Finite Group Databasebeta

← C2×C3×#5421 #11922789

Subgroups — C2×C3×#5421 #11922789

Order 1152.

937 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 937 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#5421C214yesmaximal, direct product factor
4C2×C3×#859C211yesmaximal, characteristic, complemented
5C2×#5421C311yesmaximal, characteristic, direct product factor
6C2×C3×#448—31nomaximal
7C22×C3×#67—41nomaximal
8C22×C3×SL(2,𝔽3)—22no—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yescomplemented
11C2×C3×GL(2,𝔽3)—22no—
12C2×C3×GL(2,𝔽3)C2212yescomplemented
13C3×#859C2212yescomplemented
14C23×C3×Q8D311yescharacteristic, complemented
15C23×SL(2,𝔽3)—21no—
16C23×SL(2,𝔽3)C2×C311yescharacteristic, complemented
17C22×GL(2,𝔽3)C2×C311yescharacteristic, complemented
18C22×C3×SD16—31no—
19C22×C3×SD16—31no—
20C2×C3×#121—31no—
21#5421C2×C314yesdirect product factor
22C2×#859C2×C311yescharacteristic, complemented
23C3×#448—34no—
24C3×#448—34no—
25C2×C3×#103—31no—
26C2×C3×#103—31no—
27C2×C3×C22⋊C8—31no—
28C24×C32—41no—
29C23×C3×D3—41no—
30C2×C3×#67—44no—
31C2×C3×#67—48no—
32C22×C3×Dic3—41no—
33C2×C3×SL(2,𝔽3)—22no—
34C2×C3×SL(2,𝔽3)D412yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36C3×GL(2,𝔽3)—22no—
37C2×#448—31noSylow
38C22×C3×Q8—22no—
39C22×C3×Q8—31no—
40C22×C3×Q8—32no—
41C22×C3×Q8—62no—
42C22×C3×Q8C2×D311yescharacteristic
43C22×C3×Q8C2×D312yescomplemented
44C22×C3×D4—31no—
45C22×C3×D4—61no—
46C23×C3×C4—31no—
47C22×#67—41no—
48C22×SL(2,𝔽3)—24no—
49C22×SL(2,𝔽3)—22no—
50C22×SL(2,𝔽3)—21no—
51C22×SL(2,𝔽3)—22no—
52C22×SL(2,𝔽3)C22×C311yescharacteristic
53C22×SL(2,𝔽3)C22×C312yescomplemented
54C2×GL(2,𝔽3)—22no—
55C2×GL(2,𝔽3)C22×C312yescomplemented
56C2×C3×SD16—32no—
57C2×C3×SD16—34no—
58C2×C3×SD16—32no—
59C2×C3×SD16—62no—
60C2×C3×SD16—64no—
61C2×C3×SD16—62no—
62C22×C3×C8—31no—
63C22×C3×C8—31no—
64C3×#121—34no—
65C3×#121—62no—
66C2×C3×C4⋊C4—31no—
67C2×C3×C22⋊C4—61no—
68#859C22×C312yescomplemented
69C3×#103—32no—
70C3×#103—34no—
71C3×#103—32no—
72C3×C22⋊C8—34no—
73C23×C32—44no—
74C23×C32—42no—
75C23×C32—41no—
76C23×C32—82no—
77C23×C32—82no—
78C22×C3×D3—42no—
79C22×C3×D3—44no—
80C22×C3×D3—82no—
81C22×C3×D3—82no—
82C3×#67—416no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—44no—
85C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
86C23×Q8C3×D311yescharacteristic, complemented
87C22×SD16—31no—
88C22×SD16—31no—
89C2×#121—31no—
90#448—34no—
91#448—34no—
92C2×#103—31no—
93C2×#103—31no—
94C2×C22⋊C8—31no—
95C24×C3—41no—
96C24×C3—61no—
97C24×C3—81no—
98C24×C3S411yescharacteristic
99C23×D3—41no—
100C2×C3×Q8—22no—
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—32no—
104C2×C3×Q8—34no—
105C2×C3×Q8—61no—
106C2×C3×Q8—62no—
107C2×C3×Q8—64no—
108C2×C3×Q8—62no—
109C2×C3×Q8—64no—
110C2×C3×Q8#6712yescomplemented
111C2×C3×Q8C22×D311yescharacteristic
112C2×C3×D4—32no—
113C2×C3×D4—62no—
114C2×C3×D4—62no—
115C2×C3×D4—61no—
116C2×C3×D4—64no—
117C2×C3×D4—124no—
118C22×C3×C4—31no—
119C22×C3×C4—32no—
120C22×C3×C4—32no—
121C22×C3×C4—31no—
122C22×C3×C4—62no—
123C22×C3×C4—62no—
124C22×C3×C4—61no—
