Finite Group Databasebeta

← C3×#10593515 #13166879

Subgroups — C3×#10593515 #13166879

Order 1728.

1028 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 122 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×#157C211yesmaximal, characteristic, complemented
2C2×C3×#3811C211yesmaximal, characteristic
3C2×C3×D32⋊C4C212yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic, complemented
5C3×#63737C212yesmaximal, complemented
6#10593515C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×D32C2211yescharacteristic
9C22×C3×#157—22no—
10C22×C3×#157—24no—
11C22×C3×#157C2212yes—
12C2×C32×#67—21no—
13C3×#3811—22no—
14C3×#3811C414yescomplemented
15C2×C3×#626—21no—
16C2×C3×#624C2212yes—
17C2×C3×#624C2211yescharacteristic
18C3×D32⋊C4—24no—
19C32×#217—21no—
20C3×#3722—21no—
21C23×#157C2×C311yescharacteristic, complemented
22C2×C3×D3×D4—61no—
23C2×#3811C2×C311yescharacteristic
24C2×D32⋊C4C2×C312yescomplemented
25C3×D3×C22⋊C4—61no—
26#63974C2×C311yescharacteristic, complemented
27#63737C2×C312yescomplemented
28C23×C33D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3C2×C412yes—
32C22×C3×C3⋊D3D411yescharacteristic
33C22×C3×C3⋊D3D412yes—
34C22×C3×C3⋊D3C2311yescharacteristic
35C22×C32×D3—21no—
36C2×C3×D32—22no—
37C2×C3×D32—22no—
38C2×C3×D32—21no—
39C2×C3×D32C2×C412yescomplemented
40C2×C3×#157—28no—
41C2×C3×#157—22no—
42C2×C3×#157—22no—
43C2×C3×#157D414yescomplemented
44C32×#67—24no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—22no—
47C3×#626—24no—
48C3×#624—21no—
49C3×#624—22no—
50C3×#624—22no—
51C3×#624C2×C412yescomplemented
52C3×#386—91nomaximal
53C23×C3⋊D3C22×C311yescharacteristic
54C23×C3×D3—61no—
55C23×C3×D3—61no—
56C23×C3×D3—61no—
57C22×D32C22×C311yescharacteristic
58C22×#157—22no—
59C22×#157—24no—
60C22×#157C22×C312yes—
61C2×C32×D4—61no—
62C2×C3×#67—21no—
63C2×C3×#67—21no—
64C2×C3×#67—41no—
65C2×C3×#67—61no—
66C3×D3×D4—64no—
67C3×D3×D4—64no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—62no—
71#3811—22no—
72#3811C3×C414yescomplemented
73C2×#626—21no—
74C2×#624C22×C312yes—
75C2×#624C22×C311yescharacteristic
76D32⋊C4—24no—
77C32×C22⋊C4—61no—
78C3×#217—21no—
79C3×#217—21no—
80C3×#217—41no—
81C3×(C2×D3)⋊C4—62no—
82#3722—21no—
83C22×C33—22no—
84C22×C33C22⋊C412yescomplemented
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—24no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3C22⋊C412yescomplemented
91C2×C3×C3⋊D3C22×C411yescharacteristic
92C2×C3×C3⋊D3C2×D412yes—
93C2×C3×C3⋊D3C2×D411yescharacteristic
94C2×C3×C3⋊D3#4112yes—
95C2×C32×D3—22no—
96C2×C32×D3—42no—
97C3×D32—24no—
98C3×D32—41no—
99C3×D32—41no—
100C3×#157—24no—
101C32×Dic3—22no—
102C32×Dic3—42no—
103C22×C3×D4—91no—
104C23×C3×C4—91no—
105C2×D3×D4—61no—
106C2×C3×C22⋊C4—91no—
107C2×C3×C22⋊C4—92no—
108D3×C22⋊C4—61no—
109C3×#95—92no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32—41no—
114C23×C32—61no—
115C23×C32C3×D411yescharacteristic
116C22×C3⋊D3—22no—
117C22×C3⋊D3—22no—
118C22×C3⋊D3C2×C3×C412yes—
119C22×C3⋊D3C3×D411yescharacteristic
120C22×C3⋊D3C3×D412yes—
121C22×C3⋊D3C23×C311yescharacteristic
122C22×C3×D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—41no—
125C22×C3×D3—62no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—62no—
132C22×C3×D3—61no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—122no—
136C22×C3×D3—122no—
137C22×C3×D3—122no—
138C22×C3×D3—122no—
139C22×C3×D3—122no—
140C22×C3×D3—122no—
141C2×D32—21no—
142C2×D32—22no—
143C2×D32—22no—
144C2×D32C2×C3×C412yescomplemented
145C2×#157—28no—
146C2×#157—22no—
147C2×#157—22no—
148C2×#157C3×D414yescomplemented
149C32×D4—64no—
150C2×C32×C4—61no—
151C2×C32×C4—62no—
152C3×#67—24no—
