Finite Group Databasebeta

← C2×#11723703 #13178660

Subgroups — C2×#11723703 #13178660

Order 1728.

1562 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1562 conjugacy classes of subgroups — too many to draw. Only its 199 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×#3830C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C2×C3×#3753C212yesmaximal, complemented
4#11723703C218yesmaximal, direct product factor
5C2×#88811C212yesmaximal, complemented
6C2×#88809C211yesmaximal, characteristic, complemented
7C2×#63927—31nomaximal
8C2×#63925—32nomaximal
9C22×#1113—21no—
10C2×C33×D4C2211yescharacteristic, complemented
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830C2214yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic, complemented
18C2×#6295—21no—
19C4⋊#1113—22no—
20C4⋊#1113C2214yescomplemented
21C3×#3753C2218yescomplemented
22C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
23#88811C2218yescomplemented
24#88809C2214yescomplemented
25C2×#3830D311yescharacteristic, complemented
26C2×C3×#237—32no—
27C2×C3×#237—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30C2×#3753D312yescomplemented
31C2×C3×#213—32no—
32C2×C3×#213—32no—
33C2×C3×#213—62no—
34#63927—38no—
35#63925—316no—
36C2×#3717—32no—
37C2×#3716—32no—
38C2×#3715—32no—
39C2×#3712—31no—
40C2×#3711—32no—
41C23×C33—21no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D411yescharacteristic
44C2×#1113—24no—
45C2×#1113—44no—
46C33×D4—21no—
47C33×D4C2312yescomplemented
48C2×C33×C4C2311yescharacteristic
49C3×#638—24no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—24no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4⋊(C3⋊D3)C2314yes—
55C3×C4×C3⋊D3D412yes—
56C3×C4×C3⋊D3C2312yes—
57C3×#634—21no—
58C3×#634C2312yescomplemented
59#6295—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C2×#949—91no—
62C2×#911—92no—
63C22×D32—62no—
64C22×D32—61no—
65C2×C32×D4—21no—
66C2×C32×D4—22no—
67C2×C32×D4—41no—
68C2×C32×D4C2×D311yescharacteristic, complemented
69C2×C32×D4C2×D312yescomplemented
70C2×#638—21no—
71C22×C3⋊Dic3—21no—
72#3830—22no—
73#3830C2×D314yescomplemented
74C2×C4⋊(C3⋊D3)C2×D312yes—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic, complemented
77C2×C3×#67—62no—
78C2×C3×#67—62no—
79C22×C3×Dic3—62no—
80C22×C3×Dic3—62no—
81C3×#237—38no—
82C3×#237—64no—
83C3×#237—64no—
84C3×#237—64no—
85C2×C3×D12—32no—
86C2×C3×D12—32no—
87C2×C3×D12—62no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—61no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—61no—
92C2×#626—62no—
93C2×#625—61no—
94#3802—34no—
95#3802—62no—
96D3×D12—38no—
97D3×D12—64no—
98#3753C2×D318yescomplemented
99C2×C3⋊(C3⋊C8)C2×D312yescomplemented
100C3×#213—38no—
101C3×#213—38no—
102C3×#213—68no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—32no—
105C2×C3×C3⋊C8—62no—
106#3717—38no—
107#3716—38no—
108#3715—38no—
109#3712—34no—
110#3711—38no—
111C22×C33—22no—
112C22×C33—42no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—24no—
