Finite Group Databasebeta

← C3×#10593512 #13166876

Subgroups — C3×#10593512 #13166876

Order 1728.

2060 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 2060 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C23×C3×D32C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C214yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic, complemented
5#10593512C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C23×C32×D3—21no—
8C22×C3×D32—21no—
9C22×C3×D32—22no—
10C22×C3×D32—28no—
11C22×C3×D32C414yescomplemented
12C22×C3×D32C2212yes—
13C22×C3×#157C2212yescomplemented
14C2×C3×#624C2212yescomplemented
15C3×C22⋊#157—22no—
16C3×D32⋊C4—28no—
17C32×#217—21no—
18C3×#3722—21no—
19C24×C3×D3—61no—
20C23×D32C2×C311yescharacteristic
21C2×C22⋊#157C2×C311yescharacteristic, complemented
22C2×D32⋊C4C2×C314yescomplemented
23C3×D3×C22⋊C4—61no—
24#63974C2×C311yescharacteristic, complemented
25C23×C33D411yescharacteristic
26C22×C3×C3⋊D3—24no—
27C22×C3×C3⋊D3C2×C412yes—
28C22×C3×C3⋊D3D411yescharacteristic
29C22×C3×C3⋊D3D412yes—
30C22×C3×C3⋊D3C2311yescharacteristic
31C22×C32×D3—28no—
32C22×C32×D3—24no—
33C22×C32×D3—22no—
34C2×C3×D32—28no—
35C2×C3×D32—216no—
36C2×C3×D32—24no—
37C2×C3×D32—24no—
38C2×C3×D32—22no—
39C2×C3×D32—216no—
40C2×C3×D32C2×C414yes—
41C2×C3×D32D418yes—
42C2×C3×#157—24no—
43C2×C3×#157—22no—
44C2×C32×Dic3—22no—
45C3×#624—24no—
46C3×#379—91nomaximal
47C24×C32—61no—
48C23×C3⋊D3C22×C311yescharacteristic
49C23×C3×D3—21no—
50C23×C3×D3—21no—
51C23×C3×D3—41no—
52C23×C3×D3—64no—
53C23×C3×D3—62no—
54C23×C3×D3—616no—
55C23×C3×D3—62no—
56C23×C3×D3—64no—
57C23×C3×D3—61no—
58C23×C3×D3—61no—
59C22×D32—21no—
60C22×D32—22no—
61C22×D32—28no—
62C22×D32C3×C414yescomplemented
63C22×D32C22×C312yes—
64C22×#157C22×C312yescomplemented
65C2×C3×C4×D3—62no—
66C2×#624C22×C312yescomplemented
67C22⋊#157—22no—
68D32⋊C4—28no—
69C32×C22⋊C4—61no—
70C3×#217—21no—
71C3×#217—21no—
72C3×#217—41no—
73C3×(C2×D3)⋊C4—62no—
74#3722—21no—
75C22×C33—22no—
76C22×C33C22⋊C412yescomplemented
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—24no—
80C2×C3×C3⋊D3—24no—
81C2×C3×C3⋊D3C22⋊C412yescomplemented
82C2×C3×C3⋊D3C22×C411yescharacteristic
83C2×C3×C3⋊D3C2×D414yes—
84C2×C3×C3⋊D3C2×D411yescharacteristic
85C2×C32×D3—216no—
86C2×C32×D3—24no—
87C2×C32×D3—28no—
88C3×D32—216no—
89C3×D32—24no—
90C3×D32—28no—
91C3×D32C22⋊C418yescomplemented
92C3×#157—42no—
93C32×Dic3—42no—
94C25×C3—91no—
95C24×D3—61no—
96C2×C3×C22⋊C4—91no—
97C2×C3×C22⋊C4—91no—
98C2×C3×C22⋊C4—94no—
99D3×C22⋊C4—61no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32—41no—
104C23×C32—68no—
105C23×C32—62no—
106C23×C32—64no—
107C23×C32C3×D411yescharacteristic
108C22×C3⋊D3—24no—
109C22×C3⋊D3C2×C3×C412yes—
110C22×C3⋊D3C3×D411yescharacteristic
111C22×C3⋊D3C3×D412yes—
112C22×C3⋊D3C23×C311yescharacteristic
113C22×C3×D3—28no—
114C22×C3×D3—28no—
115C22×C3×D3—22no—
116C22×C3×D3—24no—
117C22×C3×D3—24no—
118C22×C3×D3—22no—
119C22×C3×D3—44no—
120C22×C3×D3—42no—
121C22×C3×D3—48no—
122C22×C3×D3—64no—
123C22×C3×D3—68no—
124C22×C3×D3—62no—
125C22×C3×D3—61no—
126C22×C3×D3—62no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—62no—
131C22×C3×D3—62no—
132C22×C3×D3—632no—
133C22×C3×D3—68no—
134C22×C3×D3—64no—
135C22×C3×D3—68no—
136C22×C3×D3—632no—
137C22×C3×D3—616no—
138C22×C3×D3—68no—
139C22×C3×D3—124no—
140C2×D32—28no—
141C2×D32—216no—
142C2×D32—24no—
143C2×D32—24no—
144C2×D32—22no—
145C2×D32—216no—
146C2×D32C2×C3×C414yes—
147C2×D32C3×D418yes—
148C2×#157—24no—
