Finite Group Databasebeta

← C3×#10596505 #13181024

Subgroups — C3×#10596505 #13181024

Order 1728.

914 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C3×#64306C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4C3×#64239C214yesmaximal, complemented
5C3×#64228C212yesmaximal, complemented
6C3×#64227C212yesmaximal, complemented
7C3×#64226C214yesmaximal, complemented
8#10596505C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C32×#235—21no—
11C2×C32×Dic6—21no—
12C3×#3811—21no—
13C2×C3×#627C2212yescomplemented
14C2×C3×#624—21no—
15C3×#3805C2212yes—
16C3×#3802C2211yescharacteristic
17C3×C4×D32—21no—
18C3×#3798C2212yes—
19C3×#3797—21no—
20C3×#3797—21no—
21C3×#3797C2212yes—
22C3×#3797C2211yescharacteristic
23C3×#3797C2212yescomplemented
24C3×#3796—21no—
25C3×#3792C2212yescomplemented
26C3×#3788C2212yescomplemented
27C3×#3787C2212yes—
28C3×#3776C2218yescomplemented
29C3×#3775C2218yescomplemented
30C3×D3×#41—61no—
31C2×C3×D3×Q8—61no—
32#64306C2×C311yescharacteristic
33C2×#3797C2×C311yescharacteristic, complemented
34#64283C2×C311yescharacteristic, complemented
35#64239C2×C314yescomplemented
36#64228C2×C312yescomplemented
37#64227C2×C312yescomplemented
38#64226C2×C314yescomplemented
39C22×C3×C3⋊D3D411yescharacteristic
40C2×C3×D32—21no—
41C2×C33×C4D411yescharacteristic
42C2×C3×C3⋊Dic3C2311yescharacteristic
43C3×C4×C3⋊D3D411yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×C4×C3⋊D3C2311yescharacteristic
46C3×C4×C3⋊D3C2311yescharacteristic
47C32×#67—22no—
48C2×C32×Dic3—22no—
49C32×D12—21no—
50C32×C4×D3—22no—
51C32×Dic6—21no—
52C32×Dic6—22no—
53C32×Dic6—22no—
54C3×#627—22no—
55C3×#627C2314yes—
56C3×#627C2312yes—
57C3×#626—22no—
58C3×#625C2312yes—
59C3×#624—22no—
60C3×#624—22no—
61C3×#624—21no—
62C3×D3×Dic3—22no—
63C3×#622C2314yes—
64C2×C3×#137—91nomaximal
65C32×#41—61no—
66C2×C32×Q8—61no—
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C3×#239—61no—
69C3×D3×Q8—62no—
70C3×D3×Q8—62no—
71C3×D3×Q8—61no—
72C3×D3×Q8—62no—
73C3×D3×Q8—62no—
74C3×#237—61no—
75C3×#237—62no—
76C3×D3×D4—61no—
77C3×D3×D4—62no—
78C3×#235—21no—
79C3×#235—21no—
80C3×#235—41no—
81C3×#235—62no—
82C2×C3×C4×D3—62no—
83C2×C3×C4×D3—62no—
84C2×C3×C4×D3—61no—
85C2×C3×C4×D3—61no—
86C2×C3×Dic6—21no—
87C2×C3×Dic6—21no—
88C2×C3×Dic6—41no—
89C2×C3×Dic6—62no—
90#3811—21no—
91C2×#627C22×C312yescomplemented
92C2×#624—21no—
93#3805C22×C312yes—
94#3802C22×C311yescharacteristic
95C4×D32—21no—
96#3798C22×C312yes—
97#3797—21no—
98#3797—21no—
99#3797C22×C312yes—
100#3797C22×C311yescharacteristic
101#3797C22×C312yescomplemented
102#3796—21no—
103#3792C22×C312yescomplemented
104#3788C22×C312yescomplemented
105#3787C22×C312yes—
106#3776C22×C318yescomplemented
107#3775C22×C318yescomplemented
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3C2×D411yescharacteristic
111C2×C3×C3⋊D3C2×D411yescharacteristic
112C2×C32×D3—22no—
113C3×D32—42no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3C2×D411yescharacteristic
117C3×C3⋊Dic3C2411yescharacteristic
118C32×Dic3—22no—
119C32×Dic3—24no—
120C2×C3×#41—91no—
121C22×C3×Q8—91no—
122D3×#41—61no—
123C2×D3×Q8—61no—
124C3×#137—94no—
125C3×#137—94no—
126C2×C3×Q16—92no—
127C2×C3×SD16—92no—
128C2×C3×M16—91no—
129C22×C3⋊D3C3×D411yescharacteristic
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—62no—
133C2×D32—21no—
134C32×Q8—62no—
135C32×Q8—62no—
136C32×Q8—61no—
137C32×D4—61no—
138C32×D4—62no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4—62no—
144C2×C32×C4—62no—
145C2×C32×C4C3×D411yescharacteristic
146C2×C3⋊Dic3C23×C311yescharacteristic
