Finite Group Databasebeta

← C2×#11721321 #13152086

Subgroups — C2×#11721321 #13152086

Order 1728.

2553 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 2553 conjugacy classes of subgroups — too many to draw. Only its 251 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
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C22×#6235C211yesmaximal, characteristic, complemented
3C22×#6200C212yesmaximal, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5#11721321C218yesmaximal, direct product factor
6C2×#88450C212yesmaximal, complemented
7C2×#63981—31nomaximal
8C2×#63976—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×#272C2211yescharacteristic, complemented
11C23×(C3⋊D3)⋊C3—24no—
12C23×(C3⋊D3)⋊C3—28no—
13C23×(C3⋊D3)⋊C3C2212yes—
14C23×(C3⋊D3)⋊C3C2214yescomplemented
15C2×#6235—24no—
16C2×#6235C2214yescomplemented
17C22×#1084C2211yescharacteristic, complemented
18C22×(C3⋊Dic3)⋊C3C2212yescomplemented
19C2×#6200—28no—
20C2×#6200C2218yescomplemented
21#3757⋊C3C2214yescomplemented
22#88450C2218yescomplemented
23C24×C3⋊D3D311yescharacteristic, complemented
24C24×C3×D3—31no—
25C22×C3×#67—31no—
26C22×C3×#67—31no—
27C22×C3×#67—61no—
28C22×#626—32no—
29C22×#625—32no—
30C2×#3757D311yescharacteristic, complemented
31C2×C3×#217—31no—
32#63981—38no—
33#63976—38no—
34C2×#3722—32no—
35C2×(C2×D3)⋊Dic3—32no—
36C23×H(3,𝔽3)—24no—
37C23×H(3,𝔽3)D414yescomplemented
38C23×H(3,𝔽3)C2312yes—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×#272—22no—
41C22×#272—24no—
42C22×#272C2312yescomplemented
43C22×(C3⋊D3)⋊C3—232no—
44C22×(C3⋊D3)⋊C3—24no—
45C22×(C3⋊D3)⋊C3—24no—
46C22×(C3⋊D3)⋊C3—28no—
47C22×(C3⋊D3)⋊C3—216no—
48C22×(C3⋊D3)⋊C3D418yescomplemented
49C22×(C3⋊D3)⋊C3C2314yes—
50#6235—28no—
51C2×#1084—22no—
52C2×#1084C2312yescomplemented
53C2×(C3⋊Dic3)⋊C3—24no—
54C2×(C3⋊Dic3)⋊C3C2314yescomplemented
55#6200—216no—
56C2×#953—91no—
57C2×#937—91no—
58C2×#882—91no—
59C24×C32—21no—
60C24×C32C2×D311yescharacteristic
61C24×C32C2×D311yescharacteristic
62C23×C3⋊D3—24no—
63C23×C3⋊D3—28no—
64C23×C3⋊D3C2×D312yes—
65C23×C3⋊D3C2×D314yescomplemented
66C23×C3×D3—32no—
67C23×C3×D3—31no—
68C23×C3×D3—31no—
69C23×C3×D3—34no—
70C23×C3×D3—68no—
71C23×C3×D3—64no—
72C23×C3×D3—61no—
73C22×C3⋊Dic3C2×D312yescomplemented
74C2×C3×#67—34no—
75C2×C3×#67—34no—
76C2×C3×#67—64no—
77C2×C3×#67—68no—
78C2×C3×#67—64no—
79C2×C3×#67—64no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—61no—
84C2×#626—38no—
85C2×#626—68no—
86C2×#625—38no—
87C2×#625—68no—
88#3757C2×D314yescomplemented
89C3×#217—34no—
90#3722—38no—
91(C2×D3)⋊Dic3—38no—
92C22×H(3,𝔽3)—28no—
93C22×H(3,𝔽3)—24no—
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)C2×D414yes—
96C22×H(3,𝔽3)C2×D414yescomplemented
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×#272—22no—
99C2×#272—24no—
100C2×#272—44no—
101C2×(C3⋊D3)⋊C3—232no—
102C2×(C3⋊D3)⋊C3—216no—
103C2×(C3⋊D3)⋊C3—24no—
104C2×(C3⋊D3)⋊C3—24no—
105C2×(C3⋊D3)⋊C3C2×D418yescomplemented
106#1084—22no—
