Finite Group Databasebeta

← C2×C3×#64233 #13181046

Subgroups — C2×C3×#64233 #13181046

Order 1728.

1264 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1264 conjugacy classes of subgroups — too many to draw. Only its 194 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×#3811C211yesmaximal, characteristic
2C22×C3×#624C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4C2×C3×#3773C212yesmaximal, complemented
5C3×#64233C218yesmaximal, direct product factor
6C2×C3×D32⋊C4C212yesmaximal, complemented
7C2×#64233C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C3×D32C2211yescharacteristic
10C22×C3×#157C2212yes—
11C2×C32×#67—21no—
12C22×C32×Dic3—21no—
13C3×#3811—22no—
14C3×#3811C2214yes—
15C2×C3×#626—21no—
16C2×C3×#624—22no—
17C2×C3×#624—21no—
18C2×C3×#624C2211yescharacteristic
19C2×C3×#624C2212yes—
20C2×C3×#624C2214yescomplemented
21C3×C22⋊#157C2214yescomplemented
22C3×#3773C2218yescomplemented
23C3×D32⋊C4C2218yescomplemented
24C2×C3×D3×D4—61no—
25C22×C3×C4×D3—61no—
26C2×#3811C2×C311yescharacteristic
27C22×#624C2×C311yescharacteristic, complemented
28C2×C22⋊#157C2×C311yescharacteristic, complemented
29C2×#3773C2×C312yescomplemented
30#64233C2×C318yesdirect product factor
31C2×D32⋊C4C2×C312yescomplemented
32C23×C33D411yescharacteristic
33C22×C3×C3⋊D3—24no—
34C22×C3×C3⋊D3D411yescharacteristic
35C22×C3×C3⋊D3D412yes—
36C22×C3×C3⋊D3C2312yes—
37C22×C3×C3⋊D3C2311yescharacteristic
38C22×C32×D3—21no—
39C2×C3×D32—24no—
40C2×C3×D32—21no—
41C2×C3×D32C2312yes—
42C2×C3×#157—24no—
43C2×C3×#157C2314yes—
44C32×#67—24no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—24no—
47C2×C32×Dic3—21no—
48C3×#626—24no—
49C3×#624—22no—
50C3×#624—24no—
51C3×#624—21no—
52C3×#624C2314yes—
53C3×#624C2312yes—
54C2×C3×#123—91nomaximal
55C23×C3⋊D3C22×C311yescharacteristic
56C23×C3×D3—61no—
57C23×C3×D3—61no—
58C23×C3×D3—61no—
59C22×D32C22×C311yescharacteristic
60C22×#157C22×C312yes—
61C2×C32×D4—61no—
62C22×C32×C4—61no—
63C2×C3×#67—21no—
64C2×C3×#67—21no—
65C2×C3×#67—41no—
66C2×C3×#67—61no—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—21no—
69C22×C3×Dic3—41no—
70C3×D3×D4—64no—
71C3×D3×D4—64no—
72C2×C3×D12—61no—
73C2×C3×C4×D3—62no—
74C2×C3×C4×D3—64no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—64no—
77C2×C3×C4×D3—62no—
78#3811—22no—
79#3811C22×C314yes—
80C2×#626—21no—
81C2×#624—22no—
82C2×#624—21no—
83C2×#624C22×C312yes—
84C2×#624C22×C311yescharacteristic
85C2×#624C22×C314yescomplemented
86C22⋊#157C22×C314yescomplemented
87#3773C22×C318yescomplemented
88D32⋊C4C22×C318yescomplemented
89C22×C33—22no—
90C22×C33C2×D412yes—
91C22×C33C2×D411yescharacteristic
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—24no—
94C2×C3×C3⋊D3—24no—
95C2×C3×C3⋊D3C2×D412yes—
96C2×C3×C3⋊D3C2×D411yescharacteristic
97C2×C3×C3⋊D3#4114yes—
98C2×C3×C3⋊D3C2411yescharacteristic
99C2×C32×D3—22no—
