Finite Group Databasebeta

← C3×#85540 #11922820

Subgroups — C3×#85540 #11922820

Order 1152.

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

normal subgroups only

This group has 701 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×#986C211yesmaximal, characteristic, complemented
3C3×#5424C211yesmaximal, characteristic, complemented
4C3×#5421C214yesmaximal, complemented
5#85540C311yesmaximal, characteristic, direct product factor
6C3×#3010—31nomaximal
7C3×2−1+4⋊C3—22no—
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yescomplemented
10C3×#986—22no—
11C2×C3×GL(2,𝔽3)C2212yes—
12C3×#940—41nomaximal
13C3×#859C2212yescomplemented
14C2×C3×2−1+4D311yescharacteristic, complemented
15C2×2−1+4⋊C3—21no—
16C2×2−1+4⋊C3C2×C311yescharacteristic, complemented
17C2×#986C2×C311yescharacteristic, complemented
18C2×C3×AGL(1,ℤ/8)—31no—
19C22×C3×SD16—31no—
20C3×#535—31no—
21#5424C2×C311yescharacteristic, complemented
22#5421C2×C314yescomplemented
23C3×#449—34no—
24C3×#448—34no—
25C3×#419—31no—
26C3×#416—31no—
27C3×#408—31no—
28C2×C32×D4—41no—
29C2×C3×#67—44no—
30C2×C3×D12—41no—
31C3×#41⋊C3D412yescomplemented
32C2×C3×SL(2,𝔽3)—22no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3×GL(2,𝔽3)—22no—
35C3×C4×Dic3—41no—
36#3010—31noSylow
37C3×2−1+4—22no—
38C3×2−1+4—62no—
39C2×C3×#41—31no—
40C2×C3×#41—32no—
41C2×C3×#41C2×D311yescharacteristic
42C22×C3×Q8—31no—
43C22×C3×Q8C2×D312yes—
44C22×C3×D4—31no—
452−1+4⋊C3—24no—
462−1+4⋊C3—22no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3C22×C311yescharacteristic
49C22×SL(2,𝔽3)—22no—
50C22×SL(2,𝔽3)C22×C312yescomplemented
51#986—22no—
52C2×GL(2,𝔽3)C22×C312yes—
53C3×AGL(1,ℤ/8)—62no—
54C3×AGL(1,ℤ/8)—62no—
55C2×C3×SD16—34no—
56C2×C3×SD16—32no—
57C2×C3×SD16—64no—
58C2×C3×D8—32no—
59C2×C3×M16—31no—
60C22×C3×C8—31no—
61C3×#127—61no—
62C3×#124—61no—
63C3×#121—34no—
64C3×#120—62no—
65C3×#117—31no—
66#940—41no—
67#859C22×C312yescomplemented
68C3×#103—34no—
69C3×#103—32no—
70C3×#102—32no—
71C3×C22⋊C8—34no—
72C23×C32—42no—
73C22×C3×D3—42no—
74C32×D4—82no—
75C2×C32×C4—41no—
76C3×#67—88no—
77C2×C3×Dic3—42no—
78C3×D12—82no—
79C3×SL(2,𝔽3)C2×D411yescharacteristic
80C2×2−1+4C3×D311yescharacteristic, complemented
81C2×AGL(1,ℤ/8)—31no—
82C22×SD16—31no—
83#535—31no—
84#449—34no—
85#448—34no—
86#419—31no—
87#416—31no—
88#408—31no—
89C24×C3—61no—
90C3×#41—32no—
91C3×#41—62no—
92C3×#41—62no—
93C3×#41—64no—
94C3×#41—64no—
95C3×#41#6712yescomplemented
96C2×C3×Q8—22no—
97C2×C3×Q8—31no—
98C2×C3×Q8—32no—
99C2×C3×Q8—34no—
100C2×C3×Q8—62no—
101C2×C3×Q8—62no—
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—32no—
105C2×C3×D4—32no—
106C2×C3×D4—41no—
107C2×C3×D4—62no—
108C2×C3×D4—62no—
109C2×C3×D4—64no—
110C2×C3×D4—61no—
111C2×C3×D4—81no—
112C2×C3×D4S411yescharacteristic
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C22×C3×C4—61no—
116C2×#67—44no—
117C2×D12—41no—
118#41⋊C3—22no—
119#41⋊C3C3×D412yescomplemented
120C2×SL(2,𝔽3)—24no—
121C2×SL(2,𝔽3)—22no—
122C2×SL(2,𝔽3)—21no—
123C2×SL(2,𝔽3)C23×C311yescharacteristic
124GL(2,𝔽3)—22no—
125C3×SD16—68no—
126C3×SD16—62no—
127C3×SD16—62no—
128C3×D8—64no—
129C3×M16—62no—
130C2×C3×C8—32no—
131C2×C3×C8—32no—
132C2×C3×C8—62no—
133C3×C4⋊C4—32no—
134C3×C22⋊C4—64no—
135C3×C22⋊C4—61no—
136C3×C42—61no—
137C4×Dic3—41no—
138C22×C32—41no—
139C22×C32—82no—
140C22×C32—84no—
141C2×C3×D3—82no—
142C2×C3×D3—84no—
143C32×C4—42no—
144C3×Dic3—44no—
1452−1+4—22no—
1462−1+4—62no—
147C2×#41—31no—
148C2×#41—32no—
149C2×#41C2×C3×D311yescharacteristic
150C22×Q8—31no—
151C22×Q8C2×C3×D312yes—
152C22×D4—31no—
153AGL(1,ℤ/8)—62no—
154AGL(1,ℤ/8)—62no—
155C2×SD16—34no—
156C2×SD16—32no—
157C2×SD16—64no—
