Finite Group Databasebeta

← C3×#3500⋊C3 #11919750

Subgroups — C3×#3500⋊C3 #11919750

Order 1152.

860 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 158 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×#994C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C214yesmaximal, complemented
4C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
5C3×#3500C311yesmaximal, characteristic, complemented
6#3500⋊C3C313yesmaximal, direct product factor
7C3×2−1+4⋊C3—22no—
8C3×#994—22no—
9C2×C3×#41⋊C3C2212yes—
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2212yescomplemented
12C32×#124—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C2×C3×2−1+4C2×C311yescharacteristic
15C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
16C2×2−1+4⋊C3C2×C313yescomplemented
17C2×#994C2×C313yescomplemented
18C3×#548C2×C314yescomplemented
19C3×#546—34no—
20C3×#536—31no—
21C3×#535—31no—
22C3×#532—31no—
23C2×C3×#124—31no—
24C3×C4×#41C2×C311yescharacteristic, complemented
25#548⋊C3C2×C3112yescomplemented
26C4×#41⋊C3C2×C313yescomplemented
27C2×C32×Q8—41no—
28C2×C32×D4—41no—
29C3×#41⋊C3—22no—
30C3×#41⋊C3D412yescomplemented
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C32×C22⋊C4—44no—
34C32×C42—41no—
35#3500C3211yescharacteristic, complemented, Sylow
36C3×2−1+4—22no—
37C3×2−1+4—62no—
38C3×H(5,𝔽2)—22no—
39C3×H(5,𝔽2)—62no—
40C2×C3×#41—31no—
41C2×C3×#41—31no—
42C2×C3×#41—32no—
43C2×C3×#41C22×C312yes—
44C2×C3×#41C22×C311yescharacteristic
45C22×C3×Q8—31no—
46C22×C3×Q8C22×C312yes—
47C22×C3×D4—31no—
48C22×C3×D4—32no—
492−1+4⋊C3—26no—
50#994—26no—
51C2×#41⋊C3C22×C316yes—
52C2×#41⋊C3C22×C313yes—
53C22×SL(2,𝔽3)C22×C316yescomplemented
54C3×C4⋊Q8—31no—
55C3×#127—31no—
56C3×#124—32no—
57C3×#124—34no—
58C3×#124—31no—
59C3×#124—43no—
60C3×#124A411yescharacteristic
61C3×#123—34no—
62C3×C22⋊Q8—34no—
63C3×#121—34no—
64C3×#121—34no—
65C3×#120—34no—
66C3×C4×Q8C22×C312yes—
67C3×C4×D4—32no—
68C3×C4×D4—34no—
69C3×#117—31no—
70C2×C3×C22⋊C4—34no—
71C2×C3×C42—31no—
72C4×SL(2,𝔽3)C22×C316yescomplemented
73C23×C32—42no—
74C32×Q8—82no—
75C32×D4—82no—
76C2×C32×C4—41no—
77C2×C32×C4—42no—
78C2×C32×C4—42no—
79C3×SL(2,𝔽3)C2×D411yescharacteristic
80C2×2−1+4C2×C3211yescharacteristic
81C2×H(5,𝔽2)C2×C3211yescharacteristic, complemented
82#548C2×C3214yescomplemented
83#546—34no—
84#536—31no—
85#535—31no—
86#532—31no—
87C2×#124—31no—
88C4×#41C2×C3211yescharacteristic, complemented
89C24×C3—32no—
90C3×#41—22no—
91C3×#41—32no—
92C3×#41—62no—
93C3×#41—62no—
94C3×#41—62no—
95C3×#41—62no—
96C3×#41—64no—
97C3×#41—64no—
98C3×#41—62no—
99C3×#41C3×D412yes—
100C2×C3×Q8—22no—
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—43no—