147C4×C3⋊D3C3×D411yescharacteristic
148C4×C3⋊D3C3×D411yescharacteristic
149C4×C3⋊D3C23×C311yescharacteristic
150C4×C3⋊D3C23×C311yescharacteristic
151C3×#67—22no—
152C3×#67—22no—
153C3×#67—42no—
154C3×#67—62no—
155C3×#67—62no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—62no—
162C3×D12—21no—
163C3×D12—21no—
164C3×D12—41no—
165C3×D12—62no—
166C3×C4×D3—22no—
167C3×C4×D3—22no—
168C3×C4×D3—42no—
169C3×C4×D3—61no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—62no—
178C3×C4×D3—64no—
179C3×C4×D3—64no—
180C3×C4×D3—62no—
181C3×Dic6—22no—
182C3×Dic6—22no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—22no—
186C3×Dic6—22no—
187C3×Dic6—42no—
188C3×Dic6—42no—
189C3×Dic6—41no—
190C3×Dic6—64no—
191C3×Dic6—62no—
192C3×Dic6—64no—
193#627—22no—
194#627C23×C314yes—
195#627C23×C312yes—
196#626—22no—
197#625C23×C312yes—
198#624—22no—
199#624—22no—
200#624—21no—
201D3×Dic3—22no—
202#622C23×C314yes—
203C2×#137—91noSylow
204C2×C33—21no—
205C2×C33C22×D411yescharacteristic
206C3×C3⋊D3—21no—
207C3×C3⋊D3#13712yescomplemented
208C32×D3—42no—
209C3×#41—61no—
210C3×#41—61no—
211C3×#41—92no—
212C3×#41—92no—
213C3×#41—121no—
214C3×#41—181no—
215C2×C3×Q8—61no—
216C2×C3×Q8—61no—
217C2×C3×Q8—92no—
218C2×C3×Q8—92no—
219C2×C3×Q8—91no—
220C2×C3×Q8—92no—
221C2×C3×Q8—121no—
222C2×C3×Q8—181no—
223C2×C3×Q8—181no—
224C2×C3×D4—91no—
225C2×C3×D4—181no—
226C22×C3×C4—91no—
227C22×C3×C4—181no—
228C22×C3×C4—181no—
229#239—61no—
230D3×Q8—62no—
231D3×Q8—61no—
232D3×Q8—62no—
233D3×Q8—62no—
234D3×Q8—62no—
235#237—62no—
236#237—61no—
237D3×D4—61no—
238D3×D4—62no—
239#235—21no—
240#235—62no—
241C2×C4×D3—61no—
242C2×C4×D3—61no—
243C2×C4×D3—62no—
244C2×C4×D3—62no—
245C2×Dic6—21no—
246C2×Dic6—62no—
247C3×Q16—98no—
248C3×SD16—98no—
249C3×M16—92no—
250C3×M16—92no—
251C2×C3×C8—92no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32—62no—
257C22×C32C2×C3×D411yescharacteristic
258C2×C3⋊D3—22no—
259C2×C3⋊D3C2×C3×D411yescharacteristic
260C2×C3⋊D3C2×C3×D411yescharacteristic
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—42no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—64no—
269C2×C3×D3—122no—
270C2×C3×D3—124no—
271D32—42no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—64no—
281C32×C4—62no—
282C32×C4C2×C3×D411yescharacteristic
283C32×C4C2×C3×D411yescharacteristic
284C3⋊Dic3C2×C3×D411yescharacteristic
285C3⋊Dic3C24×C311yescharacteristic
286C3×Dic3—24no—
287C3×Dic3—22no—
288C3×Dic3—22no—
289C3×Dic3—24no—
290C3×Dic3—44no—
291C3×Dic3—42no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C2×#41—91no—
297C22×Q8—91no—
298#137—94no—
299#137—94no—
300C2×Q16—92no—
301C2×SD16—92no—
302C2×M16—91no—
303C33C2×#13711yescharacteristic, complemented, Sylow
304C23×C3—91no—
305C23×C3—181no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—62no—
309C3×Q8—62no—
310C3×Q8—62no—
311C3×Q8—61no—
312C3×Q8—62no—
313C3×Q8—62no—
314C3×Q8—61no—
315C3×Q8—94no—
316C3×Q8—92no—
317C3×Q8—122no—
318C3×Q8—122no—
319C3×Q8—121no—
320C3×Q8—182no—
321C3×D4—61no—
322C3×D4—62no—
323C3×D4—62no—
324C3×D4—61no—
325C3×D4—92no—
326C3×D4—121no—
327C3×D4—122no—
328C3×D4—182no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—122no—
343C2×C3×C4—122no—
344C2×C3×C4—181no—
345C2×C3×C4—182no—
346C2×C3×C4—182no—
347C2×C3×C4—182no—
348C2×C3×C4GO+(4,𝔽2)11yescharacteristic
349#67—22no—
350#67—62no—
351#67—62no—
352C2×Dic3—22no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—62no—
356D12—21no—
357D12—62no—
358C4×D3—22no—
359C4×D3—61no—
360C4×D3—61no—
361C4×D3—61no—
362C4×D3—61no—
363C4×D3—61no—
364C4×D3—61no—
365C4×D3—61no—
