Finite Group Databasebeta

← C3×#87508 #11923750

Subgroups — C3×#87508 #11923750

Order 1152.

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

normal subgroups only

This group has 542 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
1C3×#5948C211yesmaximal, characteristic
2C3×#5930C212yesmaximal, complemented
3#87508C311yesmaximal, characteristic, direct product factor
4C3×#3007—31nomaximal
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
6C3×#984—42nomaximal
7C3×A4⋊Q8—41nomaximal
8C22×C3×#41D311yescharacteristic
9#5948—21no—
10#5948C2×C311yescharacteristic
11#5930C2×C312yescomplemented
12C2×C3×#137—32no—
13C2×C3×C22⋊Q8—31no—
14C3×#452—34no—
15C3×#451—32no—
16C3×#450—32no—
17C3×#417—32no—
18C3×#407—31no—
19C3×#41⋊C3—42no—
20C3×#41⋊C3—81no—
21C3×C4×A4—41no—
22C3×SL(2,ℤ/4)—42no—
23C3×#226—44no—
24#3007—31noSylow
25C2×C3×#41—32no—
26C2×C3×#41—32no—
27C2×C3×#41—61no—
28C2×C3×#41—61no—
29C22×C3×Q8—31no—
30C22×C3×Q8C2×D311yescharacteristic
31C22×C3×D4—31no—
32C23×C3×C4—31no—
33C22⋊SL(2,𝔽3)—21no—
34C22⋊SL(2,𝔽3)C22×C311yescharacteristic
35#984—42no—
36A4⋊Q8—41no—
37C3×#137—62no—
38C3×#137—62no—
39C3×#137—62no—
40C3×#137—62no—
41C2×C3×Q16—34no—
42C2×C3×SD16—34no—
43C2×C3×M16—32no—
44C3×C22⋊Q8—32no—
45C3×C22⋊Q8—32no—
46C3×C22⋊Q8—61no—
47C3×C22⋊Q8—61no—
48C2×C3×C4⋊C4—31no—
49C2×C3×C4⋊C4—61no—
50C2×C3×C22⋊C4—61no—
51C3×#103—34no—
52C3×#102—34no—
53C3×C22⋊C8—32no—
54C3×C22⋊C8—32no—
55C2×C3×A4—41no—
56C3×Dic6—161no—
57C3×SL(2,𝔽3)—42no—
58C3×SL(2,𝔽3)—81no—
59C22×#41C3×D311yescharacteristic
60C2×#137—32no—
61C2×C22⋊Q8—31no—
62#452—34no—
63#451—32no—
64#450—32no—
65#417—32no—
66#407—31no—
67C24×C3—31no—
68C3×#41—21no—
69C3×#41—32no—
70C3×#41—32no—
71C3×#41—32no—
72C3×#41—61no—
73C3×#41—62no—
74C3×#41—61no—
75C3×#41—61no—
76C3×#41—62no—
77C3×#41S412yes—
78C2×C3×Q8—32no—
79C2×C3×Q8—32no—
80C2×C3×Q8—61no—
81C2×C3×Q8—61no—
82C2×C3×Q8—62no—
83C2×C3×Q8—62no—
84C2×C3×D4—32no—
85C2×C3×D4—61no—
86C2×C3×D4—62no—
87C2×C3×D4—62no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—62no—
94C22×C3×C4—62no—
95C22×C3×C4—61no—
96C22×C3×C4—61no—
97C22×C3×C4S411yescharacteristic
98#41⋊C3—42no—
99#41⋊C3—81no—
100#41⋊C3—82no—
101#41⋊C3—82no—
102C4×A4—41no—
103C4×A4—81no—
104SL(2,ℤ/4)—42no—
105#226—44no—
106C3×Q16—64no—
107C3×Q16—64no—
108C3×SD16—64no—
109C3×SD16—64no—
110C3×M16—62no—
111C3×M16—62no—
112C2×C3×C8—34no—
113C3×C4⋊C4—32no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—62no—
116C3×C4⋊C4—62no—
117C3×C22⋊C4—62no—
118C3×C22⋊C4—62no—
119C3×A4—41no—
120C32×C4—161no—
121C3×Dic3—162no—
122C2×#41—32no—
123C2×#41—32no—
124C2×#41—61no—
125C2×#41—61no—
126C22×Q8—31no—
127C22×Q8C2×C3×D311yescharacteristic
128C22×D4—31no—
129C23×C4—31no—
130#137—62no—
131#137—62no—
132#137—62no—
133#137—62no—
134C2×Q16—34no—
135C2×SD16—34no—
136C2×M16—32no—
137C22⋊Q8—32no—
138C22⋊Q8—32no—
139C22⋊Q8—61no—
140C22⋊Q8—61no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C22⋊C4—61no—
144#103—34no—
145#102—34no—
146C22⋊C8—32no—
147C22⋊C8—32no—
148C23×C3—31no—
149C23×C3—31no—
150C23×C3—62no—
151C23×C3—61no—