116C2×C3×C3⋊D3—44no—
117C2×C3×C3⋊D3C2×D412yes—
118#1113—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3C2×D412yes—
123C2×C3×#41—91no—
124C2×#237—32no—
125C2×#237—61no—
126C2×D3×D4—92no—
127C22×D12—91no—
128C2×C3×D8—92no—
129#949—98no—
130C2×#215—92no—
131C2×#214—92no—
132C2×#213—32no—
133C2×#213—32no—
134C2×#213—62no—
135C2×#208—91no—
136#911—916no—
137C2×#204—92no—
138C2×ΓL(1,𝔽25)—92no—
139C23×C32—21no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—42no—
143C23×C32—42no—
144C22×C3⋊D3—22no—
145C22×C3⋊D3#6711yescharacteristic
146C22×C3×D3—32no—
147C22×C3×D3—61no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C22×C3×D3—122no—
151C2×D32—68no—
152C2×D32—64no—
153C2×D32—128no—
154C2×D32—124no—
155C32×D4—21no—
156C32×D4—22no—
157C32×D4—22no—
158C32×D4—22no—
159C32×D4—24no—
160C32×D4—42no—
161C32×D4—42no—
162C32×D4—42no—
163C32×D4C22×D312yescomplemented
164C32×D4C22×D314yescomplemented
165C2×C32×C4—22no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170#638—24no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—24no—
173C2×C3⋊Dic3#6711yescharacteristic
174C4⋊(C3⋊D3)—22no—
175C4⋊(C3⋊D3)C22×D314yes—
176C4×C3⋊D3#6712yes—
177C4×C3⋊D3C22×D312yes—
178#634—21no—
179#634C22×D312yescomplemented
180C3×#67—68no—
181C3×#67—68no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—68no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—68no—
187C2×C3×Dic3—62no—
188C3×D12—34no—
189C3×D12—34no—
190C3×D12—64no—
191C3×D12—62no—
192C3×D12—62no—
193C3×D12—122no—
194C3×C4×D3—34no—
195C3×C4×D3—34no—
196C3×C4×D3—62no—
197C3×C4×D3—62no—
198C3×Dic6—34no—
199C3×Dic6—62no—
200C3×Dic6—62no—
201C3×Dic6—62no—
202#626—68no—
203#625—64no—
204C3⋊(C3⋊C8)C22×D314yescomplemented
205C3×C3⋊C8—34no—
206C3×C3⋊C8—34no—
207C3×C3⋊C8—64no—
208C2×AGL(1,ℤ/8)—271noSylow
209C2×C33—42no—
210C2×C33AGL(1,ℤ/8)12yescomplemented
211C2×C33C22×D411yescharacteristic
212C3×C3⋊D3—22no—
213C3×C3⋊D3—44no—
214C23×D3—182no—
215C23×D3—181no—
216C3×#41—94no—
217C3×#41—182no—
218C2×C3×Q8—91no—
219C2×C3×D4—21no—
220C2×C3×D4—22no—
221C2×C3×D4—41no—
222C2×C3×D4—92no—
223C2×C3×D4—181no—
224C2×C3×D4D3211yescharacteristic, complemented
225C2×C3×D4D3212yescomplemented
226C22×C3×C4—91no—
227C22×C3×C4—181no—
228C2×#67—62no—
229C2×#67—62no—
230C2×#67—182no—
231C22×Dic3—62no—
232C22×Dic3—62no—
233#237—38no—
234#237—64no—
235#237—64no—
236#237—64no—
237D3×D4—98no—
238D3×D4—184no—
239C2×D12—32no—
240C2×D12—32no—
241C2×D12—62no—
242C2×D12—94no—
243C2×D12—182no—
244C2×D12—181no—
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×Dic6—32no—
248C2×Dic6—61no—
249C3×D8—98no—
250C2×C3×C8—92no—
251#215—98no—
252#214—98no—
253#213—38no—
254#213—38no—
255#213—68no—
256#208—94no—
257C2×C3⋊C8—32no—
258C2×C3⋊C8—32no—
259C2×C3⋊C8—62no—
260#204—98no—
261ΓL(1,𝔽25)—98no—
262C22×C32—22no—
263C22×C32—24no—
264C22×C32—24no—
265C22×C32—22no—
266C22×C32—21no—
267C22×C32—41no—
268C22×C32—44no—
269C22×C32—44no—
270C22×C32—42no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32—84no—
