Finite Group Databasebeta

← C2×#11724138 #13180592

Subgroups — C2×#11724138 #13180592

Order 1728.

1104 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 116 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×#88963C211yesmaximal, characteristic, complemented
2C2×D3×#625C211yesmaximal, characteristic, complemented
3C2×#88942C211yesmaximal, characteristic, complemented
4C2×#88937C211yesmaximal, characteristic, complemented
5C2×#88935C211yesmaximal, characteristic, complemented
6C2×C3×#3774C211yesmaximal, characteristic, complemented
7C2×#88925C211yesmaximal, characteristic, complemented
8#11724138C218yesmaximal, direct product factor
9C2×#64236—31nomaximal
10C22×C3×D32—21no—
11C2×#6296C2211yescharacteristic, complemented
12C2×#6295C2211yescharacteristic
13C2×#6290—21no—
14C2×D3×C3⋊Dic3—21no—
15C2×D3×C3⋊Dic3C2211yescharacteristic
16C2×C3×#625—21no—
17C2×C3×#625C2211yescharacteristic
18C2×C3×#625C2211yescharacteristic, complemented
19C2×C3⋊#622C2211yescharacteristic, complemented
20C2×C3×#622C2211yescharacteristic, complemented
21#88963—22no—
22#88963C2214yescomplemented
23D3×#625—22no—
24D3×#625C2214yescomplemented
25#88942C2214yescomplemented
26#88937C2214yescomplemented
27#88935C2214yescomplemented
28C3×#3774C2214yescomplemented
29#88925C2214yescomplemented
30C2×D3×#67—61no—
31C22×#625—31no—
32C2×#3805—31no—
33C2×#3805—61no—
34C2×#3792—31no—
35C2×#3775—31no—
36C2×#3775—31no—
37#64236—38no—
38C2×#3774—31no—
39C2×#3774D311yescharacteristic, complemented
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C22×C32×D3D411yescharacteristic
44C2×C3×D32—24no—
45C2×C3×D32—44no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3C2311yescharacteristic
49#6296—21no—
50#6296C2312yescomplemented
51#6295—21no—
52#6295C2312yes—
53#6290—24no—
54D3×C3⋊Dic3—24no—
55D3×C3⋊Dic3D412yes—
56D3×C3⋊Dic3C2312yes—
57C3×#625—24no—
58C3×#625—21no—
59C3×#625—21no—
60C3×#625C2312yes—
61C3×#625C2312yescomplemented
62C3⋊#622C2312yescomplemented
63C3×#622C2312yescomplemented
64C2×#911—91nomaximal
65C23×C3×D3—61no—
66C23×C3×D3—61no—
67C22×D32—21no—
68C2×#638—61no—
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—61no—
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75D3×#67—64no—
76D3×#67—64no—
77C2×#627—31no—
78C2×#627—61no—
79C2×#626—61no—
80C2×#625—21no—
81C2×#625—31no—
82C2×#625—34no—
83C2×#625—62no—
84C2×#625—61no—
85C2×#625—61no—
86C2×#625—121no—
87C2×#625C2×D311yescharacteristic
88C2×#625C2×D311yescharacteristic, complemented
89#3805—34no—
90#3805—62no—
91#3805—64no—
92#3805—64no—
93C2×D3×Dic3—61no—
94C2×D3×Dic3—61no—
95C2×D3×Dic3—61no—
96C2×D3×Dic3—61no—
97C2×D3×Dic3—121no—
98#3792—34no—
99C2×#622—31no—
100C2×#622C2×D311yescharacteristic, complemented
101#3775—34no—
102#3775—34no—
103#3774—34no—
104#3774C2×D314yescomplemented
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—42no—
108C2×C32×D3—22no—
