Finite Group Databasebeta

← C3×#10596494 #13181013

Subgroups — C3×#10596494 #13181013

Order 1728.

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

normal subgroups only

This group has 938 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×#64286C211yesmaximal, characteristic, complemented
4C3×#64233C214yesmaximal, complemented
5C3×#64222C212yesmaximal, complemented
6C3×#64221C212yesmaximal, complemented
7C3×D32⋊Q8C214yesmaximal, complemented
8#10596494C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C32×#235—21no—
11C2×C32×Dic6—21no—
12C3×#3811C2212yes—
13C2×C3×#627—21no—
14C2×C3×#624C2212yes—
15C3×#3805—21no—
16C3×#3802C2211yescharacteristic
17C3×C4×D32C2212yes—
18C3×#3798—21no—
19C3×#3797—21no—
20C3×#3797—21no—
21C3×#3797C2212yes—
22C3×#3797C2212yes—
23C3×#3797C2211yescharacteristic
24C3×#3796—21no—
25C3×C22⋊#157C2212yes—
26C3×C4⋊#157C2212yes—
27C3×C4×#157C2212yes—
28C3×#3773C2218yes—
29C3×D32⋊C4C2218yes—
30C3×D3×#41—61no—
31C2×C3×D3×Q8—61no—
32#64306C2×C311yescharacteristic
33C2×#3797C2×C311yescharacteristic, complemented
34#64286C2×C311yescharacteristic, complemented
35#64233C2×C314yescomplemented
36#64222C2×C312yescomplemented
37#64221C2×C312yescomplemented
38D32⋊Q8C2×C314yescomplemented
39C22×C3×C3⋊D3C2311yescharacteristic
40C2×C3×D32C2312yes—
41C2×C3×#157C2314yes—
42C2×C33×C4D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C3×C4×C3⋊D3C2311yescharacteristic
47C3×C4×C3⋊D3C2311yescharacteristic
48C32×#67—22no—
49C2×C32×Dic3—22no—
50C32×D12—21no—
51C32×C4×D3—22no—
52C32×Dic6—21no—
53C32×Dic6—22no—
54C32×Dic6—22no—
55C3×#627—22no—
56C3×#627—22no—
57C3×#627—21no—
58C3×#626—22no—
59C3×#625—21no—
60C3×#624—22no—
61C3×#624C2314yes—
62C3×#624C2312yes—
63C3×D3×Dic3—22no—
64C3×#536—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#3811C22×C312yes—
91C2×#627—21no—
92C2×#624C22×C312yes—
93#3805—21no—
94#3802C22×C311yescharacteristic
95C4×D32C22×C312yes—
96#3798—21no—
97#3797—21no—
98#3797—21no—
99#3797C22×C312yes—
100#3797C22×C312yes—
101#3797C22×C311yescharacteristic
102#3796—21no—
103C22⋊#157C22×C312yes—
104C4⋊#157C22×C312yes—
105C4×#157C22×C312yes—
106#3773C22×C318yes—
107D32⋊C4C22×C318yes—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3C2×D411yescharacteristic
111C2×C3×C3⋊D3C2411yescharacteristic
112C2×C32×D3—22no—
113C3×D32—24no—
114C3×#157—24no—
115C33×C4C2×D411yescharacteristic
116C33×C4C2×D411yescharacteristic
117C3×C3⋊Dic3C2×D411yescharacteristic
118C3×C3⋊Dic3C2×D411yescharacteristic
119C32×Dic3—22no—
120C32×Dic3—24no—
121C2×C3×#41—91no—
122C22×C3×Q8—91no—
123D3×#41—61no—
124C2×D3×Q8—61no—
125C3×C4⋊Q8—92no—
126C3×#124—92no—
127C3×#123—94no—
128C3×C22⋊Q8—94no—
129C3×#117—91no—
130C22×C3⋊D3C23×C311yescharacteristic
131C22×C3×D3—62no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×D32C23×C312yes—
135C2×#157C23×C314yes—
136C32×Q8—62no—
137C32×Q8—62no—
138C32×Q8—61no—
139C32×D4—61no—
140C32×D4—62no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4—62no—
146C2×C32×C4—62no—
147C2×C32×C4C3×D411yescharacteristic
148C2×C3⋊Dic3C3×D411yescharacteristic
149C4×C3⋊D3C3×D411yescharacteristic
150C4×C3⋊D3C3×D411yescharacteristic
151C4×C3⋊D3C23×C311yescharacteristic
152C4×C3⋊D3C23×C311yescharacteristic
153C3×#67—22no—
154C3×#67—22no—
155C3×#67—42no—
156C3×#67—62no—
157C3×#67—62no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—62no—
