Finite Group Databasebeta

← C3×#85529 #11922813

Subgroups — C3×#85529 #11922813

Order 1152.

651 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 76 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×#994C211yesmaximal, characteristic, complemented
2C2×C3×#985C211yesmaximal, characteristic, complemented
3C3×#5427C212yesmaximal, complemented
4C3×#5426C211yesmaximal, characteristic, complemented
5C3×#5425C212yesmaximal, complemented
6#85529C311yesmaximal, characteristic, direct product factor
7C3×#3012—31nomaximal
8C3×#994—22no—
9C2×C3×#41⋊C3C2212yes—
10C2×C3×#41⋊C3C2211yescharacteristic
11C3×#985—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2211yescharacteristic, complemented
14C3×#946—41nomaximal
15C3×#859C2212yescomplemented
16C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
17C2×#994—21no—
18C2×#994C2×C311yescharacteristic, complemented
19C2×#985C2×C311yescharacteristic, complemented
20C2×C3×AGL(1,ℤ/8)—31no—
21C2×C3×#135—31no—
22C3×#537—31no—
23#5427C2×C312yescomplemented
24#5426C2×C311yescharacteristic, complemented
25#5425C2×C312yescomplemented
26C3×#452—32no—
27C3×#450—32no—
28C3×#449—32no—
29C3×#447—32no—
30C3×#417—31no—
31C2×C3×#102—31no—
32C3×#408—31no—
33C2×C32×Q8—41no—
34C2×C3×C4×D3—41no—
35C3×#41⋊C3—22no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C3×GL(2,𝔽3)—21no—
39C3×#226—21no—
40C3×(C2×D3)⋊C4—42no—
41C3×C4⋊Dic3—41no—
42C3×Dic3⋊C4—42no—
43#3012—31noSylow
44C3×H(5,𝔽2)—22no—
45C3×H(5,𝔽2)—62no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41C2×D312yes—
49C2×C3×#41C2×D311yescharacteristic
50C22×C3×D4—31no—
51C22×C3×D4—32no—
52#994—24no—
53#994—22no—
54C2×#41⋊C3—22no—
55C2×#41⋊C3—21no—
56C2×#41⋊C3C22×C312yes—
57C2×#41⋊C3C22×C311yescharacteristic
58#985—22no—
59C2×GL(2,𝔽3)C22×C311yescharacteristic, complemented
60C2×#226C22×C311yescharacteristic, complemented
61C3×AGL(1,ℤ/8)—64no—
62C3×#135—62no—
63C3×#135—62no—
64C2×C3×Q16—31no—
65C2×C3×SD16—31no—
66C2×C3×SD16—32no—
67C2×C3×SD16—31no—
68C2×C3×D8—31no—
69C2×C3×D8—32no—
70C2×C3×M16—31no—
71C22×C3×C8—31no—
72C3×C22⋊Q8—32no—
73C3×C22⋊Q8—61no—
74C3×#121—32no—
75C3×#121—61no—
76C3×#121—62no—
77C2×C3×C4⋊C4—31no—
78#946—41no—
79#859C22×C312yescomplemented
80C3×#103—32no—
81C3×#102—32no—
82C3×#102—32no—
83C3×#102—32no—
84C3×C22⋊C8—32no—
85C3×C22⋊C8—32no—
86C22×C3×D3—41no—
87C32×Q8—82no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C3×Dic3—42no—
91C2×C3×Dic3—41no—
92C3×C4×D3—82no—
93C3×SL(2,𝔽3)C2×D411yescharacteristic
94C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
95C2×AGL(1,ℤ/8)—31no—
96C2×#135—31no—
97#537—31no—
98#452—32no—
99#450—32no—
100#449—32no—
101#447—32no—
102#417—31no—
103C2×#102—31no—
104#408—31no—
105C24×C3—61no—
106C3×#41—22no—
107C3×#41—32no—
108C3×#41—62no—
109C3×#41—62no—
110C3×#41—62no—
111C3×#41—62no—
112C3×#41—62no—
