Finite Group Databasebeta

← C3×#3428⋊C3 #11919684

Subgroups — C3×#3428⋊C3 #11919684

Order 1152.

860 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 222 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×#520⋊C3C212yesmaximal, complemented
3C3×#519⋊C3C212yesmaximal, complemented
4C3×C4×#41⋊C3C212yesmaximal, complemented
5C3×#3428C311yesmaximal, characteristic, complemented
6#3428⋊C3C313yesmaximal, direct product factor
7C2×C3×#41⋊C3C414yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C32×#117—41nomaximal
12C3×C4×SL(2,𝔽3)C2214yescomplemented
13C22×C3×#41C2×C311yescharacteristic
14C22×#41⋊C3C2×C313yes—
15C3×#520—32no—
16C3×#520C2×C312yescomplemented
17C3×#519C2×C312yes—
18C3×#518—32no—
19C3×C4×#41—32no—
20C3×C4×#41C2×C312yescomplemented
21C2×C3×#117—31no—
22C2×C3×#117—31no—
23#520⋊C3C2×C316yescomplemented
24#519⋊C3C2×C316yescomplemented
25C4×#41⋊C3C2×C316yescomplemented
26C22×C32×C4—41no—
27C3×#41⋊C3C2×C412yescomplemented
28C3×#41⋊C3C2×C412yescomplemented
29C2×C3×SL(2,𝔽3)C2×C412yescomplemented
30C2×C3×SL(2,𝔽3)C2311yescharacteristic
31C32×C4⋊C4—42no—
32C32×C22⋊C4—42no—
33C32×C42—42no—
34#3428C3211yescharacteristic, complemented, Sylow
35C2×C3×#41—34no—
36C2×C3×#41—31no—
37C2×C3×#41—31no—
38C2×C3×#41C3×C414yescomplemented
39C2×C3×#41C22×C311yescharacteristic
40C2×C3×#41C22×C311yescharacteristic
41C22×C3×Q8C22×C311yescharacteristic
42C22×C3×D4—31no—
43C23×C3×C4—31no—
44C2×#41⋊C3C3×C4112yescomplemented
45C2×#41⋊C3C22×C313yes—
46C2×#41⋊C3C22×C313yes—
47C22×SL(2,𝔽3)C22×C313yes—
48C3×C4×Q8—34no—
49C3×C4×Q8C22×C314yes—
50C3×C4×D4—38no—
51C3×C4×D4—34no—
52C3×C4×D4—34no—
53C3×#117—32no—
54C3×#117—31no—
55C3×#117—32no—
56C3×#117—32no—
57C3×#117—32no—
58C3×#117—32no—
59C3×#117—32no—
60C3×#117—43no—
61C3×#117A411yescharacteristic
62C2×C3×C4⋊C4—32no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C22⋊C4—32no—
65C2×C3×C22⋊C4—32no—
66C2×C3×C42—32no—
67C2×C3×C42—32no—
68C4×SL(2,𝔽3)C22×C3112yescomplemented
69C23×C32—41no—
70C2×C32×C4—44no—
71C2×C32×C4—41no—
72C2×C32×C4—44no—
73C2×C32×C4—41no—
74C3×SL(2,𝔽3)C22×C411yescharacteristic
75C22×#41C2×C3211yescharacteristic
76#520—32no—
77#520C2×C3212yescomplemented
78#519C2×C3212yes—
79#518—32no—
80C4×#41—32no—
81C4×#41C2×C3212yescomplemented
82C2×#117—31no—
83C2×#117—31no—
84C24×C3—31no—
85C3×#41—34no—
86C3×#41—34no—
87C3×#41—34no—
88C3×#41—34no—
89C3×#41—32no—
90C3×#41—32no—
91C3×#41C2×C3×C412yes—
92C3×#41C2×C3×C412yes—
93C2×C3×Q8—31no—
94C2×C3×Q8—32no—
95C2×C3×Q8C2×C3×C412yes—
96C2×C3×Q8C23×C311yescharacteristic
97C2×C3×D4—31no—
98C2×C3×D4—31no—
99C2×C3×D4—34no—
100C2×C3×D4—32no—
