Finite Group Databasebeta

← C3×#3480⋊C3 #11919730

Subgroups — C3×#3480⋊C3 #11919730

Order 1152.

680 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 162 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C3×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C212yesmaximal, complemented
4C3×C4×#41⋊C3C212yesmaximal, complemented
5C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C3×#3480C311yesmaximal, characteristic, complemented
7#3480⋊C3C313yesmaximal, direct product factor
8C2×C3×#41⋊C3C2212yes—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C32×#117—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×#559—31no—
15C3×#557C2×C311yescharacteristic, complemented
16C3×#556—31no—
17C3×#553—32no—
18C3×#550C2×C311yescharacteristic, complemented
19C3×#548C2×C312yes—
20C3×#532—31no—
21C3×#529—32no—
22C3×C4×#41—31no—
23C3×C4×#41C2×C312yescomplemented
24C2×C3×C4×Q8C2×C311yescharacteristic
25#557⋊C3C2×C313yescomplemented
26C4⋊(#41⋊C3)C2×C313yescomplemented
27#548⋊C3C2×C316yescomplemented
28C4×#41⋊C3C2×C316yescomplemented
29C2×C4×SL(2,𝔽3)C2×C313yescomplemented
30C22×C32×C4—41no—
31C3×#41⋊C3—22no—
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C32×C4⋊C4—41no—
35C32×C4⋊C4—41no—
36C32×C22⋊C4—42no—
37C32×C42—42no—
38#3480C3211yescharacteristic, complemented, Sylow
39C2×C3×#41—31no—
40C2×C3×#41C22×C312yes—
41C22×C3×Q8C22×C311yescharacteristic
42C2×#41⋊C3C22×C316yes—
43C22×SL(2,𝔽3)C22×C313yes—
44C3×C4⋊Q8—31no—
45C3×#127—31no—
46C3×#126—34no—
47C3×#125—32no—
48C3×#124—32no—
49C3×#124—32no—
50C3×#123—34no—
51C3×C22⋊Q8—32no—
52C3×#121—32no—
53C3×#121—32no—
54C3×#121—32no—
55C3×C4×Q8—31no—
56C3×C4×Q8—31no—
57C3×C4×Q8—31no—
58C3×C4×Q8—31no—
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×Q8C22×C311yescharacteristic
61C3×C4×Q8C22×C312yes—
62C3×C4×D4—32no—
63C3×C4×D4—32no—
64C3×C4×D4—32no—
65C3×C4×D4—31no—
66C3×C4×D4—31no—
67C3×C4×D4—32no—
68C3×C4×D4—32no—
69C3×#117—32no—
70C3×#117—32no—
71C3×#117—43no—
72C3×#117A411yescharacteristic
73C2×C3×C4⋊C4—31no—
74C2×C3×C42—31no—
75C2×C3×C42—32no—
76C4×SL(2,𝔽3)C22×C313yes—
77C4×SL(2,𝔽3)C22×C313yes—
78C4×SL(2,𝔽3)C22×C316yes—
79C23×C32—41no—
80C2×C32×C4—44no—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C3×SL(2,𝔽3)#4111yescharacteristic, complemented
86#559—31no—
87#557C2×C3211yescharacteristic, complemented
88#556—31no—
89#553—32no—
90#550C2×C3211yescharacteristic, complemented
91#548C2×C3212yes—
92#532—31no—
93#529—32no—
94C4×#41—31no—
95C4×#41C2×C3212yescomplemented
96C2×C4×Q8C2×C3211yescharacteristic
97C3×#41—22no—
98C3×#41—64no—
99C3×#41—62no—
100C2×C3×Q8—21no—
101C2×C3×Q8—31no—
102C2×C3×Q8—31no—
103C2×C3×Q8—61no—
104C2×C3×Q8C23×C311yescharacteristic
105C2×C3×D4—32no—
106C2×C3×D4—32no—
107C2×C3×D4—32no—
108C2×C3×D4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4—43no—
115C22×C3×C4C2×A411yescharacteristic
116#41⋊C3—26no—
117C2×SL(2,𝔽3)—23no—
118C2×SL(2,𝔽3)C23×C313yes—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—32no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—43no—
127C3×C4⋊C4—43no—
128C3×C4⋊C4C2×A411yescharacteristic
129C3×C4⋊C4C2×A411yescharacteristic
130C3×C22⋊C4—34no—
131C3×C22⋊C4—34no—
132C3×C22⋊C4—32no—
133C3×C22⋊C4—32no—
134C3×C22⋊C4—46no—
135C3×C22⋊C4C2×A412yes—
136C3×C42—32no—
137C3×C42—32no—
138C3×C42—31no—
139C3×C42—31no—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—46no—
143C3×C42C2×A412yes—
144C22×C32—42no—
145C22×C32—41no—
146C22×C32—81no—
147C22×C32—81no—
148C32×C4—42no—
149C32×C4—42no—
150C32×C4—82no—
151C32×C4—82no—
152C2×#41—31no—
153C2×#41C22×C3212yes—
154C22×Q8C22×C3211yescharacteristic
155C4⋊Q8—31no—
156#127—31no—
157#126—34no—
158#125—32no—
159#124—32no—
160#124—32no—
161#123—34no—
162C22⋊Q8—32no—
163#121—32no—
164#121—32no—
165#121—32no—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×Q8C22×C3211yescharacteristic
171C4×Q8C22×C3211yescharacteristic
172C4×Q8C22×C3212yes—
173C4×D4—32no—
174C4×D4—32no—
175C4×D4—32no—
176C4×D4—31no—
177C4×D4—31no—
178C4×D4—32no—
179C4×D4—32no—
180#117—32no—
181#117—32no—
