Finite Group Databasebeta

← C3×#10590597 #13157615

Subgroups — C3×#10590597 #13157615

Order 1728.

952 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 136 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×#3802C211yesmaximal, characteristic
2C3×#64106C211yesmaximal, characteristic, complemented
3C2×C3×#3716C212yesmaximal, complemented
4C32×#809C212yesmaximal, complemented
5C3×#63557C211yesmaximal, characteristic, complemented
6C3×#4776—32nomaximal
7#10590597C311yesmaximal, characteristic, direct product factor
8C22×C3×D32—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic, complemented
11C2×C32×D12C2212yes—
12C2×C3×#625—21no—
13C3×#3802—24no—
14C33×C4⋊C4C2211yescharacteristic, complemented
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)—21no—
17C2×C32×C3⋊C8C2212yescomplemented
18C3×#3716—24no—
19C2×C3×D3×D4—32no—
20C2×#3802C2×C311yescharacteristic
21#64106C2×C311yescharacteristic, complemented
22C2×C3×#214—32no—
23C2×C3×#204—32no—
24C3×#897—32no—
25C3×#897—61no—
26C2×#3716C2×C312yescomplemented
27C32×#102—32no—
28C3×(C2×D3)⋊C8—32no—
29C3×#809—22no—
30C3×#809D312yescomplemented
31C3×#809C2×C312yescomplemented
32#63557C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C32×D3—22no—
35C2×C3×D32—24no—
36C2×C3×D32—42no—
37C2×C3×D32—42no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3—22no—
43C3×#634—22no—
44C32×D12—22no—
45C32×D12D414yes—
46C3×#625—24no—
47C32×C3⋊C8—22no—
48C3×#450—91no—
49#4776—32no—
50C23×C3×D3—62no—
51C22×D32—21no—
52C2×C32×D4—32no—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×#634C22×C311yescharacteristic, complemented
55C2×C3×#67—62no—
56C3×D3×D4—64no—
57C3×D3×D4—64no—
58C2×C3×D12—22no—
59C2×C3×D12C2×D312yes—
60C2×C3×D12C22×C312yes—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—32no—
64C2×C3×Dic6—61no—
65C2×#625—21no—
66#3802—24no—
67C2×C32×C8—32no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—22no—
70C32×C4⋊C4—41no—
71C32×C4⋊C4C2×D312yescomplemented
72C32×C4⋊C4C22×C311yescharacteristic, complemented
73#3752—21no—
74C4⋊(C3⋊Dic3)—21no—
75C3×#214—64no—
76C3×(C2×D3)⋊C4—62no—
77C3×(C2×D3)⋊C4—62no—
78C3×C4⋊Dic3—62no—
79C3×C4⋊Dic3—62no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8C2×D312yescomplemented
82C2×C3×C3⋊C8C22×C312yescomplemented
83C3×#204—64no—
84#3716—24no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3SD1612yescomplemented
89C2×C32×D3—24no—
90C2×C32×D3—42no—
91C2×C32×D3—42no—
92C3×D32—48no—
93C33×C4—41no—
94C33×C4C2×D412yes—
95C3×C3⋊Dic3—21no—
96C3×C3⋊Dic3—41no—
97C22×C3×D4—91no—
98C2×D3×D4—32no—
99C2×C3×SD16—92no—
100C3×C22⋊Q8—91no—
101C2×#214—32no—
102C2×#204—32no—
103#897—32no—
104#897—61no—
105C3×#102—32no—
106C3×#102—32no—
107C3×#102—62no—
108C3×C22⋊C8—91no—
109(C2×D3)⋊C8—32no—
110#809C3×D312yescomplemented
111C23×C32—62no—
112C22×C3⋊D3C3×D411yescharacteristic
113C22×C3×D3—22no—
114C22×C3×D3—22no—
115C22×C3×D3—32no—
116C22×C3×D3—42no—
117C22×C3×D3—61no—
118C22×C3×D3—68no—
119C22×C3×D3—124no—
120C22×C3×D3—124no—
121C2×D32—24no—
122C2×D32—42no—
123C2×D32—42no—
124C32×D4—34no—
125C32×D4—62no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—41no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4C22×D312yes—
135C2×C32×C4C23×C311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3C3×D411yescharacteristic
