Finite Group Databasebeta

← C2×C3×#548⋊C3 #11919714

Subgroups — C2×C3×#548⋊C3 #11919714

Order 1152.

1036 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C214yesmaximal, direct product factor
4C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×C3×#548C311yesmaximal, characteristic, complemented
6C2×#548⋊C3C313yesmaximal, direct product factor
7C2×C3×#41⋊C3—22no—
8C2×C3×#41⋊C3C2212yescomplemented
9C22×C3×SL(2,𝔽3)—22no—
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2212yescomplemented
12C2×C32×C22⋊C4—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C22×C3×#41C2×C311yescharacteristic, complemented
15C23×C3×Q8C2×C311yescharacteristic
16C22×#41⋊C3C2×C313yescomplemented
17C23×SL(2,𝔽3)C2×C313yescomplemented
18C3×#548—34no—
19C3×#548C2×C314yescomplemented
20C2×C3×#124—31no—
21C2×C3×C22⋊Q8—31no—
22C2×C3×#121—31no—
23C2×C3×C4×Q8C2×C311yescharacteristic, complemented
24C2×C3×C4×D4—31no—
25#548⋊C3C2×C3112yesdirect product factor
26C2×C4×SL(2,𝔽3)C2×C313yescomplemented
27C24×C32—41no—
28C22×C32×C4—41no—
29C22×C32×C4—41no—
30C3×#41⋊C3—22no—
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)D412yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C32×C22⋊C4—44no—
35C2×#548C3211yescharacteristic, complemented, Sylow
36C2×C3×#41—22no—
37C2×C3×#41—32no—
38C2×C3×#41—62no—
39C2×C3×#41C22×C312yescomplemented
40C22×C3×Q8—22no—
41C22×C3×Q8—31no—
42C22×C3×Q8—32no—
43C22×C3×Q8—62no—
44C22×C3×Q8C22×C312yes—
45C22×C3×Q8C22×C311yescharacteristic
46C22×C3×D4—31no—
47C22×C3×D4—31no—
48C23×C3×C4—31no—
49C23×C3×C4—31no—
50C2×#41⋊C3—26no—
51C2×#41⋊C3C22×C316yescomplemented
52C22×SL(2,𝔽3)—26no—
53C22×SL(2,𝔽3)C22×C313yes—
54C22×SL(2,𝔽3)C22×C316yescomplemented
55C3×#124—34no—
56C3×#124—34no—
57C3×C22⋊Q8—34no—
58C3×C22⋊Q8—34no—
59C3×#121—34no—
60C3×#121—34no—
61C3×C4×Q8—32no—
62C3×C4×Q8C22×C312yescomplemented
63C3×C4×D4—34no—
64C3×C4×D4—34no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—31no—
67C2×C3×C22⋊C4—32no—
68C2×C3×C22⋊C4—31no—
69C2×C3×C22⋊C4—43no—
70C2×C3×C22⋊C4A411yescharacteristic
71C2×C3×C42—31no—
72C4×SL(2,𝔽3)C22×C316yescomplemented
73C23×C32—44no—
74C23×C32—42no—
75C23×C32—41no—
76C23×C32—82no—
77C23×C32—82no—
78C2×C32×C4—42no—
79C2×C32×C4—42no—
80C2×C32×C4—82no—
81C2×C32×C4—82no—
82C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
83C22×#41C2×C3211yescharacteristic, complemented
84C23×Q8C2×C3211yescharacteristic
85#548—34no—
86#548C2×C3214yescomplemented
87C2×#124—31no—
88C2×C22⋊Q8—31no—
89C2×#121—31no—
90C2×C4×Q8C2×C3211yescharacteristic, complemented
91C2×C4×D4—31no—
92C24×C3—31no—
93C24×C3—43no—
94C24×C3C2×A411yescharacteristic
95C3×#41—22no—
96C3×#41—64no—
97C3×#41—64no—
98C3×#41—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—32no—
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—32no—
104C2×C3×Q8—62no—
105C2×C3×Q8—62no—
106C2×C3×Q8—64no—
107C2×C3×Q8—62no—
108C2×C3×Q8—64no—
109C2×C3×Q8C3×D412yes—
110C2×C3×Q8C23×C311yescharacteristic
111C2×C3×D4—32no—
112C2×C3×D4—32no—
113C2×C3×D4—34no—
114C2×C3×D4—62no—
115C2×C3×D4—62no—
116C2×C3×D4—64no—
117C22×C3×C4—31no—
118C22×C3×C4—32no—
