q8—91no—
70#887—62no—
71C3×C4⋊C8—91no—
72C3×C4⋊C8—91no—
73C3×C4⋊C8—91no—
74C3×#97—91no—
75C3×#97—91no—
76C3×C4×C8—91no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—42no—
81C2×C32×C4—42no—
82C2×C32×C4—42no—
83C2×C32×C4C2×C3×C411yescharacteristic
84C2×C32×C4C2×C3×C411yescharacteristic
85C2×C32×C4C2×C3×C411yescharacteristic
86C2×C3⋊Dic3C2×C3×C411yescharacteristic
87C2×C3⋊Dic3C2×C3×C411yescharacteristic
88C2×C3⋊Dic3C2×C3×C411yescharacteristic
89C2×C3⋊Dic3C23×C311yescharacteristic
90#634—22no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×Dic6—68no—
96#622—21no—
97#622—21no—
98#622—21no—
99#622C3×Q812yescomplemented
100#446—91noSylow
101C2×C33C4×Q811yescharacteristic
102C2×C33C2×M1611yescharacteristic
103C2×C33#13111yescharacteristic
104C2×C3×Q8—91no—
105C2×Dic6—62no—
106C2×C3×C8—91no—
107C2×C3×C8—91no—
108C2×C3×C8—91no—
109C2×C3×C8—91no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—91no—
114C3×C4⋊C4#15711yescharacteristic
115C3×C42—91no—
116C3×C42—91no—
117C3×C42—91no—
118Dic3⋊C4—62no—
119Dic3⋊C4—62no—
120C4×Dic3—62no—
121C4×Dic3—62no—
122C4×Dic3—62no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C22×C3×C411yescharacteristic
126C32×C4—21no—
127C32×C4—24no—
128C32×C4—21no—
129C32×C4—21no—
130C32×C4—44no—
131C32×C4—42no—
132C32×C4—42no—
133C32×C4—82no—
134C32×C4—82no—
135C3⋊Dic3—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3C3×M1612yes—
138C3⋊Dic3C2×C3×Q811yescharacteristic
139C3⋊Dic3C3×#4111yescharacteristic
140C3×Dic3—64no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—64no—
144C3×Dic3—122no—
145C4×Q8—91no—
146C4⋊C8—91no—
147C4⋊C8—91no—
148C4⋊C8—91no—
149#97—91no—
150#97—91no—
151C4×C8—91no—
152C33#44611yescharacteristic, complemented, Sylow
153C3×Q8—182no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4C2×#15711yescharacteristic
165C2×C3×C4C2×#15711yescharacteristic
166C2×C3×C4C2×#15711yescharacteristic
167C2×Dic3—62no—
168C2×Dic3—62no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171Dic6—68no—
172C3×C8—92no—
173C3×C8—181no—
174C3×C8—181no—
175C3×C8—181no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—42no—
180C2×C32—42no—
181C2×C32—42no—
182C2×C32C3×C4×Q811yescharacteristic
183C2×C32C2×C3×M1611yescharacteristic
184C2×C32C3×#13111yescharacteristic
185C2×Q8—91no—
186C2×C8—91no—
187C2×C8—91no—
188C2×C8—91no—
189C2×C8—91no—
190C4⋊C4—91no—
191C4⋊C4—91no—
192C4⋊C4C3×#15711yescharacteristic
193C42—91no—
194C42—91no—
195C42—91no—
196C22×C3—22no—
197C22×C3—42no—
198C22×C3C22×#15711yescharacteristic
199C3×C4—24no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—42no—
204C3×C4—42no—
205C3×C4—44no—
206C3×C4—82no—
207C3×C4—82no—
208C3×C4—92no—
209C3×C4—91no—
210C3×C4—91no—
211C3×C4—181no—
212C3×C4—181no—
213Dic3—64no—
214Dic3—62no—
215Dic3—62no—
216Dic3—64no—
217Dic3—122no—
218C32—22no—
219C32—42no—
220C32C3×#44611yescharacteristic, complemented
221Q8—182no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4C2×C3×#15711yescharacteristic
227C2×C4C2×C3×#15711yescharacteristic
228C2×C4C2×C3×#15711yescharacteristic
229C8—92no—
230C8—181no—
231C8—181no—
232C8—181no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—42no—
239C2×C3#6428311yescharacteristic
240C2×C3#6428711yescharacteristic
241C2×C3Q8×#15711yescharacteristic
242C22C22×C3×#15711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—92no—
247C4—91no—
248C4—91no—
249C4—181no—
250C4—181no—
251C3—22no—
252C3—42no—
253C3#1059375411yescharacteristic, direct product factor
254C2C3×#6428311yescharacteristic
255C2C3×#6428711yescharacteristic
256C2C3×Q8×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socle, prime radicalC22×C33C22×C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#446
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #10593754C3direct (N × H)
39. C3×#622Q8semidirect
99. #622C3×Q8semidirect
152. C33#446semidirect
220. C32C3×#446semidirect
253. C3#10593754direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#637792
C3×#637772
C3×C4⋊#6222
C3×C4×#6222
#105937543
C3×#4469