Finite Group Databasebeta

← C3×#10590598 #13157614

Subgroups — C3×#10590598 #13157614

Order 1728.

1012 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1012 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, complemented
2C3×#64104C211yesmaximal, characteristic, complemented
3C2×C3×#3714C212yesmaximal, complemented
4C32×#809C212yesmaximal, complemented
5C3×#63557C211yesmaximal, characteristic, complemented
6C3×#4773—32nomaximal
7#10590598C311yesmaximal, characteristic, direct product factor
8C22×C3×D32—21no—
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×C3×#625—21no—
14C3×#3802—24no—
15C33×C4⋊C4C2211yescharacteristic, complemented
16C3×#3752—21no—
17C2×C32×C3⋊C8C2212yescomplemented
18C3×#3714—24no—
19C2×C3×D3×D4—32no—
20C2×#3802C2×C311yescharacteristic, complemented
21#64104C2×C311yescharacteristic, complemented
22C2×C3×#213—32no—
23C2×C3×D24—32no—
24C3×#895—32no—
25C3×#895—61no—
26C2×#3714C2×C312yescomplemented
27C32×#102—32no—
28C3×(C2×D3)⋊C8—32no—
29C3×#809—22no—
30C3×#809D312yescomplemented
31C3×#809C2×C312yescomplemented
32#63557C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3—21no—
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⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—22no—
43C3×C4⋊(C3⋊D3)—42no—
44C3×C4×C3⋊D3—22no—
45C32×D12—22no—
46C32×D12D414yescomplemented
47C3×#625—24no—
48C32×C3⋊C8—22no—
49C3×#447—91no—
50#4773—32no—
51C23×C3×D3—62no—
52C22×D32—21no—
53C2×C32×D4—32no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×#67—62no—
58C3×D3×D4—64no—
59C3×D3×D4—64no—
60C2×C3×D12—22no—
61C2×C3×D12—32no—
62C2×C3×D12—62no—
63C2×C3×D12—61no—
64C2×C3×D12—62no—
65C2×C3×D12C2×D312yes—
66C2×C3×D12C22×C312yes—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—61no—
69C2×#625—21no—
70#3802—24no—
71C2×C32×C8—32no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—22no—
74C32×C4⋊C4—41no—
75C32×C4⋊C4C2×D312yescomplemented
76C32×C4⋊C4C22×C311yescharacteristic, complemented
77#3752—21no—
78C3×#213—64no—
79C3×(C2×D3)⋊C4—62no—
80C3×(C2×D3)⋊C4—62no—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2×D312yescomplemented
83C2×C3×C3⋊C8C22×C312yescomplemented
84C3×D24—64no—
85#3714—24no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3D812yescomplemented
93C2×C32×D3—24no—
94C2×C32×D3—42no—
95C2×C32×D3—42no—
96C3×D32—48no—
97C33×C4—41no—
98C33×C4C2×D412yes—
99C3×C3⋊Dic3—21no—
100C22×C3×D4—91no—
101C2×D3×D4—32no—
102C2×C3×D8—92no—
103C3×#121—91no—
104C2×#213—32no—
105C2×D24—32no—
106#895—32no—
107#895—61no—
108C3×#102—32no—
109C3×#102—32no—
110C3×#102—62no—
111C3×C22⋊C8—91no—
112(C2×D3)⋊C8—32no—
113#809C3×D312yescomplemented
114C23×C32—62no—
115C22×C3⋊D3—21no—
116C22×C3⋊D3C3×D411yescharacteristic
117C22×C3×D3—22no—
118C22×C3×D3—22no—
119C22×C3×D3—32no—
120C22×C3×D3—42no—
121C22×C3×D3—61no—
122C22×C3×D3—68no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—124no—
126C22×C3×D3—124no—
127C2×D32—24no—
128C2×D32—42no—
129C2×D32—42no—
130C32×D4—34no—
131C32×D4—62no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—41no—
138C2×C32×C4—42no—
139C2×C32×C4—42no—
140C2×C32×C4C22×D312yes—
141C2×C32×C4C23×C311yescharacteristic
142C2×C3⋊Dic3C3×D411yescharacteristic
143C4⋊(C3⋊D3)—22no—
144C4⋊(C3⋊D3)—42no—
145C4×C3⋊D3—22no—
146C3×#67—68no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—61no—
149C3×D12—22no—
150C3×D12—22no—
151C3×D12—24no—
152C3×D12—42no—
153C3×D12—62no—
154C3×D12—62no—
155C3×D12—122no—
156C3×D12—124no—
157C3×D12—124no—