149C2×#157—22no—
150C2×C32×C4—62no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—42no—
154C3×C4×D3—124no—
155#624—24no—
156#379—91noSylow
157C2×C33—22no—
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#12012yes—
160C3×C3⋊D3—22no—
161C3×C3⋊D3C2×C22⋊C412yes—
162C3×C3⋊D3#12012yes—
163C32×D3—28no—
164C24×C3—61no—
165C24×C3—61no—
166C24×C3—92no—
167C24×C3—94no—
168C24×C3—91no—
169C24×C3—121no—
170C24×C3—188no—
171C24×C3—181no—
172C24×C3—182no—
173C23×D3—21no—
174C23×D3—64no—
175C23×D3—62no—
176C23×D3—61no—
177C23×D3—61no—
178C23×D3—616no—
179C23×D3—62no—
180C23×D3—64no—
181C22×C3×C4—92no—
182C22×C3×C4—92no—
183C2×C4×D3—62no—
184C3×C22⋊C4—61no—
185C3×C22⋊C4—61no—
186C3×C22⋊C4—121no—
187C3×C22⋊C4—181no—
188C3×C22⋊C4—188no—
189C3×C22⋊C4—182no—
190#217—21no—
191(C2×D3)⋊C4—62no—
192C22×C32—24no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—42no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32—44no—
204C22×C32—616no—
205C22×C32—64no—
206C22×C32—68no—
207C22×C32—82no—
208C22×C32C3×C22⋊C412yescomplemented
209C22×C32C2×C3×D411yescharacteristic
210C2×C3⋊D3—22no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3—24no—
213C2×C3⋊D3C3×C22⋊C412yescomplemented
214C2×C3⋊D3C22×C3×C411yescharacteristic
215C2×C3⋊D3C2×C3×D414yes—
216C2×C3⋊D3C2×C3×D411yescharacteristic
217C2×C3×D3—216no—
218C2×C3×D3—216no—
219C2×C3×D3—24no—
220C2×C3×D3—28no—
221C2×C3×D3—24no—
222C2×C3×D3—28no—
223C2×C3×D3—44no—
224C2×C3×D3—48no—
225C2×C3×D3—416no—
226C2×C3×D3—68no—
227C2×C3×D3—62no—
228C2×C3×D3—61no—
229C2×C3×D3—61no—
230C2×C3×D3—64no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—64no—
234C2×C3×D3—62no—
235C2×C3×D3—68no—
236C2×C3×D3—616no—
237C2×C3×D3—632no—
238C2×C3×D3—68no—
239C2×C3×D3—68no—
240C2×C3×D3—64no—
241C2×C3×D3—124no—
242C2×C3×D3—124no—
243C2×C3×D3—122no—
244D32—216no—
245D32—24no—
246D32—28no—
247D32C3×C22⋊C418yescomplemented
248#157—42no—
249C32×C4—122no—
250C3×Dic3—42no—
251C3×Dic3—42no—
252C3×Dic3—82no—
253C25—91no—
254C2×C22⋊C4—91no—
255C2×C22⋊C4—91no—
256C2×C22⋊C4—94no—
257C33#37911yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—68no—
263C23×C3—62no—
264C23×C3—64no—
265C23×C3—68no—
266C23×C3—62no—
267C23×C3—64no—
268C23×C3—98no—
269C23×C3—94no—
270C23×C3—91no—
271C23×C3—91no—
272C23×C3—92no—
273C23×C3—92no—
274C23×C3—128no—
275C23×C3—122no—
276C23×C3—124no—
277C23×C3—184no—
278C23×C3—1816no—
279C23×C3—182no—
280C23×C3—184no—
281C23×C3—184no—
282C23×C3—1816no—
283C23×C3—188no—
284C23×C3GO+(4,𝔽2)11yescharacteristic
285C22×D3—28no—
286C22×D3—22no—
287C22×D3—24no—
288C22×D3—64no—
289C22×D3—68no—
290C22×D3—62no—
291C22×D3—61no—
292C22×D3—62no—
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—62no—
296C22×D3—61no—
297C22×D3—62no—
298C22×D3—632no—
299C22×D3—68no—
300C22×D3—68no—
301C22×D3—64no—
302C22×D3—632no—
303C22×D3—616no—
304C22×D3—68no—
305C22×D3—124no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—122no—
309C2×C3×C4—184no—
310C2×C3×C4—182no—
311C2×C3×C4—184no—
312C2×Dic3—22no—
313C4×D3—124no—
314C2×C32—24no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—42no—
323C2×C32—42no—
324C2×C32—42no—
325C2×C32—44no—
326C2×C32—68no—
327C2×C32—82no—
328C2×C32C2×C3×C22⋊C411yescharacteristic
329C2×C32C3×#12012yes—
330C3⋊D3—22no—
331C3⋊D3C2×C3×C22⋊C412yes—
332C3⋊D3C3×#12012yes—
333C3×D3—28no—
334C3×D3—28no—
335C3×D3—48no—
336C3×D3—62no—
337C3×D3—62no—
338C3×D3—62no—
339C3×D3—62no—
340C3×D3—64no—
341C3×D3—122no—
342C24—61no—
343C24—92no—