100C2×C32×D3—42no—
101C3×D32—24no—
102C3×D32—42no—
103C3×#157—24no—
104C32×Dic3—24no—
105C32×Dic3—22no—
106C22×C3×D4—91no—
107C23×C3×C4—91no—
108C2×D3×D4—61no—
109C22×C4×D3—61no—
110C3×#123—98no—
111C2×C3×C4⋊C4—92no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—92no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C23×C32—41no—
118C23×C32—61no—
119C23×C32C3×D411yescharacteristic
120C22×C3⋊D3—24no—
121C22×C3⋊D3C3×D411yescharacteristic
122C22×C3⋊D3C3×D412yes—
123C22×C3⋊D3C23×C312yes—
124C22×C3⋊D3C23×C311yescharacteristic
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C22×C3×D3—41no—
128C22×C3×D3—62no—
129C22×C3×D3—61no—
130C22×C3×D3—62no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—62no—
134C22×C3×D3—61no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—68no—
139C22×C3×D3—124no—
140C22×C3×D3—124no—
141C2×D32—21no—
142C2×D32—24no—
143C2×D32C23×C312yes—
144C2×#157—24no—
145C2×#157C23×C314yes—
146C32×D4—64no—
147C2×C32×C4—64no—
148C2×C32×C4—62no—
149C2×C32×C4—61no—
150C3×#67—24no—
151C3×#67—24no—
152C3×#67—44no—
153C3×#67—64no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3—44no—
162C2×C3×Dic3—41no—
163C3×D12—64no—
164C3×C4×D3—64no—
165C3×C4×D3—62no—
166C3×C4×D3—64no—
167C3×C4×D3—68no—
168C3×C4×D3—62no—
169#626—24no—
170#624—22no—
171#624—24no—
172#624—21no—
173#624C23×C314yes—
174#624C23×C312yes—
175C2×#123—91noSylow
176C2×C33—22no—
177C2×C33#12312yescomplemented
178C2×C33C22×D411yescharacteristic
179C3×C3⋊D3—22no—
180C3×C3⋊D3#12312yescomplemented
181C3×C3⋊D3C2×#4112yes—
182C32×D3—42no—
183C24×C3—91no—
184C24×C3—91no—
185C23×D3—61no—
186C23×D3—61no—
187C23×D3—61no—
188C2×C3×D4—61no—
189C2×C3×D4—61no—
190C2×C3×D4—94no—
191C2×C3×D4—121no—
192C2×C3×D4—181no—
193C2×C3×D4—182no—
194C22×C3×C4—61no—
195C22×C3×C4—61no—
196C22×C3×C4—92no—
197C22×C3×C4—91no—
198C22×C3×C4—94no—
199C22×C3×C4—92no—
200C22×C3×C4—121no—
201C22×C3×C4—182no—
202C22×C3×C4—181no—
203C2×#67—21no—
204C2×#67—61no—
205C22×Dic3—21no—
206D3×D4—64no—
207D3×D4—64no—
208C2×D12—61no—
209C2×C4×D3—62no—
210C2×C4×D3—61no—
211C2×C4×D3—64no—
212C2×C4×D3—64no—
213C2×C4×D3—62no—
214C3×C4⋊C4—98no—
215C3×C22⋊C4—98no—
216C3×C22⋊C4—94no—
217C22×C32—24no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32—42no—
225C22×C32—41no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—44no—
229C22×C32—62no—
230C22×C32—82no—
231C22×C32—122no—
232C22×C32C2×C3×D412yes—
233C22×C32C2×C3×D411yescharacteristic
234C2×C3⋊D3—22no—
235C2×C3⋊D3—24no—
236C2×C3⋊D3—24no—
237C2×C3⋊D3C2×C3×D412yes—
238C2×C3⋊D3C2×C3×D411yescharacteristic
239C2×C3⋊D3C3×#4114yes—
240C2×C3⋊D3C24×C311yescharacteristic
241C2×C3×D3—22no—
242C2×C3×D3—22no—
243C2×C3×D3—42no—
244C2×C3×D3—42no—
245C2×C3×D3—42no—
246C2×C3×D3—61no—
247C2×C3×D3—62no—
248C2×C3×D3—61no—
249C2×C3×D3—64no—
250C2×C3×D3—64no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—62no—