107(C3⋊Dic3)⋊C3—24no—
108C25×C3—91no—
109C24×D3—31no—
110C24×D3—91no—
111C22×C3×D4—91no—
112C22×#67—31no—
113C22×#67—31no—
114C22×#67—61no—
115C22×#67—92no—
116C22×#67—92no—
117C22×D12—92no—
118C2×C3×C22⋊C4—91no—
119#953—98no—
120C2×#217—31no—
121C2×#217—91no—
122C2×#217—92no—
123#937—98no—
124C2×(C2×D3)⋊C4—92no—
125C2×(C2×D3)⋊C4—92no—
126#882—98no—
127C23×C32—24no—
128C23×C32—24no—
129C23×C32—24no—
130C23×C32—21no—
131C23×C32—22no—
132C23×C32—44no—
133C23×C32D1214yescomplemented
134C23×C32#6714yescomplemented
135C23×C32C22×D312yes—
136C23×C32C22×D312yes—
137C23×C32C22×D311yescharacteristic
138C23×C32C22×D311yescharacteristic
139C22×C3⋊D3—232no—
140C22×C3⋊D3—24no—
141C22×C3⋊D3—24no—
142C22×C3⋊D3—28no—
143C22×C3⋊D3—216no—
144C22×C3⋊D3#6718yescomplemented
145C22×C3⋊D3C22×D314yes—
146C22×C3×D3—38no—
147C22×C3×D3—34no—
148C22×C3×D3—32no—
149C22×C3×D3—32no—
150C22×C3×D3—62no—
151C22×C3×D3—616no—
152C22×C3×D3—632no—
153C22×C3×D3—64no—
154C22×C3×D3—68no—
155C22×C3×D3—64no—
156C22×C3×D3—64no—
157C22×C3×D3—62no—
158C22×C3×D3—64no—
159C22×C3×D3—64no—
160C22×C3×D3—62no—
161C22×C3×D3—124no—
162C2×C3⋊Dic3—24no—
163C2×C3⋊Dic3C22×D314yescomplemented
164C3×#67—616no—
165C3×#67—68no—
166C3×#67—68no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—62no—
175#626—616no—
176#625—616no—
177C2×#120—271noSylow
178C2×H(3,𝔽3)—24no—
179C2×H(3,𝔽3)#12014yescomplemented
180C2×H(3,𝔽3)C22×D411yescharacteristic
181C2×H(3,𝔽3)C22×D412yes—
182#272—42no—
183(C3⋊D3)⋊C3—28no—
184C24×C3—31no—
185C24×C3—31no—
186C24×C3—61no—
187C24×C3—92no—
188C24×C3—94no—
189C24×C3—91no—
190C24×C3—184no—
191C24×C3—188no—
192C24×C3D3211yescharacteristic
193C23×D3—32no—
194C23×D3—31no—
195C23×D3—31no—
196C23×D3—34no—
197C23×D3—61no—
198C23×D3—68no—
199C23×D3—64no—
200C23×D3—92no—
201C23×D3—94no—
202C23×D3—188no—
203C23×D3—184no—
204C2×C3×D4—94no—
205C2×C3×D4—184no—
206C22×C3×C4—92no—
207C22×C3×C4—91no—
208C2×#67—34no—
209C2×#67—34no—
210C2×#67—64no—
211C2×#67—64no—
212C2×#67—64no—
213C2×#67—68no—
214C2×#67—98no—
215C2×#67—98no—
216C2×#67—188no—
217C2×#67—188no—
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—32no—
221C22×Dic3—61no—
222C22×Dic3—92no—
223C2×D12—98no—
224C2×D12—188no—
225C3×C22⋊C4—94no—
226#217—34no—
227#217—94no—
228#217—98no—
229(C2×D3)⋊C4—98no—
230(C2×D3)⋊C4—98no—
231C22×C32—28no—
232C22×C32—28no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—21no—
236C22×C32—22no—
237C22×C32—24no—
238C22×C32—24no—
239C22×C32—44no—
240C22×C32—48no—
241C22×C32C2×D1212yes—
242C22×C32C2×D1214yescomplemented
243C22×C32D3×D414yes—
244C22×C32C2×#6712yes—
245C22×C32C2×#6714yes—
246C22×C32C2×#6714yescomplemented
247C22×C32C23×D311yescharacteristic
248C22×C32C23×D311yescharacteristic
249C2×C3⋊D3—232no—
250C2×C3⋊D3—216no—
251C2×C3⋊D3—24no—
252C2×C3⋊D3—24no—
253C2×C3⋊D3C2×#6718yescomplemented
254C2×C3×D3—38no—
255C2×C3×D3—632no—
256C2×C3×D3—616no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C2×C3×D3—64no—
260C2×C3×D3—62no—
