Finite Group Databasebeta

← C2×#11724213 #13180872

Subgroups — C2×#11724213 #13180872

Order 1728.

1114 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C22×C3×C3⋊S4C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4#11724213C214yesmaximal, direct product factor
5C2×(C3⋊S4)⋊C4—31nomaximal
6C2×Dic3×S4—31nomaximal
7C2×#67⋊Dic3—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3×C3⋊S4C414yescomplemented
10C2×C3×C3⋊S4C2212yes—
11C2×#6294—41nomaximal
12A4×C3⋊Dic3C2212yescomplemented
13C3×C3⋊SL(2,ℤ/4)C2212yescomplemented
14C22×C3⋊S4D311yescharacteristic
15C22×C3×S4—31no—
16C22×C3×S4—61no—
17C23×C3⋊Dic3D311yescharacteristic, complemented
18C22×C3×#67—31no—
19C22×D3×Dic3—31no—
20C2×Dic3×A4—31no—
21C2×Dic3×A4—31no—
22C2×Dic3×A4—61no—
23C2×C3⋊SL(2,ℤ/4)D311yescharacteristic, complemented
24C2×C3×SL(2,ℤ/4)—31no—
25C2×C3×SL(2,ℤ/4)—61no—
26(C3⋊S4)⋊C4—34no—
27Dic3×S4—34no—
28C2×C3×#217—31no—
29#67⋊Dic3—34no—
30#67⋊Dic3—34no—
31C2×Dic3⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C2×Dic32—31no—
34C22×C3×C3⋊D3—41no—
35C2×C32×A4C2×C412yescomplemented
36C2×C32×A4C2311yescharacteristic
37C3×C3⋊S4C2×C414yescomplemented
38C2×C3×C3⋊Dic3—41no—
39C2×C3×C3⋊Dic3—41no—
40#6294—44no—
41C2×C4×S4—91no—
42C2×D4×Dic3—91no—
43C2×#67⋊C4—91no—
44C24×C32C2×D311yescharacteristic
45C23×C3×D3—31no—
46C22×C3×A4—21no—
47C22×C3×A4—21no—
48C22×C3×A4—21no—
49C22×C3×A4—41no—
50C22×C3×A4C2×D311yescharacteristic
51C22×C3×A4C2×D311yescharacteristic
52C2×C3⋊S4Dic314yescomplemented
53C2×C3⋊S4C2×D312yes—
54C2×C3×S4—34no—
55C2×C3×S4—32no—
56C2×C3×S4—64no—
57C2×C3×S4—62no—
58C22×C3⋊Dic3—31no—
59C22×C3⋊Dic3—31no—
60C22×C3⋊Dic3—32no—
61C22×C3⋊Dic3C2×D312yescomplemented
62C2×C3×#67—34no—
63C2×C3×#67—32no—
64C2×C3×#67—34no—
65C2×C3×#67—32no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C2×#624—121no—
70C2×D3×Dic3—34no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—62no—
73C2×D3×Dic3—61no—
74C2×D3×Dic3—121no—
75Dic3×A4—32no—
76Dic3×A4—32no—
77Dic3×A4—62no—
78C3⋊SL(2,ℤ/4)C2×D312yescomplemented
79C3×SL(2,ℤ/4)—32no—
80C3×SL(2,ℤ/4)—62no—
81C3×#217—32no—
82C3×#217—32no—
83Dic3⋊Dic3—34no—
84(C2×D3)⋊Dic3—34no—
85Dic32—34no—
86C22×C33—41no—
87C2×C3×C3⋊D3—44no—
88C2×C3×C3⋊D3—42no—
89C32×A4C22×C411yescharacteristic, complemented
90C3×C3⋊Dic3—42no—
91C3×C3⋊Dic3—42no—
92C22×S4—31no—
93C22×S4—61no—
94C22×C3×D4—91no—
95C22×#67—31no—
96C23×Dic3—31no—
97C23×Dic3—31no—
98C23×Dic3—61no—
99C23×Dic3—91no—
100C22×C4×D3—91no—
101C2×C4×A4—91no—
102C2×SL(2,ℤ/4)—31no—
103C2×SL(2,ℤ/4)—61no—
104C4×S4—94no—
105C2×C3×C22⋊C4—91no—
106C2×#217—31no—
107C2×#217—91no—
108D4×Dic3—94no—
109D4×Dic3—94no—
110C2×(C2×D3)⋊C4—91no—
111C2×C4⋊Dic3—91no—
112C2×Dic3⋊C4—91no—
113C2×C4×Dic3—91no—
114C2×C4×Dic3—91no—
115#67⋊C4—94no—
116#67⋊C4—94no—
117C23×C32—31no—
118C23×C32—31no—
119C23×C32—32no—
120C23×C32C4×D312yescomplemented
121C23×C32C22×D311yescharacteristic
122C22×C3⋊D3—41no—
123C22×C3×D3—32no—
124C22×C3×D3—34no—
125C22×C3×D3—62no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—121no—
129C22×C3×D3—241no—
130C2×C3×A4—22no—
131C2×C3×A4—22no—
132C2×C3×A4—21no—
133C2×C3×A4—21no—