104C2×C3×Q8—62no—
105C2×C3×Q8—62no—
106C2×C3×Q8—62no—
107C2×C3×Q8C2×A411yescharacteristic
108C2×C3×Q8C23×C311yescharacteristic
109C2×C3×D4—31no—
110C2×C3×D4—32no—
111C2×C3×D4—32no—
112C2×C3×D4—34no—
113C2×C3×D4—32no—
114C2×C3×D4—31no—
115C2×C3×D4—43no—
116C2×C3×D4—62no—
117C2×C3×D4—62no—
118C2×C3×D4—64no—
119C2×C3×D4—64no—
120C2×C3×D4C2×A411yescharacteristic
121C22×C3×C4—31no—
122C22×C3×C4—32no—
123C22×C3×C4—32no—
124C22×C3×C4—32no—
125C22×C3×C4—32no—
126#41⋊C3—26no—
127#41⋊C3C3×D416yescomplemented
128C2×SL(2,𝔽3)—26no—
129C2×SL(2,𝔽3)C23×C313yes—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—34no—
132C3×C22⋊C4—38no—
133C3×C22⋊C4—32no—
134C3×C22⋊C4—34no—
135C3×C22⋊C4—412no—
136C3×C22⋊C4C2×A414yes—
137C3×C42—31no—
138C3×C42—32no—
139C3×C42—43no—
140C3×C42C2×A411yescharacteristic
141C22×C32—41no—
142C22×C32—82no—
143C22×C32—84no—
144C32×C4—42no—
145C32×C4—82no—
146C32×C4—82no—
1472−1+4—22no—
1482−1+4—62no—
149H(5,𝔽2)—22no—
150H(5,𝔽2)—62no—
151C2×#41—31no—
152C2×#41—31no—
153C2×#41—32no—
154C2×#41C22×C3212yes—
155C2×#41C22×C3211yescharacteristic
156C22×Q8—31no—
157C22×Q8C22×C3212yes—
158C22×D4—31no—
159C22×D4—32no—
160C4⋊Q8—31no—
161#127—31no—
162#124—32no—
163#124—34no—
164#124—31no—
165#124C3×A411yescharacteristic
166#123—34no—
167C22⋊Q8—34no—
168#121—34no—
169#121—34no—
170#120—34no—
171C4×Q8C22×C3212yes—
172C4×D4—32no—
173C4×D4—34no—
174#117—31no—
175C2×C22⋊C4—34no—
176C2×C42—31no—
177C23×C3—31no—
178C23×C3—32no—
179C23×C3—46no—
180C23×C3—64no—
181C23×C3—124no—
182C23×C3C22×A412yes—
183C3×Q8—22no—
184C3×Q8—31no—
185C3×Q8—62no—
186C3×Q8—62no—
187C3×Q8—62no—
188C3×Q8—86no—
189C3×Q8C2×C3×D411yescharacteristic
190C3×D4—22no—
191C3×D4—32no—
192C3×D4—32no—
193C3×D4—64no—
194C3×D4—62no—
195C3×D4—62no—
196C3×D4—64no—
197C3×D4—62no—
198C3×D4—86no—
199C3×D4—124no—
200C2×C3×C4—31no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—32no—
204C2×C3×C4—32no—
205C2×C3×C4—43no—
206C2×C3×C4—46no—
207C2×C3×C4—46no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—64no—
213C2×C3×C4—122no—
214C2×C3×C4C22×A412yes—
215C2×C3×C4C22×A412yes—
216C2×C3×C4C22×A411yescharacteristic
217SL(2,𝔽3)C2×C3×D413yes—
218C2×C32—41no—
219C2×C32—42no—
220C2×C32—162no—
221C24—32no—
222#41—22no—
223#41—32no—
224#41—62no—
225#41—62no—
226#41—62no—
227#41—62no—
228#41—64no—
229#41—64no—
230#41—62no—
231#41C32×D412yes—
232C2×Q8—22no—
233C2×Q8—31no—
234C2×Q8—32no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8C2×C3×A411yescharacteristic
239C2×Q8C23×C3211yescharacteristic
240C2×D4—31no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—34no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—62no—
247C2×D4—62no—
