Finite Group Databasebeta

← C3×#73148 #11920926

Subgroups — C3×#73148 #11920926

Order 1152.

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

normal subgroups only

This group has 613 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×#859C212yesmaximal, complemented
3#73148C311yesmaximal, characteristic, direct product factor
4C3×#1787—31nomaximal
5C22×C3×SL(2,𝔽3)—22no—
6C22×C3×SL(2,𝔽3)C412yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#217—41nomaximal
9C3×#859—22no—
10C23×C3×Q8D311yescharacteristic
11C23×SL(2,𝔽3)—21no—
12C23×SL(2,𝔽3)C2×C311yescharacteristic
13C2×#859C2×C312yescomplemented
14C2×C3×#103—32no—
15C2×C3×#103—32no—
16C2×C3×C22⋊C8—31no—
17C3×#380—31no—
18C24×C32—41no—
19C22×C3×Dic3—42no—
20C2×C3×SL(2,𝔽3)—22no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C2×C3×SL(2,𝔽3)D412yes—
23C3×#217—44no—
24#1787—31noSylow
25C22×C3×Q8—22no—
26C22×C3×Q8—31no—
27C22×C3×Q8—32no—
28C22×C3×Q8—62no—
29C22×C3×Q8Dic312yescomplemented
30C22×C3×Q8C2×D311yescharacteristic
31C23×C3×C4—31no—
32C22×SL(2,𝔽3)—24no—
33C22×SL(2,𝔽3)—22no—
34C22×SL(2,𝔽3)—22no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)C3×C412yescomplemented
37C22×SL(2,𝔽3)C22×C311yescharacteristic
38C22×C3×C8—32no—
39C2×C3×C4⋊C4—32no—
40C2×C3×C22⋊C4—61no—
41C2×#217—41no—
42#859—22no—
43C3×#103—62no—
44C3×#103—64no—
45C3×#103—62no—
46C3×C22⋊C8—62no—
47C3×#95—61no—
48C23×C32—42no—
49C23×C32—42no—
50C23×C32—42no—
51C23×C32—41no—
52C23×C32—82no—
53C23×C32—82no—
54C2×C3×Dic3—44no—
55C2×C3×Dic3—82no—
56C2×C3×Dic3—82no—
57C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
58C23×Q8C3×D311yescharacteristic
59C2×#103—32no—
60C2×#103—32no—
61C2×C22⋊C8—31no—
62#380—31no—
63C24×C3—41no—
64C24×C3—81no—
65C24×C3S411yescharacteristic
66C2×C3×Q8—22no—
67C2×C3×Q8—32no—
68C2×C3×Q8—32no—
69C2×C3×Q8—31no—
70C2×C3×Q8—34no—
71C2×C3×Q8—61no—
72C2×C3×Q8—62no—
73C2×C3×Q8—64no—
74C2×C3×Q8—62no—
75C2×C3×Q8—64no—
76C2×C3×Q8C2×Dic311yescharacteristic
77C2×C3×Q8#6712yes—
78C22×C3×C4—31no—
79C22×C3×C4—32no—
80C22×C3×C4—32no—
81C22×C3×C4—31no—
82C22×C3×C4—62no—
83C22×C3×C4—62no—
84C22×C3×C4—62no—
85C22×Dic3—42no—
86C2×SL(2,𝔽3)—24no—
87C2×SL(2,𝔽3)—22no—
88C2×SL(2,𝔽3)—21no—
89C2×SL(2,𝔽3)—22no—
90C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
91C2×SL(2,𝔽3)C3×D412yes—
92C2×C3×C8—62no—
93C2×C3×C8—62no—
94C2×C3×C8—62no—
95C3×C4⋊C4—62no—
96C3×C4⋊C4—62no—
97C3×C22⋊C4—122no—
98#217—44no—
99C22×C32—44no—
100C22×C32—41no—
101C22×C32—41no—
102C22×C32—42no—
103C22×C32—41no—
104C22×C32—42no—
105C22×C32—84no—
106C22×C32—82no—
107C22×C32—84no—
108C22×C32—82no—
109C3×Dic3—84no—
110C22×Q8—22no—
111C22×Q8—31no—
112C22×Q8—32no—
113C22×Q8—62no—
114C22×Q8C3×Dic312yescomplemented
115C22×Q8C2×C3×D311yescharacteristic
116C23×C4—31no—
117C22×C8—32no—
118C2×C4⋊C4—32no—
119C2×C22⋊C4—61no—
120#103—62no—
121#103—64no—
122#103—62no—
123C22⋊C8—62no—
124#95—61no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—41no—
128C23×C3—42no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3—82no—
132C23×C3—82no—
133C23×C3—84no—
134C23×C3—84no—
135C23×C3—81no—
136C23×C3—82no—
137C23×C3—82no—
138C23×C3—82no—
139C23×C3#22612yes—
140C23×C3GL(2,𝔽3)12yes—
141C23×C3SL(2,ℤ/4)12yes—
142C23×C3C2×S411yescharacteristic
143C3×Q8—31no—
144C3×Q8—32no—
145C3×Q8—34no—
146C3×Q8—61no—
147C3×Q8—62no—
148C3×Q8—64no—
149C3×Q8#21711yescharacteristic, complemented
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—62no—
156C2×C3×C4—64no—
157C2×C3×C4—62no—
