Finite Group Databasebeta

← C3×#10590594 #13157611

Subgroups — C3×#10590594 #13157611

Order 1728.

962 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 156 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×#64101C211yesmaximal, characteristic, complemented
3C2×C3×#3712C211yesmaximal, characteristic, complemented
4C32×#809C212yesmaximal, complemented
5C3×#63563C212yesmaximal, complemented
6C3×#4770—32nomaximal
7#10590594C311yesmaximal, characteristic, direct product factor
8C22×C3×D32—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C3×#625—21no—
12C3×#3802—22no—
13C3×#3802C414yescomplemented
14C33×C4⋊C4C2211yescharacteristic, complemented
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C3×C4×C3⋊Dic3—21no—
18C2×C32×C3⋊C8C2212yescomplemented
19C3×#3712—22no—
20C2×C3×D3×D4—32no—
21C2×#3802C2×C311yescharacteristic
22#64101C2×C311yescharacteristic, complemented
23C2×C3×ΓL(1,𝔽25)—32no—
24C3×#892—32no—
25C3×#892—61no—
26C2×#3712C2×C311yescharacteristic, complemented
27C32×#102—32no—
28C3×#832—32no—
29C3×#821—32no—
30C3×#809—22no—
31C3×#809D312yescomplemented
32C3×#809C2×C312yescomplemented
33#63563C2×C312yescomplemented
34C22×C3×C3⋊D3D411yescharacteristic
35C22×C32×D3—22no—
36C2×C3×D32—24no—
37C2×C3×D32—42no—
38C2×C3×D32—42no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3C2×C412yes—
44C3×C4×C3⋊D3D412yes—
45C32×D12—22no—
46C32×D12C2×C414yes—
47C3×#625—24no—
48C32×C3⋊C8—22no—
49C3×#418—91no—
50#4770—32no—
51C23×C3×D3—62no—
52C22×D32—21no—
53C2×C32×D4—32no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×C3×#67—62no—
56C3×D3×D4—38no—
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×#625—21no—
64#3802—22no—
65#3802C3×C414yescomplemented
66C2×C32×C8—32no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4—22no—
69C32×C4⋊C4—41no—
70C32×C4⋊C4C2×D312yescomplemented
71C32×C4⋊C4C22×C311yescharacteristic, complemented
72#3752—21no—
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
74C4×C3⋊Dic3—21no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—62no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—61no—
79C3×C4×Dic3—62no—
80C3×C4×Dic3—62no—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2×D312yescomplemented
83C2×C3×C3⋊C8C22×C312yescomplemented
84C3×ΓL(1,𝔽25)—64no—
85#3712—22no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3C22⋊C412yes—
90C2×C32×D3—24no—
91C2×C32×D3—42no—
92C2×C32×D3—42no—
93C3×D32—48no—
94C33×C4—41no—
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—41no—
98C3×C3⋊Dic3C22⋊C412yes—
99C22×C3×D4—91no—
100C2×D3×D4—32no—
101C2×C3×M16—91no—
102C3×#117—91no—
103C2×ΓL(1,𝔽25)—32no—
104#892—32no—
105#892—61no—
106C3×#102—32no—
107C3×#102—32no—
108C3×#102—62no—
109C3×#102—92no—
110#832—32no—
111#821—32no—
112#809C3×D312yescomplemented
113C23×C32—62no—
114C22×C3⋊D3C3×D411yescharacteristic
115C22×C3×D3—22no—
116C22×C3×D3—22no—
117C22×C3×D3—32no—
118C22×C3×D3—42no—
119C22×C3×D3—61no—
120C22×C3×D3—68no—
121C22×C3×D3—124no—
122C22×C3×D3—124no—
123C2×D32—24no—
124C2×D32—42no—
125C2×D32—42no—
126C32×D4—34no—
127C32×D4—62no—
128C2×C32×C4—21no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—41no—
134C2×C32×C4—42no—
135C2×C32×C4—42no—
136C2×C32×C4C22×D312yes—
137C2×C32×C4C23×C311yescharacteristic