134C2×C3×A4—22no—
135C2×C3×A4—21no—
136C2×C3×A4—42no—
137C2×C3×A4—41no—
138C2×C3×A4C4×D312yescomplemented
139C2×C3×A4C2×Dic312yescomplemented
140C2×C3×A4C22×D311yescharacteristic
141C2×C3×A4C22×D311yescharacteristic
142C3⋊S4C2×Dic314yescomplemented
143C3×S4—34no—
144C3×S4—64no—
145C2×C3⋊Dic3—31no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—32no—
149C2×C3⋊Dic3—41no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3S411yescharacteristic, complemented
152C3×#67—38no—
153C3×#67—34no—
154C3×#67—34no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—121no—
163C2×C3×Dic3—121no—
164C2×C3×Dic3—121no—
165C2×C3×Dic3—241no—
166C2×C3×Dic3—241no—
167#624—124no—
168D3×Dic3—64no—
169D3×Dic3—64no—
170D3×Dic3—124no—
171C2×C4×D4—271noSylow
172C2×C33—42no—
173C2×C33—41no—
174C3×C3⋊D3—44no—
175C24×C3—21no—
176C24×C3—91no—
177C24×C3D3211yescharacteristic
178C24×C3D3211yescharacteristic
179C23×D3—31no—
180C22×A4—21no—
181C22×A4—21no—
182C22×A4—41no—
183C22×A4D3211yescharacteristic
184C2×S4—34no—
185C2×S4—32no—
186C2×S4—64no—
187C2×S4—62no—
188C2×C3×D4—92no—
189C2×C3×D4—94no—
190C2×C3×D4—94no—
191C2×C3×D4—92no—
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C2×#67—32no—
196C2×#67—34no—
197C2×#67—34no—
198C2×#67—32no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—32no—
203C22×Dic3—32no—
204C22×Dic3—62no—
205C22×Dic3—91no—
206C22×Dic3—91no—
207C22×Dic3—94no—
208C22×Dic3—92no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—92no—
212C22×Dic3—181no—
213C22×Dic3—181no—
214C22×Dic3—182no—
215C22×Dic3—181no—
216C22×Dic3—182no—
217C22×Dic3—181no—
218C2×C4×D3—94no—
219C2×C4×D3—181no—
220C2×C4×D3—182no—
221C2×C4×D3—181no—
222C2×C4×D3—361no—
223C4×A4—92no—
224SL(2,ℤ/4)—32no—
225SL(2,ℤ/4)—62no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228#217—32no—
229#217—32no—
230#217—94no—
231(C2×D3)⋊C4—94no—
232C4⋊Dic3—94no—
233Dic3⋊C4—94no—
234C4×Dic3—94no—
235C4×Dic3—94no—
236C22×C32—32no—
237C22×C32—31no—
238C22×C32—32no—
239C22×C32—32no—
240C22×C32—31no—
241C22×C32—31no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—61no—
245C22×C32—81no—
246C22×C32—81no—
247C22×C32—81no—
248C22×C32—161no—
249C22×C32C2×C4×D311yescharacteristic, complemented
250C22×C32C2×S411yescharacteristic
251C2×C3⋊D3—44no—
252C2×C3⋊D3—42no—
253C2×C3×D3—34no—
254C2×C3×D3—62no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—62no—
258C2×C3×D3—122no—
259C2×C3×D3—124no—
260C2×C3×D3—242no—
261C2×C3×D3—244no—
262C3×A4—21no—
263C3×A4—21no—
264C3×A4—21no—
265C3×A4—41no—
266C3×A4C2×C4×D311yescharacteristic, complemented
267C3×A4C22×Dic311yescharacteristic, complemented
268C3⋊Dic3—32no—
269C3⋊Dic3—42no—
270C3⋊Dic3C2×S412yescomplemented
271C3×Dic3—34no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—122no—
275C3×Dic3—122no—
276C3×Dic3—122no—
277C3×Dic3—242no—
278C3×Dic3—242no—
279C22×D4—91no—
280C23×C4—91no—
281C23×C4—271no—
282C4×D4—274no—
283C4×D4—274no—
284C2×C4⋊C4—271no—
285C2×C22⋊C4—91no—
286C2×C22⋊C4—271no—
287C2×C42—271no—
288C33—41noSylow
289C23×C3—22no—
290C23×C3—21no—
291C23×C3—31no—
292C23×C3—31no—
293C23×C3—31no—
294C23×C3—31no—
295C23×C3—32no—