101C2×C3×D4—34no—
102C22×C3×C4—31no—
103C22×C3×C4—34no—
104C22×C3×C4—34no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—34no—
110C22×C3×C4—34no—
111C22×C3×C4—43no—
112C22×C3×C4C2×A411yescharacteristic
113#41⋊C3C2×C3×C416yescomplemented
114#41⋊C3C2×C3×C416yescomplemented
115C2×SL(2,𝔽3)C2×C3×C416yescomplemented
116C2×SL(2,𝔽3)C23×C313yes—
117C3×C4⋊C4—34no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—34no—
120C3×C4⋊C4—46no—
121C3×C4⋊C4C2×A412yes—
122C3×C22⋊C4—34no—
123C3×C22⋊C4—32no—
124C3×C22⋊C4—34no—
125C3×C22⋊C4—46no—
126C3×C22⋊C4C2×A412yes—
127C3×C42—34no—
128C3×C42—34no—
129C3×C42—32no—
130C3×C42—46no—
131C3×C42C2×A412yes—
132C22×C32—42no—
133C22×C32—41no—
134C22×C32—82no—
135C32×C4—42no—
136C32×C4—42no—
137C32×C4—84no—
138C2×#41—31no—
139C2×#41—31no—
140C2×#41—34no—
141C2×#41C32×C414yescomplemented
142C2×#41C22×C3211yescharacteristic
143C2×#41C22×C3211yescharacteristic
144C22×Q8C22×C3211yescharacteristic
145C22×D4—31no—
146C23×C4—31no—
147C4×Q8—34no—
148C4×Q8C22×C3214yes—
149C4×D4—38no—
150C4×D4—34no—
151C4×D4—34no—
152#117—32no—
153#117—31no—
154#117—32no—
155#117—32no—
156#117—32no—
157#117—32no—
158#117—32no—
159#117C3×A411yescharacteristic
160C2×C4⋊C4—32no—
161C2×C4⋊C4—32no—
162C2×C22⋊C4—32no—
163C2×C22⋊C4—32no—
164C2×C42—32no—
165C2×C42—32no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—34no—
169C23×C3—43no—
170C23×C3—122no—
171C23×C3C22×A411yescharacteristic
172C3×Q8—32no—
173C3×Q8—31no—
174C3×Q8—32no—
175C3×Q8C22×C3×C411yescharacteristic
176C3×D4—32no—
177C3×D4—32no—
178C3×D4—32no—
179C3×D4—32no—
180C3×D4—38no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—34no—
187C2×C3×C4—32no—
188C2×C3×C4—34no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—34no—
192C2×C3×C4—43no—
193C2×C3×C4—43no—
194C2×C3×C4—412no—
195C2×C3×C4—412no—
196C2×C3×C4—124no—
197C2×C3×C4—124no—
198C2×C3×C4C4×A414yes—
199C2×C3×C4C22×A414yes—
200C2×C3×C4C22×A411yescharacteristic
201C2×C3×C4C22×A411yescharacteristic
202SL(2,𝔽3)C22×C3×C413yes—
203C2×C32—41no—
204C2×C32—42no—
205C2×C32—82no—
206C24—31no—
207#41—34no—
208#41—34no—
209#41—34no—
210#41—34no—
211#41—32no—
212#41—32no—
213#41C2×C32×C412yes—
214#41C2×C32×C412yes—
215C2×Q8—31no—
216C2×Q8—32no—
217C2×Q8C2×C32×C412yes—
218C2×Q8C23×C3211yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—34no—
222C2×D4—32no—
223C2×D4—34no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—34no—
230C22×C4—34no—
231C22×C4—34no—
232C22×C4—34no—
233C22×C4C2×C3×A411yescharacteristic
234C4⋊C4—34no—
235C4⋊C4—32no—
236C4⋊C4—34no—
237C4⋊C4C2×C3×A412yes—
238C22⋊C4—34no—
239C22⋊C4—32no—
240C22⋊C4—34no—
