Finite Group Databasebeta

← C3×#85537 #11922823

Subgroups — C3×#85537 #11922823

Order 1152.

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

normal subgroups only

This group has 723 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#5427C212yesmaximal, complemented
4C3×#5421C212yesmaximal, complemented
5C3×#5420C211yesmaximal, characteristic, complemented
6#85537C311yesmaximal, characteristic, direct product factor
7C3×#3010—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic, complemented
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C3×GL(2,𝔽3)—22no—
13C2×C3×GL(2,𝔽3)C2212yes—
14C3×#937—41nomaximal
15C3×#859C2212yescomplemented
16C2×C3×2−1+4D311yescharacteristic, complemented
17C2×2−1+4⋊C3—21no—
18C2×2−1+4⋊C3C2×C311yescharacteristic, complemented
19C22×GL(2,𝔽3)C2×C311yescharacteristic, complemented
20C2×C3×AGL(1,ℤ/8)—31no—
21C22×C3×SD16—31no—
22C3×#535—31no—
23#5427C2×C312yescomplemented
24#5421C2×C312yescomplemented
25#5420C2×C311yescharacteristic, complemented
26C3×#449—32no—
27C3×#449—32no—
28C3×#448—32no—
29C3×#448—32no—
30C3×#419—31no—
31C3×#416—31no—
32C3×#408—31no—
33C23×C3×D3—41no—
34C2×C32×D4—41no—
35C2×C3×#67—42no—
36C3×#41⋊C3—21no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)D412yescomplemented
39C2×C3×SL(2,𝔽3)C2311yescharacteristic
40C3×GL(2,𝔽3)—22no—
41C3×#217—41no—
42C3×(C2×D3)⋊C4—42no—
43#3010—31noSylow
44C3×2−1+4—22no—
45C3×2−1+4—62no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41C2×D311yescharacteristic, complemented
50C22×C3×Q8—31no—
51C22×C3×Q8C2×D311yescharacteristic
52C22×C3×Q8C2×D311yescharacteristic, complemented
53C22×C3×D4—31no—
542−1+4⋊C3—24no—
552−1+4⋊C3—22no—
56C2×#41⋊C3—21no—
57C2×#41⋊C3C22×C311yescharacteristic, complemented
58C22×SL(2,𝔽3)—21no—
59C22×SL(2,𝔽3)—21no—
60C22×SL(2,𝔽3)C22×C311yescharacteristic
61C22×SL(2,𝔽3)C22×C311yescharacteristic, complemented
62C2×GL(2,𝔽3)—22no—
63C2×GL(2,𝔽3)C22×C312yes—
64C3×AGL(1,ℤ/8)—64no—
65C2×C3×SD16—32no—
66C2×C3×SD16—32no—
67C2×C3×SD16—32no—
68C2×C3×SD16—62no—
69C2×C3×SD16—62no—
70C2×C3×D8—32no—
71C2×C3×M16—31no—
72C22×C3×C8—31no—
73C3×#127—61no—
74C3×#124—61no—
75C3×#121—32no—
76C3×#121—32no—
77C3×#120—61no—
78C3×#120—61no—
79C3×#117—31no—
80#937—41no—
81#859C22×C312yescomplemented
82C3×#103—32no—
83C3×#103—32no—
84C3×#103—32no—
85C3×#102—32no—
86C3×C22⋊C8—32no—
87C3×C22⋊C8—32no—
88C23×C32—41no—
89C23×C32—41no—
90C22×C3×D3—42no—
91C22×C3×D3—82no—
92C22×C3×D3—84no—
93C32×D4—82no—
94C2×C32×C4—41no—
95C3×#67—84no—
96C2×C3×Dic3—42no—
97C3×SL(2,𝔽3)C2×D411yescharacteristic
98C2×2−1+4C3×D311yescharacteristic, complemented
99C2×AGL(1,ℤ/8)—31no—
100C22×SD16—31no—
101#535—31no—
102#449—32no—
103#449—32no—
104#448—32no—
105#448—32no—
106#419—31no—
107#416—31no—
108#408—31no—
109C24×C3—61no—
110C23×D3—41no—
111C3×#41—21no—
112C3×#41—34no—
113C3×#41—62no—
114C3×#41—62no—
115C3×#41—62no—
116C3×#41—62no—
117C3×#41—62no—
118C3×#41—61no—
119C2×C3×Q8—21no—
120C2×C3×Q8—31no—
121C2×C3×Q8—31no—
122C2×C3×Q8—31no—
123C2×C3×Q8—32no—
124C2×C3×Q8—62no—
125C2×C3×Q8—62no—