119C22×C3×C4—32no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—32no—
123C22×C3×C4—32no—
124C22×C3×C4—43no—
125C22×C3×C4—43no—
126C22×C3×C4—62no—
127C22×C3×C4—62no—
128C22×C3×C4—62no—
129C22×C3×C4—62no—
130C22×C3×C4C2×A411yescharacteristic
131C22×C3×C4C2×A411yescharacteristic
132#41⋊C3—26no—
133C2×SL(2,𝔽3)—26no—
134C2×SL(2,𝔽3)C3×D416yescomplemented
135C2×SL(2,𝔽3)C23×C313yes—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—32no—
138C3×C4⋊C4—34no—
139C3×C22⋊C4—38no—
140C3×C22⋊C4—34no—
141C3×C22⋊C4—412no—
142C3×C22⋊C4C2×A414yes—
143C3×C42—32no—
144C3×C42—32no—
145C22×C32—42no—
146C22×C32—42no—
147C22×C32—44no—
148C22×C32—42no—
149C22×C32—41no—
150C22×C32—82no—
151C22×C32—88no—
152C22×C32—82no—
153C32×C4—82no—
154C32×C4—82no—
155C2×#41—22no—
156C2×#41—32no—
157C2×#41—62no—
158C2×#41C22×C3212yescomplemented
159C22×Q8—22no—
160C22×Q8—31no—
161C22×Q8—32no—
162C22×Q8—62no—
163C22×Q8C22×C3212yes—
164C22×Q8C22×C3211yescharacteristic
165C22×D4—31no—
166C22×D4—31no—
167C23×C4—31no—
168C23×C4—31no—
169#124—34no—
170#124—34no—
171C22⋊Q8—34no—
172C22⋊Q8—34no—
173#121—34no—
174#121—34no—
175C4×Q8—32no—
176C4×Q8C22×C3212yescomplemented
177C4×D4—34no—
178C4×D4—34no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C22⋊C4—32no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4C3×A411yescharacteristic
184C2×C42—31no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—32no—
188C23×C3—43no—
189C23×C3—46no—
190C23×C3—412no—
191C23×C3—62no—
192C23×C3—62no—
193C23×C3—62no—
194C23×C3—86no—
195C23×C3—86no—
196C23×C3#41⋊C314yescomplemented
197C23×C3C22×A412yes—
198C23×C3C22×A411yescharacteristic
199C3×Q8—32no—
200C3×Q8—31no—
201C3×Q8—32no—
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—64no—
205C3×Q8C2×C3×D411yescharacteristic
206C3×D4—62no—
207C3×D4—62no—
208C3×D4—64no—
209C3×D4—68no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—46no—
219C2×C3×C4—46no—
220C2×C3×C4—62no—
221C2×C3×C4—64no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—64no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—86no—
229C2×C3×C4—86no—
230C2×C3×C4C22×A412yes—
231C2×C3×C4C22×A412yes—
232SL(2,𝔽3)C2×C3×D413yescomplemented
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—44no—
237C2×C32—84no—
238C24—31no—
239C24C2×C3×A411yescharacteristic
240#41—22no—
241#41—64no—
242#41—64no—
243#41—62no—
244C2×Q8—22no—
245C2×Q8—32no—
246C2×Q8—31no—
247C2×Q8—32no—
248C2×Q8—32no—
249C2×Q8—62no—
250C2×Q8—62no—
251C2×Q8—64no—
252C2×Q8—62no—
253C2×Q8—64no—
254C2×Q8C32×D412yes—
255C2×Q8C23×C3211yescharacteristic
256C2×D4—32no—
257C2×D4—32no—
258C2×D4—34no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—64no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—32no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4C2×C3×A411yescharacteristic
274C22×C4C2×C3×A411yescharacteristic
275C4⋊C4—32no—
276C4⋊C4—32no—
277C4⋊C4—34no—
278C22⋊C4—38no—
279C22⋊C4—34no—
280C22⋊C4C2×C3×A414yes—
281C42—32no—
282C42—32no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—28no—
286C22×C3—46no—
287C22×C3—46no—
288C22×C3—412no—
289C22×C3—46no—
290C22×C3—43no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—62no—
