Finite Group Databasebeta

← C2×C3×#5431 #11922816

Subgroups — C2×C3×#5431 #11922816

Order 1152.

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

normal subgroups only

This group has 665 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×#5431C214yesmaximal, direct product factor
4C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#5431C311yesmaximal, characteristic, direct product factor
6C2×C3×#454—31nomaximal
7C3×2−1+4⋊C3—21no—
8C3×2−1+4⋊C3C2212yescomplemented
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)—21no—
11C3×#986—21no—
12C3×#986C2212yescomplemented
13C2×C3×GL(2,𝔽3)—21no—
14C2×C3×#213—41nomaximal
15C3×GU(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4D311yescharacteristic, complemented
17C2×2−1+4⋊C3—21no—
18C2×2−1+4⋊C3C2×C311yescharacteristic, complemented
19C2×#986C2×C311yescharacteristic, complemented
20C2×C3×AGL(1,ℤ/8)—31no—
21C2×C3×AGL(1,ℤ/8)—31no—
22C2×C3×#124—31no—
23#5431C2×C314yesdirect product factor
24C2×GU(2,𝔽3)C2×C311yescharacteristic, complemented
25C3×#454—34no—
26C3×#454—34no—
27C2×C3×#104—31no—
28C2×C3×#104—31no—
29C2×C3×#101—31no—
30C2×C32×D4—41no—
31C2×C3×D12—41no—
32C3×#41⋊C3D411yescharacteristic
33C3×#41⋊C3C2311yescharacteristic
34C2×C3×SL(2,𝔽3)—22no—
35C2×C3×SL(2,𝔽3)D411yescharacteristic
36C3×GL(2,𝔽3)—22no—
37C3×#213—44no—
38C2×C3×C3⋊C8—41no—
39C2×#454—31noSylow
40C3×2−1+4—21no—
41C3×2−1+4—32no—
42C3×2−1+4—61no—
43C3×2−1+4C2×D312yescomplemented
44C2×C3×#41—31no—
45C2×C3×#41—61no—
46C2×C3×#41C2×D311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8—31no—
49C22×C3×D4—31no—
502−1+4⋊C3—22no—
512−1+4⋊C3—21no—
522−1+4⋊C3—22no—
532−1+4⋊C3C22×C312yescomplemented
54C2×#41⋊C3—21no—
55C2×#41⋊C3C22×C311yescharacteristic
56C22×SL(2,𝔽3)—22no—
57C22×SL(2,𝔽3)—21no—
58#986—21no—
59#986C22×C312yescomplemented
60C2×GL(2,𝔽3)—21no—
61C3×AGL(1,ℤ/8)—32no—
62C3×AGL(1,ℤ/8)—34no—
63C3×AGL(1,ℤ/8)—32no—
64C3×AGL(1,ℤ/8)—61no—
65C3×AGL(1,ℤ/8)—62no—
66C3×AGL(1,ℤ/8)—61no—
67C2×C3×SD16—61no—
68C2×C3×SD16—61no—
69C2×C3×D8—61no—
70C2×C3×D8—61no—
71C2×C3×M16—31no—
72C2×C3×M16—31no—
73C3×#124—34no—
74C3×#124—62no—
75C2×C3×C22⋊C4—62no—
76C2×C3×C42—31no—
77C2×#213—41no—
78GU(2,𝔽3)C22×C312yescomplemented
79C3×#104—32no—
80C3×#104—34no—
81C3×#104—32no—
82C3×#101—34no—
83C23×C32—81no—
84C22×C3×D3—81no—
85C32×D4—42no—
86C32×D4—81no—
87C2×C32×C4—41no—
88C3×D12—42no—
89C3×D12—81no—
90C3×SL(2,𝔽3)C2×D411yescharacteristic
91C3×C3⋊C8—42no—
92C2×2−1+4C3×D311yescharacteristic, complemented
93C2×AGL(1,ℤ/8)—31no—
94C2×AGL(1,ℤ/8)—31no—
95C2×#124—31no—
96#454—34no—
97#454—34no—
98C2×#104—31no—
99C2×#104—31no—
100C2×#101—31no—
101C24×C3—61no—
102C3×#41—31no—
103C3×#41—31no—
104C3×#41—32no—
105C3×#41—32no—
106C3×#41—62no—
107C3×#41—61no—
108C3×#41—61no—
109C3×#41—64no—
110C3×#41—62no—
111C3×#41#6711yescharacteristic
112C3×#41C22×D311yescharacteristic
113C2×C3×Q8—22no—
114C2×C3×Q8—32no—
115C2×C3×Q8—32no—
116C2×C3×Q8—31no—