113C3×#41#6712yes—
114C2×C3×Q8—31no—
115C2×C3×Q8—41no—
116C2×C3×Q8—81no—
117C2×C3×Q8S411yescharacteristic
118C2×C3×Q8C22×D311yescharacteristic
119C2×C3×D4—31no—
120C2×C3×D4—32no—
121C2×C3×D4—34no—
122C2×C3×D4—31no—
123C2×C3×D4—32no—
124C2×C3×D4—31no—
125C2×C3×D4—62no—
126C2×C3×D4—64no—
127C2×C3×D4—64no—
128C2×C3×D4—61no—
129C2×C3×D4—62no—
130C22×C3×C4—31no—
131C22×C3×C4—32no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C2×C4×D3—41no—
135#41⋊C3—24no—
136#41⋊C3—22no—
137#41⋊C3—22no—
138#41⋊C3C3×D412yescomplemented
139C2×SL(2,𝔽3)—21no—
140C2×SL(2,𝔽3)C23×C311yescharacteristic
141GL(2,𝔽3)—21no—
142#226—21no—
143C3×Q16—61no—
144C3×Q16—61no—
145C3×SD16—62no—
146C3×SD16—64no—
147C3×SD16—61no—
148C3×SD16—61no—
149C3×D8—62no—
150C3×D8—64no—
151C3×M16—62no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—32no—
155C2×C3×C8—62no—
156C3×C4⋊C4—32no—
157C3×C4⋊C4—62no—
158C3×C4⋊C4—61no—
159C3×C22⋊C4—62no—
160C3×C22⋊C4—62no—
161(C2×D3)⋊C4—42no—
162C4⋊Dic3—41no—
163Dic3⋊C4—42no—
164C22×C32—41no—
165C2×C3×D3—42no—
166C2×C3×D3—81no—
167C2×C3×D3—81no—
168C32×C4—42no—
169C32×C4—82no—
170C3×Dic3—81no—
171C3×Dic3—82no—
172H(5,𝔽2)—22no—
173H(5,𝔽2)—62no—
174C2×#41—31no—
175C2×#41—31no—
176C2×#41C2×C3×D312yes—
177C2×#41C2×C3×D311yescharacteristic
178C22×D4—31no—
179C22×D4—32no—
180AGL(1,ℤ/8)—64no—
181#135—62no—
182#135—62no—
183C2×Q16—31no—
184C2×SD16—31no—
185C2×SD16—32no—
186C2×SD16—31no—
187C2×D8—31no—
188C2×D8—32no—
189C2×M16—31no—
190C22×C8—31no—
191C22⋊Q8—32no—
192C22⋊Q8—61no—
193#121—32no—
194#121—61no—
195#121—62no—
196C2×C4⋊C4—31no—
197#103—32no—
198#102—32no—
199#102—32no—
200#102—32no—
201C22⋊C8—32no—
202C22⋊C8—32no—
203C23×C3—31no—
204C23×C3—32no—
205C23×C3—64no—
206C23×C3—61no—
207C23×C3—241no—
208C23×C3—241no—
209C22×D3—41no—
210C3×Q8—22no—
211C3×Q8—31no—
212C3×Q8—61no—
213C3×Q8—61no—
214C3×Q8—82no—
215C3×Q8—84no—
216C3×Q8C2×#6711yescharacteristic
217C3×D4—32no—
218C3×D4—34no—
219C3×D4—64no—
220C3×D4—61no—
221C3×D4—62no—
222C3×D4—61no—
223C3×D4—61no—
224C3×D4—62no—
225C3×D4—122no—
226C3×D4—242no—
227C2×C3×C4—31no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—82no—
237C2×C3×C4—81no—
238C2×C3×C4—122no—
239C2×C3×C4—241no—
240C2×C3×C4C2×S412yes—
241C2×C3×C4C2×S411yescharacteristic
242C2×Dic3—42no—
243C2×Dic3—41no—
244C4×D3—82no—
245SL(2,𝔽3)—21no—
246SL(2,𝔽3)C2×C3×D411yescharacteristic
247C3×C8—61no—
248C3×C8—61no—
249C3×C8—62no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C3×D3—82no—
254C24—61no—
255#41—22no—
256#41—32no—
257#41—62no—
258#41—62no—
259#41—62no—
260#41—62no—
261#41—62no—
262#41C3×#6712yes—
263C2×Q8—31no—
264C2×Q8C3×S411yescharacteristic
265C2×Q8C22×C3×D311yescharacteristic
266C2×D4—31no—
267C2×D4—32no—
268C2×D4—34no—
269C2×D4—31no—
270C2×D4—32no—
271C2×D4—31no—
272C2×D4—62no—
273C2×D4—64no—
274C2×D4—64no—
275C2×D4—61no—
