Finite Group Databasebeta

← C3×#87507 #11923749

Subgroups — C3×#87507 #11923749

Order 1152.

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

normal subgroups only

This group has 554 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
1C2×C3×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#5931C211yesmaximal, characteristic, complemented
3C3×#5930C211yesmaximal, characteristic, complemented
4#87507C311yesmaximal, characteristic, direct product factor
5C3×#3004—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7C3×GL(2,ℤ/4)—41nomaximal
8C3×#983—42nomaximal
9C23×C3×Q8D311yescharacteristic, complemented
10C2×C22⋊SL(2,𝔽3)—21no—
11C2×C22⋊SL(2,𝔽3)C2×C311yescharacteristic, complemented
12#5931C2×C311yescharacteristic, complemented
13#5930C2×C311yescharacteristic, complemented
14C2×C3×#137—32no—
15C3×#525—31no—
16C3×#451—31no—
17C3×#451—32no—
18C3×#451—31no—
19C3×#448—31no—
20C3×#448—32no—
21C3×#448—31no—
22C3×#419—32no—
23C3×#407—31no—
24C22×C3×A4—41no—
25C2×C3×S4—41no—
26C2×C3×SL(2,𝔽3)—42no—
27C2×C3×SL(2,𝔽3)—81no—
28C3×SL(2,ℤ/4)—41no—
29C3×GL(2,𝔽3)—42no—
30C3×#226—42no—
31#3004—31noSylow
32C2×C3×#41—31no—
33C22×C3×Q8—31no—
34C22×C3×Q8—32no—
35C22×C3×Q8—32no—
36C22×C3×Q8—61no—
37C22×C3×Q8—61no—
38C22×C3×Q8C2×D311yescharacteristic
39C23×C3×C4—31no—
40C22⋊SL(2,𝔽3)—21no—
41C22⋊SL(2,𝔽3)C22×C311yescharacteristic
42GL(2,ℤ/4)—41no—
43#983—42no—
44C3×#137—62no—
45C3×#137—62no—
46C3×#137—62no—
47C3×#137—62no—
48C2×C3×Q16—32no—
49C2×C3×Q16—32no—
50C2×C3×SD16—32no—
51C2×C3×SD16—32no—
52C2×C3×M16—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×#103—32no—
65C3×#103—32no—
66C3×C22⋊C8—32no—
67C3×C22⋊C8—31no—
68C3×C22⋊C8—31no—
69C2×C3×A4—41no—
70C2×C3×A4—81no—
71C3×S4—81no—
72C3×#67—161no—
73C3×SL(2,𝔽3)—42no—
74C3×SL(2,𝔽3)—81no—
75C23×Q8C3×D311yescharacteristic, complemented
76C2×#137—32no—
77#525—31no—
78#451—31no—
79#451—32no—
80#451—31no—
81#448—31no—
82#448—32no—
83#448—31no—
84#419—32no—
85#407—31no—
86C24×C3S411yescharacteristic
87C22×A4—41no—
88C22×A4—81no—
89C2×S4—41no—
90C3×#41—62no—
91C3×#41—62no—
92C2×C3×Q8—21no—
93C2×C3×Q8—32no—
94C2×C3×Q8—32no—
95C2×C3×Q8—32no—
96C2×C3×Q8—32no—
97C2×C3×Q8—32no—
98C2×C3×Q8—31no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—62no—
102C2×C3×Q8—62no—
103C2×C3×Q8—62no—
104C2×C3×Q8—61no—
105C2×C3×Q8—61no—
106C2×C3×Q8—61no—
107C2×C3×Q8—62no—
108C2×C3×Q8S412yescomplemented
109C2×C3×D4—31no—
110C2×C3×D4—61no—
111C2×C3×D4—61no—
112C2×C3×D4—61no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—62no—
120C22×C3×C4—62no—
121C22×C3×C4—61no—
122C2×SL(2,𝔽3)—42no—
123C2×SL(2,𝔽3)—81no—
124C2×SL(2,𝔽3)—82no—
125C2×SL(2,𝔽3)—82no—
126SL(2,ℤ/4)—41no—
127GL(2,𝔽3)—42no—
128#226—42no—
129C3×Q16—62no—
130C3×Q16—62no—
131C3×Q16—62no—
132C3×Q16—62no—
133C3×SD16—62no—
134C3×SD16—62no—
135C3×SD16—62no—
136C3×SD16—62no—
137C3×M16—62no—
138C3×M16—62no—
139C2×C3×C8—32no—
140C2×C3×C8—32no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—61no—
147C3×C22⋊C4—61no—
148C3×C22⋊C4—61no—
149C3×C22⋊C4—61no—
150C3×C42—61no—
151C22×C32—161no—
152C2×C3×D3—161no—
153C3×A4—41no—
154C3×Dic3—161no—
155C2×#41—31no—
156C22×Q8—31no—
157C22×Q8—32no—
158C22×Q8—32no—
159C22×Q8—61no—
160C22×Q8—61no—
161C22×Q8C2×C3×D311yescharacteristic
162C23×C4—31no—
163#137—62no—
164#137—62no—
165#137—62no—
166#137—62no—
167C2×Q16—32no—
168C2×Q16—32no—
169C2×SD16—32no—
170C2×SD16—32no—
171C2×M16—32no—
172#123—61no—
173C22⋊Q8—31no—
174C22⋊Q8—31no—
175#121—31no—
176#121—31no—
177#120—61no—
178C4×D4—61no—
179C4×D4—61no—
180#117—31no—
181#103—32no—
182#103—32no—
