Finite Group Databasebeta

← C3×#10591516 #13158543

Subgroups — C3×#10591516 #13158543

Order 1728.

1092 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1092 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×#64147C211yesmaximal, characteristic, complemented
3C2×C32×#213C212yesmaximal, complemented
4C3×#63563C212yesmaximal, complemented
5C3×#63557C211yesmaximal, characteristic, complemented
6C3×#5156—32nomaximal
7#10591516C311yesmaximal, characteristic, direct product factor
8C22×C3×D32—21no—
9C2×C33×D4C2211yescharacteristic, complemented
10C2×C3×#638—21no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×C3×#625—21no—
14C3×#3802—24no—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C32×#213—24no—
18C2×C32×C3⋊C8C2212yescomplemented
19C2×C3×D3×D4—32no—
20C2×#3802C2×C311yescharacteristic, complemented
21C2×C32×D8—32no—
22#64147C2×C311yescharacteristic, complemented
23C3×#938—32no—
24C3×#938—61no—
25C2×C3×#213—22no—
26C2×C3×#213D312yescomplemented
27C2×C3×#213C2×C312yescomplemented
28C3×#832—32no—
29C3×#821—32no—
30C3×(C2×D3)⋊C8—32no—
31#63563C2×C312yescomplemented
32#63557C2×C311yescharacteristic, complemented
33C23×C33—21no—
34C22×C3×C3⋊D3D411yescharacteristic
35C22×C32×D3—22no—
36C2×C3×D32—24no—
37C2×C3×D32—42no—
38C2×C3×D32—42no—
39C33×D4—22no—
40C2×C33×C4C2311yescharacteristic
41C3×#638—42no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3—22no—
45C32×D12—22no—
46C32×D12D414yescomplemented
47C3×#625—24no—
48C32×C3⋊C8—22no—
49C3×#447—91no—
50#5156—32no—
51C23×C3×D3—62no—
52C22×D32—21no—
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—32no—
56C2×C32×D4—41no—
57C2×C32×D4C2×D312yescomplemented
58C2×C32×D4C22×C311yescharacteristic, complemented
59C2×#638—21no—
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C2×C3×#67—62no—
62C2×C3×#67—62no—
63C2×C3×#67—62no—
64C3×D3×D4—64no—
65C3×D3×D4—64no—
66C2×C3×D12—22no—
67C2×C3×D12C2×D312yes—
68C2×C3×D12C22×C312yes—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—61no—
71C2×#625—21no—
72#3802—24no—
73C32×D8—64no—
74C2×C32×C8—32no—
75#3757—21no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
77C3×#217—62no—
78C3×#217—62no—
79C3×#213—28no—
80C3×#213—24no—
81C3×#213—24no—
82C3×C4⋊Dic3—32no—
83C3×C4⋊Dic3—61no—
84C2×C3×C3⋊C8—22no—
85C2×C3×C3⋊C8C2×D312yescomplemented
86C2×C3×C3⋊C8C22×C312yescomplemented
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3D812yescomplemented
94C2×C32×D3—24no—
95C2×C32×D3—42no—
96C2×C32×D3—42no—
97C3×D32—48no—
98C33×C4C2×D412yes—
99C3×C3⋊Dic3—21no—
100C3×C3⋊Dic3—41no—
101C22×C3×D4—91no—
102C2×D3×D4—32no—
103C2×C3×D8—32no—
104C2×C3×D8—32no—
105C2×C3×D8—62no—
106C3×#121—91no—
107#938—32no—
108#938—61no—
109C2×#213C3×D312yescomplemented
110C3×#102—92no—
111C3×C22⋊C8—91no—
112#832—32no—
113#821—32no—
114(C2×D3)⋊C8—32no—
115C23×C32—21no—
116C23×C32—22no—
117C23×C32—22no—
118C23×C32—42no—
119C23×C32—42no—
120C23×C32—62no—
121C22×C3⋊D3C3×D411yescharacteristic
122C22×C3×D3—22no—
123C22×C3×D3—22no—
124C22×C3×D3—32no—
125C22×C3×D3—42no—
126C22×C3×D3—61no—
127C22×C3×D3—68no—
128C22×C3×D3—124no—
129C22×C3×D3—124no—
130C2×D32—24no—
131C2×D32—42no—
132C2×D32—42no—
133C32×D4—22no—
134C32×D4—24no—
135C32×D4—22no—
136C32×D4—22no—
137C32×D4—34no—
138C32×D4—42no—
139C32×D4—42no—
140C32×D4—44no—
141C32×D4—62no—
142C2×C32×C4—21no—
143C2×C32×C4—22no—
144C2×C32×C4—41no—
145C2×C32×C4C22×D312yes—
146C2×C32×C4C23×C311yescharacteristic
147#638—42no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3C3×D411yescharacteristic
150C4×C3⋊D3—22no—
151C3×#67—68no—
152C3×#67—124no—