248C2×D4—64no—
249C2×D4—64no—
250C2×D4C2×C3×A411yescharacteristic
251C22×C4—31no—
252C22×C4—32no—
253C22×C4—32no—
254C22×C4—32no—
255C22×C4—32no—
256C4⋊C4—32no—
257C4⋊C4—34no—
258C22⋊C4—38no—
259C22⋊C4—32no—
260C22⋊C4—34no—
261C22⋊C4C2×C3×A414yes—
262C42—31no—
263C42—32no—
264C42C2×C3×A411yescharacteristic
265C22×C3—22no—
266C22×C3—24no—
267C22×C3—32no—
268C22×C3—43no—
269C22×C3—62no—
270C22×C3—62no—
271C22×C3—64no—
272C22×C3—86no—
273C22×C3—812no—
274C22×C3—244no—
275C22×C3C23×A411yescharacteristic
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—46no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—86no—
284C3×C4—86no—
285C3×C4D4×A412yes—
286C32—41noSylow
287C23—31no—
288C23—32no—
289C23—64no—
290C23—124no—
291C23C22×C3×A412yes—
292Q8—22no—
293Q8—31no—
294Q8—62no—
295Q8—62no—
296Q8—62no—
297Q8C2×C32×D411yescharacteristic
298D4—22no—
299D4—32no—
300D4—32no—
301D4—64no—
302D4—62no—
303D4—62no—
304D4—64no—
305D4—62no—
306D4—124no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—32no—
311C2×C4—32no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—64no—
317C2×C4—122no—
318C2×C4C22×C3×A412yes—
319C2×C4C22×C3×A412yes—
320C2×C4C22×C3×A411yescharacteristic
321C2×C3—42no—
322C2×C3—43no—
323C2×C3—46no—
324C2×C3—62no—
325C2×C3—122no—
326C2×C3—166no—
327C2×C3C2×D4×A411yescharacteristic
328C2×C3#594512yes—
329C22—22no—
330C22—24no—
331C22—32no—
332C22—62no—
333C22—62no—
334C22—64no—
335C22—244no—
336C22C23×C3×A411yescharacteristic
337C4—22no—
338C4—22no—
339C4—31no—
340C4—31no—
341C4—62no—
342C4—62no—
343C4C3×D4×A412yes—
344C3—43no—
345C3#3500⋊C311yescharacteristic, direct product factor
346C2—42no—
347C2—62no—
348C2—122no—
349C2C2×C3×D4×A411yescharacteristic
350C2C3×#594512yes—
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3500C3
hypercenterC3×#124A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×C3×#994C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×C4×#41⋊C3C2semidirect
5. C3×#3500C3semidirect
6. #3500⋊C3C3direct (N × H)
11. C22×C3×SL(2,𝔽3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
15. C2×C3×H(5,𝔽2)C2×C3semidirect
16. C2×2−1+4⋊C3C2×C3semidirect
17. C2×#994C2×C3semidirect
18. C3×#548C2×C3semidirect
24. C3×C4×#41C2×C3semidirect
25. #548⋊C3C2×C3semidirect
26. C4×#41⋊C3C2×C3semidirect
30. C3×#41⋊C3D4semidirect
35. #3500C32semidirect
53. C22×SL(2,𝔽3)C22×C3semidirect
72. C4×SL(2,𝔽3)C22×C3semidirect
81. C2×H(5,𝔽2)C2×C32semidirect
82. #548C2×C32semidirect
88. C4×#41C2×C32semidirect
127. #41⋊C3C3×D4semidirect
345. C3#3500⋊C3direct (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×#9942
C3×#548⋊C32
C3×C4×#41⋊C32
C3×#35003
#3500⋊C33
C32×#1244