296C23×C3—32no—
297C23×C3—61no—
298C23×C3—61no—
299C23×C3—62no—
300C23×C3—92no—
301C23×C3—94no—
302C23×C3—181no—
303C23×C3—182no—
304C23×C3—181no—
305C23×C3D3×Dic312yescomplemented
306C23×C3#62412yescomplemented
307C23×C3C2×D3211yescharacteristic
308C23×C3C2×D3211yescharacteristic
309C22×D3—32no—
310C22×D3—34no—
311C22×D3—61no—
312C22×D3—62no—
313C22×D3—61no—
314C22×D3—121no—
315C22×D3—241no—
316C2×A4—22no—
317C2×A4—22no—
318C2×A4—21no—
319C2×A4—21no—
320C2×A4—42no—
321C2×A4—41no—
322C2×A4D3×Dic312yescomplemented
323C2×A4C2×D3211yescharacteristic
324S4—34no—
325S4—64no—
326C3×D4—98no—
327C3×D4—94no—
328C3×D4—94no—
329C2×C3×C4—92no—
330C2×C3×C4—94no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—181no—
334C2×C3×C4—182no—
335C2×C3×C4—181no—
336C2×C3×C4—361no—
337#67—38no—
338#67—34no—
339#67—34no—
340C2×Dic3—32no—
341C2×Dic3—34no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—32no—
346C2×Dic3—61no—
347C2×Dic3—62no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—92no—
357C2×Dic3—92no—
358C2×Dic3—121no—
359C2×Dic3—182no—
360C2×Dic3—182no—
361C2×Dic3—181no—
362C2×Dic3—182no—
363C2×Dic3—184no—
364C2×Dic3—184no—
365C2×Dic3—181no—
366C2×Dic3—181no—
367C2×Dic3—182no—
368C2×Dic3—241no—
369C4×D3—184no—
370C4×D3—184no—
371C4×D3—364no—
372C2×C32—31no—
373C2×C32—31no—
374C2×C32—32no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—42no—
378C2×C32—42no—
379C2×C32—81no—
380C2×C32—82no—
381C2×C32—81no—
382C2×C32—81no—
383C2×C32—82no—
384C2×C32—82no—
385C2×C32—162no—
386C2×C32—161no—
387C2×C32C4×S412yescomplemented
388C2×C32C22×S411yescharacteristic
389C3⋊D3—44no—
390C3×D3—64no—
391C3×D3—124no—
392C3×D3—244no—
393C24—91no—
394C24#111311yescharacteristic
395C2×D4—92no—
396C2×D4—94no—
397C2×D4—94no—
398C2×D4—92no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—91no—
402C22×C4—92no—
403C22×C4—271no—
404C22×C4—274no—
405C22×C4—271no—
406C22×C4—272no—
407C22×C4—541no—
408C22×C4—542no—
409C22×C4—541no—
410C4⋊C4—274no—
411C22⋊C4—92no—
412C22⋊C4—92no—
413C22⋊C4—274no—
414C42—274no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—32no—
418C22×C3—31no—
419C22×C3—32no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—32no—
426C22×C3—32no—
427C22×C3—32no—
428C22×C3—32no—
429C22×C3—41no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—62no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—94no—
442C22×C3—161no—
443C22×C3—182no—
444C22×C3—184no—
445C22×C3—182no—
446C22×C3—184no—
447C22×C3C2×D3×Dic311yescharacteristic, complemented
448C22×C3C2×#62411yescharacteristic, complemented
449C22×C3D3×S411yescharacteristic
450C22×C3D3×S411yescharacteristic
451C2×D3—34no—
452C2×D3—62no—
453C2×D3—64no—
454C2×D3—62no—
455C2×D3—64no—
456C2×D3—122no—
457C2×D3—124no—
458C2×D3—244no—
459C2×D3—242no—
460A4—21no—
461A4—21no—
462A4—41no—
463A4C2×D3×Dic311yescharacteristic, complemented
464C3×C4—94no—
465C3×C4—182no—
466C3×C4—182no—
467C3×C4—362no—
468Dic3—34no—
469Dic3—32no—
470Dic3—32no—
471Dic3—62no—
472Dic3—62no—
473Dic3—62no—
474Dic3—92no—
475Dic3—92no—
476Dic3—122no—
477Dic3—182no—
478Dic3—242no—
479C32—41no—
480C32—41no—
481C32—81no—