183#103—32no—
184#103—32no—
185C22⋊C8—32no—
186C22⋊C8—31no—
187C22⋊C8—31no—
188C23×C3—21no—
189C23×C3—31no—
190C23×C3—31no—
191C23×C3—61no—
192C23×C3—61no—
193C23×C3C2×S411yescharacteristic
194C2×A4—41no—
195C2×A4—81no—
196C2×A4—82no—
197C2×A4—81no—
198S4—81no—
199C3×Q8—21no—
200C3×Q8—32no—
201C3×Q8—32no—
202C3×Q8—32no—
203C3×Q8—61no—
204C3×Q8—62no—
205C3×Q8—61no—
206C3×Q8—62no—
207C3×Q8—62no—
208C3×Q8—61no—
209C3×Q8C2×S412yes—
210C3×D4—62no—
211C3×D4—121no—
212C3×D4—122no—
213C3×D4—121no—
214C3×D4—121no—
215C3×D4—121no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—122no—
232#67—161no—
233SL(2,𝔽3)—42no—
234SL(2,𝔽3)—81no—
235SL(2,𝔽3)—82no—
236SL(2,𝔽3)—82no—
237C3×C8—62no—
238C3×C8—62no—
239C2×C32—161no—
240C2×C32—321no—
241C3×D3—321no—
242C24C3×S411yescharacteristic
243#41—62no—
244#41—62no—
245C2×Q8—21no—
246C2×Q8—32no—
247C2×Q8—32no—
248C2×Q8—32no—
249C2×Q8—32no—
250C2×Q8—32no—
251C2×Q8—31no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—62no—
255C2×Q8—62no—
256C2×Q8—62no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—62no—
261C2×Q8C3×S412yescomplemented
262C2×D4—31no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—61no—
275Q16—62no—
276Q16—62no—
277Q16—62no—
278Q16—62no—
279SD16—62no—
280SD16—62no—
281SD16—62no—
282SD16—62no—
283M16—62no—
284M16—62no—
285C2×C8—32no—
286C2×C8—32no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C42—61no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—32no—
300C22×C3—31no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3—161no—
308C22×C3—321no—
309C22×C3#98311yescharacteristic, complemented
310C22×C3#102011yescharacteristic
311C2×D3—161no—
312A4—41no—
313A4—81no—
314C3×C4—32no—
315C3×C4—32no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—121no—
319C3×C4—121no—
320C3×C4—121no—
321Dic3—161no—
322C32—161noSylow
323C23—21no—
324C23—31no—
325C23—31no—
326C23—61no—
327C23—61no—
328C23C2×C3×S411yescharacteristic
329Q8—21no—
330Q8—32no—
331Q8—32no—
332Q8—32no—
333Q8—61no—
334Q8—62no—
335Q8—61no—
336Q8—62no—
337Q8—62no—
338Q8—61no—
339Q8C2×C3×S412yes—
340D4—62no—
341D4—121no—
342D4—121no—
343D4—122no—
344D4—121no—
345D4—121no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—122no—
362C8—62no—
363C8—62no—
364C2×C3—21no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—61no—
368C2×C3—161no—
369C2×C3—241no—
370C2×C3—321no—
371C2×C3—322no—
372C2×C3—321no—
373C2×C3C2×#102011yescharacteristic
374D3—321no—
375C22—31no—
376C22—31no—
377C22—32no—
378C22—31no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22C3×#98311yescharacteristic, complemented
386C22C3×#102011yescharacteristic
387C4—32no—
388C4—32no—
389C4—61no—
390C4—61no—
391C4—121no—
392C4—121no—
393C4—121no—
394C3—161no—
395C3—321no—
396C3#8750711yescharacteristic, direct product factor
397C2—21no—
398C2—31no—
399C2—31no—
400C2—61no—
401C2—241no—
402C2C2×C3×#102011yescharacteristic
subgroupG/H
centerC2×C3C2×#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×#1020
FittingC23×C3×Q8D3
socleC23×C3C2×S4
hypercenterC22×C3#1020
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊SL(2,𝔽3)C2semidirect
2. C3×#5931C2semidirect
3. C3×#5930C2semidirect
4. #87507C3direct (N × H)
9. C23×C3×Q8D3semidirect
11. C2×C22⋊SL(2,𝔽3)C2×C3semidirect
12. #5931C2×C3semidirect
13. #5930C2×C3semidirect
75. C23×Q8C3×D3semidirect
108. C2×C3×Q8S4semidirect
261. C2×Q8C3×S4semidirect
309. C22×C3#983semidirect
385. C22C3×#983semidirect
396. C3#87507direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3×#59312
C3×#59302
#875073
C3×#30043
C3×GL(2,ℤ/4)4
C3×#9834