Finite Group Databasebeta

← C2×#11723709 #13178732

Subgroups — C2×#11723709 #13178732

Order 1728.

1530 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1530 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×#3832C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C2×C3×#3755C212yesmaximal, complemented
4#11723709C218yesmaximal, direct product factor
5C2×#88810C212yesmaximal, complemented
6C2×#88809C211yesmaximal, characteristic, complemented
7C2×#63941—31nomaximal
8C2×#63939—32nomaximal
9C22×#1113—21no—
10C2×C33×Q8C2211yescharacteristic, complemented
11C3×#3832—22no—
12C3×#3832C2214yescomplemented
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×#6295—21no—
19C4⋊#1113—22no—
20C4⋊#1113C2214yescomplemented
21C3×#3755C2218yescomplemented
22C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
23#88810C2218yescomplemented
24#88809C2214yescomplemented
25C2×#3832D311yescharacteristic, complemented
26C2×C3×#239—32no—
27C2×C3×#239—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30C2×#3755D312yescomplemented
31C2×C3×#215—32no—
32C2×C3×#215—32no—
33C2×C3×#215—62no—
34#63941—38no—
35#63939—316no—
36C2×#3714—32no—
37C2×#3714—32no—
38C2×#3713—32no—
39C2×#3712—31no—
40C2×#3711—32no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D411yescharacteristic
44C2×#1113—24no—
45C2×#1113—44no—
46C33×Q8—21no—
47C33×Q8C2312yescomplemented
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—24no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4⋊(C3⋊D3)C2314yes—
55C3×C4⋊(C3⋊D3)C2312yescomplemented
56C3×C4×C3⋊D3—24no—
57C3×C4×C3⋊D3D412yes—
58C3×C4×C3⋊D3C2312yes—
59#6295—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C2×#949—91no—
62C2×#914—92no—
63C22×D32—62no—
64C22×D32—61no—
65C2×C32×Q8—21no—
66C2×C32×Q8—22no—
67C2×C32×Q8—41no—
68C2×C32×Q8C2×D311yescharacteristic, complemented
69C2×C32×Q8C2×D312yescomplemented
70#3832—22no—
71#3832C2×D314yescomplemented
72C2×C4⋊(C3⋊D3)—21no—
73C2×C4⋊(C3⋊D3)C2×D312yes—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
75C2×C4×C3⋊D3—21no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C3×#239—38no—
78C3×#239—64no—
79C3×#239—64no—
80C3×#239—64no—
81C2×C3×D12—32no—
82C2×C3×D12—32no—
83C2×C3×D12—32no—
84C2×C3×D12—62no—
85C2×C3×D12—61no—
86C2×C3×D12—62no—
87C2×C3×D12—62no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—62no—
90C2×C3×C4×D3—61no—
91C2×C3×C4×D3—62no—
92C2×#626—62no—
93C2×#625—61no—
94#3802—34no—
95#3802—62no—
96D3×D12—38no—
97D3×D12—64no—
98#3755C2×D318yescomplemented
99C2×C3⋊(C3⋊C8)C2×D312yescomplemented
100C3×#215—38no—
101C3×#215—38no—
102C3×#215—68no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—32no—
105C2×C3×C3⋊C8—62no—
106#3714—38no—
107#3714—38no—
108#3713—38no—
109#3712—34no—
110#3711—38no—
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—24no—
115C2×C3×C3⋊D3—42no—
116C2×C3×C3⋊D3—44no—
117C2×C3×C3⋊D3C2×D412yes—
