Finite Group Databasebeta

← C2×C3×#5423 #11922795

Subgroups — C2×C3×#5423 #11922795

Order 1152.

669 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 124 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
2C3×#5423C214yesmaximal, direct product factor
3C2×C3×GU(2,𝔽3)C212yesmaximal, complemented
4C2×#5423C311yesmaximal, characteristic, direct product factor
5C2×C3×#421—31nomaximal
6C2×C3×#41⋊C3C411yescharacteristic, complemented
7C2×C3×#41⋊C3C412yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2212yes—
10C22×C3×SL(2,𝔽3)C411yescharacteristic, complemented
11C2×C3×#208—41nomaximal
12C3×GU(2,𝔽3)C2214yescomplemented
13C22×C3×#41D311yescharacteristic
14C22×#41⋊C3—21no—
15C22×#41⋊C3C2×C311yescharacteristic
16C22×C3×M16—31no—
17C2×C3×#117—31no—
18#5423C2×C314yesdirect product factor
19C2×GU(2,𝔽3)C2×C312yescomplemented
20C3×#421—34no—
21C3×#421—34no—
22C2×C3×#104—32no—
23C2×C3×#104—32no—
24C22×C32×C4—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3C2×C412yescomplemented
27C3×#41⋊C3C2311yescharacteristic
28C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
29C2×C3×SL(2,𝔽3)C2×C412yescomplemented
30C3×#208—44no—
31C2×C3×C3⋊C8—42no—
32C2×#421—31noSylow
33C2×C3×#41—31no—
34C2×C3×#41—32no—
35C2×C3×#41—31no—
36C2×C3×#41—32no—
37C2×C3×#41Dic311yescharacteristic
38C2×C3×#41Dic312yes—
39C2×C3×#41C2×D311yescharacteristic
40C2×C3×#41C2×D312yes—
41C22×C3×Q8Dic311yescharacteristic
42C22×C3×D4—31no—
43C23×C3×C4—31no—
44C2×#41⋊C3—21no—
45C2×#41⋊C3—22no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3—22no—
48C2×#41⋊C3C3×C411yescharacteristic, complemented
49C2×#41⋊C3C3×C412yescomplemented
50C2×#41⋊C3C22×C311yescharacteristic
51C2×#41⋊C3C22×C312yes—
52C22×SL(2,𝔽3)—21no—
53C22×SL(2,𝔽3)C3×C411yescharacteristic, complemented
54C2×C3×M16—32no—
55C2×C3×M16—34no—
56C2×C3×M16—62no—
57C2×C3×M16—61no—
58C22×C3×C8—61no—
59C3×#117—34no—
60C3×#117—62no—
61C2×C3×C4⋊C4—61no—
62C2×C3×C22⋊C4—61no—
63C2×C3×C42—32no—
64C2×#208—41no—
65GU(2,𝔽3)C22×C314yescomplemented
66C3×#104—38no—
67C3×#104—34no—
68C3×#104—34no—
69C23×C32—41no—
70C2×C32×C4—42no—
71C2×C32×C4—41no—
72C2×C32×C4—42no—
73C2×C32×C4—41no—
74C3×SL(2,𝔽3)C22×C411yescharacteristic
75C3×C3⋊C8—44no—
76C22×#41C3×D311yescharacteristic
77C22×M16—31no—
78C2×#117—31no—
79#421—34no—
80#421—34no—
81C2×#104—32no—
82C2×#104—32no—
83C24×C3—31no—
84C3×#41—32no—
85C3×#41—31no—
86C3×#41—32no—
87C3×#41—32no—
88C3×#41—32no—
89C3×#41—31no—
90C3×#41—32no—
91C3×#41—61no—
92C3×#41—62no—
93C3×#41—61no—
94C3×#41C2×Dic311yescharacteristic
95C3×#41C2×Dic312yes—
96C3×#41C22×D311yescharacteristic
97C2×C3×Q8—31no—
98C2×C3×Q8—32no—
99C2×C3×Q8C2×Dic311yescharacteristic
100C2×C3×Q8C2×Dic312yes—
101C2×C3×D4—31no—
102C2×C3×D4—32no—
103C2×C3×D4—31no—
104C2×C3×D4—32no—
105C2×C3×D4—61no—
106C2×C3×D4—62no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—32no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—41no—