153C3×#67—124no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C3×D12—22no—
159C3×D12—22no—
160C3×D12—24no—
161C3×D12—42no—
162C3×D12#6714yescomplemented
163C3×D12C3×D414yescomplemented
164C3×C4×D3—62no—
165C3×C4×D3—62no—
166C3×C4×D3—122no—
167#625—24no—
168C32×C8—62no—
169C3×C3⋊C8—24no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—22no—
172#447—91noSylow
173C2×C33—81no—
174C2×C33#12011yescharacteristic
175C2×C33C2×D811yescharacteristic
176C2×C33AGL(1,ℤ/8)11yescharacteristic
177C3×C3⋊D3—22no—
178C32×D3—44no—
179C24×C3—181no—
180C23×D3—62no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—32no—
184C2×C3×D4—32no—
185C2×C3×D4—41no—
186C2×C3×D4—62no—
187C2×C3×D4—181no—
188C2×C3×D4—184no—
189C2×C3×D4—181no—
190C2×C3×D4D3211yescharacteristic, complemented
191C2×C3×D4C2×C3×D312yescomplemented
192C22×C3×C4—91no—
193C2×#67—62no—
194C2×#67—62no—
195C2×#67—62no—
196D3×D4—64no—
197D3×D4—64no—
198C2×D12C2×C3×D312yes—
199C2×C4×D3—32no—
200C2×C4×D3—61no—
201C3×D8—64no—
202C3×D8—64no—
203C3×D8—68no—
204C2×C3×C8—32no—
205C2×C3×C8—32no—
206C2×C3×C8—62no—
207C3×C4⋊C4—91no—
208C3×C22⋊C4—181no—
209#217—62no—
210#217—62no—
211#213—24no—
212C4⋊Dic3—32no—
213C4⋊Dic3—61no—
214C2×C3⋊C8C2×C3×D312yescomplemented
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—41no—
218C22×C32—44no—
219C22×C32—42no—
220C22×C32—44no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—64no—
230C22×C32—82no—
231C22×C32—82no—
232C22×C32—82no—
233C22×C32—82no—
234C22×C32—122no—
235C22×C32—122no—
236C22×C32D3×D412yes—
237C22×C32C2×C3×D411yescharacteristic
238C2×C3⋊D3—21no—
239C2×C3⋊D3—21no—
240C2×C3⋊D3C3×D812yescomplemented
241C2×C3×D3—24no—
242C2×C3×D3—24no—
243C2×C3×D3—34no—
244C2×C3×D3—44no—
245C2×C3×D3—42no—
246C2×C3×D3—42no—
247C2×C3×D3—42no—
248C2×C3×D3—42no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—82no—
253C2×C3×D3—82no—
254C2×C3×D3—121no—
255C2×C3×D3—121no—
256C2×C3×D3—1216no—
257D32—48no—
258C32×C4—22no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—42no—
262C32×C4D3×D412yes—
263C32×C4C2×#6712yes—
264C32×C4C2×C3×D412yes—
265C3⋊Dic3—21no—
266C3⋊Dic3—41no—
267C3×Dic3—62no—
268C3×Dic3—121no—
269C3×Dic3—122no—
270C3×Dic3—122no—
271C22×D4—91no—
272C2×D8—32no—
273#121—91no—
274#102—92no—
275C22⋊C8—91no—
276C33#44711yescharacteristic, complemented, Sylow
277C23×C3—22no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—62no—
283C23×C3—62no—
284C23×C3—91no—
285C23×C3—122no—
286C23×C3—184no—
287C23×C3—362no—
288C23×C3—362no—
289C22×D3—22no—
290C22×D3—32no—
291C22×D3—61no—
292C22×D3—68no—
293C22×D3—124no—
294C22×D3—124no—
295C3×D4—24no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—34no—
300C3×D4—34no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—44no—
304C3×D4—64no—
305C3×D4—62no—
306C3×D4—62no—
307C3×D4—122no—
308C3×D4—184no—
309C3×D4—362no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—91no—
314C2×C3×C4—182no—
315C2×C3×C4—181no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C22×C3×D312yes—
318#67—68no—
319#67—124no—
320#67—124no—
321C2×Dic3—32no—
322C2×Dic3—61no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325D12—22no—
326D12C3×#6714yescomplemented
327C4×D3—62no—
328C4×D3—62no—
329C4×D3—122no—
330C3×C8—64no—
331C3×C8—62no—
332C3×C8—62no—
333C3⋊C8—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—82no—
344C2×C32—84no—
345C2×C32—84no—
346C2×C32—82no—
347C2×C32—81no—
348C2×C32—124no—
349C2×C32D3×D812yes—
350C2×C32#91112yes—
351C2×C32#93712yes—
352C2×C32C3×#12011yescharacteristic
353C2×C32C2×C3×D811yescharacteristic
354C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
