Finite Group Databasebeta

← C3×#85530 #11922814

Subgroups — C3×#85530 #11922814

Order 1152.

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

normal subgroups only

This group has 673 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
1C2×C3×#994C211yesmaximal, characteristic, complemented
2C2×C3×#986C211yesmaximal, characteristic, complemented
3C3×#5427C214yesmaximal, complemented
4C3×#5424C211yesmaximal, characteristic, complemented
5#85530C311yesmaximal, characteristic, direct product factor
6C3×#3008—31nomaximal
7C3×#994—22no—
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2212yescomplemented
10C3×#986—22no—
11C2×C3×GL(2,𝔽3)C2212yescomplemented
12C3×#947—41nomaximal
13C3×#859C2212yescomplemented
14C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
15C2×#994—21no—
16C2×#994C2×C311yescharacteristic, complemented
17C2×#986C2×C311yescharacteristic, complemented
18C2×C3×AGL(1,ℤ/8)—31no—
19C22×C3×D8—31no—
20C3×#535—31no—
21#5427C2×C314yescomplemented
22#5424C2×C311yescharacteristic, complemented
23C3×#449—34no—
24C3×#447—34no—
25C3×#418—31no—
26C3×#416—31no—
27C3×#408—31no—
28C2×C32×Q8—41no—
29C2×C3×D12—41no—
30C3×#41⋊C3—22no—
31C3×#41⋊C3D412yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3×GL(2,𝔽3)—22no—
34C3×(C2×D3)⋊C4—44no—
35C3×C4×Dic3—41no—
36#3008—31noSylow
37C3×H(5,𝔽2)—22no—
38C3×H(5,𝔽2)—62no—
39C2×C3×#41—31no—
40C2×C3×#41C2×D312yes—
41C2×C3×#41C2×D311yescharacteristic
42C22×C3×D4—31no—
43C22×C3×D4—32no—
44C22×C3×D4—31no—
45#994—24no—
46#994—22no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3—22no—
49C2×#41⋊C3C22×C311yescharacteristic
50C2×#41⋊C3C22×C312yescomplemented
51#986—22no—
52C2×GL(2,𝔽3)C22×C312yescomplemented
53C3×AGL(1,ℤ/8)—62no—
54C3×AGL(1,ℤ/8)—62no—
55C2×C3×SD16—32no—
56C2×C3×D8—34no—
57C2×C3×D8—32no—
58C2×C3×D8—64no—
59C2×C3×M16—31no—
60C22×C3×C8—31no—
61C3×#127—61no—
62C3×#124—61no—
63C3×#121—34no—
64C3×#120—62no—
65C3×#117—31no—
66#947—41no—
67#859C22×C312yescomplemented
68C3×#103—32no—
69C3×#102—32no—
70C3×#102—34no—
71C3×C22⋊C8—34no—
72C22×C3×D3—42no—
73C32×Q8—82no—
74C2×C32×C4—41no—
75C2×C32×C4—42no—
76C2×C3×Dic3—42no—
77C3×D12—82no—
78C3×SL(2,𝔽3)C2×D411yescharacteristic
79C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
80C2×AGL(1,ℤ/8)—31no—
81C22×D8—31no—
82#535—31no—
83#449—34no—
84#447—34no—
85#418—31no—
86#416—31no—
87#408—31no—
88C24×C3—61no—
89C24×C3—61no—
90C3×#41—22no—
91C3×#41—32no—
92C3×#41—62no—
93C3×#41—62no—
94C3×#41—62no—
95C3×#41#6712yes—
96C2×C3×Q8—41no—
97C2×C3×Q8—81no—
98C2×C3×Q8S411yescharacteristic
99C2×C3×Q8C22×D311yescharacteristic
100C2×C3×D4—31no—
101C2×C3×D4—32no—
102C2×C3×D4—34no—
103C2×C3×D4—32no—
104C2×C3×D4—32no—
105C2×C3×D4—31no—
106C2×C3×D4—62no—
107C2×C3×D4—62no—
108C2×C3×D4—62no—
109C2×C3×D4—64no—
110C2×C3×D4—64no—
111C2×C3×D4—64no—
112C2×C3×D4—61no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C2×D12—41no—
116#41⋊C3—24no—
117#41⋊C3—22no—
118#41⋊C3—22no—
119#41⋊C3C3×D412yescomplemented
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)C23×C311yescharacteristic
122GL(2,𝔽3)—22no—
123C3×SD16—62no—
124C3×SD16—62no—
125C3×D8—68no—
126C3×D8—64no—
127C3×M16—62no—
128C2×C3×C8—32no—
129C2×C3×C8—32no—
130C2×C3×C8—62no—
131C3×C4⋊C4—32no—
132C3×C22⋊C4—61no—
133C3×C22⋊C4—64no—
134C3×C42—61no—
135(C2×D3)⋊C4—44no—
136C4×Dic3—41no—
137C22×C32—41no—
138C2×C3×D3—82no—
139C2×C3×D3—84no—
140C32×C4—42no—
141C32×C4—82no—
142C3×Dic3—82no—
143H(5,𝔽2)—22no—
144H(5,𝔽2)—62no—
145C2×#41—31no—
146C2×#41C2×C3×D312yes—
147C2×#41C2×C3×D311yescharacteristic
148C22×D4—31no—
149C22×D4—32no—
150C22×D4—31no—
151AGL(1,ℤ/8)—62no—
152AGL(1,ℤ/8)—62no—
153C2×SD16—32no—
154C2×D8—34no—
