Finite Group Databasebeta

← C3×#73151 #11920929

Subgroups — C3×#73151 #11920929

Order 1152.

497 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3×#859C212yesmaximal, complemented
3#73151C311yesmaximal, characteristic, direct product factor
4C3×#1788—31nomaximal
5C2×C3×#41⋊C3—22no—
6C2×C3×#41⋊C3C412yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C3×#859—22no—
9C3×C3⋊#95—41nomaximal
10C22×C3×#41D311yescharacteristic
11C22×#41⋊C3—21no—
12C22×#41⋊C3C2×C311yescharacteristic
13C2×#859C2×C312yescomplemented
14C2×C3×#103—32no—
15C2×C3×#102—32no—
16C2×C3×C22⋊C8—31no—
17C3×#380—31no—
18C22×C32×C4—41no—
19C22×C3×Dic3—42no—
20C3×#41⋊C3—22no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C2×C3×SL(2,𝔽3)D412yes—
23#1788—31noSylow
24C2×C3×#41—22no—
25C2×C3×#41—32no—
26C2×C3×#41—62no—
27C2×C3×#41Dic312yescomplemented
28C22×C3×Q8C2×D311yescharacteristic
29C22×C3×D4—31no—
30C23×C3×C4—31no—
31C2×#41⋊C3—24no—
32C2×#41⋊C3—22no—
33C2×#41⋊C3—22no—
34C2×#41⋊C3C3×C412yescomplemented
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)C22×C311yescharacteristic
37C22×C3×C8—32no—
38C2×C3×C4⋊C4—32no—
39C2×C3×C22⋊C4—61no—
40#859—22no—
41C3×#103—62no—
42C3×#103—62no—
43C3×#102—64no—
44C3×C22⋊C8—62no—
45C3×#95—61no—
46C3⋊#95—41no—
47C23×C32—41no—
48C2×C32×C4—42no—
49C2×C32×C4—82no—
50C2×C3×Dic3—44no—
51C2×C3×Dic3—82no—
52C2×C3×Dic3—82no—
53C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
54C22×#41C3×D311yescharacteristic
55C2×#103—32no—
56C2×#102—32no—
57C2×C22⋊C8—31no—
58#380—31no—
59C24×C3—31no—
60C3×#41—22no—
61C3×#41—64no—
62C3×#41—62no—
63C3×#41—64no—
64C2×C3×Q8—32no—
65C2×C3×Q8—31no—
66C2×C3×Q8C2×Dic311yescharacteristic
67C2×C3×Q8#6712yes—
68C2×C3×D4—32no—
69C2×C3×D4—34no—
70C2×C3×D4—61no—
71C2×C3×D4—62no—
72C22×C3×C4—31no—
73C22×C3×C4—32no—
74C22×C3×C4—32no—
75C22×C3×C4—41no—
76C22×C3×C4—62no—
77C22×C3×C4—62no—
78C22×C3×C4—62no—
79C22×C3×C4—81no—
80C22×C3×C4S411yescharacteristic
81C22×Dic3—42no—
82#41⋊C3—24no—
83#41⋊C3—22no—
84C2×SL(2,𝔽3)—21no—
85C2×SL(2,𝔽3)—22no—
86C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
87C2×SL(2,𝔽3)C3×D412yes—
88C2×C3×C8—62no—
89C2×C3×C8—62no—
90C2×C3×C8—62no—
91C3×C4⋊C4—62no—
92C3×C4⋊C4—62no—
93C3×C22⋊C4—122no—
94C22×C32—41no—
95C22×C32—42no—
96C22×C32—41no—
97C22×C32—41no—
98C22×C32—42no—
99C32×C4—82no—
100C3×Dic3—84no—
101C2×#41—22no—
102C2×#41—32no—
103C2×#41—62no—
104C2×#41C3×Dic312yescomplemented
105C22×Q8C2×C3×D311yescharacteristic
106C22×D4—31no—
107C23×C4—31no—
108C22×C8—32no—
109C2×C4⋊C4—32no—
110C2×C22⋊C4—61no—
111#103—62no—
112#103—62no—
113#102—64no—
114C22⋊C8—62no—
115#95—61no—
116C23×C3—32no—
117C23×C3—41no—
118C23×C3—61no—
119C23×C3—62no—
120C23×C3—61no—
121C23×C3—62no—
122C23×C3—81no—
123C23×C3C2×S411yescharacteristic
124C3×Q8—31no—
125C3×Q8—32no—
126C3×Q8—61no—
127C3×Q8#21711yescharacteristic
128C3×D4—34no—
129C3×D4—62no—
130C3×D4—64no—
131C2×C3×C4—22no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—32no—
135C2×C3×C4—32no—
136C2×C3×C4—42no—
137C2×C3×C4—62no—
138C2×C3×C4—64no—
139C2×C3×C4—62no—
140C2×C3×C4—82no—
