Finite Group Databasebeta

← C2×C3×#63741 #13166868

Subgroups — C2×C3×#63741 #13166868

Order 1728.

720 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3805C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic, complemented
3C2×C3×Dic32C211yesmaximal, characteristic, complemented
4C3×#63741C214yesmaximal, direct product factor
5C2×#63741C311yesmaximal, characteristic, direct product factor
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C32×#67—21no—
8C22×C32×Dic3—21no—
9C2×C3×#627C411yescharacteristic, complemented
10C2×C3×#625C411yescharacteristic, complemented
11C3×#3805—21no—
12C3×#3805—21no—
13C3×#3805C412yescomplemented
14C3×#3805C2212yes—
15C2×C3×D3×Dic3—21no—
16C3×#3792—21no—
17C3×#3792C2212yescomplemented
18C2×C3×#622—21no—
19C3×Dic32—21no—
20C3×Dic32C2212yescomplemented
21C2×C3×#237—61no—
22C2×#3805C2×C311yescharacteristic
23C2×#3792C2×C311yescharacteristic, complemented
24C2×C3×C4×Dic3—61no—
25C2×Dic32C2×C311yescharacteristic, complemented
26#63741C2×C314yesdirect product factor
27C23×C33D411yescharacteristic
28C22×C32×D3—21no—
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C2×C3×C3⋊Dic3D411yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C32×#67—22no—
36C32×#67—22no—
37C2×C32×Dic3—24no—
38C2×C32×Dic3—21no—
39C2×C32×Dic3—22no—
40C3×#627—21no—
41C3×#627C2×C412yescomplemented
42C3×#625—21no—
43C3×#625C2×C412yescomplemented
44C3×D3×Dic3—22no—
45C3×D3×Dic3—22no—
46C3×#622—22no—
47C2×C3×#104—91nomaximal
48C2×C32×D4—61no—
49C22×C32×C4—61no—
50C22×C3⋊Dic3C22×C311yescharacteristic
51C2×C3×#67—21no—
52C2×C3×#67—21no—
53C2×C3×#67—41no—
54C2×C3×#67—61no—
55C22×C3×Dic3—21no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3—41no—
58C22×C3×Dic3—61no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C3×#237—62no—
62C3×#237—62no—
63C3×#237—62no—
64C3×#237—62no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—61no—
67C2×#627C3×C411yescharacteristic, complemented
68C2×#625C3×C411yescharacteristic, complemented
69#3805—21no—
70#3805—21no—
71#3805C3×C412yescomplemented
72#3805C22×C312yes—
73C2×D3×Dic3—21no—
74#3792—21no—
75#3792C22×C312yescomplemented
76C2×#622—21no—
77C3×C4×Dic3—62no—
78C3×C4×Dic3—62no—
79Dic32—21no—
80Dic32C22×C312yescomplemented
81C22×C33—22no—
82C22×C33C22⋊C411yescharacteristic
83C22×C33C22⋊C411yescharacteristic
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—22no—
86C2×C32×D3—42no—
87C3×C3⋊Dic3C22⋊C411yescharacteristic
88C3×C3⋊Dic3C22⋊C411yescharacteristic
89C3×C3⋊Dic3C22×C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C32×Dic3—24no—
92C32×Dic3—22no—
93C2×C3×#41—91no—
94C2×#237—61no—
95C2×C3×M16—91no—
96C2×C3×C42—91no—
97C2×C4×Dic3—61no—
98C3×#104—94no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—41no—
102C23×C32—41no—
103C23×C32—61no—
104C23×C32C3×D411yescharacteristic
105C22×C3×D3—21no—
106C22×C3×D3—21no—
107C22×C3×D3—41no—
108C32×D4—62no—
109C32×D4—62no—
110C2×C32×C4—62no—
111C2×C32×C4—61no—
112C2×C32×C4—64no—
113C2×C3⋊Dic3C2×C3×C411yescharacteristic
114C2×C3⋊Dic3C2×C3×C411yescharacteristic
115C2×C3⋊Dic3C3×D411yescharacteristic