482C32—81no—
483C32—81no—
484C32—161no—
485C32C2×C4×S411yescharacteristic, complemented
486C23—31no—
487C23—31no—
488C23—32no—
489C23—92no—
490C23—94no—
491C23—181no—
492C23—182no—
493C23—181no—
494C23#629412yescomplemented
495C23C2×#111311yescharacteristic
496D4—98no—
497D4—94no—
498D4—94no—
499C2×C4—92no—
500C2×C4—94no—
501C2×C4—91no—
502C2×C4—92no—
503C2×C4—92no—
504C2×C4—181no—
505C2×C4—182no—
506C2×C4—181no—
507C2×C4—272no—
508C2×C4—272no—
509C2×C4—271no—
510C2×C4—272no—
511C2×C4—544no—
512C2×C4—544no—
513C2×C4—541no—
514C2×C3—22no—
515C2×C3—21no—
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—32no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—32no—
522C2×C3—42no—
523C2×C3—41no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—62no—
527C2×C3—82no—
528C2×C3—82no—
529C2×C3—81no—
530C2×C3—81no—
531C2×C3—162no—
532C2×C3—161no—
533C2×C3—184no—
534C2×C3Dic3×S412yescomplemented
535C2×C3(C3⋊S4)⋊C412yescomplemented
536C2×C3C2×D3×S411yescharacteristic
537C2×C3C2×D3×S411yescharacteristic
538D3—64no—
539D3—124no—
540D3—244no—
541C22—32no—
542C22—31no—
543C22—32no—
544C22—32no—
545C22—31no—
546C22—31no—
547C22—61no—
548C22—94no—
549C22—182no—
550C22—184no—
551C22—184no—
552C22—182no—
553C22C2×#629411yescharacteristic, complemented
554C22#8910111yescharacteristic
555C4—94no—
556C4—92no—
557C4—182no—
558C4—182no—
559C4—272no—
560C3—21no—
561C3—41no—
562C3—81no—
563C3—81no—
564C3—161no—
565C3C2×Dic3×S411yescharacteristic, complemented
566C3C2×(C3⋊S4)⋊C411yescharacteristic, complemented
567C2—31no—
568C2—31no—
569C2—32no—
570C2—184no—
571C2#1172421312yesdirect product factor
572C2C2×#8910111yescharacteristic
subgroupG/H
center, hypercenterC22#89101
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×#89101
Fitting, socleC24×C32C2×D3
prime radicalC22×C3×C3⋊S4C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×C3⋊Dic3C2semidirect
3. C2×C3×C3⋊SL(2,ℤ/4)C2semidirect
4. #11724213C2direct (N × H)
9. C2×C3×C3⋊S4C4semidirect
12. A4×C3⋊Dic3C22semidirect
13. C3×C3⋊SL(2,ℤ/4)C22semidirect
17. C23×C3⋊Dic3D3semidirect
23. C2×C3⋊SL(2,ℤ/4)D3semidirect
35. C2×C32×A4C2×C4semidirect
37. C3×C3⋊S4C2×C4semidirect
52. C2×C3⋊S4Dic3semidirect
61. C22×C3⋊Dic3C2×D3semidirect
78. C3⋊SL(2,ℤ/4)C2×D3semidirect
89. C32×A4C22×C4semidirect
120. C23×C32C4×D3semidirect
138. C2×C3×A4C4×D3semidirect
139. C2×C3×A4C2×Dic3semidirect
142. C3⋊S4C2×Dic3semidirect
151. C2×C3⋊Dic3S4semidirect
249. C22×C32C2×C4×D3semidirect
266. C3×A4C2×C4×D3semidirect
267. C3×A4C22×Dic3semidirect
270. C3⋊Dic3C2×S4semidirect
305. C23×C3D3×Dic3semidirect
306. C23×C3#624semidirect
322. C2×A4D3×Dic3semidirect
387. C2×C32C4×S4semidirect
447. C22×C3C2×D3×Dic3semidirect
448. C22×C3C2×#624semidirect
463. A4C2×D3×Dic3semidirect
485. C32C2×C4×S4semidirect
494. C23#6294semidirect
534. C2×C3Dic3×S4semidirect
535. C2×C3(C3⋊S4)⋊C4semidirect
553. C22C2×#6294semidirect
565. C3C2×Dic3×S4semidirect
566. C3C2×(C3⋊S4)⋊C4semidirect
571. C2#11724213direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊S42
C2×A4×C3⋊Dic32
C2×C3×C3⋊SL(2,ℤ/4)2
#117242132
C2×(C3⋊S4)⋊C43
C2×Dic3×S43
C2×#67⋊Dic33
C2×#62944