126C2×C3×Q8—61no—
127C2×C3×Q8#6712yescomplemented
128C2×C3×Q8C22×D311yescharacteristic
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C2×C3×D4—31no—
132C2×C3×D4—32no—
133C2×C3×D4—41no—
134C2×C3×D4—62no—
135C2×C3×D4—62no—
136C2×C3×D4—61no—
137C2×C3×D4—62no—
138C2×C3×D4—62no—
139C2×C3×D4—81no—
140C2×C3×D4S411yescharacteristic
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—61no—
145C2×#67—42no—
146#41⋊C3—22no—
147#41⋊C3—21no—
148C2×SL(2,𝔽3)—22no—
149C2×SL(2,𝔽3)—21no—
150C2×SL(2,𝔽3)—22no—
151C2×SL(2,𝔽3)—21no—
152C2×SL(2,𝔽3)C3×D412yescomplemented
153C2×SL(2,𝔽3)C23×C311yescharacteristic
154GL(2,𝔽3)—22no—
155C3×SD16—64no—
156C3×SD16—62no—
157C3×SD16—64no—
158C3×SD16—62no—
159C3×D8—64no—
160C3×M16—62no—
161C2×C3×C8—32no—
162C2×C3×C8—32no—
163C2×C3×C8—62no—
164C3×C4⋊C4—32no—
165C3×C22⋊C4—62no—
166C3×C22⋊C4—61no—
167C3×C22⋊C4—62no—
168C3×C42—61no—
169#217—41no—
170(C2×D3)⋊C4—42no—
171C22×C32—42no—
172C22×C32—41no—
173C22×C32—81no—
174C22×C32—82no—
175C22×C32—82no—
176C2×C3×D3—44no—
177C2×C3×D3—82no—
178C2×C3×D3—82no—
179C2×C3×D3—88no—
180C32×C4—81no—
181C3×Dic3—82no—
1822−1+4—22no—
1832−1+4—62no—
184C2×#41—31no—
185C2×#41—31no—
186C2×#41—31no—
187C2×#41C2×C3×D311yescharacteristic, complemented
188C22×Q8—31no—
189C22×Q8C2×C3×D311yescharacteristic
190C22×Q8C2×C3×D311yescharacteristic, complemented
191C22×D4—31no—
192AGL(1,ℤ/8)—64no—
193C2×SD16—32no—
194C2×SD16—32no—
195C2×SD16—32no—
196C2×SD16—62no—
197C2×SD16—62no—
198C2×D8—32no—
199C2×M16—31no—
200C22×C8—31no—
201#127—61no—
202#124—61no—
203#121—32no—
204#121—32no—
205#120—61no—
206#120—61no—
207#117—31no—
208#103—32no—
209#103—32no—
210#103—32no—
211#102—32no—
212C22⋊C8—32no—
213C22⋊C8—32no—
214C23×C3—31no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—62no—
218C23×C3—81no—
219C23×C3—81no—
220C23×C3—122no—
221C23×C3—242no—
222C23×C3C2×S411yescharacteristic
223C23×C3C2×S411yescharacteristic
224C22×D3—42no—
225C22×D3—82no—
226C22×D3—84no—
227C3×Q8—32no—
228C3×Q8—31no—
229C3×Q8—32no—
230C3×Q8—62no—
231C3×Q8—61no—
232C3×Q8—61no—
233C3×Q8C2×#6711yescharacteristic
234C3×D4—22no—
235C3×D4—32no—
236C3×D4—62no—
237C3×D4—61no—
238C3×D4—62no—
239C3×D4—62no—
240C3×D4—62no—
241C3×D4—82no—
242C3×D4—84no—
243C3×D4—122no—
244C3×D4—242no—
245C3×D4—242no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—41no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—64no—
257C2×C3×C4—62no—
258C2×C3×C4—81no—
259C2×C3×C4C2×S411yescharacteristic
260#67—84no—
261C2×Dic3—42no—
262SL(2,𝔽3)—21no—
263SL(2,𝔽3)C2×C3×D411yescharacteristic
264C3×C8—62no—
265C3×C8—62no—
266C2×C32—42no—
267C2×C32—41no—
268C2×C32—82no—
269C2×C32—161no—
270C3×D3—84no—
271C24—61no—
272#41—21no—
273#41—34no—
274#41—62no—
275#41—62no—
276#41—62no—
277#41—62no—
278#41—62no—
279#41—61no—
280C2×Q8—21no—
281C2×Q8—31no—
282C2×Q8—31no—
283C2×Q8—31no—
284C2×Q8—32no—
285C2×Q8—62no—
286C2×Q8—62no—
287C2×Q8—61no—
288C2×Q8C3×#6712yescomplemented
289C2×Q8C22×C3×D311yescharacteristic
290C2×D4—31no—
291C2×D4—31no—
292C2×D4—31no—