155C2×D8—32no—
156C2×D8—64no—
157C2×M16—31no—
158C22×C8—31no—
159#127—61no—
160#124—61no—
161#121—34no—
162#120—62no—
163#117—31no—
164#103—32no—
165#102—32no—
166#102—34no—
167C22⋊C8—34no—
168C23×C3—31no—
169C23×C3—32no—
170C23×C3—64no—
171C23×C3—62no—
172C23×C3—122no—
173C23×C3—242no—
174C23×C3—242no—
175C22×D3—42no—
176C3×Q8—22no—
177C3×Q8—31no—
178C3×Q8—82no—
179C3×Q8—84no—
180C3×Q8C2×#6711yescharacteristic
181C3×D4—32no—
182C3×D4—34no—
183C3×D4—64no—
184C3×D4—61no—
185C3×D4—62no—
186C3×D4—62no—
187C3×D4—62no—
188C3×D4—62no—
189C3×D4—122no—
190C3×D4—244no—
191C2×C3×C4—31no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—62no—
199C2×C3×C4—82no—
200C2×C3×C4—81no—
201C2×C3×C4C2×S412yes—
202C2×C3×C4C2×S411yescharacteristic
203C2×Dic3—42no—
204D12—82no—
205SL(2,𝔽3)—21no—
206SL(2,𝔽3)C2×C3×D411yescharacteristic
207C3×C8—62no—
208C3×C8—62no—
209C2×C32—42no—
210C2×C32—41no—
211C3×D3—162no—
212C24—61no—
213C24—61no—
214#41—22no—
215#41—32no—
216#41—62no—
217#41—62no—
218#41—62no—
219#41C3×#6712yes—
220C2×Q8C3×S411yescharacteristic
221C2×Q8C22×C3×D311yescharacteristic
222C2×D4—31no—
223C2×D4—32no—
224C2×D4—34no—
225C2×D4—32no—
226C2×D4—32no—
227C2×D4—31no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—62no—
231C2×D4—64no—
232C2×D4—64no—
233C2×D4—64no—
234C2×D4—61no—
235C22×C4—31no—
236C22×C4—32no—
237SD16—62no—
238SD16—62no—
239D8—68no—
240D8—64no—
241M16—62no—
242C2×C8—32no—
243C2×C8—32no—
244C2×C8—62no—
245C4⋊C4—32no—
246C22⋊C4—61no—
247C22⋊C4—64no—
248C42—61no—
249C22×C3—32no—
250C22×C3—41no—
251C22×C3—62no—
252C22×C3—64no—
253C22×C3—62no—
254C22×C3—81no—
255C22×C3—124no—
256C22×C3—122no—
257C22×C3—244no—
258C22×C3—244no—
259C22×C3C22×S411yescharacteristic
260C2×D3—82no—
261C2×D3—84no—
262C3×C4—22no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—42no—
266C3×C4—82no—
267C3×C4—84no—
268C3×C4—82no—
269C3×C4—122no—
270C3×C4GL(2,ℤ/4)12yes—
271Dic3—82no—
272C32—41noSylow
273C23—31no—
274C23—32no—
275C23—64no—
276C23—62no—
277C23—122no—
278C23—242no—
279C23—242no—
280Q8—22no—
281Q8—31no—
282Q8C2×C3×#6711yescharacteristic
283D4—32no—
284D4—34no—
285D4—64no—
286D4—61no—
287D4—62no—
288D4—62no—
289D4—62no—
290D4—62no—
291D4—122no—
292D4—244no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4C2×C3×S412yes—
300C2×C4C2×C3×S411yescharacteristic
301C8—62no—
302C8—62no—
303C2×C3—42no—
304C2×C3—41no—
305C2×C3—62no—
306C2×C3—82no—
307C2×C3—81no—
308C2×C3—122no—
309C2×C3—242no—
310C2×C3#592512yes—
311C2×C3C2×GL(2,ℤ/4)11yescharacteristic
312D3—162no—
313C22—32no—
314C22—62no—
315C22—64no—
316C22—62no—
317C22—124no—
318C22—122no—
319C22—244no—
320C22—244no—
321C22C22×C3×S411yescharacteristic
322C4—22no—
323C4—31no—
324C4—31no—
325C4—122no—
326C4C3×GL(2,ℤ/4)12yes—
327C3—41no—
328C3—81no—
329C3#8553011yescharacteristic, direct product factor
330C2—62no—
331C2—122no—
332C2—242no—
333C2C3×#592512yes—
334C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×H(5,𝔽2)D3
hypercenterC2×C3×Q8S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2semidirect
2. C2×C3×#986C2semidirect
3. C3×#5427C2semidirect
4. C3×#5424C2semidirect
5. #85530C3direct (N × H)
9. C2×C3×#41⋊C3C22semidirect
11. C2×C3×GL(2,𝔽3)C22semidirect
13. C3×#859C22semidirect
14. C2×C3×H(5,𝔽2)D3semidirect
16. C2×#994C2×C3semidirect
17. C2×#986C2×C3semidirect
21. #5427C2×C3semidirect
22. #5424C2×C3semidirect
31. C3×#41⋊C3D4semidirect
50. C2×#41⋊C3C22×C3semidirect
52. C2×GL(2,𝔽3)C22×C3semidirect
67. #859C22×C3semidirect
79. C2×H(5,𝔽2)C3×D3semidirect
119. #41⋊C3C3×D4semidirect
329. C3#85530direct (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×#9862
C3×#54272
C3×#54242
#855303
C3×#30083
C3×#9474