153C3×#67—24no—
154C3×#67—44no—
155C3×#67—64no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—42no—
162C3×D12—64no—
163C3×C4×D3—62no—
164C3×C4×D3—62no—
165C3×C4×D3—122no—
166C3×C4×D3—122no—
167#626—24no—
168#624—21no—
169#624—22no—
170#624—22no—
171#624C2×C3×C412yescomplemented
172#386—91noSylow
173C2×C33—22no—
174C2×C33C2×C22⋊C411yescharacteristic
175C2×C33#12112yes—
176C3×C3⋊D3—22no—
177C3×C3⋊D3C4×D412yes—
178C3×C3⋊D3#12011yescharacteristic
179C3×C3⋊D3#12311yescharacteristic
180C32×D3—42no—
181C24×C3—91no—
182C24×C3—91no—
183C23×D3—61no—
184C23×D3—61no—
185C23×D3—61no—
186C2×C3×D4—61no—
187C2×C3×D4—61no—
188C2×C3×D4—94no—
189C2×C3×D4—121no—
190C2×C3×D4—181no—
191C2×C3×D4—182no—
192C22×C3×C4—91no—
193C22×C3×C4—92no—
194C22×C3×C4—92no—
195C22×C3×C4—184no—
196C22×C3×C4—182no—
197C2×#67—21no—
198C2×#67—61no—
199D3×D4—64no—
200D3×D4—64no—
201C2×D12—61no—
202C2×C4×D3—61no—
203C2×C4×D3—62no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—121no—
207C3×C22⋊C4—181no—
208C3×C22⋊C4—184no—
209#217—21no—
210(C2×D3)⋊C4—62no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—62no—
223C22×C32—82no—
224C22×C32—82no—
225C22×C32—122no—
226C22×C32C3×C22⋊C412yescomplemented
227C22×C32C2×C3×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—24no—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—22no—
232C2×C3⋊D3C3×C22⋊C412yescomplemented
233C2×C3⋊D3C22×C3×C411yescharacteristic
234C2×C3⋊D3C2×C3×D412yes—
235C2×C3⋊D3C2×C3×D411yescharacteristic
236C2×C3⋊D3C3×#4112yes—
237C2×C3×D3—22no—
238C2×C3×D3—22no—
239C2×C3×D3—42no—
240C2×C3×D3—42no—
241C2×C3×D3—42no—
242C2×C3×D3—61no—
243C2×C3×D3—61no—
244C2×C3×D3—61no—
245C2×C3×D3—62no—
246C2×C3×D3—61no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—82no—
253C2×C3×D3—124no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—124no—
257C2×C3×D3—124no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264D32—24no—
265D32—41no—
266D32—41no—
267#157—24no—
268C32×C4—62no—
269C32×C4—122no—
270C3×Dic3—22no—
271C3×Dic3—22no—
272C3×Dic3—42no—
273C3×Dic3—42no—
274C3×Dic3—42no—
275C3×Dic3—82no—
276C22×D4—91no—
277C23×C4—91no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—92no—
280#95—92no—
281C33#38611yescharacteristic, complemented, Sylow
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—61no—
287C23×C3—61no—
288C23×C3—92no—
289C23×C3—91no—
290C23×C3—91no—
291C23×C3—92no—
292C23×C3—92no—
293C23×C3—121no—
294C23×C3—182no—
295C23×C3—182no—
296C23×C3—181no—
297C23×C3—182no—
298C23×C3—182no—
299C23×C3GO+(4,𝔽2)11yescharacteristic
300C22×D3—21no—
301C22×D3—62no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—122no—
312C22×D3—122no—
313C22×D3—122no—
314C22×D3—122no—
315C22×D3—122no—
316C22×D3—122no—
317C3×D4—64no—
318C3×D4—64no—
319C3×D4—124no—
320C3×D4—184no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—92no—
326C2×C3×C4—94no—
327C2×C3×C4—121no—
328C2×C3×C4—122no—
329C2×C3×C4—181no—
330C2×C3×C4—182no—
331C2×C3×C4—182no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—188no—
335#67—24no—
336#67—64no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339D12—64no—
340C4×D3—62no—
341C4×D3—62no—
342C4×D3—122no—
343C4×D3—122no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—42no—
352C2×C32—42no—
353C2×C32—42no—
354C2×C32—42no—
355C2×C32—82no—
356C2×C32—82no—
357C2×C32—122no—
358C2×C32C2×C3×C22⋊C411yescharacteristic
359C2×C32C3×#12112yes—
360C3⋊D3—22no—
361C3⋊D3C3×C4×D412yes—
362C3⋊D3C3×#12011yescharacteristic
363C3⋊D3C3×#12311yescharacteristic
364C3×D3—42no—
365C3×D3—42no—
366C3×D3—62no—
367C3×D3—62no—
368C3×D3—61no—
369C3×D3—61no—
370C3×D3—61no—
371C3×D3—61no—
372C3×D3—82no—
373C3×D3—122no—