274C22×C32—84no—
275C22×C32D3×D412yes—
276C22×C32C2×#6711yescharacteristic
277C2×C3⋊D3—22no—
278C2×C3⋊D3—24no—
279C2×C3⋊D3—44no—
280C2×C3⋊D3C2×#6712yes—
281C2×C3×D3—34no—
282C2×C3×D3—62no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C2×C3×D3—64no—
286C2×C3×D3—122no—
287C2×C3×D3—124no—
288C2×C3×D3—124no—
289C2×C3×D3—124no—
290C2×C3×D3—244no—
291D32—1216no—
292D32—128no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4D3×D412yes—
300C32×C4C2×#6711yescharacteristic
301C32×C4C23×D312yes—
302C32×C4C23×D311yescharacteristic
303C3⋊Dic3—22no—
304C3⋊Dic3C2×#6712yes—
305C3×Dic3—34no—
306C3×Dic3—62no—
307C3×Dic3—64no—
308C3×Dic3—64no—
309C2×#41—91no—
310C22×D4—271no—
311AGL(1,ℤ/8)—278no—
312C2×SD16—272no—
313C2×D8—92no—
314C2×M16—271no—
315C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—91no—
322C23×C3—182no—
323C22×D3—32no—
324C22×D3—61no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—122no—
328C22×D3—188no—
329C22×D3—184no—
330C22×D3—368no—
331C22×D3—364no—
332C3×Q8—92no—
333C3×Q8—181no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—21no—
337C3×D4—22no—
338C3×D4—24no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—94no—
343C3×D4—184no—
344C3×D4—182no—
345C3×D4C2×D3212yescomplemented
346C3×D4C2×D3214yescomplemented
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—91no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—184no—
354C2×C3×C4—181no—
355C2×C3×C4C2×D3212yes—
356C2×C3×C4C2×D3211yescharacteristic
357#67—68no—
358#67—68no—
359#67—188no—
360C2×Dic3—32no—
361C2×Dic3—61no—
362C2×Dic3—68no—
363C2×Dic3—62no—
364C2×Dic3—68no—
365C2×Dic3—62no—
366D12—34no—
367D12—34no—
368D12—64no—
369D12—62no—
370D12—62no—
371D12—122no—
372D12—184no—
373C4×D3—34no—
374C4×D3—34no—
375C4×D3—62no—
376C4×D3—62no—
377Dic6—34no—
378Dic6—62no—
379Dic6—62no—
380Dic6—62no—
381C3×C8—94no—
382C3⋊C8—34no—
383C3⋊C8—34no—
384C3⋊C8—64no—
385C2×C32—22no—
386C2×C32—21no—
387C2×C32—24no—
388C2×C32—22no—
389C2×C32—42no—
390C2×C32—41no—
391C2×C32—42no—
392C2×C32—44no—
393C2×C32—44no—
394C2×C32—84no—
395C2×C32—84no—
396C2×C32#91114yescomplemented
397C2×C32#94912yescomplemented
398C2×C32C2×D3×D412yes—
399C2×C32C22×#6711yescharacteristic
400C3⋊D3—22no—
401C3⋊D3—44no—
402C3×D3—64no—
403C3×D3—122no—
404C3×D3—124no—
405C3×D3—124no—
406C3×D3—244no—
407C24—541no—
408#41—94no—
409#41—182no—
410C2×Q8—91no—
411C2×D4—92no—
412C2×D4—181no—
413C2×D4—274no—
414C2×D4—542no—
415C2×D4—541no—
416C2×D4#111311yescharacteristic, complemented
417C22×C4—91no—
418C22×C4—181no—
419SD16—278no—
420D8—98no—
421M16—274no—
422C2×C8—92no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—22no—
426C22×C3—21no—
427C22×C3—22no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—84no—
435C22×C3—84no—
436C22×C3—92no—
437C22×C3—182no—
438C22×C3—184no—
439C22×C3—364no—
440C22×C3D3×#6712yes—
441C22×C3#381111yescharacteristic
442C2×D3—34no—
443C2×D3—62no—
444C2×D3—64no—
445C2×D3—64no—
446C2×D3—64no—
447C2×D3—122no—
448C2×D3—124no—
