Finite Group Databasebeta

← C3×#87509 #11923751

Subgroups — C3×#87509 #11923751

Order 1152.

560 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 34 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×#5948C211yesmaximal, characteristic, complemented
2C3×#5931C211yesmaximal, characteristic, complemented
3C3×#5930C211yesmaximal, characteristic, complemented
4#87509C311yesmaximal, characteristic, direct product factor
5C3×#3002—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7C3×#985—42nomaximal
8C3×C4×S4—41nomaximal
9C22×C3×#41D311yescharacteristic, complemented
10#5948—21no—
11#5948C2×C311yescharacteristic, complemented
12#5931C2×C311yescharacteristic, complemented
13#5930C2×C311yescharacteristic, complemented
14C2×C3×#135—32no—
15C3×#525—31no—
16C3×#452—32no—
17C3×#451—31no—
18C3×#450—31no—
19C3×#449—32no—
20C3×#448—31no—
21C3×#447—31no—
22C3×#416—32no—
23C2×C3×C22⋊C8—31no—
24C2×C3×S4—41no—
25C3×#41⋊C3—42no—
26C3×#41⋊C3—81no—
27C3×C4×A4—41no—
28C3×SL(2,ℤ/4)—41no—
29C3×GL(2,𝔽3)—42no—
30C3×#226—42no—
31#3002—31noSylow
32C2×C3×#41—32no—
33C2×C3×#41—32no—
34C2×C3×#41—31no—
35C2×C3×#41—61no—
36C2×C3×#41—61no—
37C22×C3×Q8C2×D311yescharacteristic
38C22×C3×D4—31no—
39C23×C3×C4—31no—
40C22⋊SL(2,𝔽3)—21no—
41C22⋊SL(2,𝔽3)C22×C311yescharacteristic
42#985—42no—
43C4×S4—41no—
44C3×#135—62no—
45C3×#135—62no—
46C3×#135—62no—
47C3×#135—62no—
48C2×C3×Q16—32no—
49C2×C3×SD16—32no—
50C2×C3×SD16—32no—
51C2×C3×D8—32no—
52C22×C3×C8—32no—
53C3×#123—61no—
54C3×C22⋊Q8—31no—
55C3×C22⋊Q8—31no—
56C3×#121—31no—
57C3×#121—31no—
58C3×#120—61no—
59C3×C4×D4—61no—
60C3×C4×D4—61no—
61C3×#117—31no—
62C3×#103—32no—
63C3×#103—32no—
64C3×#102—32no—
65C3×#102—32no—
66C3×C22⋊C8—31no—
67C3×C22⋊C8—31no—
68C3×C22⋊C8—32no—
69C2×C3×A4—41no—
70C3×S4—42no—
71C3×C4×D3—161no—
72C3×SL(2,𝔽3)—42no—
73C3×SL(2,𝔽3)—81no—
74C22×#41C3×D311yescharacteristic, complemented
75C2×#135—32no—
76#525—31no—
77#452—32no—
78#451—31no—
79#450—31no—
80#449—32no—
81#448—31no—
82#447—31no—
83#416—32no—
84C2×C22⋊C8—31no—
85C24×C3—31no—
86C2×S4—41no—
87C3×#41—21no—
88C3×#41—32no—
89C3×#41—32no—
90C3×#41—32no—
91C3×#41—61no—
92C3×#41—62no—
93C3×#41—62no—
94C3×#41—62no—
95C3×#41—61no—
96C3×#41—61no—
97C3×#41—62no—
98C3×#41S412yescomplemented
99C2×C3×Q8—32no—
100C2×C3×Q8—31no—
101C2×C3×Q8—61no—
102C2×C3×D4—32no—
103C2×C3×D4—31no—
104C2×C3×D4—61no—
105C2×C3×D4—62no—
106C2×C3×D4—62no—
107C2×C3×D4—61no—
108C2×C3×D4—61no—
109C2×C3×D4—61no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—62no—
116C22×C3×C4—62no—
117C22×C3×C4—61no—
118C22×C3×C4S411yescharacteristic
119#41⋊C3—42no—
120#41⋊C3—81no—
121#41⋊C3—82no—
122#41⋊C3—82no—
123C4×A4—41no—
124C4×A4—81no—
125SL(2,ℤ/4)—41no—
126GL(2,𝔽3)—42no—
127#226—42no—
128C3×Q16—62no—
129C3×Q16—62no—
130C3×SD16—62no—
131C3×SD16—62no—
132C3×SD16—62no—
133C3×SD16—62no—
134C3×D8—62no—
135C3×D8—62no—
136C2×C3×C8—32no—
137C2×C3×C8—32no—
138C2×C3×C8—62no—
139C2×C3×C8—62no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—61no—
143C3×C4⋊C4—61no—
144C3×C22⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—61no—
147C3×C22⋊C4—61no—
148C3×C22⋊C4—61no—
149C3×C42—61no—
150C2×C3×D3—161no—
151C3×A4—41no—
152C32×C4—161no—
153C3×Dic3—161no—
154C2×#41—32no—
155C2×#41—32no—
156C2×#41—31no—
157C2×#41—61no—
158C2×#41—61no—
159C22×Q8C2×C3×D311yescharacteristic
160C22×D4—31no—
161C23×C4—31no—
162#135—62no—
163#135—62no—
164#135—62no—
165#135—62no—
166C2×Q16—32no—
167C2×SD16—32no—
168C2×SD16—32no—
169C2×D8—32no—
170C22×C8—32no—
171#123—61no—
172C22⋊Q8—31no—
173C22⋊Q8—31no—
174#121—31no—
175#121—31no—
176#120—61no—
177C4×D4—61no—
178C4×D4—61no—
179#117—31no—
180#103—32no—
181#103—32no—
182#102—32no—
183#102—32no—