261C2×C3×D3—64no—
262C2×C3×D3—64no—
263C2×C3×D3—62no—
264C2×C3×D3—124no—
265C2×C3×D3—128no—
266C2×C3×D3—124no—
267C2×C3×D3—124no—
268C3⋊Dic3—24no—
269C3×Dic3—64no—
270C3×Dic3—62no—
271C3×Dic3—64no—
272C3×Dic3—62no—
273C25—91no—
274C22×D4—91no—
275C22×D4—272no—
276#120—278no—
277C2×C22⋊C4—91no—
278C2×C22⋊C4—272no—
279H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
280C23×C3—24no—
281C23×C3—34no—
282C23×C3—34no—
283C23×C3—32no—
284C23×C3—32no—
285C23×C3—31no—
286C23×C3—31no—
287C23×C3—64no—
288C23×C3—64no—
289C23×C3—64no—
290C23×C3—62no—
291C23×C3—61no—
292C23×C3—94no—
293C23×C3—98no—
294C23×C3—92no—
295C23×C3—124no—
296C23×C3—184no—
297C23×C3—188no—
298C23×C3—184no—
299C23×C3—1816no—
300C23×C3—1832no—
301C23×C3—182no—
302C23×C3—184no—
303C23×C3#62614yes—
304C23×C3C2×D3212yes—
305C23×C3C2×D3211yescharacteristic
306C22×D3—34no—
307C22×D3—38no—
308C22×D3—32no—
309C22×D3—32no—
310C22×D3—62no—
311C22×D3—616no—
312C22×D3—632no—
313C22×D3—64no—
314C22×D3—68no—
315C22×D3—64no—
316C22×D3—64no—
317C22×D3—62no—
318C22×D3—62no—
319C22×D3—64no—
320C22×D3—64no—
321C22×D3—98no—
322C22×D3—94no—
323C22×D3—124no—
324C22×D3—1816no—
325C22×D3—184no—
326C22×D3—188no—
327C22×D3—184no—
328C22×D3—1832no—
329C3×D4—188no—
330C2×C3×C4—94no—
331C2×C3×C4—92no—
332C2×C3×C4—184no—
333C2×C3×C4—182no—
334#67—68no—
335#67—68no—
336#67—616no—
337#67—1816no—
338#67—1816no—
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—34no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—64no—
346C2×Dic3—64no—
347C2×Dic3—94no—
348C2×Dic3—184no—
349D12—1816no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—21no—
353C2×C32—22no—
354C2×C32—24no—
355C2×C32—44no—
356C2×C32#88214yescomplemented
357C2×C32#95314yescomplemented
358C2×C32C22×D1211yescharacteristic
359C2×C32C2×D3×D412yes—
360C2×C32C22×#6711yescharacteristic
361C2×C32C22×#6712yes—
362C3⋊D3—28no—
363C3×D3—68no—
364C3×D3—124no—
365C3×D3—122no—
366C3×D3—122no—
367C24—92no—
368C24—94no—
369C24—91no—
370C24—188no—
371C24—184no—
372C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C2×D4—94no—
374C2×D4—184no—
375C2×D4—278no—
376C2×D4—548no—
377C22×C4—92no—
378C22×C4—91no—
379C22⋊C4—94no—
380C22⋊C4—278no—
381C22×C3—24no—
382C22×C3—28no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—32no—
386C22×C3—32no—
387C22×C3—34no—
388C22×C3—34no—
389C22×C3—34no—
390C22×C3—34no—
391C22×C3—64no—
392C22×C3—68no—
393C22×C3—64no—
394C22×C3—68no—
395C22×C3—61no—
396C22×C3—62no—
397C22×C3—64no—
398C22×C3—64no—
399C22×C3—98no—
400C22×C3—124no—
401C22×C3—128no—
402C22×C3—184no—
403C22×C3—184no—
404C22×C3—1816no—
405C22×C3—1832no—
406C22×C3—182no—
407C22×C3—184no—
408C22×C3—364no—
409C22×C3C2×#62612yes—
410C22×C3C2×#62614yes—
411C22×C3D3×#6714yes—
412C22×C3C22×D3211yescharacteristic
413C2×D3—38no—
414C2×D3—632no—
415C2×D3—64no—
416C2×D3—64no—
417C2×D3—616no—
418C2×D3—64no—
419C2×D3—62no—
420C2×D3—62no—
421C2×D3—64no—
422C2×D3—64no—
423C2×D3—98no—
424C2×D3—124no—