449C2×D3—124no—
450C2×D3—124no—
451C2×D3—244no—
452C2×D3—3616no—
453C2×D3—368no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—92no—
461C3×C4—182no—
462C3×C4D3×#6712yes—
463C3×C4#381111yescharacteristic
464C3×C4C22×D3212yes—
465C3×C4C22×D3211yescharacteristic
466Dic3—34no—
467Dic3—62no—
468Dic3—64no—
469Dic3—64no—
470C32—22no—
471C32—21no—
472C32—41no—
473C32C2×#91112yescomplemented
474C32C2×#94911yescharacteristic, complemented
475C23—21no—
476C23—91no—
477C23—182no—
478C23—544no—
479C23—1084no—
480Q8—92no—
481Q8—181no—
482D4—21no—
483D4—94no—
484D4—184no—
485D4—182no—
486D4—544no—
487D4C2×#111312yescomplemented
488C2×C4—91no—
489C2×C4—92no—
490C2×C4—92no—
491C2×C4—184no—
492C2×C4—181no—
493C2×C4C2×#111311yescharacteristic
494C8—94no—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—24no—
499C2×C3—44no—
500C2×C3—44no—
501C2×C3—42no—
502C2×C3—41no—
503C2×C3—42no—
504C2×C3—84no—
505C2×C3—84no—
506C2×C3—182no—
507C2×C3—364no—
508C2×C3#6392514yescomplemented
509C2×C3#6392712yescomplemented
510C2×C3C2×D3×#6712yes—
511C2×C3C2×#381111yescharacteristic
512D3—64no—
513D3—122no—
514D3—124no—
515D3—124no—
516D3—244no—
517C22—22no—
518C22—42no—
519C22—92no—
520C22—182no—
521C22—184no—
522C22—364no—
523C22—1088no—
524C22#8904911yescharacteristic
525C4—92no—
526C4—182no—
527C4#8904911yescharacteristic
528C4C22×#111311yescharacteristic
529C3—21no—
530C3—22no—
531C3—41no—
532C3C2×#6392512yescomplemented
533C3C2×#6392711yescharacteristic, complemented
534C2—42no—
535C2—182no—
536C2—364no—
537C2#1172370312yesdirect product factor
538C2C2×#8904911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×C3×#3753C2semidirect
4. #11723703C2direct (N × H)
5. C2×#88811C2semidirect
6. C2×#88809C2semidirect
10. C2×C33×D4C22semidirect
14. C3×#3830C22semidirect
17. C2×C3×#634C22semidirect
20. C4⋊#1113C22semidirect
21. C3×#3753C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. #88811C22semidirect
24. #88809C22semidirect
25. C2×#3830D3semidirect
30. C2×#3753D3semidirect
47. C33×D4C23semidirect
58. C3×#634C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
68. C2×C32×D4C2×D3semidirect
69. C2×C32×D4C2×D3semidirect
73. #3830C2×D3semidirect
76. C2×#634C2×D3semidirect
98. #3753C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
163. C32×D4C22×D3semidirect
164. C32×D4C22×D3semidirect
179. #634C22×D3semidirect
204. C3⋊(C3⋊C8)C22×D3semidirect
210. C2×C33AGL(1,ℤ/8)semidirect
224. C2×C3×D4D32semidirect
225. C2×C3×D4D32semidirect
315. C33C2×AGL(1,ℤ/8)semidirect
345. C3×D4C2×D32semidirect
346. C3×D4C2×D32semidirect
396. C2×C32#911semidirect
397. C2×C32#949semidirect
416. C2×D4#1113semidirect
473. C32C2×#911semidirect
474. C32C2×#949semidirect
487. D4C2×#1113semidirect
508. C2×C3#63925semidirect
509. C2×C3#63927semidirect
532. C3C2×#63925semidirect
533. C3C2×#63927semidirect
537. C2#11723703direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C4⋊#11132
C2×C3×#37532
#117237032
C2×#888112
C2×#888092
C2×#639273
C2×#639253