125C2×#67—44no—
126C2×#67—48no—
127C22×Dic3—41no—
128C2×SL(2,𝔽3)—24no—
129C2×SL(2,𝔽3)—22no—
130C2×SL(2,𝔽3)—22no—
131C2×SL(2,𝔽3)—21no—
132C2×SL(2,𝔽3)C3×D412yescomplemented
133C2×SL(2,𝔽3)C23×C311yescharacteristic
134GL(2,𝔽3)—22no—
135C3×SD16—64no—
136C3×SD16—62no—
137C3×SD16—68no—
138C3×SD16—62no—
139C2×C3×C8—32no—
140C2×C3×C8—32no—
141C2×C3×C8—62no—
142C2×C3×C8—62no—
143C3×C4⋊C4—32no—
144C3×C4⋊C4—61no—
145C3×C22⋊C4—64no—
146C22×C32—42no—
147C22×C32—44no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32—41no—
151C22×C32—82no—
152C22×C32—88no—
153C22×C32—82no—
154C2×C3×D3—44no—
155C2×C3×D3—82no—
156C2×C3×D3—82no—
157C2×C3×D3—88no—
158C3×Dic3—44no—
159C22×Q8—22no—
160C22×Q8—31no—
161C22×Q8—32no—
162C22×Q8—62no—
163C22×Q8C2×C3×D311yescharacteristic
164C22×Q8C2×C3×D312yescomplemented
165C22×D4—31no—
166C22×D4—61no—
167C23×C4—31no—
168C2×SD16—32no—
169C2×SD16—34no—
170C2×SD16—32no—
171C2×SD16—62no—
172C2×SD16—64no—
173C2×SD16—62no—
174C22×C8—31no—
175C22×C8—31no—
176#121—34no—
177#121—62no—
178C2×C4⋊C4—31no—
179C2×C22⋊C4—61no—
180#103—32no—
181#103—34no—
182#103—32no—
183C22⋊C8—34no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—41no—
187C23×C3—42no—
188C23×C3—44no—
189C23×C3—62no—
190C23×C3—82no—
191C23×C3—82no—
192C23×C3—84no—
193C23×C3—84no—
194C23×C3—82no—
195C23×C3—81no—
196C23×C3—84no—
197C23×C3—122no—
198C23×C3—122no—
199C23×C3—122no—
200C23×C3GL(2,𝔽3)14yescomplemented
201C23×C3C2×S412yes—
202C23×C3C2×S411yescharacteristic
203C22×D3—42no—
204C22×D3—44no—
205C22×D3—82no—
206C22×D3—82no—
207C3×Q8—32no—
208C3×Q8—31no—
209C3×Q8—34no—
210C3×Q8—61no—
211C3×Q8—62no—
212C3×Q8—64no—
213C3×Q8C2×#6711yescharacteristic, complemented
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—122no—
217C3×D4—128no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—32no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—64no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—122no—
229#67—416no—
230C2×Dic3—42no—
231C2×Dic3—44no—
232SL(2,𝔽3)—21no—
233SL(2,𝔽3)C2×C3×D411yescharacteristic, complemented
234C3×C8—62no—
235C3×C8—62no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—44no—
240C2×C32—84no—
241C3×D3—84no—
242C24—61no—
243C24C3×S411yescharacteristic
244C2×Q8—22no—
245C2×Q8—31no—
246C2×Q8—32no—
247C2×Q8—32no—
248C2×Q8—34no—
249C2×Q8—61no—
250C2×Q8—62no—
251C2×Q8—64no—
252C2×Q8—62no—
253C2×Q8—64no—
254C2×Q8C3×#6712yescomplemented
255C2×Q8C22×C3×D311yescharacteristic
256C2×D4—32no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—61no—
260C2×D4—64no—
261C2×D4—124no—
262C22×C4—31no—
263C22×C4—32no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—61no—
269SD16—64no—
270SD16—62no—
271SD16—68no—
272SD16—62no—
273C2×C8—32no—
274C2×C8—32no—
275C2×C8—62no—
276C2×C8—62no—
277C4⋊C4—32no—
278C4⋊C4—61no—
279C22⋊C4—64no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—28no—
283C22×C3—42no—
284C22×C3—41no—
285C22×C3—42no—
286C22×C3—44no—
287C22×C3—42no—
288C22×C3—88no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—816no—
292C22×C3—84no—
293C22×C3—84no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—82no—
