Finite Group Databasebeta

← C3×#87510 #11923752

Subgroups — C3×#87510 #11923752

Order 1152.

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

normal subgroups only

This group has 602 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, complemented
2C3×#5931C212yesmaximal, complemented
3#87510C311yesmaximal, characteristic, direct product factor
4C3×#3006—31nomaximal
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
6C3×#986—42nomaximal
7C3×C4⋊S4—41nomaximal
8C22×C3×#41D311yescharacteristic, complemented
9#5948—21no—
10#5948C2×C311yescharacteristic, complemented
11#5931C2×C312yescomplemented
12C2×C3×AGL(1,ℤ/8)—32no—
13C2×C3×#121—31no—
14C3×#449—34no—
15C3×#448—32no—
16C3×#447—32no—
17C3×#417—32no—
18C3×#407—31no—
19C2×C3×S4—42no—
20C3×#41⋊C3—42no—
21C3×#41⋊C3—81no—
22C3×C4×A4—41no—
23C3×GL(2,𝔽3)—44no—
24#3006—31noSylow
25C2×C3×#41—32no—
26C2×C3×#41—32no—
27C2×C3×#41—61no—
28C2×C3×#41—61no—
29C22×C3×Q8C2×D311yescharacteristic
30C22×C3×D4—31no—
31C22×C3×D4—31no—
32C22×C3×D4—61no—
33C23×C3×C4—31no—
34C22⋊SL(2,𝔽3)—21no—
35C22⋊SL(2,𝔽3)C22×C311yescharacteristic, complemented
36#986—42no—
37C4⋊S4—41no—
38C3×AGL(1,ℤ/8)—62no—
39C3×AGL(1,ℤ/8)—62no—
40C3×AGL(1,ℤ/8)—62no—
41C3×AGL(1,ℤ/8)—62no—
42C2×C3×SD16—34no—
43C2×C3×D8—34no—
44C2×C3×M16—32no—
45C3×#121—32no—
46C3×#121—32no—
47C3×#121—61no—
48C3×#121—61no—
49C2×C3×C4⋊C4—31no—
50C2×C3×C22⋊C4—61no—
51C3×#103—34no—
52C3×#102—34no—
53C3×C22⋊C8—32no—
54C3×C22⋊C8—32no—
55C2×C3×A4—41no—
56C3×S4—82no—
57C3×D12—161no—
58C3×SL(2,𝔽3)—42no—
59C3×SL(2,𝔽3)—81no—
60C22×#41C3×D311yescharacteristic, complemented
61C2×AGL(1,ℤ/8)—32no—
62C2×#121—31no—
63#449—34no—
64#448—32no—
65#447—32no—
66#417—32no—
67#407—31no—
68C24×C3—31no—
69C24×C3—61no—
70C2×S4—42no—
71C3×#41—21no—
72C3×#41—32no—
73C3×#41—32no—
74C3×#41—32no—
75C3×#41—61no—
76C3×#41—62no—
77C3×#41—61no—
78C3×#41—61no—
79C3×#41—62no—
80C3×#41S412yescomplemented
81C2×C3×Q8—32no—
82C2×C3×Q8—61no—
83C2×C3×D4—32no—
84C2×C3×D4—32no—
85C2×C3×D4—61no—
86C2×C3×D4—62no—
87C2×C3×D4—62no—
88C2×C3×D4—61no—
89C2×C3×D4—62no—
90C2×C3×D4—62no—
91C2×C3×D4—62no—
92C2×C3×D4—62no—
93C2×C3×D4—122no—
94C2×C3×D4—122no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—62no—
101C22×C3×C4—62no—
102C22×C3×C4—61no—
103C22×C3×C4S411yescharacteristic
104#41⋊C3—42no—
105#41⋊C3—81no—
106#41⋊C3—82no—
107#41⋊C3—82no—
108C4×A4—41no—
109C4×A4—81no—
110GL(2,𝔽3)—44no—
111C3×SD16—64no—
112C3×SD16—64no—
113C3×D8—64no—
114C3×D8—64no—
115C3×M16—62no—
116C3×M16—62no—
117C2×C3×C8—34no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—61no—
120C3×C22⋊C4—62no—
121C3×C22⋊C4—62no—
122C2×C3×D3—162no—
123C3×A4—41no—
124C32×C4—161no—
125C2×#41—32no—
126C2×#41—32no—
127C2×#41—61no—
128C2×#41—61no—
129C22×Q8C2×C3×D311yescharacteristic
130C22×D4—31no—
131C22×D4—31no—
132C22×D4—61no—
133C23×C4—31no—
134AGL(1,ℤ/8)—62no—
135AGL(1,ℤ/8)—62no—
136AGL(1,ℤ/8)—62no—
137AGL(1,ℤ/8)—62no—
138C2×SD16—34no—
139C2×D8—34no—
140C2×M16—32no—
141#121—32no—
142#121—32no—
143#121—61no—
144#121—61no—
145C2×C4⋊C4—31no—
146C2×C22⋊C4—61no—
147#103—34no—
148#102—34no—
149C22⋊C8—32no—
150C22⋊C8—32no—
151C23×C3—31no—
152C23×C3—31no—
153C23×C3—62no—
154C23×C3—61no—
155C23×C3—62no—
156C23×C3—61no—
157C23×C3—121no—
158C23×C3—122no—
159C23×C3—122no—
160C23×C3—122no—
161C23×C3C2×S411yescharacteristic
162C2×A4—41no—
163C2×A4—81no—
164S4—82no—