138C4×C3⋊D3—22no—
139#634—22no—
140C3×#67—68no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C3×D12—22no—
146C3×D12—22no—
147C3×D12—24no—
148C3×D12—42no—
149C3×D12D1214yes—
150C3×D12C3×D414yes—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—122no—
154C3×Dic6—62no—
155C3×Dic6—62no—
156C3×Dic6—122no—
157#625—24no—
158C32×C8—62no—
159C3×C3⋊C8—22no—
160C3×C3⋊C8—22no—
161C3×C3⋊C8—42no—
162#450—91noSylow
163C2×C33#12011yescharacteristic
164C2×C33C2×SD1611yescharacteristic
165C2×C33AGL(1,ℤ/8)11yescharacteristic
166C3×C3⋊D3—22no—
167C32×D3—44no—
168C24×C3—181no—
169C23×D3—62no—
170C2×C3×Q8—91no—
171C2×C3×D4—32no—
172C2×C3×D4—32no—
173C2×C3×D4—62no—
174C2×C3×D4—181no—
175C2×C3×D4—184no—
176C22×C3×C4—91no—
177C2×#67—62no—
178D3×D4—64no—
179D3×D4—64no—
180C2×D12C2×C3×D312yes—
181C2×C4×D3—32no—
182C2×C4×D3—61no—
183C2×Dic6—32no—
184C2×Dic6—61no—
185C3×SD16—184no—
186C2×C3×C8—32no—
187C2×C3×C8—32no—
188C2×C3×C8—62no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—181no—
193C3×C4⋊C4D3211yescharacteristic, complemented
194C3×C4⋊C4C2×C3×D312yescomplemented
195C3×C22⋊C4—181no—
196#214—64no—
197(C2×D3)⋊C4—62no—
198(C2×D3)⋊C4—62no—
199C4⋊Dic3—62no—
200C4⋊Dic3—62no—
201C2×C3⋊C8C2×C3×D312yescomplemented
202#204—64no—
203C22×C32—21no—
204C22×C32—22no—
205C22×C32—41no—
206C22×C32—64no—
207C22×C32—122no—
208C22×C32—122no—
209C22×C32D3×D412yes—
210C22×C32C2×C3×D411yescharacteristic
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3C3×SD1612yescomplemented
214C2×C3×D3—24no—
215C2×C3×D3—24no—
216C2×C3×D3—34no—
217C2×C3×D3—44no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—42no—
221C2×C3×D3—42no—
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—82no—
226C2×C3×D3—82no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—1216no—
230D32—48no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—42no—
235C32×C4—41no—
236C32×C4—42no—
237C32×C4—42no—
238C32×C4—82no—
239C32×C4—82no—
240C32×C4C2×D1212yes—
241C32×C4D3×D412yes—
242C32×C4C2×C3×D412yes—
243C3⋊Dic3—21no—
244C3⋊Dic3—41no—
245C3×Dic3—62no—
246C3×Dic3—121no—
247C3×Dic3—122no—
248C3×Dic3—122no—
249C22×D4—91no—
250C2×SD16—92no—
251C22⋊Q8—91no—
252#102—32no—
253C22⋊C8—91no—
254C33#45011yescharacteristic, complemented, Sylow
255C23×C3—62no—
256C23×C3—62no—
257C23×C3—91no—
258C23×C3—122no—
259C23×C3—184no—
260C23×C3—362no—
261C23×C3—362no—
262C22×D3—22no—
263C22×D3—32no—
264C22×D3—61no—
265C22×D3—68no—
266C22×D3—124no—
267C22×D3—124no—
268C3×Q8—182no—
269C3×D4—34no—
270C3×D4—34no—
271C3×D4—64no—
272C3×D4—62no—
273C3×D4—62no—
274C3×D4—122no—
275C3×D4—184no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—91no—
285C2×C3×C4—182no—
286C2×C3×C4—181no—
287C2×C3×C4C2×D3211yescharacteristic
288C2×C3×C4C22×C3×D312yes—
289#67—68no—
290C2×Dic3—32no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294D12—22no—
295D12C3×D1214yes—
296C4×D3—62no—
297C4×D3—62no—
298C4×D3—122no—
299Dic6—62no—
300Dic6—62no—
301Dic6—122no—
302C3×C8—62no—
303C3×C8—62no—
304C3×C8—122no—
305C3⋊C8—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—124no—
316C2×C32#88212yes—
317C2×C32#91112yes—
318C2×C32D3×SD1612yes—
319C2×C32C3×#12011yescharacteristic
320C2×C32C2×C3×SD1611yescharacteristic