118#1113—48no—
119C33×C4—22no—
120C33×C4C2×D411yescharacteristic
121C33×C4C2411yescharacteristic
122C3×C3⋊Dic3C2×D412yes—
123C2×C3×#41—91no—
124C2×#239—32no—
125C2×#239—61no—
126C2×D3×D4—92no—
127C22×D12—91no—
128C2×C3×SD16—92no—
129#949—98no—
130C2×#215—32no—
131C2×#215—32no—
132C2×#215—62no—
133C2×#213—92no—
134C2×#213—92no—
135C2×#208—91no—
136#914—916no—
137C2×D24—92no—
138C2×ΓL(1,𝔽25)—92no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—122no—
149C2×D32—68no—
150C2×D32—64no—
151C2×D32—128no—
152C2×D32—124no—
153C32×Q8—21no—
154C32×Q8—22no—
155C32×Q8—22no—
156C32×Q8—22no—
157C32×Q8—24no—
158C32×Q8—42no—
159C32×Q8—42no—
160C32×Q8—42no—
161C32×Q8C22×D312yescomplemented
162C32×Q8C22×D314yescomplemented
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4—42no—
170C2×C32×C4—42no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3#6711yescharacteristic
174C4⋊(C3⋊D3)—24no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—22no—
177C4⋊(C3⋊D3)C22×D314yes—
178C4⋊(C3⋊D3)C22×D312yescomplemented
179C4×C3⋊D3—24no—
180C4×C3⋊D3#6712yes—
181C4×C3⋊D3C22×D312yes—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—61no—
184C3×D12—34no—
185C3×D12—34no—
186C3×D12—34no—
187C3×D12—62no—
188C3×D12—64no—
189C3×D12—68no—
190C3×D12—68no—
191C3×D12—62no—
192C3×D12—62no—
193C3×D12—62no—
194C3×D12—62no—
195C3×D12—122no—
196C3×C4×D3—34no—
197C3×C4×D3—34no—
198C3×C4×D3—62no—
199C3×C4×D3—62no—
200C3×C4×D3—68no—
201C3×C4×D3—68no—
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×C33AGL(1,ℤ/8)12yescomplemented
210C2×C33C22×D411yescharacteristic
211C3×C3⋊D3—22no—
212C3×C3⋊D3—42no—
213C3×C3⋊D3—44no—
214C23×D3—182no—
215C23×D3—181no—
216C3×#41—94no—
217C3×#41—182no—
218C2×C3×Q8—21no—
219C2×C3×Q8—22no—
220C2×C3×Q8—41no—
221C2×C3×Q8D3211yescharacteristic, complemented
222C2×C3×Q8D3212yescomplemented
223C2×C3×D4—91no—
224C2×C3×D4—92no—
225C2×C3×D4—181no—
226C22×C3×C4—91no—
227C22×C3×C4—181no—
228C2×#67—182no—
229#239—38no—
230#239—64no—
231#239—64no—
232#239—64no—
233D3×D4—98no—
234D3×D4—184no—
235C2×D12—32no—
236C2×D12—32no—
237C2×D12—32no—
238C2×D12—61no—
239C2×D12—62no—
240C2×D12—62no—
241C2×D12—62no—
242C2×D12—94no—
243C2×D12—182no—
244C2×D12—181no—
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×C4×D3—62no—
248C2×C4×D3—62no—
249C3×SD16—98no—
250C2×C3×C8—92no—
251#215—38no—
252#215—38no—
253#215—68no—
254#213—98no—
255#213—98no—
256#208—94no—
257C2×C3⋊C8—32no—
258C2×C3⋊C8—32no—
259C2×C3⋊C8—62no—
260D24—98no—
261ΓL(1,𝔽25)—98no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32D3×D412yes—
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—24no—
270C2×C3⋊D3—42no—
271C2×C3⋊D3—44no—
272C2×C3⋊D3C2×#6712yes—
273C2×C3×D3—34no—
274C2×C3×D3—62no—
275C2×C3×D3—64no—
276C2×C3×D3—64no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—122no—
281C2×C3×D3—124no—
282C2×C3×D3—124no—
283C2×C3×D3—124no—
284C2×C3×D3—124no—
