Finite Group Databasebeta

← C3×#85534 #11922818

Subgroups — C3×#85534 #11922818

Order 1152.

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

normal subgroups only

This group has 681 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
1C2×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C3×#5426C211yesmaximal, characteristic, complemented
3C3×#5424C211yesmaximal, characteristic, complemented
4C3×#5420C212yesmaximal, complemented
5C2×C3×#859C212yesmaximal, complemented
6#85534C311yesmaximal, characteristic, direct product factor
7C3×#2898—31nomaximal
8C3×2−1+4⋊C3C414yescomplemented
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yes—
11C3×D4×Dic3—41nomaximal
12C3×#859C2211yescharacteristic
13C3×#859C2212yes—
14C3×#859C2211yescharacteristic
15C2×C3×2−1+4D311yescharacteristic
16C2×2−1+4⋊C3—21no—
17C2×2−1+4⋊C3C2×C311yescharacteristic
18C2×C3×#131—31no—
19C3×#520—31no—
20#5426C2×C311yescharacteristic, complemented
21#5424C2×C311yescharacteristic, complemented
22#5420C2×C312yescomplemented
23C2×#859C2×C312yescomplemented
24C3×#419—31no—
25C3×#419—32no—
26C3×#417—32no—
27C3×#417—31no—
28C3×#416—32no—
29C3×#416—31no—
30C2×C3×#103—31no—
31C2×C3×#103—32no—
32C2×C32×D4—41no—
33C22×C3×Dic3—42no—
34C3×#41⋊C3C2×C412yescomplemented
35C2×C3×SL(2,𝔽3)C2×C414yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C3×#217—42no—
38C3×C4⋊Dic3—41no—
39C3×C4×Dic3—41no—
40#2898—31noSylow
41C3×2−1+4—34no—
42C3×2−1+4Dic314yescomplemented
43C2×C3×#41—31no—
44C2×C3×#41—32no—
45C2×C3×#41C2×D311yescharacteristic
46C22×C3×Q8—31no—
47C22×C3×Q8C2×D312yes—
482−1+4⋊C3—24no—
492−1+4⋊C3C3×C414yescomplemented
50C2×#41⋊C3—21no—
51C2×#41⋊C3C22×C311yescharacteristic
52C22×SL(2,𝔽3)—22no—
53C22×SL(2,𝔽3)C22×C312yes—
54C3×#131—64no—
55C2×C3×M16—32no—
56C2×C3×M16—31no—
57C22×C3×C8—32no—
58C22×C3×C8—31no—
59C3×C4×Q8—61no—
60C3×C4×D4—62no—
61C3×C4×D4—61no—
62C3×#117—32no—
63C3×#117—31no—
64C2×C3×C4⋊C4—32no—
65C2×C3×C4⋊C4—31no—
66D4×Dic3—41no—
67#859C22×C311yescharacteristic
68#859C22×C312yes—
69#859C22×C311yescharacteristic
70C3×#103—32no—
71C3×#103—32no—
72C3×#103—32no—
73C3×#103—32no—
74C3×#103—31no—
75C3×#103—32no—
76C3×#103—31no—
77C3×#102—31no—
78C3×#102—32no—
79C3×#102—31no—
80C23×C32—42no—
81C32×D4—44no—
82C2×C32×C4—41no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—84no—
87C3×SL(2,𝔽3)C22×C411yescharacteristic
88C2×2−1+4C3×D311yescharacteristic
89C2×#131—31no—
90#520—31no—
91#419—31no—
92#419—32no—
93#417—32no—
94#417—31no—
95#416—32no—
96#416—31no—
97C2×#103—31no—
98C2×#103—32no—
99C3×#41—34no—
100C3×#41—34no—
101C3×#41—32no—
102C3×#41—38no—
103C3×#41—64no—
104C3×#41C2×Dic312yescomplemented
105C2×C3×Q8—31no—
106C2×C3×Q8—32no—
107C2×C3×Q8—34no—
108C2×C3×Q8—34no—
109C2×C3×Q8—62no—
110C2×C3×Q8C2×Dic314yescomplemented
111C2×C3×Q8C22×D311yescharacteristic
112C2×C3×D4—32no—
113C2×C3×D4—31no—
114C2×C3×D4—41no—
115C2×C3×D4—81no—
116C2×C3×D4S411yescharacteristic
117C22×C3×C4—32no—
118C22×C3×C4—31no—
119C22×C3×C4—61no—
120C22×C3×C4—61no—
121C22×C3×C4—62no—
122C22×Dic3—42no—
123#41⋊C3—22no—
124#41⋊C3C2×C3×C412yescomplemented
125C2×SL(2,𝔽3)—24no—
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)C2×C3×C414yescomplemented
128C2×SL(2,𝔽3)C23×C311yescharacteristic
129C3×M16—64no—
130C3×M16—62no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—32no—
134C2×C3×C8—64no—
135C2×C3×C8—62no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—62no—
140C3×C4⋊C4—61no—
141C3×C22⋊C4—61no—
142C3×C22⋊C4—62no—
143C3×C42—62no—
144C3×C42—61no—
145#217—42no—
146C4⋊Dic3—41no—
147C4×Dic3—41no—
148C22×C32—44no—
149C22×C32—41no—
150C22×C32—82no—
151C22×C32—82no—
152C32×C4—42no—
153C3×Dic3—42no—
154C3×Dic3—81no—
155C3×Dic3—82no—
1562−1+4—34no—
1572−1+4C3×Dic314yescomplemented
158C2×#41—31no—
159C2×#41—32no—
160C2×#41C2×C3×D311yescharacteristic
161C22×Q8—31no—
162C22×Q8C2×C3×D312yes—
163#131—64no—
164C2×M16—32no—
165C2×M16—31no—
166C22×C8—32no—
167C22×C8—31no—
168C4×Q8—61no—
169C4×D4—62no—
170C4×D4—61no—
171#117—32no—
172#117—31no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—31no—