152C23×C3—61no—
153C23×C3—121no—
154C23×C3C2×S411yescharacteristic
155C2×A4—41no—
156C2×A4—81no—
157C3×Q8—21no—
158C3×Q8—32no—
159C3×Q8—61no—
160C3×Q8—64no—
161C3×Q8—121no—
162C3×Q8—121no—
163C3×Q8C2×S412yes—
164C3×D4—32no—
165C3×D4—32no—
166C3×D4—62no—
167C3×D4—61no—
168C3×D4—62no—
169C3×D4—61no—
170C2×C3×C4—31no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—32no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—122no—
184C2×C3×C4—122no—
185Dic6—161no—
186SL(2,𝔽3)—42no—
187SL(2,𝔽3)—81no—
188SL(2,𝔽3)—82no—
189SL(2,𝔽3)—82no—
190C3×C8—64no—
191C2×C32—161no—
192C24—31no—
193#41—21no—
194#41—32no—
195#41—32no—
196#41—32no—
197#41—61no—
198#41—62no—
199#41—61no—
200#41—61no—
201#41—62no—
202#41C3×S412yes—
203C2×Q8—32no—
204C2×Q8—32no—
205C2×Q8—61no—
206C2×Q8—61no—
207C2×Q8—62no—
208C2×Q8—62no—
209C2×D4—32no—
210C2×D4—61no—
211C2×D4—62no—
212C2×D4—62no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C3×S411yescharacteristic
223Q16—64no—
224Q16—64no—
225SD16—64no—
226SD16—64no—
227M16—62no—
228M16—62no—
229C2×C8—34no—
230C4⋊C4—32no—
231C4⋊C4—61no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C22⋊C4—62no—
235C22⋊C4—62no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—32no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3—122no—
245C22×C3—122no—
246C22×C3#98411yescharacteristic, complemented
247A4—41no—
248A4—81no—
249C3×C4—32no—
250C3×C4—31no—
251C3×C4—61no—
252C3×C4—122no—
253C3×C4—122no—
254C3×C4—161no—
255C3×C4—321no—
256C3×C4#102011yescharacteristic
257Dic3—162no—
258C32—161noSylow
259C23—31no—
260C23—31no—
261C23—62no—
262C23—61no—
263C23—61no—
264C23—121no—
265C23C2×C3×S411yescharacteristic
266Q8—21no—
267Q8—32no—
268Q8—61no—
269Q8—64no—
270Q8—121no—
271Q8—121no—
272Q8C2×C3×S412yes—
273D4—32no—
274D4—32no—
275D4—62no—
276D4—61no—
277D4—62no—
278D4—61no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—31no—
282C2×C4—32no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—122no—
293C2×C4—122no—
294C8—64no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—62no—
298C2×C3—121no—
299C2×C3—161no—
300C2×C3—321no—
301C2×C3C2×#102011yescharacteristic
302C22—31no—
303C22—31no—
304C22—32no—
305C22—61no—
306C22—61no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22—122no—
311C22—122no—
312C22C3×#98411yescharacteristic, complemented
313C4—32no—
314C4—31no—
315C4—61no—
316C4—122no—
317C4—122no—
318C4C3×#102011yescharacteristic
319C3—161no—
320C3—321no—
321C3#8750811yescharacteristic, direct product factor
322C2—31no—
323C2—31no—
324C2—62no—
325C2—121no—
326C2C2×C3×#102011yescharacteristic
subgroupG/H
centerC2×C3C2×#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×#1020
FittingC22×C3×#41D3
socleC23×C3C2×S4
prime radicalC3×#5948C2
hypercenterC3×C4#1020
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#5930C2semidirect
3. #87508C3direct (N × H)
11. #5930C2×C3semidirect
246. C22×C3#984semidirect
312. C22C3×#984semidirect
321. C3#87508direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#59482
C3×#59302
#875083
C3×#30073
C3×#9844
C3×A4⋊Q84