116C2×C3⋊Dic3C3×D411yescharacteristic
117C2×C3⋊Dic3C3×D411yescharacteristic
118C2×C3⋊Dic3C23×C311yescharacteristic
119C3×#67—22no—
120C3×#67—22no—
121C3×#67—22no—
122C3×#67—22no—
123C3×#67—42no—
124C3×#67—42no—
125C3×#67—62no—
126C3×#67—62no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—44no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×Dic6—62no—
153C3×Dic6—62no—
154#627—21no—
155#627C2×C3×C412yescomplemented
156#625—21no—
157#625C2×C3×C412yescomplemented
158D3×Dic3—22no—
159D3×Dic3—22no—
160#622—22no—
161C2×#104—91noSylow
162C2×C33—21no—
163C2×C33—21no—
164C2×C33#10412yescomplemented
165C2×C33C2×C22⋊C411yescharacteristic
166C32×D3—42no—
167C3×#41—92no—
168C3×#41—92no—
169C3×#41—181no—
170C3×#41—181no—
171C2×C3×Q8—91no—
172C2×C3×D4—61no—
173C2×C3×D4—61no—
174C2×C3×D4—91no—
175C2×C3×D4—121no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4—91no—
179C22×C3×C4—121no—
180C22×C3×C4—181no—
181C2×#67—21no—
182C2×#67—61no—
183C22×Dic3—21no—
184C22×Dic3—61no—
185C22×Dic3—61no—
186C22×Dic3—61no—
187#237—62no—
188#237—62no—
189#237—62no—
190#237—62no—
191C2×C4×D3—61no—
192C2×Dic6—61no—
193C3×M16—92no—
194C3×M16—181no—
195C2×C3×C8—181no—
196C3×C42—92no—
197C3×C42—181no—
198C4×Dic3—62no—
199C4×Dic3—62no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—62no—
219C22×C32—82no—
220C22×C32—122no—
221C22×C32C3×C22⋊C411yescharacteristic
222C22×C32C3×C22⋊C411yescharacteristic
223C22×C32C2×C3×D411yescharacteristic
224C2×C3×D3—22no—
225C2×C3×D3—22no—
226C2×C3×D3—42no—
227C2×C3×D3—42no—
228C2×C3×D3—42no—
229C2×C3×D3—82no—
230C32×C4—62no—
231C32×C4—64no—
232C3⋊Dic3C3×C22⋊C411yescharacteristic
233C3⋊Dic3C3×C22⋊C411yescharacteristic
234C3⋊Dic3C22×C3×C411yescharacteristic
235C3⋊Dic3C2×C3×D411yescharacteristic
236C3×Dic3—22no—
237C3×Dic3—24no—
238C3×Dic3—24no—
239C3×Dic3—22no—
240C3×Dic3—44no—
241C3×Dic3—42no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C3×Dic3—61no—
249C3×Dic3—61no—
250C2×#41—91no—
251C2×M16—91no—
252C2×C42—91no—
253#104—94no—
254C33C2×#10411yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—121no—
262C23×C3GO+(4,𝔽2)11yescharacteristic
263C22×D3—21no—
264C3×Q8—92no—
265C3×Q8—181no—
266C3×D4—62no—
267C3×D4—62no—
268C3×D4—62no—
269C3×D4—62no—
270C3×D4—92no—
271C3×D4—122no—
272C3×D4—122no—
273C3×D4—181no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—64no—
277C2×C3×C4—64no—
278C2×C3×C4—62no—
279C2×C3×C4—61no—
280C2×C3×C4—91no—
281C2×C3×C4—91no—
282C2×C3×C4—91no—
283C2×C3×C4—91no—
284C2×C3×C4—91no—
285C2×C3×C4—91no—
286C2×C3×C4—124no—
287C2×C3×C4—122no—
288C2×C3×C4—121no—
289C2×C3×C4—182no—
290C2×C3×C4—182no—
291#67—22no—
292#67—22no—
293#67—62no—
294#67—62no—
295C2×Dic3—22no—
296C2×Dic3—21no—
297C2×Dic3—24no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C4×D3—62no—
313C4×D3—62no—
314Dic6—62no—
315Dic6—62no—
316C3×C8—182no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—22no—
325C2×C32—22no—
326C2×C32—42no—
327C2×C32—42no—
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—41no—
334C2×C32—42no—
335C2×C32—81no—
336C2×C32—81no—
337C2×C32—122no—
338C2×C32C3×#10412yescomplemented