164C3×D12—21no—
165C3×D12—21no—
166C3×D12—41no—
167C3×D12—62no—
168C3×C4×D3—22no—
169C3×C4×D3—22no—
170C3×C4×D3—42no—
171C3×C4×D3—62no—
172C3×C4×D3—64no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—61no—
181C3×C4×D3—64no—
182C3×C4×D3—62no—
183C3×Dic6—22no—
184C3×Dic6—22no—
185C3×Dic6—21no—
186C3×Dic6—21no—
187C3×Dic6—22no—
188C3×Dic6—22no—
189C3×Dic6—42no—
190C3×Dic6—42no—
191C3×Dic6—41no—
192C3×Dic6—64no—
193C3×Dic6—64no—
194C3×Dic6—62no—
195#627—22no—
196#627—22no—
197#627—21no—
198#626—22no—
199#625—21no—
200#624—22no—
201#624C23×C314yes—
202#624C23×C312yes—
203D3×Dic3—22no—
204#536—91noSylow
205C2×C33—21no—
206C2×C33C22×D411yescharacteristic
207C3×C3⋊D3—21no—
208C3×C3⋊D32−1+412yes—
209C32×D3—42no—
210C3×#41—61no—
211C3×#41—61no—
212C3×#41—121no—
213C3×#41—181no—
214C3×#41—181no—
215C3×#41—181no—
216C2×C3×Q8—61no—
217C2×C3×Q8—61no—
218C2×C3×Q8—92no—
219C2×C3×Q8—92no—
220C2×C3×Q8—91no—
221C2×C3×Q8—121no—
222C2×C3×Q8—181no—
223C2×C3×Q8—181no—
224C2×C3×Q8—181no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C22×C3×C4—92no—
228C22×C3×C4—92no—
229C22×C3×C4—91no—
230#239—61no—
231D3×Q8—62no—
232D3×Q8—62no—
233D3×Q8—61no—
234D3×Q8—62no—
235D3×Q8—62no—
236#237—61no—
237#237—62no—
238D3×D4—61no—
239D3×D4—62no—
240#235—21no—
241#235—62no—
242C2×C4×D3—62no—
243C2×C4×D3—62no—
244C2×C4×D3—61no—
245C2×C4×D3—61no—
246C2×Dic6—21no—
247C2×Dic6—62no—
248C3×C4⋊C4—98no—
249C3×C4⋊C4—92no—
250C3×C22⋊C4—98no—
251C3×C22⋊C4—92no—
252C3×C42—92no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—62no—
258C22×C32C2×C3×D411yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3C2×C3×D411yescharacteristic
261C2×C3⋊D3C24×C311yescharacteristic
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—42no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—64no—
270C2×C3×D3—124no—
271C2×C3×D3—122no—
272D32—24no—
273#157—24no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—64no—
283C32×C4—62no—
284C32×C4C2×C3×D411yescharacteristic
285C32×C4C2×C3×D411yescharacteristic
286C3⋊Dic3C2×C3×D411yescharacteristic
287C3⋊Dic3C2×C3×D411yescharacteristic
288C3×Dic3—24no—
289C3×Dic3—22no—
290C3×Dic3—22no—
291C3×Dic3—24no—
292C3×Dic3—44no—
293C3×Dic3—42no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C2×#41—91no—
299C22×Q8—91no—
300C4⋊Q8—92no—
301#124—92no—
302#123—94no—
303C22⋊Q8—94no—
304#117—91no—
305C33#53611yescharacteristic, complemented, Sylow
306C23×C3—92no—
307C23×C3—91no—
308C22×D3—62no—
309C22×D3—61no—
310C22×D3—61no—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—61no—
314C3×Q8—62no—
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—122no—
318C3×Q8—122no—
319C3×Q8—121no—
320C3×Q8—182no—
321C3×Q8—182no—
322C3×Q8—181no—
323C3×D4—61no—
324C3×D4—62no—
325C3×D4—62no—
326C3×D4—61no—
327C3×D4—121no—
328C3×D4—122no—
329C3×D4—182no—
330C3×D4—181no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—94no—
340C2×C3×C4—92no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4—91no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—94no—
347C2×C3×C4—122no—
348C2×C3×C4—122no—
349C2×C3×C4—182no—
350C2×C3×C4—182no—
351C2×C3×C4GO+(4,𝔽2)11yescharacteristic
352#67—22no—
353#67—62no—
354#67—62no—
355C2×Dic3—22no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—62no—
359D12—21no—
360D12—62no—
361C4×D3—22no—
362C4×D3—62no—
363C4×D3—64no—
364C4×D3—61no—
365C4×D3—61no—
366C4×D3—61no—