158C2×C3×C4—62no—
159C2×C3×C4—122no—
160C2×C3×C4—122no—
161C2×C3×C4—122no—
162C2×Dic3—44no—
163C2×Dic3—82no—
164C2×Dic3—82no—
165SL(2,𝔽3)—21no—
166SL(2,𝔽3)C3×C22⋊C411yescharacteristic, complemented
167C3×C8—122no—
168C2×C32—41no—
169C2×C32—42no—
170C2×C32—41no—
171C2×C32—42no—
172C2×C32—41no—
173C2×C32—84no—
174C24C3×S411yescharacteristic
175C2×Q8—22no—
176C2×Q8—32no—
177C2×Q8—32no—
178C2×Q8—31no—
179C2×Q8—34no—
180C2×Q8—61no—
181C2×Q8—62no—
182C2×Q8—64no—
183C2×Q8—62no—
184C2×Q8—64no—
185C2×Q8C2×C3×Dic311yescharacteristic
186C2×Q8C3×#6712yes—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—32no—
190C22×C4—32no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—62no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—62no—
197C4⋊C4—62no—
198C4⋊C4—62no—
199C22⋊C4—122no—
200C22×C3—24no—
201C22×C3—22no—
202C22×C3—24no—
203C22×C3—22no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3—44no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—42no—
210C22×C3—84no—
211C22×C3—82no—
212C22×C3—84no—
213C22×C3—82no—
214C22×C3—88no—
215C22×C3—84no—
216C22×C3—88no—
217C22×C3—84no—
218C22×C3—81no—
219C22×C3—82no—
220C22×C3—84no—
221C22×C3—81no—
222C22×C3—81no—
223C22×C3—82no—
224C22×C3#85914yescomplemented
225C22×C3C2×#22611yescharacteristic
226C22×C3C2×GL(2,𝔽3)11yescharacteristic
227C22×C3#98312yes—
228C22×C3C2×SL(2,ℤ/4)11yescharacteristic
229C22×C3GL(2,ℤ/4)12yes—
230C3×C4—32no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—62no—
234C3×C4—242no—
235Dic3—84no—
236C32—41noSylow
237C23—22no—
238C23—22no—
239C23C3×#22612yes—
240C23C3×GL(2,𝔽3)12yes—
241C23C3×SL(2,ℤ/4)12yes—
242C23C2×C3×S411yescharacteristic
243Q8—31no—
244Q8—32no—
245Q8—34no—
246Q8—61no—
247Q8—62no—
248Q8—64no—
249Q8C3×#21711yescharacteristic, complemented
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—62no—
256C2×C4—64no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C8—122no—
263C2×C3—24no—
264C2×C3—41no—
265C2×C3—42no—
266C2×C3—41no—
267C2×C3—42no—
268C2×C3—41no—
269C2×C3—84no—
270C2×C3—88no—
271C2×C3—81no—
272C2×C3—82no—
273C2×C3—81no—
274C2×C3—82no—
275C2×C3—81no—
276C2×C3C2×#85911yescharacteristic
277C2×C3#541912yes—
278C2×C3#542011yescharacteristic
279C2×C3#542112yes—
280C2×C3#543211yescharacteristic
281C22—24no—
282C22—22no—
283C22—24no—
284C22—22no—
285C22C3×#85914yescomplemented
286C22C2×C3×#22611yescharacteristic
287C22C2×C3×GL(2,𝔽3)11yescharacteristic
288C22C3×#98312yes—
289C22C2×C3×SL(2,ℤ/4)11yescharacteristic
290C22C3×GL(2,ℤ/4)12yes—
291C4—32no—
292C4—31no—
293C4—31no—
294C4—62no—
295C4—242no—
296C3—41no—
297C3—81no—
298C3#7314811yescharacteristic, direct product factor
299C2—24no—
300C2C2×C3×#85911yescharacteristic
301C2C3×#541912yes—
302C2C3×#542011yescharacteristic
303C2C3×#542112yes—
304C2C3×#543211yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC23C2×C3×S4
FittingC23×C3×Q8D3
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC22×SL(2,𝔽3)C22×C3
hypercenterC24×C3S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#859C2semidirect
3. #73148C3direct (N × H)
6. C22×C3×SL(2,𝔽3)C4semidirect
13. C2×#859C2×C3semidirect
29. C22×C3×Q8Dic3semidirect
36. C22×SL(2,𝔽3)C3×C4semidirect
57. C3×SL(2,𝔽3)C22⋊C4semidirect
114. C22×Q8C3×Dic3semidirect
149. C3×Q8#217semidirect
166. SL(2,𝔽3)C3×C22⋊C4semidirect
224. C22×C3#859semidirect
249. Q8C3×#217semidirect
285. C22C3×#859semidirect
298. C3#73148direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×C3×#8592
#731483
C3×#17873
C2×C3×#2174