Finite Group Databasebeta

← C3×C3⋊#5933 #13181424

Subgroups — C3×C3⋊#5933 #13181424

Order 1728.

432 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 34 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
1C32×#997C211yesmaximal, characteristic
2C3⋊#5933C311yesmaximal, characteristic, direct product factor
3C3×#5933—33nomaximal
4C3×#5242—31nomaximal
5C3×C3⋊SL(2,ℤ/4)—41nomaximal
6C3×C3⋊#226—42nomaximal
7C32×H(5,𝔽2)D311yescharacteristic
8C3×#997—21no—
9C3×#997—23no—
10C3×#997D313yes—
11C3×#997C2×C311yescharacteristic
12C3×#951—32no—
13C3×#936—31no—
14C3×#837—32no—
15C3×#833—31no—
16C2×C32×A4—41no—
17C2×C32×A4—81no—
18C32×SL(2,𝔽3)—42no—
19#5933—33no—
20C3×#455—91no—
21#5242—31no—
22C32×#41—32no—
23C2×C32×D4—31no—
24C2×C32×D4—61no—
25C2×C3×Dic6—31no—
26C3⋊SL(2,ℤ/4)—41no—
27C3⋊#226—42no—
28C3×SL(2,ℤ/4)—123no—
29C3×#226—126no—
30C3×#217—61no—
31C3×#217—61no—
32C3×#216—62no—
33C3×#214—62no—
34C3×C4×Dic3—31no—
35C3×#208—32no—
36C32×A4—41no—
37C32×A4—81no—
38C3×C3⋊Dic3—161no—
39C3×H(5,𝔽2)—21no—
40C3×H(5,𝔽2)C3×D311yescharacteristic
41C3×H(5,𝔽2)C3⋊D311yescharacteristic
42#997—23no—
43#997C3×D313yes—
44C3×#137—92no—
45C3×#124—91no—
46#951—32no—
47#936—31no—
48C3×#104—92no—
49C3×#100—91no—
50#837—32no—
51#833—31no—
52C23×C32—21no—
53C23×C32—31no—
54C23×C32S411yescharacteristic
55C2×C3×A4—41no—
56C2×C3×A4—43no—
57C2×C3×A4—83no—
58C2×C3×A4—81no—
59C2×C3×A4—83no—
60C2×C3×A4—86no—
61C2×C3×A4—82no—
62C2×C3×A4—81no—
63C32×Q8S412yes—
64C32×D4—32no—
65C32×D4—62no—
66C2×C32×C4—31no—
67C2×C32×C4—61no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—61no—
70C3×Dic6—62no—
71C3×SL(2,𝔽3)—42no—
72C3×SL(2,𝔽3)—46no—
73C3×SL(2,𝔽3)—86no—
74C3×SL(2,𝔽3)—82no—
75C3×C3⋊C8—62no—
76#455—91noSylow
77C2×C33—161no—
78C3×#41—32no—
79C3×#41—32no—
80C3×#41—62no—
81C2×C3×Q8—91no—
82C2×C3×D4—31no—
83C2×C3×D4—31no—
84C2×C3×D4—61no—
85C2×C3×D4—61no—
86C2×C3×D4—61no—
87C2×C3×D4—62no—
88C2×Dic6—31no—
89SL(2,ℤ/4)—123no—
90#226—126no—
91C3×Q16—182no—
92C3×SD16—182no—
93C3×M16—92no—
94C3×C22⋊C4—181no—
95C3×C22⋊C4—181no—
96C3×C42—91no—
97#217—61no—
98#217—61no—
99#216—62no—
100#214—62no—
101C4×Dic3—31no—
102#208—32no—
103C22×C32—31no—
104C22×C32—41no—
105C22×C32—61no—
106C22×C32—81no—
107C22×C32—121no—
108C3×A4—41no—
109C3×A4—43no—
110C3×A4—81no—
111C3×A4—83no—
112C3×A4—83no—
113C3×A4—81no—
114C3×A4—86no—
115C3×A4—82no—
116C32×C4—32no—
117C3⋊Dic3—161no—
118C3×Dic3—121no—
119C3×Dic3—121no—
120C3×Dic3—483no—
121H(5,𝔽2)C3×C3⋊D311yescharacteristic
122#137—92no—
123#124—91no—
124#104—92no—
125#100—91no—
126C33—161noSylow