344C24—94no—
345C24—91no—
346C24—188no—
347C24—181no—
348C24—182no—
349C22×C4—92no—
350C22×C4—92no—
351C22⋊C4—61no—
352C22⋊C4—181no—
353C22⋊C4—188no—
354C22⋊C4—182no—
355C22×C3—24no—
356C22×C3—22no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—42no—
362C22×C3—44no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—616no—
368C22×C3—68no—
369C22×C3—64no—
370C22×C3—616no—
371C22×C3—64no—
372C22×C3—68no—
373C22×C3—82no—
374C22×C3—98no—
375C22×C3—91no—
376C22×C3—91no—
377C22×C3—94no—
378C22×C3—92no—
379C22×C3—1216no—
380C22×C3—128no—
381C22×C3—124no—
382C22×C3—184no—
383C22×C3—184no—
384C22×C3—188no—
385C22×C3—184no—
386C22×C3—1816no—
387C22×C3—182no—
388C22×C3D32⋊C412yescomplemented
389C22×C3C2×GO+(4,𝔽2)11yescharacteristic
390C2×D3—216no—
391C2×D3—28no—
392C2×D3—24no—
393C2×D3—68no—
394C2×D3—64no—
395C2×D3—62no—
396C2×D3—61no—
397C2×D3—64no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—62no—
402C2×D3—68no—
403C2×D3—616no—
404C2×D3—632no—
405C2×D3—68no—
406C2×D3—68no—
407C2×D3—64no—
408C2×D3—124no—
409C2×D3—124no—
410C2×D3—122no—
411C3×C4—122no—
412C3×C4—122no—
413C3×C4—242no—
414C3×C4—362no—
415Dic3—42no—
416C32—21no—
417C32—21no—
418C32—41no—
419C32—41no—
420C32C3×#37911yescharacteristic, complemented
421C23—68no—
422C23—62no—
423C23—64no—
424C23—98no—
425C23—94no—
426C23—91no—
427C23—91no—
428C23—92no—
429C23—92no—
430C23—184no—
431C23—1816no—
432C23—184no—
433C23—182no—
434C23—184no—
435C23—1816no—
436C23—188no—
437C23C3×GO+(4,𝔽2)11yescharacteristic
438C2×C4—62no—
439C2×C4—184no—
440C2×C4—184no—
441C2×C4—182no—
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—42no—
449C2×C3—44no—
450C2×C3—42no—
451C2×C3—42no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—68no—
455C2×C3—68no—
456C2×C3—82no—
457C2×C3—92no—
458C2×C3—92no—
459C2×C3—128no—
460C2×C3—182no—
461C2×C3C2×D32⋊C411yescharacteristic
462C2×C3#6424112yes—
463D3—28no—
464D3—62no—
465D3—62no—
466D3—62no—
467D3—62no—
468D3—64no—
469D3—122no—
470C22—22no—
471C22—616no—
472C22—68no—
473C22—64no—
474C22—98no—
475C22—94no—
476C22—91no—
477C22—91no—
478C22—92no—
479C22—184no—
480C22—184no—
481C22—188no—
482C22—184no—
483C22—1816no—
484C22—182no—
485C22C3×D32⋊C412yescomplemented
486C22C2×C3×GO+(4,𝔽2)11yescharacteristic
487C4—122no—
488C4—362no—
489C3—21no—
490C3—21no—
491C3—41no—
492C3—41no—
493C3#1059351211yescharacteristic, direct product factor
494C2—22no—
495C2—68no—
496C2—92no—
497C2—92no—
498C2—182no—
499C2C2×C3×D32⋊C411yescharacteristic
500C2C3×#6424112yes—
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×D32C2
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#379
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊#157C2semidirect
3. C2×C3×D32⋊C4C2semidirect
4. C3×#63974C2semidirect
5. #10593512C3direct (N × H)
11. C22×C3×D32C4semidirect
13. C22×C3×#157C22semidirect
14. C2×C3×#624C22semidirect
21. C2×C22⋊#157C2×C3semidirect
22. C2×D32⋊C4C2×C3semidirect
24. #63974C2×C3semidirect
62. C22×D32C3×C4semidirect
64. C22×#157C22×C3semidirect
66. C2×#624C22×C3semidirect
76. C22×C33C22⋊C4semidirect
81. C2×C3×C3⋊D3C22⋊C4semidirect
91. C3×D32C22⋊C4semidirect
208. C22×C32C3×C22⋊C4semidirect
213. C2×C3⋊D3C3×C22⋊C4semidirect
247. D32C3×C22⋊C4semidirect
257. C33#379semidirect
388. C22×C3D32⋊C4semidirect
420. C32C3×#379semidirect
485. C22C3×D32⋊C4semidirect
493. C3#10593512direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C2×C3×C22⋊#1572
C2×C3×D32⋊C42
C3×#639742
#105935123
C3×#3799