285C2×C3×D3—124no—
286C2×C3×D3—244no—
287D32—1216no—
288D32—128no—
289C32×C4—22no—
290C32×C4—24no—
291C32×C4—24no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—22no—
295C32×C4—21no—
296C32×C4—41no—
297C32×C4—41no—
298C32×C4—44no—
299C32×C4—44no—
300C32×C4D3×D412yes—
301C32×C4C2×#6711yescharacteristic
302C32×C4C23×D312yes—
303C32×C4C23×D311yescharacteristic
304C3⋊Dic3C2×#6712yes—
305C3×Dic3—34no—
306C3×Dic3—62no—
307C2×#41—91no—
308C22×D4—271no—
309AGL(1,ℤ/8)—278no—
310C2×SD16—92no—
311C2×D8—272no—
312C2×M16—271no—
313C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
314C23×C3—91no—
315C23×C3—182no—
316C23×C3—181no—
317C22×D3—32no—
318C22×D3—61no—
319C22×D3—62no—
320C22×D3—62no—
321C22×D3—62no—
322C22×D3—62no—
323C22×D3—122no—
324C22×D3—188no—
325C22×D3—184no—
326C22×D3—368no—
327C22×D3—364no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—21no—
331C3×Q8—22no—
332C3×Q8—24no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8C2×D3212yescomplemented
337C3×Q8C2×D3214yescomplemented
338C3×D4—92no—
339C3×D4—94no—
340C3×D4—184no—
341C3×D4—182no—
342C3×D4—181no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—91no—
352C2×C3×C4—92no—
353C2×C3×C4—92no—
354C2×C3×C4—184no—
355C2×C3×C4C2×D3212yes—
356C2×C3×C4C2×D3211yescharacteristic
357#67—188no—
358C2×Dic3—32no—
359C2×Dic3—61no—
360D12—34no—
361D12—34no—
362D12—34no—
363D12—62no—
364D12—64no—
365D12—68no—
366D12—68no—
367D12—62no—
368D12—62no—
369D12—62no—
370D12—62no—
371D12—122no—
372D12—184no—
373C4×D3—34no—
374C4×D3—34no—
375C4×D3—62no—
376C4×D3—62no—
377C4×D3—68no—
378C4×D3—68no—
379C3×C8—94no—
380C3⋊C8—34no—
381C3⋊C8—34no—
382C3⋊C8—64no—
383C2×C32—22no—
384C2×C32—21no—
385C2×C32—24no—
386C2×C32—22no—
387C2×C32—42no—
388C2×C32—41no—
389C2×C32#91414yescomplemented
390C2×C32#94912yescomplemented
391C2×C32C2×D3×D412yes—
392C2×C32C22×#6711yescharacteristic
393C3⋊D3—22no—
394C3⋊D3—42no—
395C3⋊D3—44no—
396C3×D3—64no—
397C3×D3—122no—
398C3×D3—124no—
399C3×D3—124no—
400C3×D3—124no—
401C3×D3—124no—
402C3×D3—244no—
403C24—541no—
404#41—94no—
405#41—182no—
406C2×Q8#111311yescharacteristic, complemented
407C2×D4—91no—
408C2×D4—92no—
409C2×D4—181no—
410C2×D4—274no—
411C2×D4—542no—
412C2×D4—541no—
413C22×C4—91no—
414C22×C4—181no—
415SD16—98no—
416D8—278no—
417M16—274no—
418C2×C8—92no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—41no—
422C22×C3—92no—
423C22×C3—182no—
424C22×C3—184no—
425C22×C3—182no—
426C22×C3—364no—
427C22×C3—362no—
428C22×C3D3×#6712yes—
429C22×C3#381111yescharacteristic
430C2×D3—34no—
431C2×D3—62no—
432C2×D3—64no—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—122no—
438C2×D3—124no—
439C2×D3—124no—
440C2×D3—124no—
441C2×D3—124no—
442C2×D3—124no—
443C2×D3—244no—
444C2×D3—3616no—
445C2×D3—368no—
446C3×C4—24no—
447C3×C4—24no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—22no—
453C3×C4—44no—