117C2×C3×Q8—61no—
118C2×C3×Q8—61no—
119C2×C3×Q8—61no—
120C2×C3×Q8—62no—
121C2×C3×Q8#6711yescharacteristic
122C2×C3×D4—31no—
123C2×C3×D4—32no—
124C2×C3×D4—41no—
125C2×C3×D4—61no—
126C2×C3×D4—61no—
127C2×C3×D4—61no—
128C2×C3×D4—61no—
129C2×C3×D4—61no—
130C2×C3×D4—61no—
131C2×C3×D4—81no—
132C2×C3×D4S411yescharacteristic
133C22×C3×C4—31no—
134C22×C3×C4—32no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C2×D12—41no—
138#41⋊C3—21no—
139#41⋊C3—21no—
140#41⋊C3C3×D411yescharacteristic
141#41⋊C3C23×C311yescharacteristic
142C2×SL(2,𝔽3)—24no—
143C2×SL(2,𝔽3)—22no—
144C2×SL(2,𝔽3)—21no—
145C2×SL(2,𝔽3)C3×D411yescharacteristic
146GL(2,𝔽3)—22no—
147C3×SD16—62no—
148C3×SD16—64no—
149C3×SD16—62no—
150C3×D8—62no—
151C3×D8—62no—
152C3×D8—64no—
153C3×M16—32no—
154C3×M16—32no—
155C3×M16—61no—
156C3×M16—61no—
157C2×C3×C8—61no—
158C2×C3×C8—61no—
159C3×C22⋊C4—68no—
160C3×C42—32no—
161C3×C42—61no—
162#213—44no—
163C2×C3⋊C8—41no—
164C22×C32—41no—
165C22×C32—82no—
166C22×C32—162no—
167C2×C3×D3—82no—
168C2×C3×D3—162no—
169C32×C4—41no—
170C32×C4—41no—
1712−1+4—21no—
1722−1+4—32no—
1732−1+4—61no—
1742−1+4C2×C3×D312yescomplemented
175C2×#41—31no—
176C2×#41—61no—
177C2×#41C2×C3×D311yescharacteristic
178C22×Q8—21no—
179C22×Q8—31no—
180C22×D4—31no—
181AGL(1,ℤ/8)—32no—
182AGL(1,ℤ/8)—34no—
183AGL(1,ℤ/8)—32no—
184AGL(1,ℤ/8)—61no—
185AGL(1,ℤ/8)—62no—
186AGL(1,ℤ/8)—61no—
187C2×SD16—61no—
188C2×SD16—61no—
189C2×D8—61no—
190C2×D8—61no—
191C2×M16—31no—
192C2×M16—31no—
193#124—34no—
194#124—62no—
195C2×C22⋊C4—62no—
196C2×C42—31no—
197#104—32no—
198#104—34no—
199#104—32no—
200#101—34no—
201C23×C3—21no—
202C23×C3—31no—
203C23×C3—62no—
204C23×C3—81no—
205C23×C3—82no—
206C23×C3—121no—
207C23×C3—124no—
208C23×C3—121no—
209C22×D3—81no—
210C3×Q8—31no—
211C3×Q8—32no—
212C3×Q8—62no—
213C3×Q8—62no—
214C3×Q8—61no—
215C3×Q8C2×#6711yescharacteristic
216C3×D4—21no—
217C3×D4—31no—
218C3×D4—31no—
219C3×D4—42no—
220C3×D4—61no—
221C3×D4—62no—
222C3×D4—62no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—81no—
226C3×D4—82no—
227C3×D4—82no—
228C3×D4—121no—
229C3×D4—121no—
230C3×D4C2×S412yes—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—34no—
237C2×C3×C4—41no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—62no—
243C2×C3×C4—81no—
244C2×C3×C4—122no—
245C2×C3×C4C2×S411yescharacteristic
246D12—42no—
247D12—81no—
248SL(2,𝔽3)—21no—
249SL(2,𝔽3)C2×C3×D411yescharacteristic
250C3×C8—62no—
251C3×C8—62no—
252C3⋊C8—42no—
253C2×C32—41no—
254C2×C32—42no—
255C2×C32—162no—
256C3×D3—162no—
257C24—61no—
258#41—31no—
259#41—31no—
260#41—32no—
261#41—32no—
262#41—62no—
263#41—61no—
264#41—61no—
265#41—64no—
266#41—62no—
267#41C3×#6711yescharacteristic
268#41C22×C3×D311yescharacteristic
269C2×Q8—22no—
270C2×Q8—32no—
271C2×Q8—32no—
272C2×Q8—31no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—62no—
277C2×Q8C3×#6711yescharacteristic
278C2×D4—31no—