175#103—32no—
176#103—32no—
177#103—32no—
178#103—32no—
179#103—31no—
180#103—32no—
181#103—31no—
182#102—31no—
183#102—32no—
184#102—31no—
185C23×C3—31no—
186C23×C3—42no—
187C23×C3—82no—
188C23×C3C2×S412yes—
189C3×Q8—34no—
190C3×Q8—31no—
191C3×Q8—34no—
192C3×Q8—62no—
193C3×Q8C22×Dic311yescharacteristic
194C3×D4—32no—
195C3×D4—38no—
196C3×D4—44no—
197C3×D4—61no—
198C3×D4—84no—
199C3×D4SL(2,ℤ/4)14yes—
200C2×C3×C4—34no—
201C2×C3×C4—34no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—41no—
207C2×C3×C4—64no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—81no—
212C2×C3×C4—242no—
213C2×C3×C4—241no—
214C2×C3×C4C2×S411yescharacteristic
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—84no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)C22×C3×C411yescharacteristic
221C3×C8—61no—
222C3×C8—61no—
223C3×C8—62no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—84no—
228#41—34no—
229#41—34no—
230#41—32no—
231#41—38no—
232#41—64no—
233#41C2×C3×Dic312yescomplemented
234C2×Q8—31no—
235C2×Q8—32no—
236C2×Q8—34no—
237C2×Q8—34no—
238C2×Q8—62no—
239C2×Q8C2×C3×Dic314yescomplemented
240C2×Q8C22×C3×D311yescharacteristic
241C2×D4—32no—
242C2×D4—31no—
243C2×D4C3×S411yescharacteristic
244C22×C4—32no—
245C22×C4—31no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—62no—
249M16—64no—
250M16—62no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—32no—
254C2×C8—64no—
255C2×C8—62no—
256C4⋊C4—32no—
257C4⋊C4—31no—
258C4⋊C4—31no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—62no—
263C42—62no—
264C42—61no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—32no—
268C22×C3—44no—
269C22×C3—41no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—84no—
275C22×C3—81no—
276C22×C3—162no—
277C22×C3—162no—
278C22×C3C2×SL(2,ℤ/4)14yes—
279C22×C3C22×S411yescharacteristic
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—34no—
283C3×C4—42no—
284C3×C4—82no—
285C3×C4—121no—
286C3×C4—122no—
287C3×C4—121no—
288C3×C4C2×SL(2,ℤ/4)12yes—
289Dic3—42no—
290Dic3—82no—
291Dic3—81no—
292C32—41noSylow
293C23—31no—
294C23C2×C3×S412yes—
295Q8—34no—
296Q8—31no—
297Q8—34no—
298Q8—62no—
299Q8C22×C3×Dic311yescharacteristic
300D4—32no—
301D4—38no—
302D4—61no—
303D4C3×SL(2,ℤ/4)14yes—
304C2×C4—34no—
305C2×C4—34no—
306C2×C4—32no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—64no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—62no—
314C2×C4—242no—
315C2×C4—241no—
316C2×C4C2×C3×S411yescharacteristic
317C8—61no—
318C8—61no—
319C8—62no—
320C2×C3—24no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—62no—
325C2×C3—84no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3—81no—
329C2×C3—164no—
330C2×C3#592611yescharacteristic
331C2×C3#592711yescharacteristic
332C2×C3C22×SL(2,ℤ/4)11yescharacteristic
333C22—22no—
334C22—22no—
335C22—32no—
336C22—61no—
337C22—61no—
338C22C2×C3×SL(2,ℤ/4)14yes—
339C22C22×C3×S411yescharacteristic
340C4—31no—
341C4—31no—
342C4—34no—
343C4—121no—
344C4—122no—
345C4—121no—
346C4C2×C3×SL(2,ℤ/4)12yes—
347C3—41no—
348C3—81no—
349C3#8553411yescharacteristic, direct product factor
350C2—24no—
351C2—62no—
352C2C3×#592611yescharacteristic
353C2C3×#592711yescharacteristic
354C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×SL(2,𝔽3)C23×C3
hypercenterC2×C3×D4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5426C2semidirect
3. C3×#5424C2semidirect
4. C3×#5420C2semidirect
5. C2×C3×#859C2semidirect
6. #85534C3direct (N × H)
8. C3×2−1+4⋊C3C4semidirect
20. #5426C2×C3semidirect
21. #5424C2×C3semidirect
22. #5420C2×C3semidirect
23. C2×#859C2×C3semidirect
34. C3×#41⋊C3C2×C4semidirect
35. C2×C3×SL(2,𝔽3)C2×C4semidirect
42. C3×2−1+4Dic3semidirect
49. 2−1+4⋊C3C3×C4semidirect
104. C3×#41C2×Dic3semidirect
110. C2×C3×Q8C2×Dic3semidirect
124. #41⋊C3C2×C3×C4semidirect
127. C2×SL(2,𝔽3)C2×C3×C4semidirect
157. 2−1+4C3×Dic3semidirect
233. #41C2×C3×Dic3semidirect
239. C2×Q8C2×C3×Dic3semidirect
349. C3#85534direct (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
C3×#54262
C3×#54242
C3×#54202
C2×C3×#8592
#855343
C3×#28983
C3×D4×Dic34