454C3×C4—44no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4—92no—
458C3×C4D3×#6712yes—
459C3×C4#381111yescharacteristic
460C3×C4C22×D3212yes—
461C3×C4C22×D3211yescharacteristic
462Dic3—34no—
463Dic3—62no—
464C32—22no—
465C32—21no—
466C32—41no—
467C32C2×#91412yescomplemented
468C32C2×#94911yescharacteristic, complemented
469C23—91no—
470C23—181no—
471C23—182no—
472C23—544no—
473C23—1084no—
474Q8—21no—
475Q8C2×#111312yescomplemented
476D4—92no—
477D4—94no—
478D4—184no—
479D4—182no—
480D4—181no—
481D4—544no—
482C2×C4—21no—
483C2×C4—91no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—184no—
487C2×C4C2×#111311yescharacteristic
488C8—94no—
489C2×C3—21no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—24no—
493C2×C3—41no—
494C2×C3—42no—
495C2×C3—182no—
496C2×C3—364no—
497C2×C3—362no—
498C2×C3#6393914yescomplemented
499C2×C3#6394112yescomplemented
500C2×C3C2×D3×#6712yes—
501C2×C3C2×#381111yescharacteristic
502D3—64no—
503D3—122no—
504D3—124no—
505D3—124no—
506D3—124no—
507D3—124no—
508D3—244no—
509C22—92no—
510C22—182no—
511C22—182no—
512C22—184no—
513C22—364no—
514C22—362no—
515C22—1088no—
516C22#8904911yescharacteristic
517C4—22no—
518C4—92no—
519C4#8904911yescharacteristic
520C4C22×#111311yescharacteristic
521C3—21no—
522C3—22no—
523C3—41no—
524C3C2×#6393912yescomplemented
525C3C2×#6394111yescharacteristic, complemented
526C2—182no—
527C2—364no—
528C2—362no—
529C2#1172370912yesdirect product factor
530C2C2×#8904911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×C3×#3755C2semidirect
4. #11723709C2direct (N × H)
5. C2×#88810C2semidirect
6. C2×#88809C2semidirect
10. C2×C33×Q8C22semidirect
12. C3×#3832C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C4⋊#1113C22semidirect
21. C3×#3755C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. #88810C22semidirect
24. #88809C22semidirect
25. C2×#3832D3semidirect
30. C2×#3755D3semidirect
47. C33×Q8C23semidirect
55. C3×C4⋊(C3⋊D3)C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
68. C2×C32×Q8C2×D3semidirect
69. C2×C32×Q8C2×D3semidirect
71. #3832C2×D3semidirect
74. C2×C4⋊(C3⋊D3)C2×D3semidirect
98. #3755C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
161. C32×Q8C22×D3semidirect
162. C32×Q8C22×D3semidirect
178. C4⋊(C3⋊D3)C22×D3semidirect
204. C3⋊(C3⋊C8)C22×D3semidirect
209. C2×C33AGL(1,ℤ/8)semidirect
221. C2×C3×Q8D32semidirect
222. C2×C3×Q8D32semidirect
313. C33C2×AGL(1,ℤ/8)semidirect
336. C3×Q8C2×D32semidirect
337. C3×Q8C2×D32semidirect
389. C2×C32#914semidirect
390. C2×C32#949semidirect
406. C2×Q8#1113semidirect
467. C32C2×#914semidirect
468. C32C2×#949semidirect
475. Q8C2×#1113semidirect
498. C2×C3#63939semidirect
499. C2×C3#63941semidirect
524. C3C2×#63939semidirect
525. C3C2×#63941semidirect
529. C2#11723709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C4⋊#11132
C2×C3×#37552
#117237092
C2×#888102
C2×#888092
C2×#639413
C2×#639393