321C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
322C3⋊D3—22no—
323C3×D3—44no—
324C3×D3—44no—
325C3×D3—64no—
326C3×D3—84no—
327C3×D3—122no—
328C24—181no—
329C2×Q8—91no—
330C2×D4—32no—
331C2×D4—181no—
332C2×D4—184no—
333C22×C4—91no—
334SD16—184no—
335C2×C8—32no—
336C4⋊C4—181no—
337C4⋊C4C3×D3211yescharacteristic, complemented
338C22⋊C4—181no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—41no—
342C22×C3—64no—
343C22×C3—64no—
344C22×C3—92no—
345C22×C3—124no—
346C22×C3—122no—
347C22×C3—122no—
348C22×C3—122no—
349C22×C3—122no—
350C22×C3—181no—
351C22×C3—181no—
352C22×C3—242no—
353C22×C3—242no—
354C22×C3—368no—
355C22×C3#381111yescharacteristic
356C22×C3C3×D3×D412yes—
357C2×D3—24no—
358C2×D3—34no—
359C2×D3—42no—
360C2×D3—42no—
361C2×D3—62no—
362C2×D3—62no—
363C2×D3—62no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—1216no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—41no—
373C3×C4—42no—
374C3×C4—82no—
375C3×C4—82no—
376C3×C4—181no—
377C3×C4—361no—
378C3×C4D3×D1212yes—
379C3×C4C2×C3×D1212yes—
380C3×C4C3×D3×D412yes—
381Dic3—62no—
382Dic3—121no—
383Dic3—122no—
384Dic3—122no—
385C32—22no—
386C32—21no—
387C32—41no—
388C32#477612yescomplemented
389C32C3×#45011yescharacteristic, complemented
390C23—62no—
391C23—91no—
392C23—184no—
393C23—362no—
394C23—362no—
395Q8—182no—
396D4—34no—
397D4—62no—
398D4—184no—
399C2×C4—21no—
400C2×C4—91no—
401C2×C4—182no—
402C2×C4—181no—
403C2×C4C2×C3×D3211yescharacteristic
404C8—62no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—124no—
415C2×C3—124no—
416C2×C3—182no—
417C2×C3—244no—
418C2×C3#6392311yescharacteristic
419C2×C3#6392711yescharacteristic
420C2×C3#6394011yescharacteristic
421C2×C3C3×#88212yes—
422C2×C3C3×#91112yes—
423C2×C3C3×D3×SD1612yes—
424D3—44no—
425D3—64no—
426D3—122no—
427C22—64no—
428C22—92no—
429C22—122no—
430C22—122no—
431C22—181no—
432C22—181no—
433C22—368no—
434C22C3×#381111yescharacteristic
435C4—41no—
436C4—181no—
437C4—361no—
438C4C3×D3×D1212yes—
439C3—21no—
440C3—22no—
441C3—41no—
442C3#1059059711yescharacteristic, direct product factor
443C3C3×#477612yescomplemented
444C2—124no—
445C2—182no—
446C2C3×#6392311yescharacteristic
447C2C3×#6392711yescharacteristic
448C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#450
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64106C2semidirect
3. C2×C3×#3716C2semidirect
4. C32×#809C2semidirect
5. C3×#63557C2semidirect
7. #10590597C3direct (N × H)
10. C2×C3×#634C22semidirect
14. C33×C4⋊C4C22semidirect
17. C2×C32×C3⋊C8C22semidirect
21. #64106C2×C3semidirect
26. C2×#3716C2×C3semidirect
30. C3×#809D3semidirect
31. C3×#809C2×C3semidirect
32. #63557C2×C3semidirect
54. C2×#634C22×C3semidirect
71. C32×C4⋊C4C2×D3semidirect
72. C32×C4⋊C4C22×C3semidirect
81. C2×C3×C3⋊C8C2×D3semidirect
82. C2×C3×C3⋊C8C22×C3semidirect
88. C2×C3×C3⋊D3SD16semidirect
110. #809C3×D3semidirect
193. C3×C4⋊C4D32semidirect
194. C3×C4⋊C4C2×C3×D3semidirect
201. C2×C3⋊C8C2×C3×D3semidirect
213. C2×C3⋊D3C3×SD16semidirect
254. C33#450semidirect
337. C4⋊C4C3×D32semidirect
388. C32#4776semidirect
389. C32C3×#450semidirect
442. C3#10590597direct (N × H)
443. C3C3×#4776semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C3×#641062
C2×C3×#37162
C32×#8092
C3×#635572
C3×#47763
#105905973