374C3×D3—122no—
375C24—91no—
376C24—91no—
377C2×D4—61no—
378C2×D4—94no—
379C2×D4—181no—
380C2×D4—182no—
381C22×C4—91no—
382C22×C4—92no—
383C22×C4—92no—
384C22×C4—184no—
385C22×C4—182no—
386C22⋊C4—61no—
387C22⋊C4—181no—
388C22⋊C4—184no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—62no—
401C22×C3—62no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—91no—
405C22×C3—91no—
406C22×C3—92no—
407C22×C3—92no—
408C22×C3—92no—
409C22×C3—122no—
410C22×C3—122no—
411C22×C3—122no—
412C22×C3—182no—
413C22×C3—184no—
414C22×C3—182no—
415C22×C3—184no—
416C22×C3—182no—
417C22×C3—242no—
418C22×C3—361no—
419C22×C3—361no—
420C22×C3D32⋊C412yescomplemented
421C22×C3C2×GO+(4,𝔽2)11yescharacteristic
422C2×D3—22no—
423C2×D3—42no—
424C2×D3—61no—
425C2×D3—61no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—124no—
435C2×D3—122no—
436C2×D3—122no—
437C2×D3—124no—
438C2×D3—122no—
439C2×D3—124no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—122no—
443C2×D3—122no—
444C2×D3—122no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—122no—
448C3×C4—122no—
449C3×C4—122no—
450C3×C4—184no—
451C3×C4—242no—
452Dic3—22no—
453Dic3—42no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32—41no—
458C32C3×#38611yescharacteristic, complemented
459C23—61no—
460C23—92no—
461C23—91no—
462C23—91no—
463C23—92no—
464C23—92no—
465C23—182no—
466C23—182no—
467C23—181no—
468C23—182no—
469C23—182no—
470C23C3×GO+(4,𝔽2)11yescharacteristic
471D4—64no—
472D4—184no—
473C2×C4—61no—
474C2×C4—62no—
475C2×C4—92no—
476C2×C4—94no—
477C2×C4—181no—
478C2×C4—182no—
479C2×C4—182no—
480C2×C4—182no—
481C2×C4—182no—
482C2×C4—188no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—42no—
489C2×C3—42no—
490C2×C3—42no—
491C2×C3—42no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—82no—
495C2×C3—82no—
496C2×C3—91no—
497C2×C3—92no—
498C2×C3—91no—
499C2×C3—122no—
500C2×C3—122no—
501C2×C3—182no—
502C2×C3—242no—
503C2×C3C2×D32⋊C411yescharacteristic
504C2×C3#6424212yes—
505D3—42no—
506D3—62no—
507D3—61no—
508D3—61no—
509D3—62no—
510D3—61no—
511D3—61no—
512D3—122no—
513D3—122no—
514C22—22no—
515C22—62no—
516C22—92no—
517C22—92no—
518C22—91no—
519C22—91no—
520C22—92no—
521C22—122no—
522C22—182no—
523C22—184no—
524C22—182no—
525C22—184no—
526C22—182no—
527C22—361no—
528C22—361no—
529C22C3×D32⋊C412yescomplemented
530C22C2×C3×GO+(4,𝔽2)11yescharacteristic
531C4—62no—
532C4—122no—
533C4—184no—
534C3—21no—
535C3—21no—
536C3—41no—
537C3—41no—
538C3#1059351511yescharacteristic, direct product factor
539C2—22no—
540C2—91no—
541C2—92no—
542C2—91no—
543C2—122no—
544C2—182no—
545C2C2×C3×D32⋊C411yescharacteristic
546C2C3×#6424212yes—
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3811C2
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#386
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×#157C2semidirect
3. C2×C3×D32⋊C4C2semidirect
4. C3×#63974C2semidirect
5. C3×#63737C2semidirect
6. #10593515C3direct (N × H)
14. C3×#3811C4semidirect
21. C23×#157C2×C3semidirect
24. C2×D32⋊C4C2×C3semidirect
26. #63974C2×C3semidirect
27. #63737C2×C3semidirect
39. C2×C3×D32C2×C4semidirect
43. C2×C3×#157D4semidirect
51. C3×#624C2×C4semidirect
72. #3811C3×C4semidirect
84. C22×C33C22⋊C4semidirect
90. C2×C3×C3⋊D3C22⋊C4semidirect
144. C2×D32C2×C3×C4semidirect
148. C2×#157C3×D4semidirect
171. #624C2×C3×C4semidirect
226. C22×C32C3×C22⋊C4semidirect
232. C2×C3⋊D3C3×C22⋊C4semidirect
281. C33#386semidirect
420. C22×C3D32⋊C4semidirect
458. C32C3×#386semidirect
529. C22C3×D32⋊C4semidirect
538. C3#10593515direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×C3×#38112
C2×C3×D32⋊C42
C3×#639742
C3×#637372
#105935153
C3×#3869