Finite Group Databasebeta

← C3×#85527 #11922811

Subgroups — C3×#85527 #11922811

Order 1152.

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

normal subgroups only

This group has 633 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
2C2×C3×#984C211yesmaximal, characteristic, complemented
3C3×#5425C214yesmaximal, complemented
4C3×#5424C211yesmaximal, characteristic, complemented
5#85527C311yesmaximal, characteristic, direct product factor
6C3×#3009—31nomaximal
7C3×#994—22no—
8C2×C3×#41⋊C3C2212yes—
9C2×C3×#41⋊C3C2211yescharacteristic
10C3×#984—22no—
11C2×C3×#226C2212yescomplemented
12C3×#944—41nomaximal
13C3×#859C2212yescomplemented
14C2×C3×H(5,𝔽2)D311yescharacteristic
15C2×#994—21no—
16C2×#994C2×C311yescharacteristic
17C2×#984C2×C311yescharacteristic, complemented
18C2×C3×#137—31no—
19C22×C3×SD16—31no—
20C3×#536—31no—
21#5425C2×C314yescomplemented
22#5424C2×C311yescharacteristic, complemented
23C3×#452—34no—
24C3×#450—34no—
25C3×#418—31no—
26C3×#416—31no—
27C3×#408—31no—
28C2×C32×Q8—41no—
29C2×C3×Dic6—41no—
30C3×#41⋊C3—22no—
31C3×#41⋊C3D412yes—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3×#226—22no—
34C3×Dic3⋊C4—44no—
35C3×C4×Dic3—41no—
36#3009—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×Q8—31no—
43C22×C3×D4—31no—
44C22×C3×D4—32no—
45#994—24no—
46#994—22no—
47C2×#41⋊C3—22no—
48C2×#41⋊C3—21no—
49C2×#41⋊C3C22×C312yes—
50C2×#41⋊C3C22×C311yescharacteristic
51#984—22no—
52C2×#226C22×C312yescomplemented
53C3×#137—62no—
54C3×#137—62no—
55C2×C3×Q16—32no—
56C2×C3×SD16—34no—
57C2×C3×SD16—32no—
58C2×C3×SD16—64no—
59C2×C3×M16—31no—
60C22×C3×C8—31no—
61C3×C4⋊Q8—61no—
62C3×#124—61no—
63C3×#123—62no—
64C3×C22⋊Q8—34no—
65C3×#117—31no—
66#944—41no—
67#859C22×C312yescomplemented
68C3×#103—32no—
69C3×#102—32no—
70C3×#102—34no—
71C3×C22⋊C8—34no—
72C32×Q8—82no—
73C2×C32×C4—41no—
74C2×C32×C4—42no—
75C2×C3×Dic3—42no—
76C2×C3×Dic3—42no—
77C3×Dic6—82no—
78C3×SL(2,𝔽3)C2×D411yescharacteristic
79C2×H(5,𝔽2)C3×D311yescharacteristic
80C2×#137—31no—
81C22×SD16—31no—
82#536—31no—
83#452—34no—
84#450—34no—
85#418—31no—
86#416—31no—
87#408—31no—
88C24×C3—61no—
89C3×#41—22no—
90C3×#41—32no—
91C3×#41—62no—
92C3×#41—62no—
93C3×#41—62no—
94C3×#41#6712yes—
95C2×C3×Q8—32no—
96C2×C3×Q8—41no—
97C2×C3×Q8—62no—
98C2×C3×Q8—62no—
99C2×C3×Q8—61no—
100C2×C3×Q8—81no—
101C2×C3×Q8S411yescharacteristic
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—32no—
105C2×C3×D4—34no—
106C2×C3×D4—32no—
107C2×C3×D4—31no—
108C2×C3×D4—62no—
109C2×C3×D4—64no—
110C2×C3×D4—64no—
111C22×C3×C4—31no—
112C22×C3×C4—32no—
113C22×C3×C4—61no—
114C2×Dic6—41no—
115#41⋊C3—24no—
116#41⋊C3—22no—
117#41⋊C3—22no—
118#41⋊C3C3×D412yes—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)C23×C311yescharacteristic
121#226—22no—
122C3×Q16—62no—
123C3×Q16—62no—
124C3×SD16—68no—
125C3×SD16—64no—
126C3×M16—62no—
127C2×C3×C8—32no—
128C2×C3×C8—32no—
129C2×C3×C8—62no—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—64no—
132C3×C22⋊C4—61no—
133C3×C22⋊C4—64no—
134C3×C42—61no—
135Dic3⋊C4—44no—
136C4×Dic3—41no—
137C22×C32—41no—
138C32×C4—42no—
139C32×C4—82no—
140C3×Dic3—44no—
141C3×Dic3—82no—
142H(5,𝔽2)—22no—
143H(5,𝔽2)—62no—
144C2×#41—31no—
145C2×#41C2×C3×D312yes—
146C2×#41C2×C3×D311yescharacteristic
147C22×Q8—31no—
148C22×D4—31no—
149C22×D4—32no—
150#137—62no—