117C22×C3×C4—62no—
118C22×C3×C4—81no—
119C22×C3×C4S411yescharacteristic
120#41⋊C3—21no—
121#41⋊C3—22no—
122#41⋊C3—21no—
123#41⋊C3C2×C3×C411yescharacteristic
124#41⋊C3C2×C3×C412yescomplemented
125#41⋊C3C23×C311yescharacteristic
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)—22no—
128C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
129C2×SL(2,𝔽3)C2×C3×C412yescomplemented
130C3×M16—34no—
131C3×M16—62no—
132C3×M16—64no—
133C2×C3×C8—64no—
134C2×C3×C8—62no—
135C3×C4⋊C4—64no—
136C3×C22⋊C4—64no—
137C3×C42—34no—
138C3×C42—62no—
139#208—44no—
140C2×C3⋊C8—42no—
141C22×C32—41no—
142C22×C32—42no—
143C22×C32—82no—
144C32×C4—41no—
145C32×C4—42no—
146C32×C4—41no—
147C2×#41—31no—
148C2×#41—32no—
149C2×#41—31no—
150C2×#41—32no—
151C2×#41C3×Dic311yescharacteristic
152C2×#41C3×Dic312yes—
153C2×#41C2×C3×D311yescharacteristic
154C2×#41C2×C3×D312yes—
155C22×Q8C3×Dic311yescharacteristic
156C22×D4—31no—
157C23×C4—31no—
158C2×M16—32no—
159C2×M16—34no—
160C2×M16—62no—
161C2×M16—61no—
162C22×C8—61no—
163#117—34no—
164#117—62no—
165C2×C4⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—32no—
168#104—38no—
169#104—34no—
170#104—34no—
171C23×C3—31no—
172C23×C3—31no—
173C23×C3—32no—
174C23×C3—32no—
175C23×C3—41no—
176C23×C3—81no—
177C23×C3—122no—
178C23×C3SL(2,ℤ/4)11yescharacteristic
179C3×Q8—31no—
180C3×Q8—32no—
181C3×Q8—61no—
182C3×Q8C22×Dic311yescharacteristic
183C3×D4—31no—
184C3×D4—32no—
185C3×D4—31no—
186C3×D4—61no—
187C3×D4—62no—
188C3×D4—61no—
189C3×D4—62no—
190C2×C3×C4—32no—
191C2×C3×C4—31no—
192C2×C3×C4—32no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—31no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—31no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—64no—
209C2×C3×C4—81no—
210C2×C3×C4—82no—
211C2×C3×C4—82no—
212C2×C3×C4—81no—
213C2×C3×C4—124no—
214C2×C3×C4SL(2,ℤ/4)11yescharacteristic
215C2×C3×C4SL(2,ℤ/4)12yes—
216C2×C3×C4C2×S412yes—
217C2×C3×C4C2×S411yescharacteristic
218SL(2,𝔽3)—21no—
219SL(2,𝔽3)C22×C3×C411yescharacteristic
220C3×C8—64no—
221C3⋊C8—44no—
222C2×C32—41no—
223C2×C32—42no—
224C2×C32—82no—
225C24—31no—
226#41—32no—
227#41—31no—
228#41—32no—
229#41—32no—
230#41—32no—
231#41—31no—
232#41—32no—
233#41—61no—
234#41—62no—
235#41—61no—
236#41C2×C3×Dic311yescharacteristic
237#41C2×C3×Dic312yes—
238#41C22×C3×D311yescharacteristic
239C2×Q8—31no—
240C2×Q8—32no—
241C2×Q8C2×C3×Dic311yescharacteristic
242C2×Q8C2×C3×Dic312yes—
243C2×D4—31no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—32no—
247C2×D4—61no—
248C2×D4—62no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—32no—
252C22×C4—31no—
253C22×C4—32no—
254C22×C4—31no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—32no—
258C22×C4—62no—
259C22×C4C3×S411yescharacteristic
260M16—34no—
261M16—62no—
262M16—64no—
263C2×C8—64no—
264C2×C8—62no—
265C4⋊C4—64no—
266C22⋊C4—64no—
267C42—34no—
268C42—62no—
269C22×C3—22no—