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3C3×D411yescharacteristic
140C4×C3⋊D3C2×C3×C412yes—
141C4×C3⋊D3C3×D412yes—
142C3×#67—68no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C3×D12—22no—
148C3×D12—22no—
149C3×D12—24no—
150C3×D12—42no—
151C3×D12C4×D314yes—
152C3×D12C2×C3×C414yes—
153C3×C4×D3—34no—
154C3×C4×D3—34no—
155C3×C4×D3—62no—
156C3×C4×D3—62no—
157#625—24no—
158C32×C8—62no—
159C3×C3⋊C8—22no—
160C3×C3⋊C8—22no—
161C3×C3⋊C8—42no—
162#418—91noSylow
163C2×C33C2×C22⋊C411yescharacteristic
164C2×C33AGL(1,ℤ/8)11yescharacteristic
165C2×C33AGL(1,ℤ/8)11yescharacteristic
166C3×C3⋊D3—22no—
167C32×D3—44no—
168C24×C3—181no—
169C23×D3—62no—
170C2×C3×D4—32no—
171C2×C3×D4—32no—
172C2×C3×D4—62no—
173C2×C3×D4—94no—
174C2×C3×D4—182no—
175C2×C3×D4—181no—
176C22×C3×C4—91no—
177C2×#67—62no—
178D3×D4—38no—
179D3×D4—64no—
180C2×D12C2×C3×D312yes—
181C2×C4×D3—32no—
182C2×C4×D3—61no—
183C3×M16—182no—
184C2×C3×C8—32no—
185C2×C3×C8—32no—
186C2×C3×C8—62no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4—91no—
191C3×C4⋊C4D3211yescharacteristic, complemented
192C3×C4⋊C4C2×C3×D312yescomplemented
193C3×C22⋊C4—181no—
194C3×C42—181no—
195(C2×D3)⋊C4—62no—
196(C2×D3)⋊C4—62no—
197C4⋊Dic3—32no—
198C4⋊Dic3—61no—
199C4×Dic3—62no—
200C4×Dic3—62no—
201C2×C3⋊C8C2×C3×D312yescomplemented
202ΓL(1,𝔽25)—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×C22⋊C412yes—
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—21no—
232C32×C4—21no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—41no—
236C32×C4—41no—
237C32×C4—41no—
238C32×C4—42no—
239C32×C4—42no—
240C32×C4—82no—
241C32×C4—82no—
242C32×C4C2×C4×D312yes—
243C32×C4D3×D412yes—
244C32×C4C22×C3×C411yescharacteristic
245C32×C4C2×C3×D411yescharacteristic
246C3⋊Dic3—41no—
247C3⋊Dic3C3×C22⋊C412yes—
248C3×Dic3—34no—
249C3×Dic3—62no—
250C3×Dic3—122no—
251C3×Dic3—122no—
252C22×D4—91no—
253C2×M16—91no—
254#117—91no—
255#102—32no—
256#102—92no—
257C33#41811yescharacteristic, complemented, Sylow
258C23×C3—62no—
259C23×C3—62no—
260C23×C3—91no—
261C23×C3—122no—
262C23×C3—184no—
263C23×C3—362no—
264C23×C3—362no—
265C22×D3—22no—
266C22×D3—32no—
267C22×D3—61no—
268C22×D3—68no—
269C22×D3—124no—
270C22×D3—124no—
271C3×D4—34no—
272C3×D4—34no—
273C3×D4—64no—
274C3×D4—62no—
275C3×D4—62no—
276C3×D4—122no—
277C3×D4—184no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—91no—
287C2×C3×C4—92no—
288C2×C3×C4—92no—
289C2×C3×C4—181no—
290C2×C3×C4C2×D3211yescharacteristic
291C2×C3×C4C22×C3×D312yes—
292#67—68no—
293C2×Dic3—32no—
294C2×Dic3—61no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297D12—22no—
298D12C3×C4×D314yes—
299C4×D3—34no—
300C4×D3—34no—
301C4×D3—62no—
302C4×D3—62no—
303C3×C8—62no—
304C3×C8—62no—
305C3×C8—122no—
306C3⋊C8—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—22no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—124no—
317C2×C32D3×C22⋊C412yes—
318C2×C32#91112yes—
319C2×C32#91412yes—
320C2×C32C2×C3×C22⋊C411yescharacteristic
321C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
322C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
323C3⋊D3—22no—
324C3×D3—44no—
325C3×D3—44no—
326C3×D3—64no—
327C3×D3—84no—
328C3×D3—122no—
329C24—181no—
330C2×D4—32no—