141C2×C3×C4—84no—
142C2×C3×C4—82no—
143C2×C3×C4—122no—
144C2×C3×C4—122no—
145C2×C3×C4—122no—
146C2×C3×C4SL(2,ℤ/4)12yes—
147C2×Dic3—44no—
148C2×Dic3—82no—
149C2×Dic3—82no—
150SL(2,𝔽3)—21no—
151SL(2,𝔽3)C3×C22⋊C411yescharacteristic, complemented
152C3×C8—122no—
153C2×C32—41no—
154C2×C32—42no—
155C2×C32—41no—
156C2×C32—42no—
157C2×C32—41no—
158C24—31no—
159#41—22no—
160#41—64no—
161#41—62no—
162#41—64no—
163C2×Q8—32no—
164C2×Q8—31no—
165C2×Q8C2×C3×Dic311yescharacteristic
166C2×Q8C3×#6712yes—
167C2×D4—32no—
168C2×D4—34no—
169C2×D4—61no—
170C2×D4—62no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4C3×S411yescharacteristic
178C2×C8—62no—
179C2×C8—62no—
180C2×C8—62no—
181C4⋊C4—62no—
182C4⋊C4—62no—
183C22⋊C4—122no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3—62no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—81no—
193C22×C3—81no—
194C22×C3—82no—
195C22×C3—81no—
196C22×C3—82no—
197C22×C3—122no—
198C22×C3—122no—
199C22×C3#98411yescharacteristic
200C22×C3#98512yes—
201C22×C3#98611yescharacteristic
202C22×C3C2×SL(2,ℤ/4)11yescharacteristic
203C22×C3GL(2,ℤ/4)12yes—
204C3×C4—22no—
205C3×C4—32no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—82no—
209C3×C4—84no—
210C3×C4—242no—
211Dic3—84no—
212C32—41noSylow
213C23—32no—
214C23—61no—
215C23—62no—
216C23—61no—
217C23—62no—
218C23C2×C3×S411yescharacteristic
219Q8—31no—
220Q8—32no—
221Q8—61no—
222Q8C3×#21711yescharacteristic
223D4—34no—
224D4—62no—
225D4—64no—
226C2×C4—22no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—32no—
230C2×C4—32no—
231C2×C4—62no—
232C2×C4—64no—
233C2×C4—62no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4C3×SL(2,ℤ/4)12yes—
238C8—122no—
239C2×C3—41no—
240C2×C3—42no—
241C2×C3—41no—
242C2×C3—42no—
243C2×C3—41no—
244C2×C3—81no—
245C2×C3—82no—
246C2×C3—81no—
247C2×C3—82no—
248C2×C3—81no—
249C2×C3—122no—
250C2×C3#542411yescharacteristic
251C2×C3#542512yes—
252C2×C3#542611yescharacteristic
253C2×C3#542712yes—
254C2×C3#543211yescharacteristic
255C22—62no—
256C22—62no—
257C22—62no—
258C22—122no—
259C22—122no—
260C22C3×#98411yescharacteristic
261C22C3×#98512yes—
262C22C3×#98611yescharacteristic
263C22C2×C3×SL(2,ℤ/4)11yescharacteristic
264C22C3×GL(2,ℤ/4)12yes—
265C4—22no—
266C4—32no—
267C4—31no—
268C4—31no—
269C4—242no—
270C3—41no—
271C3—81no—
272C3#7315111yescharacteristic, direct product factor
273C2—122no—
274C2C3×#542411yescharacteristic
275C2C3×#542512yes—
276C2C3×#542611yescharacteristic
277C2C3×#542712yes—
278C2C3×#543211yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC23C2×C3×S4
FittingC22×C3×#41D3
prime radicalC22×C3×#41⋊C3C2
cosocleC22×SL(2,𝔽3)C22×C3
hypercenterC22×C3×C4S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#859C2semidirect
3. #73151C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
13. C2×#859C2×C3semidirect
27. C2×C3×#41Dic3semidirect
34. C2×#41⋊C3C3×C4semidirect
53. C3×SL(2,𝔽3)C22⋊C4semidirect
104. C2×#41C3×Dic3semidirect
151. SL(2,𝔽3)C3×C22⋊C4semidirect
272. C3#73151direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3×#8592
#731513
C3×#17883
C3×C3⋊#954