158C2×D8—32no—
159C2×M16—31no—
160C22×C8—31no—
161#127—61no—
162#124—61no—
163#121—34no—
164#120—62no—
165#117—31no—
166#103—34no—
167#103—32no—
168#102—32no—
169C22⋊C8—34no—
170C23×C3—31no—
171C23×C3—42no—
172C23×C3—62no—
173C23×C3—82no—
174C23×C3—122no—
175C23×C3—242no—
176C23×C3C2×S412yes—
177C22×D3—42no—
178C3×Q8—31no—
179C3×Q8—34no—
180C3×Q8—62no—
181C3×Q8—62no—
182C3×Q8C2×#6711yescharacteristic
183C3×D4—22no—
184C3×D4—32no—
185C3×D4—61no—
186C3×D4—62no—
187C3×D4—62no—
188C3×D4—64no—
189C3×D4—82no—
190C3×D4—84no—
191C3×D4—122no—
192C3×D4—244no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—41no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—64no—
201C2×C3×C4—62no—
202C2×C3×C4—81no—
203C2×C3×C4C2×S411yescharacteristic
204#67—88no—
205C2×Dic3—42no—
206D12—82no—
207SL(2,𝔽3)—21no—
208SL(2,𝔽3)C2×C3×D411yescharacteristic
209C3×C8—62no—
210C3×C8—62no—
211C2×C32—42no—
212C2×C32—41no—
213C2×C32—162no—
214C3×D3—162no—
215C24—61no—
216#41—32no—
217#41—62no—
218#41—62no—
219#41—64no—
220#41—64no—
221#41C3×#6712yescomplemented
222C2×Q8—22no—
223C2×Q8—31no—
224C2×Q8—32no—
225C2×Q8—34no—
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8C22×C3×D311yescharacteristic
229C2×D4—31no—
230C2×D4—32no—
231C2×D4—32no—
232C2×D4—62no—
233C2×D4—62no—
234C2×D4—64no—
235C2×D4—61no—
236C2×D4C3×S411yescharacteristic
237C22×C4—31no—
238C22×C4—32no—
239C22×C4—61no—
240SD16—68no—
241SD16—62no—
242SD16—62no—
243D8—64no—
244M16—62no—
245C2×C8—32no—
246C2×C8—32no—
247C2×C8—62no—
248C4⋊C4—32no—
249C22⋊C4—64no—
250C22⋊C4—61no—
251C42—61no—
252C22×C3—24no—
253C22×C3—22no—
254C22×C3—32no—
255C22×C3—41no—
256C22×C3—62no—
257C22×C3—84no—
258C22×C3—82no—
259C22×C3—84no—
260C22×C3—81no—
261C22×C3—124no—
262C22×C3—122no—
263C22×C3—164no—
264C22×C3—244no—
265C22×C3C22×S411yescharacteristic
266C2×D3—82no—
267C2×D3—84no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—42no—
271C3×C4—62no—
272C3×C4—82no—
273C3×C4—122no—
274C3×C4GL(2,ℤ/4)12yes—
275Dic3—44no—
276C32—41noSylow
277C23—31no—
278C23—62no—
279C23—122no—
280C23—242no—
281C23C2×C3×S412yes—
282Q8—31no—
283Q8—34no—
284Q8—62no—
285Q8—62no—
286Q8C2×C3×#6711yescharacteristic
287D4—22no—
288D4—32no—
289D4—61no—
290D4—62no—
291D4—62no—
292D4—64no—
293D4—122no—
294D4—244no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—64no—
302C2×C4—62no—
303C2×C4C2×C3×S411yescharacteristic
304C8—62no—
305C8—62no—
306C2×C3—42no—
307C2×C3—42no—
308C2×C3—41no—
309C2×C3—62no—
310C2×C3—82no—
311C2×C3—81no—
312C2×C3—162no—
313C2×C3—164no—
314C2×C3—242no—
315C2×C3#592612yes—
316C2×C3C2×GL(2,ℤ/4)11yescharacteristic
317D3—162no—
318C22—24no—
319C22—22no—
320C22—32no—
321C22—62no—
322C22—124no—
323C22—122no—
324C22—244no—
325C22C22×C3×S411yescharacteristic
326C4—31no—
327C4—31no—
328C4—62no—
329C4—122no—
330C4C3×GL(2,ℤ/4)12yes—
331C3—41no—
332C3—81no—
333C3#8554011yescharacteristic, direct product factor
334C2—42no—
335C2—62no—
336C2—242no—
337C2C3×#592612yes—
338C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×C3×#986C2semidirect
3. C3×#5424C2semidirect
4. C3×#5421C2semidirect
5. #85540C3direct (N × H)
9. C22×C3×SL(2,𝔽3)C22semidirect
13. C3×#859C22semidirect
14. C2×C3×2−1+4D3semidirect
16. C2×2−1+4⋊C3C2×C3semidirect
17. C2×#986C2×C3semidirect
21. #5424C2×C3semidirect
22. #5421C2×C3semidirect
31. C3×#41⋊C3D4semidirect
50. C22×SL(2,𝔽3)C22×C3semidirect
67. #859C22×C3semidirect
80. C2×2−1+4C3×D3semidirect
95. C3×#41#67semidirect
119. #41⋊C3C3×D4semidirect
221. #41C3×#67semidirect
333. C3#85540direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×C3×#9862
C3×#54242
C3×#54212
#855403
C3×#30103
C3×#9404