109C2×C32×D3—22no—
110C2×C32×D3—22no—
111C2×C32×D3—42no—
112C2×C32×D3—42no—
113C2×C32×D3C2×D412yes—
114C3×D32—48no—
115C3⋊(C3⋊Dic3)C2×D412yes—
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3C2×D411yescharacteristic
118C3×C3⋊Dic3C2411yescharacteristic
119C22×#67—181no—
120C2×#237—91no—
121C2×#237—181no—
122C2×D3×D4—91no—
123C2×C3×D8—91no—
124C2×#214—91no—
125C2×#213—91no—
126#911—98no—
127C2×#204—91no—
128C2×ΓL(1,𝔽25)—91no—
129C23×C32—31no—
130C23×C32—61no—
131C23×C32—61no—
132C22×C3⋊D3—21no—
133C22×C3×D3—21no—
134C22×C3×D3—21no—
135C22×C3×D3—21no—
136C22×C3×D3—21no—
137C22×C3×D3—21no—
138C22×C3×D3—21no—
139C22×C3×D3—41no—
140C22×C3×D3—41no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—121no—
146C22×C3×D3—124no—
147C22×C3×D3—124no—
148C2×D32—24no—
149C2×D32—44no—
150#638—64no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—31no—
153C2×C3⋊Dic3—32no—
154C2×C3⋊Dic3—32no—
155C2×C3⋊Dic3—61no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—64no—
158C2×C3⋊Dic3—121no—
159C2×C3⋊Dic3—121no—
160C2×C3⋊Dic3C22×D311yescharacteristic
161C3×#67—62no—
162C3×#67—62no—
163C3×#67—64no—
164C3×#67—62no—
165C3×#67—64no—
166C3×#67—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—121no—
172#627—32no—
173#627—62no—
174#627—62no—
175#627—61no—
176#626—62no—
177#626—62no—
178#625—24no—
179#625—21no—
180#625—21no—
181#625—32no—
182#625—64no—
183#625—64no—
184#625—61no—
185#625—124no—
186#625C22×D312yes—
187#625C22×D312yescomplemented
188D3×Dic3—62no—
189D3×Dic3—62no—
190D3×Dic3—62no—
191D3×Dic3—62no—
192D3×Dic3—64no—
193D3×Dic3—64no—
194D3×Dic3—124no—
195#622—32no—
196#622C22×D312yescomplemented
197C2×AGL(1,ℤ/8)—271noSylow
198C2×C33AGL(1,ℤ/8)12yescomplemented
199C2×C33C22×D411yescharacteristic
200C3×C3⋊D3—42no—
201C32×D3—22no—
202C32×D3—42no—
203C32×D3—42no—
204C24×C3—181no—
205C23×D3—61no—
206C23×D3—61no—
207C2×C3×D4—91no—
208C2×C3×D4—91no—
209C2×C3×D4—181no—
210C2×#67—61no—
211C2×#67—61no—
212C2×#67—61no—
213C2×#67—61no—
214C2×#67—181no—
215C2×#67—184no—
216C2×#67—181no—
217C2×#67—184no—
218C2×#67—181no—
219C2×#67—181no—
220C2×#67—361no—
221C22×Dic3—181no—
222C22×Dic3—181no—
223C22×Dic3—181no—
224C22×Dic3—181no—
225C22×Dic3—361no—
226#237—94no—
227#237—184no—
228#237—182no—
229#237—184no—
230D3×D4—94no—
231D3×D4—182no—
232C2×D12—91no—
233C2×C4×D3—91no—
234C2×C4×D3—181no—
235C2×Dic6—91no—
236C2×Dic6—181no—
237C3×D8—94no—
238C2×C3×C8—91no—
239#214—94no—
240#213—94no—
241C2×C3⋊C8—91no—
242#204—94no—
243ΓL(1,𝔽25)—94no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—32no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—62no—
250C22×C32—62no—
251C22×C32—62no—
252C22×C32—122no—
253C22×C32—122no—
254C22×C32D3×D411yescharacteristic
255C2×C3⋊D3—22no—
256C2×C3⋊D3—42no—
257C2×C3×D3—24no—
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—22no—
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—42no—