367C4×D3—61no—
368C4×D3—61no—
369C4×D3—61no—
370C4×D3—61no—
371C4×D3—61no—
372C4×D3—64no—
373C4×D3—62no—
374Dic6—22no—
375Dic6—22no—
376Dic6—21no—
377Dic6—64no—
378Dic6—64no—
379Dic6—62no—
380C2×C32—22no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—42no—
388C2×C32—81no—
389C2×C32—122no—
390C2×C32C22×C3×D411yescharacteristic
391C3⋊D3—21no—
392C3⋊D3C3×2−1+412yes—
393C3×D3—42no—
394C3×D3—42no—
395C3×D3—62no—
396C3×D3—62no—
397C3×D3—62no—
398C3×D3—82no—
399C3×D3—121no—
400#41—61no—
401#41—181no—
402#41—181no—
403#41—181no—
404C2×Q8—61no—
405C2×Q8—92no—
406C2×Q8—92no—
407C2×Q8—91no—
408C2×Q8—181no—
409C2×Q8—181no—
410C2×Q8—181no—
411C2×D4—92no—
412C2×D4—91no—
413C22×C4—92no—
414C22×C4—92no—
415C22×C4—91no—
416C4⋊C4—92no—
417C4⋊C4—98no—
418C22⋊C4—92no—
419C22⋊C4—98no—
420C42—92no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—91no—
428C22×C3—91no—
429C22×C3—122no—
430C22×C3—182no—
431C22×C3—184no—
432C22×C3C2×GO+(4,𝔽2)11yescharacteristic
433C2×D3—22no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—64no—
439C2×D3—124no—
440C2×D3—122no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—64no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—64no—
453C3×C4—91no—
454C3×C4—91no—
455C3×C4—122no—
456C3×C4—124no—
457C3×C4—184no—
458C3×C4C2×GO+(4,𝔽2)11yescharacteristic
459C3×C4C2×GO+(4,𝔽2)11yescharacteristic
460Dic3—24no—
461Dic3—22no—
462Dic3—61no—
463Dic3—61no—
464Dic3—61no—
465Dic3—61no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32—41no—
470C32C3×#53611yescharacteristic, complemented
471C23—92no—
472C23—91no—
473Q8—62no—
474Q8—62no—
475Q8—61no—
476Q8—182no—
477Q8—182no—
478Q8—181no—
479D4—62no—
480D4—61no—
481D4—182no—
482D4—181no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—94no—
486C2×C4—92no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—94no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4C3×GO+(4,𝔽2)11yescharacteristic
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—41no—
501C2×C3—42no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—81no—
505C2×C3—92no—
506C2×C3—122no—
507C2×C3—122no—
508C2×C3—181no—
509C2×C3—242no—
510C2×C3C22×GO+(4,𝔽2)11yescharacteristic
511D3—42no—
512D3—62no—
513D3—62no—
514D3—62no—
515D3—121no—
516C22—62no—
517C22—91no—
518C22—91no—
519C22—182no—
520C22—184no—
521C22C2×C3×GO+(4,𝔽2)11yescharacteristic
522C4—62no—
523C4—64no—
524C4—91no—
525C4—91no—
526C4—184no—
527C4C2×C3×GO+(4,𝔽2)11yescharacteristic
528C4C2×C3×GO+(4,𝔽2)11yescharacteristic
529C3—21no—
530C3—21no—
531C3—41no—
532C3—41no—
533C3#1059649411yescharacteristic, direct product factor
534C2—21no—
535C2—92no—
536C2—122no—
537C2—181no—
538C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
prime radicalC3×#64306C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#536
LCSG ▷ C2×C3⋊D3 ▷ 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×#64286C2semidirect
4. C3×#64233C2semidirect
5. C3×#64222C2semidirect
6. C3×#64221C2semidirect
7. C3×D32⋊Q8C2semidirect
8. #10596494C3direct (N × H)
33. C2×#3797C2×C3semidirect
34. #64286C2×C3semidirect
35. #64233C2×C3semidirect
36. #64222C2×C3semidirect
37. #64221C2×C3semidirect
38. D32⋊Q8C2×C3semidirect
305. C33#536semidirect
470. C32C3×#536semidirect
533. C3#10596494direct (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×#642862
C3×#642332
C3×#642222
C3×#642212
C3×D32⋊Q82
#105964943
C3×#5369