151#137—62no—
152C2×Q16—32no—
153C2×SD16—34no—
154C2×SD16—32no—
155C2×SD16—64no—
156C2×M16—31no—
157C22×C8—31no—
158C4⋊Q8—61no—
159#124—61no—
160#123—62no—
161C22⋊Q8—34no—
162#117—31no—
163#103—32no—
164#102—32no—
165#102—34no—
166C22⋊C8—34no—
167C23×C3—31no—
168C23×C3—32no—
169C23×C3—64no—
170C23×C3—242no—
171C3×Q8—22no—
172C3×Q8—31no—
173C3×Q8—62no—
174C3×Q8—62no—
175C3×Q8—82no—
176C3×Q8—84no—
177C3×Q8—122no—
178C3×Q8C2×#6711yescharacteristic
179C3×D4—32no—
180C3×D4—34no—
181C3×D4—64no—
182C3×D4—61no—
183C3×D4—62no—
184C3×D4—62no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—82no—
195C2×C3×C4—81no—
196C2×C3×C4—122no—
197C2×C3×C4C2×S412yes—
198C2×C3×C4C2×S411yescharacteristic
199C2×Dic3—42no—
200C2×Dic3—42no—
201Dic6—82no—
202SL(2,𝔽3)—21no—
203SL(2,𝔽3)C2×C3×D411yescharacteristic
204C3×C8—62no—
205C3×C8—62no—
206C2×C32—42no—
207C2×C32—41no—
208C24—61no—
209#41—22no—
210#41—32no—
211#41—62no—
212#41—62no—
213#41—62no—
214#41C3×#6712yes—
215C2×Q8—32no—
216C2×Q8—62no—
217C2×Q8—62no—
218C2×Q8—61no—
219C2×Q8C3×S411yescharacteristic
220C2×Q8C22×C3×D311yescharacteristic
221C2×D4—31no—
222C2×D4—32no—
223C2×D4—34no—
224C2×D4—32no—
225C2×D4—31no—
226C2×D4—62no—
227C2×D4—64no—
228C2×D4—64no—
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—61no—
232Q16—62no—
233Q16—62no—
234SD16—68no—
235SD16—64no—
236M16—62no—
237C2×C8—32no—
238C2×C8—32no—
239C2×C8—62no—
240C4⋊C4—32no—
241C4⋊C4—64no—
242C22⋊C4—61no—
243C22⋊C4—64no—
244C42—61no—
245C22×C3—32no—
246C22×C3—41no—
247C22×C3—62no—
248C22×C3—64no—
249C22×C3—62no—
250C22×C3—81no—
251C22×C3—244no—
252C22×C3C22×S411yescharacteristic
253C3×C4—22no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—42no—
257C3×C4—82no—
258C3×C4—84no—
259C3×C4—82no—
260C3×C4—122no—
261C3×C4—122no—
262C3×C4GL(2,ℤ/4)12yes—
263Dic3—44no—
264Dic3—82no—
265C32—41noSylow
266C23—31no—
267C23—32no—
268C23—64no—
269C23—242no—
270Q8—22no—
271Q8—31no—
272Q8—62no—
273Q8—62no—
274Q8—122no—
275Q8C2×C3×#6711yescharacteristic
276D4—32no—
277D4—34no—
278D4—64no—
279D4—61no—
280D4—62no—
281D4—62no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—32no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—122no—
290C2×C4C2×C3×S412yes—
291C2×C4C2×C3×S411yescharacteristic
292C8—62no—
293C8—62no—
294C2×C3—42no—
295C2×C3—41no—
296C2×C3—62no—
297C2×C3—82no—
298C2×C3—81no—
299C2×C3—122no—
300C2×C3#592412yes—
301C2×C3C2×GL(2,ℤ/4)11yescharacteristic
302C22—32no—
303C22—62no—
304C22—64no—
305C22—62no—
306C22—244no—
307C22C22×C3×S411yescharacteristic
308C4—22no—
309C4—31no—
310C4—31no—
311C4—122no—
312C4—122no—
313C4C3×GL(2,ℤ/4)12yes—
314C3—41no—
315C3—81no—
316C3#8552711yescharacteristic, direct product factor
317C2—62no—
318C2—122no—
319C2C3×#592412yes—
320C2C2×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
prime radicalC2×C3×#994C2
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
2. C2×C3×#984C2semidirect
3. C3×#5425C2semidirect
4. C3×#5424C2semidirect
5. #85527C3direct (N × H)
11. C2×C3×#226C22semidirect
13. C3×#859C22semidirect
17. C2×#984C2×C3semidirect
21. #5425C2×C3semidirect
22. #5424C2×C3semidirect
52. C2×#226C22×C3semidirect
67. #859C22×C3semidirect
316. C3#85527direct (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×#9842
C3×#54252
C3×#54242
#855273
C3×#30093
C3×#9444