355C3⋊D3—22no—
356C3×D3—44no—
357C3×D3—44no—
358C3×D3—64no—
359C3×D3—84no—
360C3×D3—122no—
361C24—181no—
362C2×D4—32no—
363C2×D4—181no—
364C2×D4—184no—
365C2×D4—181no—
366C2×D4C3×D3211yescharacteristic, complemented
367C22×C4—91no—
368D8—64no—
369C2×C8—32no—
370C4⋊C4—91no—
371C22⋊C4—181no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—42no—
375C22×C3—44no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—44no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—64no—
387C22×C3—64no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—92no—
393C22×C3—124no—
394C22×C3—122no—
395C22×C3—122no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—242no—
401C22×C3—242no—
402C22×C3—368no—
403C22×C3#381111yescharacteristic
404C22×C3C3×D3×D412yes—
405C2×D3—24no—
406C2×D3—34no—
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—1216no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—181no—
420C3×C4—361no—
421C3×C4D3×#6712yes—
422C3×C4C3×D3×D412yes—
423C3×C4C2×C3×#6712yes—
424Dic3—62no—
425Dic3—121no—
426Dic3—122no—
427Dic3—122no—
428C32—21no—
429C32—22no—
430C32—41no—
431C32#515612yescomplemented
432C32C3×#44711yescharacteristic, complemented
433C23—21no—
434C23—62no—
435C23—91no—
436C23—184no—
437C23—362no—
438C23—362no—
439D4—22no—
440D4—34no—
441D4—62no—
442D4—184no—
443D4—362no—
444C2×C4—91no—
445C2×C4—182no—
446C2×C4—181no—
447C2×C4C2×C3×D3211yescharacteristic
448C8—62no—
449C2×C3—21no—
450C2×C3—22no—
451C2×C3—21no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—82no—
459C2×C3—84no—
460C2×C3—84no—
461C2×C3—81no—
462C2×C3—82no—
463C2×C3—124no—
464C2×C3—124no—
465C2×C3—182no—
466C2×C3—244no—
467C2×C3#6392611yescharacteristic
468C2×C3#6392711yescharacteristic
469C2×C3#6398311yescharacteristic
470C2×C3C3×D3×D812yes—
471C2×C3C3×#91112yes—
472C2×C3C3×#93712yes—
473D3—44no—
474D3—64no—
475D3—122no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—64no—
480C22—92no—
481C22—122no—
482C22—122no—
483C22—181no—
484C22—181no—
485C22—368no—
486C22C3×#381111yescharacteristic
487C4—181no—
488C4—361no—
489C4C3×D3×#6712yes—
490C3—21no—
491C3—22no—
492C3—41no—
493C3#1059151611yescharacteristic, direct product factor
494C3C3×#515612yescomplemented
495C2—81no—
496C2—124no—
497C2—182no—
498C2C3×#6392611yescharacteristic
499C2C3×#6392711yescharacteristic
500C2C3×#6398311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#447
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3802C2semidirect
2. C3×#64147C2semidirect
3. C2×C32×#213C2semidirect
4. C3×#63563C2semidirect
5. C3×#63557C2semidirect
7. #10591516C3direct (N × H)
9. C2×C33×D4C22semidirect
16. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C32×C3⋊C8C22semidirect
20. C2×#3802C2×C3semidirect
22. #64147C2×C3semidirect
26. C2×C3×#213D3semidirect
27. C2×C3×#213C2×C3semidirect
31. #63563C2×C3semidirect
32. #63557C2×C3semidirect
46. C32×D12D4semidirect
57. C2×C32×D4C2×D3semidirect
58. C2×C32×D4C22×C3semidirect
76. C4⋊(C3⋊Dic3)C22×C3semidirect
85. C2×C3×C3⋊C8C2×D3semidirect
86. C2×C3×C3⋊C8C22×C3semidirect
93. C2×C3×C3⋊D3D8semidirect
109. C2×#213C3×D3semidirect
162. C3×D12#67semidirect
163. C3×D12C3×D4semidirect
190. C2×C3×D4D32semidirect
191. C2×C3×D4C2×C3×D3semidirect
214. C2×C3⋊C8C2×C3×D3semidirect
240. C2×C3⋊D3C3×D8semidirect
276. C33#447semidirect
326. D12C3×#67semidirect
366. C2×D4C3×D32semidirect
431. C32#5156semidirect
432. C32C3×#447semidirect
493. C3#10591516direct (N × H)
494. C3C3×#5156semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C3×#641472
C2×C32×#2132
C3×#635632
C3×#635572
C3×#51563
#105915163