297C22×C3—84no—
298C22×C3—82no—
299C22×C3—122no—
300C22×C3—122no—
301C22×C3—122no—
302C22×C3—244no—
303C22×C3C2×GL(2,𝔽3)12yes—
304C22×C3C2×GL(2,𝔽3)14yescomplemented
305C22×C3#98312yes—
306C22×C3GL(2,ℤ/4)12yes—
307C22×C3C22×S411yescharacteristic
308C2×D3—44no—
309C2×D3—82no—
310C2×D3—82no—
311C2×D3—88no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—32no—
315C3×C4—62no—
316C3×C4—122no—
317Dic3—44no—
318C32—41noSylow
319C23—22no—
320C23—22no—
321C23—62no—
322C23—122no—
323C23—122no—
324C23—122no—
325C23C3×GL(2,𝔽3)14yescomplemented
326C23C2×C3×S412yes—
327C23C2×C3×S411yescharacteristic
328Q8—32no—
329Q8—31no—
330Q8—34no—
331Q8—61no—
332Q8—62no—
333Q8—64no—
334Q8C2×C3×#6711yescharacteristic, complemented
335D4—62no—
336D4—62no—
337D4—122no—
338D4—128no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—64no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—122no—
350C8—62no—
351C8—62no—
352C2×C3—24no—
353C2×C3—44no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—84no—
358C2×C3—88no—
359C2×C3—84no—
360C2×C3—81no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3—242no—
364C2×C3#542114yesdirect product factor
365C2×C3C22×GL(2,𝔽3)11yescharacteristic
366C2×C3C2×#98311yescharacteristic
367C2×C3C2×GL(2,ℤ/4)11yescharacteristic
368D3—84no—
369C22—28no—
370C22—22no—
371C22—22no—
372C22—122no—
373C22—122no—
374C22—122no—
375C22—244no—
376C22C2×C3×GL(2,𝔽3)12yes—
377C22C2×C3×GL(2,𝔽3)14yescomplemented
378C22C3×#98312yes—
379C22C3×GL(2,ℤ/4)12yes—
380C22C22×C3×S411yescharacteristic
381C4—31no—
382C4—31no—
383C4—32no—
384C4—62no—
385C4—122no—
386C3—41no—
387C3—81no—
388C3C2×#542111yescharacteristic, direct product factor
389C2—24no—
390C2—242no—
391C2C3×#542114yesdirect product factor
392C2C22×C3×GL(2,𝔽3)11yescharacteristic
393C2C2×C3×#98311yescharacteristic
394C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
hypercenterC24×C3S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C22×C3×GL(2,𝔽3)C2semidirect
3. C3×#5421C2direct (N × H)
4. C2×C3×#859C2semidirect
5. C2×#5421C3direct (N × H)
10. C22×C3×SL(2,𝔽3)C22semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C3×#859C22semidirect
14. C23×C3×Q8D3semidirect
16. C23×SL(2,𝔽3)C2×C3semidirect
17. C22×GL(2,𝔽3)C2×C3semidirect
21. #5421C2×C3direct (N × H)
22. C2×#859C2×C3semidirect
34. C2×C3×SL(2,𝔽3)D4semidirect
43. C22×C3×Q8C2×D3semidirect
53. C22×SL(2,𝔽3)C22×C3semidirect
55. C2×GL(2,𝔽3)C22×C3semidirect
68. #859C22×C3semidirect
85. C3×SL(2,𝔽3)C2×D4semidirect
86. C23×Q8C3×D3semidirect
110. C2×C3×Q8#67semidirect
132. C2×SL(2,𝔽3)C3×D4semidirect
164. C22×Q8C2×C3×D3semidirect
200. C23×C3GL(2,𝔽3)semidirect
213. C3×Q8C2×#67semidirect
233. SL(2,𝔽3)C2×C3×D4semidirect
254. C2×Q8C3×#67semidirect
304. C22×C3C2×GL(2,𝔽3)semidirect
325. C23C3×GL(2,𝔽3)semidirect
334. Q8C2×C3×#67semidirect
364. C2×C3#5421direct (N × H)
377. C22C2×C3×GL(2,𝔽3)semidirect
388. C3C2×#5421direct (N × H)
391. C2C3×#5421direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C22×C3×GL(2,𝔽3)2
C3×#54212
C2×C3×#8592
C2×#54213
C2×C3×#4483
C22×C3×#674