276C2×D4—62no—
277C22×C4—31no—
278C22×C4—32no—
279C22×C4—61no—
280C22×C4—61no—
281Q16—61no—
282Q16—61no—
283SD16—62no—
284SD16—64no—
285SD16—61no—
286SD16—61no—
287D8—62no—
288D8—64no—
289M16—62no—
290C2×C8—31no—
291C2×C8—31no—
292C2×C8—32no—
293C2×C8—62no—
294C4⋊C4—32no—
295C4⋊C4—61no—
296C4⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22×C3—32no—
300C22×C3—41no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—81no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—244no—
311C22×C3C22×S411yescharacteristic
312C2×D3—42no—
313C2×D3—81no—
314C2×D3—81no—
315C3×C4—22no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—42no—
319C3×C4—82no—
320C3×C4—84no—
321C3×C4—82no—
322C3×C4—122no—
323C3×C4—121no—
324C3×C4GL(2,ℤ/4)12yes—
325Dic3—81no—
326Dic3—82no—
327C32—41noSylow
328C23—31no—
329C23—32no—
330C23—64no—
331C23—61no—
332C23—241no—
333C23—241no—
334Q8—22no—
335Q8—31no—
336Q8—61no—
337Q8—61no—
338Q8C2×C3×#6711yescharacteristic
339D4—32no—
340D4—34no—
341D4—64no—
342D4—61no—
343D4—62no—
344D4—61no—
345D4—61no—
346D4—62no—
347D4—122no—
348D4—242no—
349C2×C4—31no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—61no—
356C2×C4—122no—
357C2×C4—241no—
358C2×C4C2×C3×S412yes—
359C2×C4C2×C3×S411yescharacteristic
360C8—61no—
361C8—61no—
362C8—62no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—62no—
367C2×C3—81no—
368C2×C3—81no—
369C2×C3—81no—
370C2×C3—122no—
371C2×C3—241no—
372C2×C3#592411yescharacteristic
373C2×C3#592511yescharacteristic
374C2×C3C2×GL(2,ℤ/4)11yescharacteristic
375D3—82no—
376C22—32no—
377C22—61no—
378C22—61no—
379C22—62no—
380C22—62no—
381C22—62no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—244no—
386C22C22×C3×S411yescharacteristic
387C4—22no—
388C4—31no—
389C4—31no—
390C4—122no—
391C4—121no—
392C4C3×GL(2,ℤ/4)12yes—
393C3—41no—
394C3—81no—
395C3#8552911yescharacteristic, direct product factor
396C2—62no—
397C2—122no—
398C2—241no—
399C2C3×#592411yescharacteristic
400C2C3×#592511yescharacteristic
401C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×H(5,𝔽2)D3
hypercenterC2×C3×Q8S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2semidirect
2. C2×C3×#985C2semidirect
3. C3×#5427C2semidirect
4. C3×#5426C2semidirect
5. C3×#5425C2semidirect
6. #85529C3direct (N × H)
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C3×#859C22semidirect
16. C2×C3×H(5,𝔽2)D3semidirect
18. C2×#994C2×C3semidirect
19. C2×#985C2×C3semidirect
23. #5427C2×C3semidirect
24. #5426C2×C3semidirect
25. #5425C2×C3semidirect
36. C3×#41⋊C3D4semidirect
59. C2×GL(2,𝔽3)C22×C3semidirect
60. C2×#226C22×C3semidirect
79. #859C22×C3semidirect
94. C2×H(5,𝔽2)C3×D3semidirect
138. #41⋊C3C3×D4semidirect
395. C3#85529direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C2×C3×#9852
C3×#54272
C3×#54262
C3×#54252
#855293
C3×#30123
C3×#9464