254C2×C3×D3—68no—
255C2×C3×D3—68no—
256C2×C3×D3—64no—
257C2×C3×D3—82no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C2×C3×D3—122no—
262C2×C3×D3—124no—
263D32—24no—
264D32—42no—
265#157—24no—
266C32×C4—64no—
267C32×C4—62no—
268C3×Dic3—24no—
269C3×Dic3—22no—
270C3×Dic3—24no—
271C3×Dic3—22no—
272C3×Dic3—44no—
273C3×Dic3—42no—
274C22×D4—91no—
275C23×C4—91no—
276#123—98no—
277C2×C4⋊C4—92no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—92no—
280C33C2×#12311yescharacteristic, complemented, Sylow
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—61no—
286C23×C3—61no—
287C23×C3—92no—
288C23×C3—91no—
289C23×C3—92no—
290C23×C3—92no—
291C23×C3—91no—
292C23×C3—121no—
293C23×C3—184no—
294C23×C3—181no—
295C23×C3—184no—
296C23×C3GO+(4,𝔽2)11yescharacteristic
297C22×D3—21no—
298C22×D3—62no—
299C22×D3—61no—
300C22×D3—62no—
301C22×D3—61no—
302C22×D3—62no—
303C22×D3—61no—
304C22×D3—61no—
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—62no—
308C22×D3—68no—
309C22×D3—124no—
310C22×D3—124no—
311C3×D4—64no—
312C3×D4—64no—
313C3×D4—124no—
314C3×D4—184no—
315C2×C3×C4—62no—
316C2×C3×C4—64no—
317C2×C3×C4—61no—
318C2×C3×C4—64no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—94no—
322C2×C3×C4—92no—
323C2×C3×C4—94no—
324C2×C3×C4—124no—
325C2×C3×C4—122no—
326C2×C3×C4—121no—
327C2×C3×C4—182no—
328C2×C3×C4—184no—
329C2×C3×C4—181no—
330C2×C3×C4—184no—
331#67—24no—
332#67—64no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—21no—
336D12—64no—
337C4×D3—64no—
338C4×D3—62no—
339C4×D3—64no—
340C4×D3—68no—
341C4×D3—62no—
342C2×C32—24no—
343C2×C32—22no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—42no—
349C2×C32—42no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—42no—
353C2×C32—44no—
354C2×C32—82no—
355C2×C32—122no—
356C2×C32C3×#12312yescomplemented
357C2×C32C22×C3×D411yescharacteristic
358C3⋊D3—22no—
359C3⋊D3C3×#12312yescomplemented
360C3⋊D3C2×C3×#4112yes—
361C3×D3—42no—
362C3×D3—42no—
363C3×D3—62no—
364C3×D3—62no—
365C3×D3—62no—
366C3×D3—62no—
367C3×D3—64no—
368C3×D3—82no—
369C3×D3—122no—
370C24—91no—
371C24—91no—
372C2×D4—61no—
373C2×D4—94no—
374C2×D4—181no—
375C2×D4—182no—
376C22×C4—61no—
377C22×C4—92no—
378C22×C4—91no—
379C22×C4—94no—
380C22×C4—92no—
381C22×C4—182no—
382C22×C4—181no—
383C4⋊C4—98no—
384C22⋊C4—98no—
385C22⋊C4—94no—
386C22×C3—24no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—42no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—42no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—82no—
401C22×C3—91no—
402C22×C3—91no—
403C22×C3—94no—
404C22×C3—92no—
405C22×C3—122no—
406C22×C3—122no—
407C22×C3—122no—
408C22×C3—184no—
409C22×C3—184no—
410C22×C3—184no—
411C22×C3—182no—
412C22×C3—242no—
413C22×C3—362no—
414C22×C3C2×GO+(4,𝔽2)12yes—
415C22×C3C2×GO+(4,𝔽2)11yescharacteristic
416C2×D3—22no—
417C2×D3—42no—
418C2×D3—61no—