265C2×C3×D3—42no—
266C2×C3×D3—42no—
267C2×C3×D3—42no—
268C2×C3×D3—42no—
269C2×C3×D3—44no—
270C2×C3×D3—42no—
271C2×C3×D3—42no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—82no—
275C2×C3×D3—122no—
276C2×C3×D3—128no—
277C2×C3×D3—128no—
278C2×C3×D3—122no—
279C2×C3×D3—122no—
280C2×C3×D3—242no—
281D32—48no—
282C3⋊Dic3—22no—
283C3⋊Dic3—32no—
284C3⋊Dic3—62no—
285C3⋊Dic3—62no—
286C3⋊Dic3—122no—
287C3⋊Dic3—122no—
288C3⋊Dic3D3×D411yescharacteristic
289C3⋊Dic3C23×D311yescharacteristic
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—122no—
297C2×#41—271no—
298C22×D4—271no—
299AGL(1,ℤ/8)—278no—
300C2×SD16—271no—
301C2×SD16—271no—
302C2×D8—91no—
303C2×D8—271no—
304C2×M16—271no—
305C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
306C23×C3—61no—
307C23×C3—61no—
308C23×C3—61no—
309C23×C3—61no—
310C23×C3—61no—
311C23×C3—61no—
312C23×C3—121no—
313C23×C3—121no—
314C23×C3—181no—
315C23×C3—184no—
316C23×C3—364no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—64no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—64no—
323C22×D3—121no—
324C22×D3—124no—
325C22×D3—124no—
326C22×D3GO+(4,𝔽2)11yescharacteristic
327C3×D4—92no—
328C3×D4—92no—
329C3×D4—181no—
330C3×D4—181no—
331C3×D4—184no—
332C2×C3×C4—91no—
333C2×C3×C4—181no—
334#67—62no—
335#67—62no—
336#67—62no—
337#67—64no—
338#67—62no—
339#67—64no—
340#67—182no—
341#67—182no—
342#67—184no—
343#67—184no—
344#67—184no—
345#67—364no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—91no—
351C2×Dic3—121no—
352C2×Dic3—182no—
353C2×Dic3—182no—
354C2×Dic3—182no—
355C2×Dic3—182no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—184no—
359C2×Dic3—184no—
360C2×Dic3—361no—
361C2×Dic3—361no—
362C2×Dic3—364no—
363D12—92no—
364D12—181no—
365C4×D3—92no—
366C4×D3—92no—
367C4×D3—184no—
368Dic6—92no—
369Dic6—182no—
370Dic6—181no—
371Dic6—182no—
372C3×C8—92no—
373C3⋊C8—92no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—22no—
377C2×C32—22no—
378C2×C32—42no—
379C2×C32—42no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32—62no—
383C2×C32—122no—
384C2×C32—122no—
385C2×C32#91112yescomplemented
386C2×C32C2×D3×D411yescharacteristic
387C3⋊D3—42no—
388C3×D3—24no—
389C3×D3—42no—
390C3×D3—42no—
391C3×D3—42no—
392C3×D3—44no—
393C3×D3—42no—
394C3×D3—42no—
395C3×D3—82no—
396C3×D3—122no—
397C3×D3—122no—
398C3×D3—242no—
399C24—181no—
400#41—274no—
401#41—542no—
402C2×Q8—271no—
403C2×D4—91no—
404C2×D4—91no—
405C2×D4—181no—
406C2×D4—271no—
407C2×D4—274no—
408C2×D4—542no—
409C2×D4—541no—
410C22×C4—271no—
411C22×C4—541no—
412SD16—274no—
413SD16—274no—
414D8—94no—
415D8—274no—
416M16—274no—
417C2×C8—91no—
418C2×C8—271no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—64no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—122no—
431C22×C3—122no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—122no—
435C22×C3—124no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—182no—
