Finite Group Databasebeta

← C3×#2897⋊C3 #11919544

Subgroups — C3×#2897⋊C3 #11919544

Order 1152.

462 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×C3×#994C211yesmaximal, characteristic
2C2×C3×#131⋊C3C211yesmaximal, characteristic, complemented
3C3×#520⋊C3C211yesmaximal, characteristic, complemented
4C3×#2897C311yesmaximal, characteristic, complemented
5#2897⋊C3C313yesmaximal, direct product factor
6C3×#994—21no—
7C3×#994C412yescomplemented
8C2×C3×#41⋊C3—21no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C3×#131⋊C3—21no—
11C3×C4×SL(2,𝔽3)—21no—
12C32×#103—41nomaximal
13C2×C3×H(5,𝔽2)C2×C311yescharacteristic
14C2×#994C2×C313yes—
15C2×C3×#131C2×C311yescharacteristic
16C3×#520C2×C311yescharacteristic
17C2×#131⋊C3C2×C313yescomplemented
18#520⋊C3C2×C313yescomplemented
19C3×#418—31no—
20C3×#417—31no—
21C3×#416—31no—
22C2×C3×#102—31no—
23C2×C32×Q8—41no—
24C3×#41⋊C3—22no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C2×C32×C8—41no—
29C32×C4⋊C4—41no—
30#2897C3211yescharacteristic, complemented, Sylow
31C3×H(5,𝔽2)—21no—
32C3×H(5,𝔽2)—32no—
33C3×H(5,𝔽2)—61no—
34C3×H(5,𝔽2)C3×C412yescomplemented
35C2×C3×#41—21no—
36C2×C3×#41—31no—
37C2×C3×#41C22×C311yescharacteristic
38C22×C3×D4—31no—
39C22×C3×D4—61no—
40#994—23no—
41#994C3×C416yescomplemented
42C2×#41⋊C3—23no—
43C2×#41⋊C3C22×C313yes—
44C3×#131—21no—
45C3×#131—61no—
46C2×C3×M16—31no—
47C22×C3×C8—31no—
48C3×C4×Q8—21no—
49C3×C4×D4—61no—
50C3×#117—31no—
51C2×C3×C4⋊C4—31no—
52#131⋊C3—23no—
53C4×SL(2,𝔽3)—23no—
54C3×#103—31no—
55C3×#103—43no—
56C3×#103A411yescharacteristic
57C3×#102—32no—
58C3×#102—31no—
59C3×#102—31no—
60C32×Q8—42no—
61C32×Q8—81no—
62C2×C32×C4—41no—
63C2×C32×C4—81no—
64C2×C32×C4—81no—
65C3×SL(2,𝔽3)C22⋊C411yescharacteristic
66C32×C8—81no—
67C2×H(5,𝔽2)C2×C3211yescharacteristic
68C2×#131C2×C3211yescharacteristic
69#520C2×C3211yescharacteristic
70#418—31no—
71#417—31no—
72#416—31no—
73C2×#102—31no—
74C24×C3—61no—
75C3×#41—22no—
76C3×#41—31no—
77C3×#41—31no—
78C3×#41—32no—
79C3×#41—32no—
80C3×#41—61no—
81C3×#41—61no—
82C3×#41—62no—
83C3×#41C2×C3×C411yescharacteristic
84C3×#41C3×D411yescharacteristic
85C2×C3×Q8—43no—
86C2×C3×Q8C3×D411yescharacteristic
87C2×C3×Q8C2×A411yescharacteristic
88C2×C3×D4—31no—
89C2×C3×D4—32no—
90C2×C3×D4—32no—
91C2×C3×D4—31no—
92C2×C3×D4—61no—
93C2×C3×D4—61no—
94C2×C3×D4—61no—
95C2×C3×D4—61no—
96C2×C3×D4—64no—
97C2×C3×D4—62no—
98C2×C3×D4—62no—
99C22×C3×C4—31no—
100C22×C3×C4—61no—
101C22×C3×C4—61no—
102#41⋊C3—26no—
103#41⋊C3C2×C3×C413yes—
104#41⋊C3C3×D413yes—
105C2×SL(2,𝔽3)C3×D413yes—
106C3×M16—61no—
107C3×M16—61no—
108C2×C3×C8—31no—
109C2×C3×C8—43no—
110C2×C3×C8—61no—
111C2×C3×C8—61no—
112C2×C3×C8C2×A411yescharacteristic
113C3×C4⋊C4—31no—
114C3×C4⋊C4—43no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4C2×A411yescharacteristic
117C3×C22⋊C4—61no—
118C3×C42—61no—
119C22×C32—41no—
120C32×C4—41no—
121C32×C4—41no—
122C32×C4—82no—
123C32×C4—161no—
124H(5,𝔽2)—21no—
125H(5,𝔽2)—32no—
126H(5,𝔽2)—61no—
127H(5,𝔽2)C32×C412yescomplemented
128C2×#41—21no—
129C2×#41—31no—
130C2×#41C22×C3211yescharacteristic
131C22×D4—31no—
132C22×D4—61no—
133#131—21no—
134#131—61no—
135C2×M16—31no—
136C22×C8—31no—
137C4×Q8—21no—
138C4×D4—61no—
139#117—31no—
140C2×C4⋊C4—31no—
141#103—31no—
142#103C3×A411yescharacteristic
143#102—32no—
144#102—31no—
145#102—31no—
146C23×C3—31no—
147C23×C3—62no—
148C23×C3—62no—
149C23×C3—61no—
150C23×C3—241no—
151C23×C3—241no—
152C3×Q8—21no—
153C3×Q8—31no—
154C3×Q8—46no—