182#117C3×A411yescharacteristic
183C2×C4⋊C4—31no—
184C2×C42—31no—
185C2×C42—32no—
186C23×C3—32no—
187C23×C3—43no—
188C23×C3C22×A411yescharacteristic
189C3×Q8—31no—
190C3×Q8—61no—
191C3×Q8—61no—
192C3×Q8—62no—
193C3×Q8C3×#4111yescharacteristic
194C3×D4—62no—
195C3×D4—62no—
196C3×D4—64no—
197C3×D4—62no—
198C3×D4—122no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—43no—
205C2×C3×C4—43no—
206C2×C3×C4—46no—
207C2×C3×C4—46no—
208C2×C3×C4—412no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—64no—
214C2×C3×C4—122no—
215C2×C3×C4—121no—
216C2×C3×C4#41⋊C314yescomplemented
217C2×C3×C4C22×A412yes—
218C2×C3×C4C22×A412yes—
219C2×C3×C4C22×A411yescharacteristic
220C2×C3×C4C22×A411yescharacteristic
221SL(2,𝔽3)C3×#4113yescomplemented
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—82no—
226#41—22no—
227#41—64no—
228#41—62no—
229C2×Q8—21no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—61no—
233C2×Q8C23×C3211yescharacteristic
234C2×D4—32no—
235C2×D4—32no—
236C2×D4—32no—
237C2×D4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—32no—
241C22×C4—32no—
242C22×C4—32no—
243C22×C4C2×C3×A411yescharacteristic
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—32no—
247C4⋊C4—32no—
248C4⋊C4—32no—
249C4⋊C4—32no—
250C4⋊C4—32no—
251C4⋊C4C2×C3×A411yescharacteristic
252C4⋊C4C2×C3×A411yescharacteristic
253C22⋊C4—34no—
254C22⋊C4—34no—
255C22⋊C4—32no—
256C22⋊C4—32no—
257C22⋊C4C2×C3×A412yes—
258C42—31no—
259C42—31no—
260C42—32no—
261C42—32no—
262C42—31no—
263C42—31no—
264C42C2×C3×A412yes—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—46no—
268C22×C3—43no—
269C22×C3—62no—
270C22×C3—62no—
271C22×C3—62no—
272C22×C3—83no—
273C22×C3—83no—
274C22×C3C2×#41⋊C312yes—
275C22×C3C23×A411yescharacteristic
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—46no—
281C3×C4—46no—
282C3×C4—62no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—86no—
287C3×C4—86no—
288C3×C4C2×#41⋊C312yes—
289C3×C4C2×#41⋊C312yes—
290C32—41noSylow
291C23—32no—
292C23C22×C3×A411yescharacteristic
293Q8—31no—
294Q8—61no—
295Q8—61no—
296Q8—62no—
297Q8C32×#4111yescharacteristic
298D4—62no—
299D4—62no—
300D4—64no—
301D4—62no—
302D4—122no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—32no—
307C2×C4—31no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—64no—
313C2×C4—122no—
314C2×C4—121no—
315C2×C4C3×#41⋊C314yescomplemented
316C2×C4C22×C3×A412yes—
317C2×C4C22×C3×A412yes—
318C2×C4C22×C3×A411yescharacteristic
319C2×C4C22×C3×A411yescharacteristic
320C2×C3—22no—
321C2×C3—43no—
322C2×C3—43no—
323C2×C3—43no—
324C2×C3—86no—
325C2×C3—122no—
326C2×C3C22×#41⋊C311yescharacteristic
327C2×C3A4×#4111yescharacteristic
328C2×C3#594511yescharacteristic
329C22—21no—
330C22—21no—
331C22—62no—
332C22—62no—
333C22—62no—
334C22C2×C3×#41⋊C312yes—
335C22C23×C3×A411yescharacteristic
336C4—22no—
337C4—22no—
338C4—31no—
339C4—31no—
340C4—62no—
341C4—61no—
342C4—61no—
343C4—61no—
344C4C2×C3×#41⋊C312yes—
345C4C2×C3×#41⋊C312yes—
346C3—43no—
347C3#3480⋊C311yescharacteristic, direct product factor
348C2—22no—
349C2—122no—
350C2C22×C3×#41⋊C311yescharacteristic
351C2C3×A4×#4111yescharacteristic
352C2C3×#594511yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3480C3
socleC22×C3C23×A4
hypercenterC3×#117A4
hypocenterQ8C32×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×C4×#41⋊C3C2semidirect
5. C2×C3×C4×SL(2,𝔽3)C2semidirect
6. C3×#3480C3semidirect
7. #3480⋊C3C3direct (N × H)
15. C3×#557C2×C3semidirect
18. C3×#550C2×C3semidirect
23. C3×C4×#41C2×C3semidirect
25. #557⋊C3C2×C3semidirect
26. C4⋊(#41⋊C3)C2×C3semidirect
27. #548⋊C3C2×C3semidirect
28. C4×#41⋊C3C2×C3semidirect
29. C2×C4×SL(2,𝔽3)C2×C3semidirect
38. #3480C32semidirect
85. C3×SL(2,𝔽3)#41semidirect
87. #557C2×C32semidirect
90. #550C2×C32semidirect
95. C4×#41C2×C32semidirect
216. C2×C3×C4#41⋊C3semidirect
221. SL(2,𝔽3)C3×#41semidirect
315. C2×C4C3×#41⋊C3semidirect
347. C3#3480⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C3×C4⋊(#41⋊C3)2
C3×#548⋊C32
C3×C4×#41⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×#34803
#3480⋊C33
C32×#1174