331C2×D4—94no—
332C2×D4—182no—
333C2×D4—181no—
334C22×C4—91no—
335M16—182no—
336C2×C8—32no—
337C4⋊C4—91no—
338C4⋊C4C3×D3211yescharacteristic, complemented
339C22⋊C4—181no—
340C42—181no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—41no—
344C22×C3—64no—
345C22×C3—64no—
346C22×C3—92no—
347C22×C3—124no—
348C22×C3—122no—
349C22×C3—122no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—242no—
355C22×C3—242no—
356C22×C3—368no—
357C22×C3#381111yescharacteristic
358C22×C3C3×D3×D412yes—
359C2×D3—24no—
360C2×D3—34no—
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—62no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—1216no—
369C3×C4—21no—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—82no—
379C3×C4—82no—
380C3×C4—92no—
381C3×C4—361no—
382C3×C4C4×D3211yescharacteristic
383C3×C4#380211yescharacteristic
384C3×C4C2×C3×C4×D312yes—
385C3×C4C3×D3×D412yes—
386Dic3—34no—
387Dic3—62no—
388Dic3—122no—
389Dic3—122no—
390C32—22no—
391C32—21no—
392C32—41no—
393C32#477012yescomplemented
394C32C3×#41811yescharacteristic, complemented
395C23—62no—
396C23—91no—
397C23—184no—
398C23—362no—
399C23—362no—
400D4—34no—
401D4—62no—
402D4—184no—
403C2×C4—21no—
404C2×C4—91no—
405C2×C4—92no—
406C2×C4—92no—
407C2×C4—181no—
408C2×C4C2×C3×D3211yescharacteristic
409C8—62no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—22no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—124no—
420C2×C3—124no—
421C2×C3—182no—
422C2×C3—244no—
423C2×C3#6392111yescharacteristic
424C2×C3#6392711yescharacteristic
425C2×C3#6394111yescharacteristic
426C2×C3C3×D3×C22⋊C412yes—
427C2×C3C3×#91112yes—
428C2×C3C3×#91412yes—
429D3—44no—
430D3—64no—
431D3—122no—
432C22—64no—
433C22—92no—
434C22—122no—
435C22—122no—
436C22—181no—
437C22—181no—
438C22—368no—
439C22C3×#381111yescharacteristic
440C4—41no—
441C4—92no—
442C4—361no—
443C4C3×C4×D3211yescharacteristic
444C4C3×#380211yescharacteristic
445C3—21no—
446C3—22no—
447C3—41no—
448C3#1059059411yescharacteristic, direct product factor
449C3C3×#477012yescomplemented
450C2—124no—
451C2—182no—
452C2C3×#6392111yescharacteristic
453C2C3×#6392711yescharacteristic
454C2C3×#6394111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#418
LCSG ▷ 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×#64101C2semidirect
3. C2×C3×#3712C2semidirect
4. C32×#809C2semidirect
5. C3×#63563C2semidirect
7. #10590594C3direct (N × H)
13. C3×#3802C4semidirect
14. C33×C4⋊C4C22semidirect
16. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C32×C3⋊C8C22semidirect
22. #64101C2×C3semidirect
26. C2×#3712C2×C3semidirect
31. C3×#809D3semidirect
32. C3×#809C2×C3semidirect
33. #63563C2×C3semidirect
65. #3802C3×C4semidirect
70. C32×C4⋊C4C2×D3semidirect
71. C32×C4⋊C4C22×C3semidirect
73. C4⋊(C3⋊Dic3)C22×C3semidirect
82. C2×C3×C3⋊C8C2×D3semidirect
83. C2×C3×C3⋊C8C22×C3semidirect
112. #809C3×D3semidirect
191. C3×C4⋊C4D32semidirect
192. C3×C4⋊C4C2×C3×D3semidirect
201. C2×C3⋊C8C2×C3×D3semidirect
257. C33#418semidirect
338. C4⋊C4C3×D32semidirect
393. C32#4770semidirect
394. C32C3×#418semidirect
448. C3#10590594direct (N × H)
449. C3C3×#4770semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C3×#641012
C2×C3×#37122
C32×#8092
C3×#635632
C3×#47703
#105905943