165C3×Q8—21no—
166C3×Q8—32no—
167C3×Q8—61no—
168C3×Q8C2×S412yes—
169C3×D4—32no—
170C3×D4—32no—
171C3×D4—62no—
172C3×D4—64no—
173C3×D4—61no—
174C3×D4—62no—
175C3×D4—61no—
176C3×D4—121no—
177C3×D4—122no—
178C3×D4—124no—
179C3×D4—122no—
180C3×D4—121no—
181C2×C3×C4—31no—
182C2×C3×C4—32no—
183C2×C3×C4—31no—
184C2×C3×C4—32no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—62no—
193C2×C3×C4—122no—
194D12—161no—
195SL(2,𝔽3)—42no—
196SL(2,𝔽3)—81no—
197SL(2,𝔽3)—82no—
198SL(2,𝔽3)—82no—
199C3×C8—64no—
200C2×C32—161no—
201C3×D3—322no—
202C24—31no—
203C24—61no—
204#41—21no—
205#41—32no—
206#41—32no—
207#41—32no—
208#41—61no—
209#41—62no—
210#41—61no—
211#41—61no—
212#41—62no—
213#41C3×S412yescomplemented
214C2×Q8—32no—
215C2×Q8—61no—
216C2×D4—32no—
217C2×D4—32no—
218C2×D4—61no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—61no—
222C2×D4—62no—
223C2×D4—62no—
224C2×D4—62no—
225C2×D4—62no—
226C2×D4—122no—
227C2×D4—122no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—61no—
236C22×C4C3×S411yescharacteristic
237SD16—64no—
238SD16—64no—
239D8—64no—
240D8—64no—
241M16—62no—
242M16—62no—
243C2×C8—34no—
244C4⋊C4—32no—
245C4⋊C4—61no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3—122no—
259C22×C3—122no—
260C22×C3—122no—
261C22×C3—244no—
262C22×C3#98611yescharacteristic, complemented
263C2×D3—162no—
264A4—41no—
265A4—81no—
266C3×C4—32no—
267C3×C4—31no—
268C3×C4—61no—
269C3×C4—122no—
270C3×C4—161no—
271C3×C4—321no—
272C3×C4#102011yescharacteristic
273C32—161noSylow
274C23—31no—
275C23—31no—
276C23—62no—
277C23—61no—
278C23—62no—
279C23—61no—
280C23—121no—
281C23—122no—
282C23—122no—
283C23—122no—
284C23C2×C3×S411yescharacteristic
285Q8—21no—
286Q8—32no—
287Q8—61no—
288Q8C2×C3×S412yes—
289D4—32no—
290D4—32no—
291D4—62no—
292D4—64no—
293D4—61no—
294D4—62no—
295D4—61no—
296D4—121no—
297D4—122no—
298D4—124no—
299D4—122no—
300D4—121no—
301C2×C4—31no—
302C2×C4—32no—
303C2×C4—31no—
304C2×C4—32no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—62no—
313C2×C4—122no—
314C8—64no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—62no—
318C2×C3—121no—
319C2×C3—161no—
320C2×C3—242no—
321C2×C3—321no—
322C2×C3C2×#102011yescharacteristic
323D3—322no—
324C22—31no—
325C22—31no—
326C22—32no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22—61no—
331C22—61no—
332C22—122no—
333C22—122no—
334C22—122no—
335C22—122no—
336C22—122no—
337C22—244no—
338C22C3×#98611yescharacteristic, complemented
339C4—32no—
340C4—31no—
341C4—61no—
342C4—122no—
343C4C3×#102011yescharacteristic
344C3—161no—
345C3—321no—
346C3#8751011yescharacteristic, direct product factor
347C2—31no—
348C2—31no—
349C2—62no—
350C2—121no—
351C2—242no—
352C2C2×C3×#102011yescharacteristic
subgroupG/H
centerC2×C3C2×#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×#1020
FittingC22×C3×#41D3
socleC23×C3C2×S4
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
1. C3×#5948C2semidirect
2. C3×#5931C2semidirect
3. #87510C3direct (N × H)
5. C3×C22⋊SL(2,𝔽3)C22semidirect
8. C22×C3×#41D3semidirect
10. #5948C2×C3semidirect
11. #5931C2×C3semidirect
35. C22⋊SL(2,𝔽3)C22×C3semidirect
60. C22×#41C3×D3semidirect
80. C3×#41S4semidirect
213. #41C3×S4semidirect
262. C22×C3#986semidirect
338. C22C3×#986semidirect
346. C3#87510direct (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×#59312
#875103
C3×#30063
C3×#9864
C3×C4⋊S44