419C2×D3—62no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—62no—
426C2×D3—68no—
427C2×D3—68no—
428C2×D3—64no—
429C2×D3—124no—
430C2×D3—124no—
431C2×D3—124no—
432C2×D3—122no—
433C2×D3—124no—
434C3×C4—64no—
435C3×C4—62no—
436C3×C4—64no—
437C3×C4—62no—
438C3×C4—124no—
439C3×C4—122no—
440C3×C4—184no—
441Dic3—24no—
442Dic3—22no—
443C32—21no—
444C32—21no—
445C32—41no—
446C32—41no—
447C32C2×C3×#12311yescharacteristic, complemented
448C23—61no—
449C23—92no—
450C23—91no—
451C23—92no—
452C23—92no—
453C23—91no—
454C23—184no—
455C23—181no—
456C23—184no—
457C23C3×GO+(4,𝔽2)11yescharacteristic
458D4—64no—
459D4—184no—
460C2×C4—64no—
461C2×C4—62no—
462C2×C4—61no—
463C2×C4—94no—
464C2×C4—92no—
465C2×C4—94no—
466C2×C4—182no—
467C2×C4—184no—
468C2×C4—181no—
469C2×C4—184no—
470C2×C3—24no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—22no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—42no—
477C2×C3—44no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—82no—
483C2×C3—92no—
484C2×C3—92no—
485C2×C3—122no—
486C2×C3—122no—
487C2×C3—182no—
488C2×C3—242no—
489C2×C3#6423312yesdirect product factor
490C2×C3C22×GO+(4,𝔽2)11yescharacteristic
491D3—42no—
492D3—62no—
493D3—62no—
494D3—62no—
495D3—62no—
496D3—64no—
497D3—122no—
498C22—22no—
499C22—62no—
500C22—94no—
501C22—91no—
502C22—91no—
503C22—92no—
504C22—122no—
505C22—184no—
506C22—184no—
507C22—184no—
508C22—182no—
509C22—362no—
510C22C2×C3×GO+(4,𝔽2)12yes—
511C22C2×C3×GO+(4,𝔽2)11yescharacteristic
512C4—64no—
513C4—62no—
514C4—184no—
515C3—21no—
516C3—21no—
517C3—41no—
518C3—41no—
519C3C2×#6423311yescharacteristic, direct product factor
520C2—22no—
521C2—92no—
522C2—92no—
523C2—122no—
524C2—182no—
525C2C3×#6423312yesdirect product factor
526C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3811C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×#123
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. C22×C3×#624C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C2×C3×#3773C2semidirect
5. C3×#64233C2direct (N × H)
6. C2×C3×D32⋊C4C2semidirect
7. C2×#64233C3direct (N × H)
20. C2×C3×#624C22semidirect
21. C3×C22⋊#157C22semidirect
22. C3×#3773C22semidirect
23. C3×D32⋊C4C22semidirect
27. C22×#624C2×C3semidirect
28. C2×C22⋊#157C2×C3semidirect
29. C2×#3773C2×C3semidirect
30. #64233C2×C3direct (N × H)
31. C2×D32⋊C4C2×C3semidirect
85. C2×#624C22×C3semidirect
86. C22⋊#157C22×C3semidirect
87. #3773C22×C3semidirect
88. D32⋊C4C22×C3semidirect
177. C2×C33#123semidirect
180. C3×C3⋊D3#123semidirect
280. C33C2×#123semidirect
356. C2×C32C3×#123semidirect
359. C3⋊D3C3×#123semidirect
447. C32C2×C3×#123semidirect
489. C2×C3#64233direct (N × H)
519. C3C2×#64233direct (N × H)
525. C2C3×#64233direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38112
C22×C3×#6242
C2×C3×C22⋊#1572
C2×C3×#37732
C3×#642332
C2×C3×D32⋊C42
C2×#642333
C2×C3×#1239