293C2×D4—32no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—61no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4C3×S411yescharacteristic
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—61no—
304SD16—64no—
305SD16—62no—
306SD16—64no—
307SD16—62no—
308D8—64no—
309M16—62no—
310C2×C8—32no—
311C2×C8—32no—
312C2×C8—62no—
313C4⋊C4—32no—
314C22⋊C4—62no—
315C22⋊C4—61no—
316C22⋊C4—62no—
317C42—61no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—62no—
324C22×C3—61no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—81no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—81no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—162no—
335C22×C3—162no—
336C22×C3—244no—
337C22×C3GL(2,ℤ/4)12yes—
338C22×C3C22×S411yescharacteristic
339C2×D3—44no—
340C2×D3—82no—
341C2×D3—82no—
342C2×D3—88no—
343C3×C4—21no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—32no—
347C3×C4—61no—
348C3×C4—81no—
349C3×C4—82no—
350C3×C4—122no—
351Dic3—82no—
352C32—41noSylow
353C23—31no—
354C23—62no—
355C23—122no—
356C23—242no—
357C23C2×C3×S411yescharacteristic
358C23C2×C3×S411yescharacteristic
359Q8—32no—
360Q8—31no—
361Q8—32no—
362Q8—62no—
363Q8—61no—
364Q8—61no—
365Q8C2×C3×#6711yescharacteristic
366D4—22no—
367D4—32no—
368D4—62no—
369D4—61no—
370D4—62no—
371D4—62no—
372D4—62no—
373D4—122no—
374D4—242no—
375D4—242no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—64no—
386C2×C4—62no—
387C2×C4C2×C3×S411yescharacteristic
388C8—62no—
389C8—62no—
390C2×C3—22no—
391C2×C3—41no—
392C2×C3—42no—
393C2×C3—41no—
394C2×C3—82no—
395C2×C3—82no—
396C2×C3—81no—
397C2×C3—121no—
398C2×C3—162no—
399C2×C3—161no—
400C2×C3—162no—
401C2×C3—242no—
402C2×C3#592612yes—
403C2×C3C2×GL(2,ℤ/4)11yescharacteristic
404D3—84no—
405C22—22no—
406C22—22no—
407C22—21no—
408C22—62no—
409C22—61no—
410C22—122no—
411C22—122no—
412C22—122no—
413C22—244no—
414C22C3×GL(2,ℤ/4)12yes—
415C22C22×C3×S411yescharacteristic
416C4—21no—
417C4—31no—
418C4—31no—
419C4—32no—
420C4—61no—
421C4—122no—
422C3—41no—
423C3—81no—
424C3#8553711yescharacteristic, direct product factor
425C2—22no—
426C2—41no—
427C2—121no—
428C2—242no—
429C2C3×#592612yes—
430C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C22×C3×GL(2,𝔽3)C2semidirect
3. C3×#5427C2semidirect
4. C3×#5421C2semidirect
5. C3×#5420C2semidirect
6. #85537C3direct (N × H)
9. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
15. C3×#859C22semidirect
16. C2×C3×2−1+4D3semidirect
18. C2×2−1+4⋊C3C2×C3semidirect
19. C22×GL(2,𝔽3)C2×C3semidirect
23. #5427C2×C3semidirect
24. #5421C2×C3semidirect
25. #5420C2×C3semidirect
38. C2×C3×SL(2,𝔽3)D4semidirect
49. C2×C3×#41C2×D3semidirect
52. C22×C3×Q8C2×D3semidirect
57. C2×#41⋊C3C22×C3semidirect
61. C22×SL(2,𝔽3)C22×C3semidirect
81. #859C22×C3semidirect
98. C2×2−1+4C3×D3semidirect
127. C2×C3×Q8#67semidirect
152. C2×SL(2,𝔽3)C3×D4semidirect
187. C2×#41C2×C3×D3semidirect
190. C22×Q8C2×C3×D3semidirect
288. C2×Q8C3×#67semidirect
424. C3#85537direct (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
C22×C3×GL(2,𝔽3)2
C3×#54272
C3×#54212
C3×#54202
#855373
C3×#30103
C3×#9374