439C22×C3—242no—
440C22×C3—362no—
441C22×C3—368no—
442C22×C3C2×GO+(4,𝔽2)11yescharacteristic
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—122no—
451C2×D3—128no—
452C2×D3—128no—
453C2×D3—122no—
454C2×D3—122no—
455C2×D3—242no—
456C2×D3C2×GO+(4,𝔽2)12yes—
457C3×C4—91no—
458C3×C4—91no—
459C3×C4—182no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—62no—
465Dic3—62no—
466Dic3—92no—
467Dic3—122no—
468Dic3—182no—
469Dic3—182no—
470Dic3—362no—
471Dic3—362no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32—41no—
476C32C2×#91111yescharacteristic, complemented
477C23—31no—
478C23—61no—
479C23—61no—
480C23—184no—
481C23—181no—
482C23—364no—
483Q8—272no—
484Q8—541no—
485D4—92no—
486D4—92no—
487D4—181no—
488D4—181no—
489D4—184no—
490D4—272no—
491D4—541no—
492D4—544no—
493C2×C4—91no—
494C2×C4—181no—
495C2×C4—272no—
496C2×C4—272no—
497C2×C4—271no—
498C2×C4—544no—
499C8—92no—
500C8—272no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—42no—
506C2×C3—42no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—64no—
510C2×C3—122no—
511C2×C3—122no—
512C2×C3—122no—
513C2×C3—124no—
514C2×C3—122no—
515C2×C3—122no—
516C2×C3—242no—
517C2×C3—362no—
518C2×C3#6423612yescomplemented
519C2×C3C22×GO+(4,𝔽2)11yescharacteristic
520D3—22no—
521D3—42no—
522D3—42no—
523D3—122no—
524D3—122no—
525D3—242no—
526C22—32no—
527C22—62no—
528C22—62no—
529C22—62no—
530C22—122no—
531C22—122no—
532C22—182no—
533C22—368no—
534C22—362no—
535C22D3×GO+(4,𝔽2)11yescharacteristic
536C4—91no—
537C4—91no—
538C4—182no—
539C4—272no—
540C3—21no—
541C3—21no—
542C3—41no—
543C3—41no—
544C3C2×#6423611yescharacteristic, complemented
545C2—62no—
546C2—122no—
547C2—122no—
548C2—362no—
549C2#1172413812yesdirect product factor
550C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88963C2semidirect
2. C2×D3×#625C2semidirect
3. C2×#88942C2semidirect
4. C2×#88937C2semidirect
5. C2×#88935C2semidirect
6. C2×C3×#3774C2semidirect
7. C2×#88925C2semidirect
8. #11724138C2direct (N × H)
11. C2×#6296C22semidirect
18. C2×C3×#625C22semidirect
19. C2×C3⋊#622C22semidirect
20. C2×C3×#622C22semidirect
22. #88963C22semidirect
24. D3×#625C22semidirect
25. #88942C22semidirect
26. #88937C22semidirect
27. #88935C22semidirect
28. C3×#3774C22semidirect
29. #88925C22semidirect
39. C2×#3774D3semidirect
50. #6296C23semidirect
61. C3×#625C23semidirect
62. C3⋊#622C23semidirect
63. C3×#622C23semidirect
88. C2×#625C2×D3semidirect
100. C2×#622C2×D3semidirect
104. #3774C2×D3semidirect
187. #625C22×D3semidirect
196. #622C22×D3semidirect
198. C2×C33AGL(1,ℤ/8)semidirect
305. C33C2×AGL(1,ℤ/8)semidirect
385. C2×C32#911semidirect
476. C32C2×#911semidirect
518. C2×C3#64236semidirect
544. C3C2×#64236semidirect
549. C2#11724138direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889632
C2×D3×#6252
C2×#889422
C2×#889372
C2×#889352
C2×C3×#37742
C2×#889252
#117241382
C2×#642363
C2×#9119