158C3×D12D1214yescomplemented
159C3×D12C3×D414yescomplemented
160C3×C4×D3—62no—
161C3×C4×D3—62no—
162C3×C4×D3—122no—
163#625—24no—
164C32×C8—62no—
165C3×C3⋊C8—22no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8—42no—
168#447—91noSylow
169C2×C33#12011yescharacteristic
170C2×C33C2×D811yescharacteristic
171C2×C33AGL(1,ℤ/8)11yescharacteristic
172C3×C3⋊D3—22no—
173C3×C3⋊D3—81no—
174C32×D3—44no—
175C24×C3—181no—
176C23×D3—62no—
177C2×C3×D4—32no—
178C2×C3×D4—32no—
179C2×C3×D4—62no—
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C2×C3×D4—181no—
183C2×C3×D4—184no—
184C22×C3×C4—91no—
185C2×#67—62no—
186D3×D4—64no—
187D3×D4—64no—
188C2×D12—32no—
189C2×D12—61no—
190C2×D12—62no—
191C2×D12—62no—
192C2×D12C2×C3×D312yes—
193C2×C4×D3—32no—
194C2×C4×D3—61no—
195C3×D8—184no—
196C2×C3×C8—32no—
197C2×C3×C8—32no—
198C2×C3×C8—62no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4D3211yescharacteristic, complemented
203C3×C4⋊C4C2×C3×D312yescomplemented
204C3×C22⋊C4—181no—
205#213—64no—
206(C2×D3)⋊C4—62no—
207(C2×D3)⋊C4—62no—
208C2×C3⋊C8C2×C3×D312yescomplemented
209D24—64no—
210C22×C32—21no—
211C22×C32—22no—
212C22×C32—41no—
213C22×C32—64no—
214C22×C32—122no—
215C22×C32—122no—
216C22×C32D3×D412yes—
217C22×C32C2×C3×D411yescharacteristic
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3C3×D812yescomplemented
224C2×C3×D3—24no—
225C2×C3×D3—24no—
226C2×C3×D3—34no—
227C2×C3×D3—44no—
228C2×C3×D3—42no—
229C2×C3×D3—42no—
230C2×C3×D3—42no—
231C2×C3×D3—42no—
232C2×C3×D3—62no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—82no—
236C2×C3×D3—82no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—1216no—
240C2×C3×D3—122no—
241C2×C3×D3—122no—
242C2×C3×D3—122no—
243C2×C3×D3—122no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246D32—48no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—42no—
251C32×C4—41no—
252C32×C4—42no—
253C32×C4—42no—
254C32×C4—82no—
255C32×C4—82no—
256C32×C4C2×D1212yes—
257C32×C4D3×D412yes—
258C32×C4C2×C3×D412yes—
259C3⋊Dic3—21no—
260C3×Dic3—62no—
261C3×Dic3—121no—
262C22×D4—91no—
263C2×D8—92no—
264#121—91no—
265#102—32no—
266C22⋊C8—91no—
267C33#44711yescharacteristic, complemented, Sylow
268C23×C3—62no—
269C23×C3—62no—
270C23×C3—91no—
271C23×C3—122no—
272C23×C3—184no—
273C23×C3—181no—
274C23×C3—362no—
275C23×C3—362no—
276C22×D3—22no—
277C22×D3—32no—
278C22×D3—61no—
279C22×D3—68no—
280C22×D3—62no—
281C22×D3—62no—
282C22×D3—124no—
283C22×D3—124no—
284C3×D4—34no—
285C3×D4—34no—
286C3×D4—64no—
287C3×D4—62no—
288C3×D4—62no—
289C3×D4—122no—
290C3×D4—184no—
291C3×D4—182no—
292C3×D4—362no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—91no—
302C2×C3×C4—182no—
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C22×C3×D312yes—
305#67—68no—
306C2×Dic3—32no—
307C2×Dic3—61no—
308D12—22no—
309D12—62no—
310D12—62no—
311D12—122no—
312D12—124no—
313D12—124no—
314D12C3×D1214yescomplemented
315C4×D3—62no—
316C4×D3—62no—
317C4×D3—122no—
318C3×C8—62no—
319C3×C8—62no—
320C3×C8—122no—
321C3⋊C8—22no—
322C2×C32—22no—
323C2×C32—22no—
324C2×C32—22no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—124no—
332C2×C32#88212yes—
333C2×C32D3×D812yes—
334C2×C32#91412yes—
335C2×C32C3×#12011yescharacteristic
336C2×C32C2×C3×D811yescharacteristic
337C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
338C3⋊D3—22no—
339C3⋊D3—81no—
340C3×D3—44no—
341C3×D3—44no—
342C3×D3—64no—
343C3×D3—84no—
344C3×D3—122no—
345C3×D3—242no—
346C3×D3—242no—
347C24—181no—
348C2×D4—32no—
349C2×D4—91no—
350C2×D4—181no—