155C3×Q8—83no—
156C3×Q8C3×C22⋊C411yescharacteristic
157C3×Q8C4×A412yes—
158C3×D4—31no—
159C3×D4—31no—
160C3×D4—32no—
161C3×D4—32no—
162C3×D4—62no—
163C3×D4—62no—
164C3×D4—62no—
165C3×D4—62no—
166C3×D4—61no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—43no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—61no—
178C2×C3×C4—83no—
179C2×C3×C4—83no—
180C2×C3×C4—241no—
181C2×C3×C4C22×A411yescharacteristic
182SL(2,𝔽3)C3×C22⋊C413yes—
183C3×C8—21no—
184C3×C8—61no—
185C3×C8—83no—
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—41no—
189C24—61no—
190#41—22no—
191#41—31no—
192#41—31no—
193#41—32no—
194#41—32no—
195#41—61no—
196#41—61no—
197#41—62no—
198#41C2×C32×C411yescharacteristic
199#41C32×D411yescharacteristic
200C2×Q8C32×D411yescharacteristic
201C2×Q8C2×C3×A411yescharacteristic
202C2×D4—31no—
203C2×D4—32no—
204C2×D4—32no—
205C2×D4—31no—
206C2×D4—61no—
207C2×D4—61no—
208C2×D4—61no—
209C2×D4—61no—
210C2×D4—64no—
211C2×D4—62no—
212C2×D4—62no—
213C22×C4—31no—
214C22×C4—61no—
215C22×C4—61no—
216M16—61no—
217M16—61no—
218C2×C8—31no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8C2×C3×A411yescharacteristic
222C4⋊C4—31no—
223C4⋊C4—61no—
224C4⋊C4C2×C3×A411yescharacteristic
225C22⋊C4—61no—
226C42—61no—
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—43no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—62no—
233C22×C3—121no—
234C22×C3—121no—
235C22×C3—242no—
236C22×C3—242no—
237C22×C3D4×A411yescharacteristic
238C3×C4—22no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—41no—
242C3×C4—43no—
243C3×C4—43no—
244C3×C4—86no—
245C3×C4—121no—
246C3×C4—163no—
247C3×C4C2×C4×A411yescharacteristic
248C3×C4D4×A411yescharacteristic
249C32—41noSylow
250C23—31no—
251C23—62no—
252C23—62no—
253C23—61no—
254C23—241no—
255C23—241no—
256Q8—21no—
257Q8—31no—
258Q8C32×C22⋊C411yescharacteristic
259Q8C3×C4×A412yes—
260D4—31no—
261D4—31no—
262D4—32no—
263D4—32no—
264D4—62no—
265D4—62no—
266D4—62no—
267D4—62no—
268D4—61no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—241no—
280C2×C4C22×C3×A411yescharacteristic
281C8—21no—
282C8—61no—
283C2×C3—43no—
284C2×C3—43no—
285C2×C3—43no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—122no—
289C2×C3A4×C22⋊C411yescharacteristic
290C2×C3#576⋊C311yescharacteristic
291C2×C3#577⋊C311yescharacteristic
292C22—31no—
293C22—31no—
294C22—61no—
295C22—61no—
296C22—62no—
297C22—121no—
298C22—121no—
299C22—242no—
300C22—242no—
301C22C3×D4×A411yescharacteristic
302C4—22no—
303C4—31no—
304C4—31no—
305C4—41no—
306C4—121no—
307C4C2×C3×C4×A411yescharacteristic
308C4C3×D4×A411yescharacteristic
309C3—43no—
310C3#2897⋊C311yescharacteristic, direct product factor
311C2—61no—
312C2—61no—
313C2—122no—
314C2C3×A4×C22⋊C411yescharacteristic
315C2C3×#576⋊C311yescharacteristic
316C2C3×#577⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3D4×A4
commutator#41C2×C32×C4
FrattiniC2×C4C22×C3×A4
FittingC3×#2897C3
prime radicalC2×C3×#994C2
cosocleC2×#41C22×C32
hypercenterC3×#103A4
hypocenterQ8C32×C22⋊C4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#131⋊C3C2semidirect
3. C3×#520⋊C3C2semidirect
4. C3×#2897C3semidirect
5. #2897⋊C3C3direct (N × H)
7. C3×#994C4semidirect
17. C2×#131⋊C3C2×C3semidirect
18. #520⋊C3C2×C3semidirect
30. #2897C32semidirect
34. C3×H(5,𝔽2)C3×C4semidirect
41. #994C3×C4semidirect
127. H(5,𝔽2)C32×C4semidirect
310. C3#2897⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C2×C3×#131⋊C32
C3×#520⋊C32
C3×#28973
#2897⋊C33
C32×#1034