279C2×D4—32no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4C3×S411yescharacteristic
287C22×C4—31no—
288C22×C4—32no—
289C22×C4—61no—
290C22×C4—61no—
291SD16—62no—
292SD16—64no—
293SD16—62no—
294D8—62no—
295D8—62no—
296D8—64no—
297M16—32no—
298M16—32no—
299M16—61no—
300M16—61no—
301C2×C8—61no—
302C2×C8—61no—
303C22⋊C4—68no—
304C42—32no—
305C42—61no—
306C22×C3—22no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—62no—
312C22×C3—82no—
313C22×C3—84no—
314C22×C3—81no—
315C22×C3—124no—
316C22×C3—122no—
317C22×C3—162no—
318C22×C3—164no—
319C22×C3—242no—
320C22×C3—242no—
321C22×C3GL(2,ℤ/4)11yescharacteristic
322C2×D3—82no—
323C2×D3—162no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—62no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4—122no—
332C3×C4GL(2,ℤ/4)11yescharacteristic
333C3×C4C22×S411yescharacteristic
334C32—41noSylow
335C23—21no—
336C23—31no—
337C23—62no—
338C23—121no—
339C23—124no—
340C23—121no—
341Q8—31no—
342Q8—32no—
343Q8—62no—
344Q8—62no—
345Q8—61no—
346Q8C2×C3×#6711yescharacteristic
347D4—21no—
348D4—31no—
349D4—31no—
350D4—61no—
351D4—62no—
352D4—62no—
353D4—62no—
354D4—62no—
355D4—121no—
356D4—121no—
357D4C2×C3×S412yes—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—34no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—122no—
370C2×C4C2×C3×S411yescharacteristic
371C8—62no—
372C8—62no—
373C2×C3—42no—
374C2×C3—42no—
375C2×C3—41no—
376C2×C3—61no—
377C2×C3—61no—
378C2×C3—82no—
379C2×C3—81no—
380C2×C3—162no—
381C2×C3—164no—
382C2×C3—242no—
383C2×C3#543112yesdirect product factor
384C2×C3C2×GL(2,ℤ/4)11yescharacteristic
385D3—162no—
386C22—22no—
387C22—31no—
388C22—31no—
389C22—42no—
390C22—62no—
391C22—124no—
392C22—122no—
393C22—242no—
394C22—242no—
395C22C3×GL(2,ℤ/4)11yescharacteristic
396C4—31no—
397C4—31no—
398C4—62no—
399C4—122no—
400C4C3×GL(2,ℤ/4)11yescharacteristic
401C4C22×C3×S411yescharacteristic
402C3—41no—
403C3—81no—
404C3C2×#543111yescharacteristic, direct product factor
405C2—42no—
406C2—61no—
407C2—61no—
408C2—242no—
409C2C3×#543112yesdirect product factor
410C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23×C3
FrattiniC4C22×C3×S4
FittingC2×C3×2−1+4D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ 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×#5431C2direct (N × H)
4. C2×C3×GU(2,𝔽3)C2semidirect
5. C2×#5431C3direct (N × H)
8. C3×2−1+4⋊C3C22semidirect
12. C3×#986C22semidirect
15. C3×GU(2,𝔽3)C22semidirect
16. C2×C3×2−1+4D3semidirect
18. C2×2−1+4⋊C3C2×C3semidirect
19. C2×#986C2×C3semidirect
23. #5431C2×C3direct (N × H)
24. C2×GU(2,𝔽3)C2×C3semidirect
43. C3×2−1+4C2×D3semidirect
53. 2−1+4⋊C3C22×C3semidirect
59. #986C22×C3semidirect
78. GU(2,𝔽3)C22×C3semidirect
92. C2×2−1+4C3×D3semidirect
174. 2−1+4C2×C3×D3semidirect
383. C2×C3#5431direct (N × H)
404. C3C2×#5431direct (N × H)
409. C2C3×#5431direct (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×#54312
C2×C3×GU(2,𝔽3)2
C2×#54313
C2×C3×#4543
C2×C3×#2134