366C4×D3—61no—
367C4×D3—62no—
368C4×D3—64no—
369C4×D3—64no—
370C4×D3—62no—
371Dic6—22no—
372Dic6—22no—
373Dic6—21no—
374Dic6—64no—
375Dic6—62no—
376Dic6—64no—
377C3×C8—94no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—42no—
386C2×C32—81no—
387C2×C32—122no—
388C2×C32C22×C3×D411yescharacteristic
389C3⋊D3—21no—
390C3⋊D3C3×#13712yescomplemented
391C3×D3—42no—
392C3×D3—42no—
393C3×D3—62no—
394C3×D3—62no—
395C3×D3—62no—
396C3×D3—82no—
397C3×D3—121no—
398#41—61no—
399#41—92no—
400#41—92no—
401#41—181no—
402C2×Q8—61no—
403C2×Q8—92no—
404C2×Q8—92no—
405C2×Q8—91no—
406C2×Q8—92no—
407C2×Q8—181no—
408C2×Q8—181no—
409C2×D4—91no—
410C2×D4—181no—
411C22×C4—91no—
412C22×C4—181no—
413C22×C4—181no—
414Q16—98no—
415SD16—98no—
416M16—92no—
417M16—92no—
418C2×C8—92no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—91no—
426C22×C3—91no—
427C22×C3—122no—
428C22×C3—182no—
429C22×C3—362no—
430C22×C3C2×GO+(4,𝔽2)11yescharacteristic
431C2×D3—22no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—64no—
437C2×D3—122no—
438C2×D3—124no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—64no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—64no—
451C3×C4—91no—
452C3×C4—91no—
453C3×C4—122no—
454C3×C4—124no—
455C3×C4C2×GO+(4,𝔽2)11yescharacteristic
456C3×C4C2×GO+(4,𝔽2)11yescharacteristic
457Dic3—24no—
458Dic3—22no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463C32—21no—
464C32—21no—
465C32—41no—
466C32—41no—
467C32C2×C3×#13711yescharacteristic, complemented
468C23—91no—
469C23—181no—
470Q8—62no—
471Q8—62no—
472Q8—61no—
473Q8—94no—
474Q8—92no—
475Q8—182no—
476D4—62no—
477D4—61no—
478D4—92no—
479D4—182no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—181no—
488C2×C4—182no—
489C2×C4—182no—
490C2×C4—182no—
491C2×C4C3×GO+(4,𝔽2)11yescharacteristic
492C8—94no—
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—81no—
502C2×C3—92no—
503C2×C3—122no—
504C2×C3—122no—
505C2×C3—181no—
506C2×C3—242no—
507C2×C3C22×GO+(4,𝔽2)11yescharacteristic
508D3—42no—
509D3—62no—
510D3—62no—
511D3—62no—
512D3—121no—
513C22—62no—
514C22—91no—
515C22—91no—
516C22—182no—
517C22—362no—
518C22C2×C3×GO+(4,𝔽2)11yescharacteristic
519C4—62no—
520C4—64no—
521C4—91no—
522C4—91no—
523C4C2×C3×GO+(4,𝔽2)11yescharacteristic
524C4C2×C3×GO+(4,𝔽2)11yescharacteristic
525C3—21no—
526C3—21no—
527C3—41no—
528C3—41no—
529C3#1059650511yescharacteristic, direct product factor
530C2—21no—
531C2—92no—
532C2—122no—
533C2—181no—
534C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
prime radicalC3×#64306C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#137
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C3×#64283C2semidirect
4. C3×#64239C2semidirect
5. C3×#64228C2semidirect
6. C3×#64227C2semidirect
7. C3×#64226C2semidirect
8. #10596505C3direct (N × H)
13. C2×C3×#627C22semidirect
23. C3×#3797C22semidirect
25. C3×#3792C22semidirect
26. C3×#3788C22semidirect
28. C3×#3776C22semidirect
29. C3×#3775C22semidirect
33. C2×#3797C2×C3semidirect
34. #64283C2×C3semidirect
35. #64239C2×C3semidirect
36. #64228C2×C3semidirect
37. #64227C2×C3semidirect
38. #64226C2×C3semidirect
91. C2×#627C22×C3semidirect
101. #3797C22×C3semidirect
103. #3792C22×C3semidirect
104. #3788C22×C3semidirect
106. #3776C22×C3semidirect
107. #3775C22×C3semidirect
207. C3×C3⋊D3#137semidirect
303. C33C2×#137semidirect
390. C3⋊D3C3×#137semidirect
467. C32C2×C3×#137semidirect
529. C3#10596505direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×#37972
C3×#642832
C3×#642392
C3×#642282
C3×#642272
C3×#642262
#105965053
C2×C3×#1379