127C23×C3—22no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—31no—
132C23×C3—31no—
133C23×C3—61no—
134C23×C3C3×S411yescharacteristic
135C23×C3C3⋊S411yescharacteristic
136C2×A4—43no—
137C2×A4—83no—
138C2×A4—86no—
139C2×A4—83no—
140C3×Q8—22no—
141C3×Q8—182no—
142C3×Q8C3×S412yes—
143C3×Q8C3⋊S412yes—
144C3×D4—32no—
145C3×D4—32no—
146C3×D4—62no—
147C3×D4—62no—
148C3×D4—62no—
149C3×D4—64no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—62no—
156C2×C3×C4—181no—
157C2×C3×C4—181no—
158C2×Dic3—61no—
159C2×Dic3—61no—
160Dic6—62no—
161SL(2,𝔽3)—46no—
162SL(2,𝔽3)—86no—
163C3×C8—182no—
164C3⋊C8—62no—
165C2×C32—61no—
166C2×C32—121no—
167C2×C32—161no—
168C2×C32—163no—
169C2×C32—323no—
170C2×C32—321no—
171C2×C32#102011yescharacteristic
172#41—32no—
173C2×Q8—91no—
174C2×D4—31no—
175C2×D4—61no—
176Q16—182no—
177SD16—182no—
178M16—92no—
179C22⋊C4—181no—
180C22⋊C4—181no—
181C42—91no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—61no—
187C22×C3—62no—
188C22×C3—61no—
189C22×C3—61no—
190C22×C3—81no—
191C22×C3—82no—
192C22×C3—81no—
193C22×C3—81no—
194C22×C3—121no—
195C22×C3—121no—
196C22×C3—241no—
197A4—43no—
198A4—83no—
199A4—83no—
200A4—86no—
201C3×C4—32no—
202C3×C4—32no—
203C3×C4—62no—
204C3×C4—361no—
205C3×C4—361no—
206Dic3—121no—
207Dic3—121no—
208Dic3—483no—
209C32—161no—
210C32—163no—
211C32—323no—
212C32—321no—
213C32#593311yescharacteristic, complemented
214C23—21no—
215C23—31no—
216C23C3×C3⋊S411yescharacteristic
217Q8—182no—
218Q8C3×C3⋊S412yes—
219D4—32no—
220D4—62no—
221C2×C4—31no—
222C2×C4—61no—
223C2×C4—181no—
224C2×C4—181no—
225C8—182no—
226C2×C3—21no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3—121no—
230C2×C3—122no—
231C2×C3—121no—
232C2×C3—121no—
233C2×C3—163no—
234C2×C3—323no—
235C2×C3C3×#102011yescharacteristic
236C2×C3C3⋊#102011yescharacteristic
237C22—31no—
238C22—41no—
239C22—61no—
240C22—81no—
241C22—121no—
242C4—32no—
243C4—361no—
244C4—361no—
245C3—21no—
246C3—163no—
247C3—323no—
248C3C3×#593311yescharacteristic, complemented
249C3C3⋊#593311yescharacteristic, direct product factor
250C2—61no—
251C2—121no—
252C2C3×C3⋊#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C3C3⋊#1020
commutator, cosocle, hypocenterC3×#997C2×C3
FrattiniC2C3×C3⋊#1020
FittingC32×H(5,𝔽2)D3
socleC2×C32#1020
prime radicalC32×#997C2
LCSG ▷ C3×#997
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3⋊#5933C3direct (N × H)
213. C32#5933semidirect
248. C3C3×#5933semidirect
249. C3C3⋊#5933direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#9972
C3⋊#59333
C3×#59333
C3×#52423
C3×C3⋊SL(2,ℤ/4)4
C3×C3⋊#2264