270C22×C3—31no—
271C22×C3—32no—
272C22×C3—31no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—62no—
276C22×C3—62no—
277C22×C3—82no—
278C22×C3—84no—
279C22×C3—82no—
280C22×C3—81no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3C2×SL(2,ℤ/4)12yes—
284C22×C3C2×SL(2,ℤ/4)11yescharacteristic
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—32no—
288C3×C4—41no—
289C3×C4—42no—
290C3×C4—41no—
291C3×C4—81no—
292C3×C4—82no—
293C3×C4—81no—
294C3×C4—124no—
295C3×C4C2×SL(2,ℤ/4)12yes—
296C3×C4C2×SL(2,ℤ/4)11yescharacteristic
297C3×C4C22×S411yescharacteristic
298C32—41noSylow
299C23—31no—
300C23—31no—
301C23—32no—
302C23—32no—
303C23—122no—
304C23C3×SL(2,ℤ/4)11yescharacteristic
305Q8—31no—
306Q8—32no—
307Q8—61no—
308Q8C22×C3×Dic311yescharacteristic
309D4—31no—
310D4—32no—
311D4—31no—
312D4—61no—
313D4—62no—
314D4—61no—
315D4—62no—
316C2×C4—32no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—32no—
322C2×C4—31no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—31no—
328C2×C4—32no—
329C2×C4—31no—
330C2×C4—64no—
331C2×C4—124no—
332C2×C4C3×SL(2,ℤ/4)12yes—
333C2×C4C3×SL(2,ℤ/4)11yescharacteristic
334C2×C4C2×C3×S411yescharacteristic
335C2×C4C2×C3×S412yes—
336C8—64no—
337C2×C3—22no—
338C2×C3—42no—
339C2×C3—41no—
340C2×C3—61no—
341C2×C3—61no—
342C2×C3—62no—
343C2×C3—82no—
344C2×C3—84no—
345C2×C3—82no—
346C2×C3—81no—
347C2×C3#542312yesdirect product factor
348C2×C3C22×SL(2,ℤ/4)11yescharacteristic
349C22—22no—
350C22—31no—
351C22—32no—
352C22—31no—
353C22—62no—
354C22—62no—
355C22—122no—
356C22—122no—
357C22C2×C3×SL(2,ℤ/4)12yes—
358C22C2×C3×SL(2,ℤ/4)11yescharacteristic
359C4—31no—
360C4—31no—
361C4—32no—
362C4—124no—
363C4C2×C3×SL(2,ℤ/4)11yescharacteristic
364C4C2×C3×SL(2,ℤ/4)12yes—
365C4C22×C3×S411yescharacteristic
366C3—41no—
367C3—81no—
368C3C2×#542311yescharacteristic, direct product factor
369C2—22no—
370C2—61no—
371C2—61no—
372C2—62no—
373C2C3×#542312yesdirect product factor
374C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC4C22×C3×S4
FittingC22×C3×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC22×C3×#41⋊C3C2
cosocle#41⋊C3C23×C3
hypercenterC22×C3×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#5423C2direct (N × H)
3. C2×C3×GU(2,𝔽3)C2semidirect
4. C2×#5423C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
7. C2×C3×#41⋊C3C4semidirect
10. C22×C3×SL(2,𝔽3)C4semidirect
12. C3×GU(2,𝔽3)C22semidirect
18. #5423C2×C3direct (N × H)
19. C2×GU(2,𝔽3)C2×C3semidirect
26. C3×#41⋊C3C2×C4semidirect
29. C2×C3×SL(2,𝔽3)C2×C4semidirect
48. C2×#41⋊C3C3×C4semidirect
49. C2×#41⋊C3C3×C4semidirect
53. C22×SL(2,𝔽3)C3×C4semidirect
65. GU(2,𝔽3)C22×C3semidirect
124. #41⋊C3C2×C3×C4semidirect
129. C2×SL(2,𝔽3)C2×C3×C4semidirect
347. C2×C3#5423direct (N × H)
368. C3C2×#5423direct (N × H)
373. C2C3×#5423direct (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
C3×#54232
C2×C3×GU(2,𝔽3)2
C2×#54233
C2×C3×#4213
C2×C3×#2084