241C22⋊C4C2×C3×A412yes—
242C42—34no—
243C42—32no—
244C42—34no—
245C42C2×C3×A412yes—
246C22×C3—22no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—46no—
250C22×C3—43no—
251C22×C3—62no—
252C22×C3—62no—
253C22×C3—86no—
254C22×C3—124no—
255C22×C3C2×C4×A412yes—
256C22×C3C23×A411yescharacteristic
257C3×C4—24no—
258C3×C4—31no—
259C3×C4—32no—
260C3×C4—31no—
261C3×C4—46no—
262C3×C4—46no—
263C3×C4—64no—
264C3×C4—812no—
265C3×C4C2×C4×A412yes—
266C3×C4C2×C4×A412yes—
267C32—41noSylow
268C23—31no—
269C23—31no—
270C23—34no—
271C23—122no—
272C23C22×C3×A411yescharacteristic
273Q8—32no—
274Q8—31no—
275Q8—32no—
276Q8C22×C32×C411yescharacteristic
277D4—32no—
278D4—32no—
279D4—32no—
280D4—32no—
281D4—38no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—34no—
290C2×C4—32no—
291C2×C4—34no—
292C2×C4—34no—
293C2×C4—124no—
294C2×C4—124no—
295C2×C4C3×C4×A414yes—
296C2×C4C22×C3×A414yes—
297C2×C4C22×C3×A411yescharacteristic
298C2×C4C22×C3×A411yescharacteristic
299C2×C3—22no—
300C2×C3—43no—
301C2×C3—46no—
302C2×C3—62no—
303C2×C3—62no—
304C2×C3—86no—
305C2×C3C22×C4×A411yescharacteristic
306C2×C3#594512yes—
307C22—22no—
308C22—32no—
309C22—32no—
310C22—62no—
311C22—62no—
312C22—124no—
313C22C2×C3×C4×A412yes—
314C22C23×C3×A411yescharacteristic
315C4—24no—
316C4—31no—
317C4—32no—
318C4—31no—
319C4—64no—
320C4C2×C3×C4×A412yes—
321C4C2×C3×C4×A412yes—
322C3—43no—
323C3#3428⋊C311yescharacteristic, direct product factor
324C2—22no—
325C2—62no—
326C2—62no—
327C2C22×C3×C4×A411yescharacteristic
328C2C3×#594512yes—
subgroupG/H
centerC2×C3×C4C22×A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC22C23×C3×A4
FittingC3×#3428C3
socleC22×C3C23×A4
prime radicalC22×C3×#41⋊C3C2
cosocleC2×Q8C23×C32
hypercenterC3×#117A4
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C3×#519⋊C3C2semidirect
4. C3×C4×#41⋊C3C2semidirect
5. C3×#3428C3semidirect
6. #3428⋊C3C3direct (N × H)
7. C2×C3×#41⋊C3C4semidirect
12. C3×C4×SL(2,𝔽3)C22semidirect
16. C3×#520C2×C3semidirect
20. C3×C4×#41C2×C3semidirect
23. #520⋊C3C2×C3semidirect
24. #519⋊C3C2×C3semidirect
25. C4×#41⋊C3C2×C3semidirect
27. C3×#41⋊C3C2×C4semidirect
28. C3×#41⋊C3C2×C4semidirect
29. C2×C3×SL(2,𝔽3)C2×C4semidirect
34. #3428C32semidirect
38. C2×C3×#41C3×C4semidirect
44. C2×#41⋊C3C3×C4semidirect
68. C4×SL(2,𝔽3)C22×C3semidirect
77. #520C2×C32semidirect
81. C4×#41C2×C32semidirect
113. #41⋊C3C2×C3×C4semidirect
114. #41⋊C3C2×C3×C4semidirect
115. C2×SL(2,𝔽3)C2×C3×C4semidirect
141. C2×#41C32×C4semidirect
323. C3#3428⋊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
C3×#520⋊C32
C3×#519⋊C32
C3×C4×#41⋊C32
C3×#34283
#3428⋊C33
C32×#1174