425C2×D3—124no—
426C2×D3—124no—
427C2×D3—128no—
428C2×D3—184no—
429C2×D3—184no—
430C2×D3—1816no—
431C2×D3—1832no—
432C3×C4—184no—
433C3×C4—182no—
434Dic3—62no—
435Dic3—62no—
436Dic3—64no—
437Dic3—64no—
438Dic3—184no—
439C32—21no—
440C32C2×#88211yescharacteristic, complemented
441C32C2×#95311yescharacteristic, complemented
442C23—24no—
443C23—94no—
444C23—98no—
445C23—92no—
446C23—1816no—
447C23—1832no—
448C23—184no—
449C23—188no—
450C23—184no—
451C23—182no—
452C23—184no—
453C23#620014yescomplemented
454C23C2×((C3⋊D3)⋊C3)⋊C212yes—
455C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
456D4—188no—
457D4—5416no—
458C2×C4—94no—
459C2×C4—92no—
460C2×C4—182no—
461C2×C4—184no—
462C2×C3—24no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—32no—
466C2×C3—32no—
467C2×C3—34no—
468C2×C3—34no—
469C2×C3—64no—
470C2×C3—64no—
471C2×C3—61no—
472C2×C3—62no—
473C2×C3—64no—
474C2×C3—124no—
475C2×C3—188no—
476C2×C3—362no—
477C2×C3#6397714yes—
478C2×C3C22×#62611yescharacteristic
479C2×C3C2×D3×#6712yes—
480D3—68no—
481D3—122no—
482D3—122no—
483D3—124no—
484D3—188no—
485C22—24no—
486C22—28no—
487C22—98no—
488C22—1832no—
489C22—184no—
490C22—184no—
491C22—1816no—
492C22—182no—
493C22—184no—
494C22—364no—
495C22C2×#620012yes—
496C22C2×#620014yescomplemented
497C22#8867614yes—
498C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
499C4—182no—
500C4—184no—
501C3—31no—
502C3—31no—
503C3—61no—
504C3C2×#6397711yescharacteristic
505C2—24no—
506C2—188no—
507C2—362no—
508C2#1172132114yesdirect product factor
509C2C22×#620011yescharacteristic
510C2C2×#8867612yes—
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×(C3⋊D3)⋊C3C2semidirect
2. C22×#6235C2semidirect
3. C22×#6200C2semidirect
4. C2×#3757⋊C3C2semidirect
5. #11721321C2direct (N × H)
6. C2×#88450C2semidirect
10. C23×#272C22semidirect
14. C23×(C3⋊D3)⋊C3C22semidirect
16. C2×#6235C22semidirect
17. C22×#1084C22semidirect
18. C22×(C3⋊Dic3)⋊C3C22semidirect
20. C2×#6200C22semidirect
21. #3757⋊C3C22semidirect
22. #88450C22semidirect
23. C24×C3⋊D3D3semidirect
30. C2×#3757D3semidirect
37. C23×H(3,𝔽3)D4semidirect
42. C22×#272C23semidirect
48. C22×(C3⋊D3)⋊C3D4semidirect
52. C2×#1084C23semidirect
54. C2×(C3⋊Dic3)⋊C3C23semidirect
65. C23×C3⋊D3C2×D3semidirect
73. C22×C3⋊Dic3C2×D3semidirect
88. #3757C2×D3semidirect
96. C22×H(3,𝔽3)C2×D4semidirect
105. C2×(C3⋊D3)⋊C3C2×D4semidirect
133. C23×C32D12semidirect
134. C23×C32#67semidirect
144. C22×C3⋊D3#67semidirect
163. C2×C3⋊Dic3C22×D3semidirect
179. C2×H(3,𝔽3)#120semidirect
242. C22×C32C2×D12semidirect
246. C22×C32C2×#67semidirect
253. C2×C3⋊D3C2×#67semidirect
279. H(3,𝔽3)C2×#120semidirect
356. C2×C32#882semidirect
357. C2×C32#953semidirect
440. C32C2×#882semidirect
441. C32C2×#953semidirect
453. C23#6200semidirect
496. C22C2×#6200semidirect
508. C2#11721321direct (N × H)

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C22×#62352
C22×#62002
C2×#3757⋊C32
#117213212
C2×#884502
C2×#639813
C2×#639763