339C2×C32C2×C3×C22⋊C411yescharacteristic
340C3×D3—42no—
341C3×D3—42no—
342C3×D3—82no—
343#41—92no—
344#41—92no—
345#41—181no—
346#41—181no—
347C2×Q8—91no—
348C2×D4—61no—
349C2×D4—91no—
350C22×C4—61no—
351C22×C4—91no—
352C22×C4—181no—
353M16—92no—
354M16—181no—
355C2×C8—181no—
356C42—92no—
357C42—181no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—62no—
377C22×C3—62no—
378C22×C3—82no—
379C22×C3—122no—
380C22×C3—122no—
381C22×C3—122no—
382C22×C3—242no—
383C22×C3D32⋊C411yescharacteristic
384C22×C3D32⋊C411yescharacteristic
385C22×C3C2×GO+(4,𝔽2)11yescharacteristic
386C2×D3—22no—
387C2×D3—42no—
388C3×C4—62no—
389C3×C4—64no—
390C3×C4—64no—
391C3×C4—62no—
392C3×C4—91no—
393C3×C4—91no—
394C3×C4—91no—
395C3×C4—91no—
396C3×C4—124no—
397C3×C4—122no—
398Dic3—22no—
399Dic3—24no—
400Dic3—61no—
401Dic3—61no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—61no—
407Dic3—61no—
408C32—21no—
409C32—21no—
410C32—41no—
411C32—41no—
412C32C2×C3×#10411yescharacteristic, complemented
413C23—61no—
414C23C3×GO+(4,𝔽2)11yescharacteristic
415Q8—92no—
416Q8—181no—
417D4—62no—
418D4—62no—
419D4—92no—
420D4—181no—
421C2×C4—64no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—91no—
430C2×C4—182no—
431C2×C4—182no—
432C8—182no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—22no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—42no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—81no—
452C2×C3—81no—
453C2×C3—122no—
454C2×C3—122no—
455C2×C3—242no—
456C2×C3#6374112yesdirect product factor
457C2×C3C2×D32⋊C411yescharacteristic
458D3—42no—
459C22—22no—
460C22—62no—
461C22—122no—
462C22C3×D32⋊C411yescharacteristic
463C22C3×D32⋊C411yescharacteristic
464C22C2×C3×GO+(4,𝔽2)11yescharacteristic
465C4—64no—
466C4—62no—
467C4—91no—
468C4—91no—
469C4—91no—
470C4—91no—
471C3—21no—
472C3—21no—
473C3—41no—
474C3—41no—
475C3C2×#6374111yescharacteristic, direct product factor
476C2—21no—
477C2—21no—
478C2—122no—
479C2C3×#6374112yesdirect product factor
480C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3D32⋊C4
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C22⋊C4
prime radicalC2×C3×#3805C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×#104
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3792C2semidirect
3. C2×C3×Dic32C2semidirect
4. C3×#63741C2direct (N × H)
5. C2×#63741C3direct (N × H)
9. C2×C3×#627C4semidirect
10. C2×C3×#625C4semidirect
13. C3×#3805C4semidirect
17. C3×#3792C22semidirect
20. C3×Dic32C22semidirect
23. C2×#3792C2×C3semidirect
25. C2×Dic32C2×C3semidirect
26. #63741C2×C3direct (N × H)
41. C3×#627C2×C4semidirect
43. C3×#625C2×C4semidirect
67. C2×#627C3×C4semidirect
68. C2×#625C3×C4semidirect
71. #3805C3×C4semidirect
75. #3792C22×C3semidirect
80. Dic32C22×C3semidirect
155. #627C2×C3×C4semidirect
157. #625C2×C3×C4semidirect
164. C2×C33#104semidirect
254. C33C2×#104semidirect
338. C2×C32C3×#104semidirect
412. C32C2×C3×#104semidirect
456. C2×C3#63741direct (N × H)
475. C3C2×#63741direct (N × H)
479. C2C3×#63741direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×C3×#37922
C2×C3×Dic322
C3×#637412
C2×#637413
C2×C3×#1049