184C22⋊C8—31no—
185C22⋊C8—31no—
186C22⋊C8—32no—
187C23×C3—31no—
188C23×C3—31no—
189C23×C3—62no—
190C23×C3—61no—
191C23×C3—61no—
192C23×C3—61no—
193C23×C3—121no—
194C23×C3C2×S411yescharacteristic
195C2×A4—41no—
196C2×A4—81no—
197S4—42no—
198C3×Q8—21no—
199C3×Q8—32no—
200C3×Q8—61no—
201C3×Q8—62no—
202C3×Q8C2×S412yes—
203C3×D4—32no—
204C3×D4—32no—
205C3×D4—62no—
206C3×D4—62no—
207C3×D4—61no—
208C3×D4—62no—
209C3×D4—61no—
210C3×D4—121no—
211C3×D4—122no—
212C3×D4—122no—
213C3×D4—121no—
214C2×C3×C4—31no—
215C2×C3×C4—32no—
216C2×C3×C4—31no—
217C2×C3×C4—32no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C4×D3—161no—
231SL(2,𝔽3)—42no—
232SL(2,𝔽3)—81no—
233SL(2,𝔽3)—82no—
234SL(2,𝔽3)—82no—
235C3×C8—62no—
236C3×C8—62no—
237C2×C32—161no—
238C3×D3—162no—
239C24—31no—
240#41—21no—
241#41—32no—
242#41—32no—
243#41—32no—
244#41—61no—
245#41—62no—
246#41—62no—
247#41—62no—
248#41—61no—
249#41—61no—
250#41—62no—
251#41C3×S412yescomplemented
252C2×Q8—32no—
253C2×Q8—31no—
254C2×Q8—61no—
255C2×D4—32no—
256C2×D4—31no—
257C2×D4—61no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4C3×S411yescharacteristic
272Q16—62no—
273Q16—62no—
274SD16—62no—
275SD16—62no—
276SD16—62no—
277SD16—62no—
278D8—62no—
279D8—62no—
280C2×C8—32no—
281C2×C8—32no—
282C2×C8—62no—
283C2×C8—62no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C42—61no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—32no—
297C22×C3—61no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—122no—
303C22×C3—122no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3#98511yescharacteristic, complemented
308C2×D3—161no—
309A4—41no—
310A4—81no—
311C3×C4—32no—
312C3×C4—31no—
313C3×C4—61no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4—161no—
318C3×C4—321no—
319C3×C4#102011yescharacteristic
320Dic3—161no—
321C32—161noSylow
322C23—31no—
323C23—31no—
324C23—62no—
325C23—61no—
326C23—61no—
327C23—61no—
328C23—121no—
329C23C2×C3×S411yescharacteristic
330Q8—21no—
331Q8—32no—
332Q8—61no—
333Q8—62no—
334Q8C2×C3×S412yes—
335D4—32no—
336D4—32no—
337D4—62no—
338D4—62no—
339D4—61no—
340D4—62no—
341D4—61no—
342D4—122no—
343D4—121no—
344D4—122no—
345D4—121no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—31no—
349C2×C4—32no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—121no—
361C2×C4—121no—
362C8—62no—
363C8—62no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—62no—
367C2×C3—121no—
368C2×C3—161no—
369C2×C3—241no—
370C2×C3—321no—
371C2×C3C2×#102011yescharacteristic
372D3—162no—
373C22—31no—
374C22—31no—
375C22—32no—
376C22—61no—
377C22—61no—
378C22—61no—
379C22—61no—
380C22—61no—
381C22—122no—
382C22—122no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22C3×#98511yescharacteristic, complemented
387C4—32no—
388C4—31no—
389C4—61no—
390C4—121no—
391C4—121no—
392C4—121no—
393C4C3×#102011yescharacteristic
394C3—161no—
395C3—321no—
396C3#8750911yescharacteristic, direct product factor
397C2—31no—
398C2—31no—
399C2—62no—
400C2—121no—
401C2—241no—
402C2C2×C3×#102011yescharacteristic
subgroupG/H
center, hypercenterC3×C4#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×#1020
FittingC22×C3×#41D3
socleC23×C3C2×S4
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#5948C2semidirect
2. C3×#5931C2semidirect
3. C3×#5930C2semidirect
4. #87509C3direct (N × H)
9. C22×C3×#41D3semidirect
11. #5948C2×C3semidirect
12. #5931C2×C3semidirect
13. #5930C2×C3semidirect
74. C22×#41C3×D3semidirect
98. C3×#41S4semidirect
251. #41C3×S4semidirect
307. C22×C3#985semidirect
386. C22C3×#985semidirect
396. C3#87509direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#59482
C3×#59312
C3×#59302
#875093
C3×#30023
C3×#9854
C3×C4×S44