294C22×C3—86no—
295C22×C3—824no—
296C22×C3—86no—
297C22×C3—124no—
298C22×C3D4×A412yes—
299C22×C3C2×#41⋊C312yes—
300C22×C3C2×#41⋊C314yescomplemented
301C22×C32−1+4⋊C312yes—
302C22×C3C23×A411yescharacteristic
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—31no—
306C3×C4—32no—
307C3×C4—31no—
308C3×C4—62no—
309C3×C4—62no—
310C3×C4—86no—
311C3×C4—86no—
312C32—41noSylow
313C23—22no—
314C23—22no—
315C23—32no—
316C23—62no—
317C23—62no—
318C23—62no—
319C23C3×#41⋊C314yescomplemented
320C23C22×C3×A412yes—
321C23C22×C3×A411yescharacteristic
322Q8—32no—
323Q8—31no—
324Q8—32no—
325Q8—62no—
326Q8—62no—
327Q8—64no—
328Q8C2×C32×D411yescharacteristic
329D4—62no—
330D4—62no—
331D4—64no—
332D4—68no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—64no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—64no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4C22×C3×A412yes—
350C2×C4C22×C3×A412yes—
351C2×C3—24no—
352C2×C3—43no—
353C2×C3—43no—
354C2×C3—43no—
355C2×C3—412no—
356C2×C3—812no—
357C2×C3—122no—
358C2×C3#548⋊C314yesdirect product factor
359C2×C3C2×D4×A411yescharacteristic
360C2×C3C22×#41⋊C311yescharacteristic
361C2×C3C2×2−1+4⋊C311yescharacteristic
362C22—22no—
363C22—22no—
364C22—28no—
365C22—62no—
366C22—62no—
367C22—62no—
368C22—124no—
369C22C3×D4×A412yes—
370C22C2×C3×#41⋊C312yes—
371C22C2×C3×#41⋊C314yescomplemented
372C22C3×2−1+4⋊C312yes—
373C22C23×C3×A411yescharacteristic
374C4—22no—
375C4—22no—
376C4—31no—
377C4—32no—
378C4—31no—
379C4—62no—
380C4—62no—
381C3—43no—
382C3C2×#548⋊C311yescharacteristic, direct product factor
383C2—24no—
384C2—122no—
385C2C3×#548⋊C314yesdirect product factor
386C2C2×C3×D4×A411yescharacteristic
387C2C22×C3×#41⋊C311yescharacteristic
388C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC2×C3×#548C3
hypercenterC2×C3×C22⋊C4A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C23×C3×SL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2direct (N × H)
4. C2×C3×C4×SL(2,𝔽3)C2semidirect
5. C2×C3×#548C3semidirect
6. C2×#548⋊C3C3direct (N × H)
8. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. C22×C3×#41C2×C3semidirect
16. C22×#41⋊C3C2×C3semidirect
17. C23×SL(2,𝔽3)C2×C3semidirect
19. C3×#548C2×C3semidirect
23. C2×C3×C4×Q8C2×C3semidirect
25. #548⋊C3C2×C3direct (N × H)
26. C2×C4×SL(2,𝔽3)C2×C3semidirect
32. C2×C3×SL(2,𝔽3)D4semidirect
35. C2×#548C32semidirect
39. C2×C3×#41C22×C3semidirect
51. C2×#41⋊C3C22×C3semidirect
54. C22×SL(2,𝔽3)C22×C3semidirect
62. C3×C4×Q8C22×C3semidirect
72. C4×SL(2,𝔽3)C22×C3semidirect
82. C3×SL(2,𝔽3)C2×D4semidirect
83. C22×#41C2×C32semidirect
86. #548C2×C32semidirect
90. C2×C4×Q8C2×C32semidirect
134. C2×SL(2,𝔽3)C3×D4semidirect
158. C2×#41C22×C32semidirect
176. C4×Q8C22×C32semidirect
196. C23×C3#41⋊C3semidirect
232. SL(2,𝔽3)C2×C3×D4semidirect
300. C22×C3C2×#41⋊C3semidirect
319. C23C3×#41⋊C3semidirect
358. C2×C3#548⋊C3direct (N × H)
371. C22C2×C3×#41⋊C3semidirect
382. C3C2×#548⋊C3direct (N × H)
385. C2C3×#548⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C23×C3×SL(2,𝔽3)2
C3×#548⋊C32
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#5483
C2×#548⋊C33
C2×C32×C22⋊C44