351C2×D4—181no—
352C2×D4—184no—
353C22×C4—91no—
354D8—184no—
355C2×C8—32no—
356C4⋊C4C3×D3211yescharacteristic, complemented
357C22⋊C4—181no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—41no—
361C22×C3—64no—
362C22×C3—64no—
363C22×C3—92no—
364C22×C3—124no—
365C22×C3—122no—
366C22×C3—122no—
367C22×C3—122no—
368C22×C3—122no—
369C22×C3—181no—
370C22×C3—181no—
371C22×C3—242no—
372C22×C3—242no—
373C22×C3—368no—
374C22×C3—361no—
375C22×C3—361no—
376C22×C3—361no—
377C22×C3#381111yescharacteristic
378C22×C3C3×D3×D412yes—
379C2×D3—24no—
380C2×D3—34no—
381C2×D3—42no—
382C2×D3—42no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—121no—
387C2×D3—121no—
388C2×D3—1216no—
389C2×D3—122no—
390C2×D3—122no—
391C2×D3—122no—
392C2×D3—122no—
393C2×D3—122no—
394C2×D3—122no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—41no—
401C3×C4—42no—
402C3×C4—82no—
403C3×C4—82no—
404C3×C4—181no—
405C3×C4D3×D1212yes—
406C3×C4C2×C3×D1212yes—
407C3×C4C3×D3×D412yes—
408Dic3—62no—
409Dic3—121no—
410C32—22no—
411C32—21no—
412C32—41no—
413C32#477312yescomplemented
414C32C3×#44711yescharacteristic, complemented
415C23—62no—
416C23—91no—
417C23—184no—
418C23—181no—
419C23—362no—
420C23—362no—
421D4—34no—
422D4—62no—
423D4—184no—
424D4—182no—
425D4—362no—
426C2×C4—21no—
427C2×C4—91no—
428C2×C4—182no—
429C2×C4C2×C3×D3211yescharacteristic
430C8—62no—
431C2×C3—21no—
432C2×C3—22no—
433C2×C3—21no—
434C2×C3—22no—
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—124no—
441C2×C3—124no—
442C2×C3—182no—
443C2×C3—244no—
444C2×C3—721no—
445C2×C3#6392311yescharacteristic
446C2×C3#6392611yescharacteristic
447C2×C3#6394111yescharacteristic
448C2×C3C3×#88212yes—
449C2×C3C3×D3×D812yes—
450C2×C3C3×#91412yes—
451D3—44no—
452D3—64no—
453D3—122no—
454D3—242no—
455D3—242no—
456C22—64no—
457C22—92no—
458C22—122no—
459C22—122no—
460C22—181no—
461C22—181no—
462C22—368no—
463C22—361no—
464C22—361no—
465C22—361no—
466C22C3×#381111yescharacteristic
467C4—41no—
468C4—181no—
469C4C3×D3×D1212yes—
470C3—21no—
471C3—22no—
472C3—41no—
473C3#1059059811yescharacteristic, direct product factor
474C3C3×#477312yescomplemented
475C2—124no—
476C2—182no—
477C2—721no—
478C2C3×#6392311yescharacteristic
479C2C3×#6392611yescharacteristic
480C2C3×#6394111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#447
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
1. C2×C3×#3802C2semidirect
2. C3×#64104C2semidirect
3. C2×C3×#3714C2semidirect
4. C32×#809C2semidirect
5. C3×#63557C2semidirect
7. #10590598C3direct (N × H)
10. C2×C3×C4⋊(C3⋊D3)C22semidirect
15. C33×C4⋊C4C22semidirect
17. C2×C32×C3⋊C8C22semidirect
20. C2×#3802C2×C3semidirect
21. #64104C2×C3semidirect
26. C2×#3714C2×C3semidirect
30. C3×#809D3semidirect
31. C3×#809C2×C3semidirect
32. #63557C2×C3semidirect
46. C32×D12D4semidirect
55. C2×C4⋊(C3⋊D3)C22×C3semidirect
75. C32×C4⋊C4C2×D3semidirect
76. C32×C4⋊C4C22×C3semidirect
82. C2×C3×C3⋊C8C2×D3semidirect
83. C2×C3×C3⋊C8C22×C3semidirect
92. C2×C3×C3⋊D3D8semidirect
113. #809C3×D3semidirect
158. C3×D12D12semidirect
159. C3×D12C3×D4semidirect
202. C3×C4⋊C4D32semidirect
203. C3×C4⋊C4C2×C3×D3semidirect
208. C2×C3⋊C8C2×C3×D3semidirect
223. C2×C3⋊D3C3×D8semidirect
267. C33#447semidirect
314. D12C3×D12semidirect
356. C4⋊C4C3×D32semidirect
413. C32#4773semidirect
414. C32C3×#447semidirect
473. C3#10590598direct (N × H)
474. C3C3×#4773semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C3×#641042
C2×C3×#37142
C32×#8092
C3×#635572
C3×#47733
#105905983