Finite Group Databasebeta

← C2×#11721316 #13152077

Subgroups — C2×#11721316 #13152077

Order 1728.

2449 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 2449 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×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×#6198C212yesmaximal, complemented
4C2×#88552C211yesmaximal, characteristic, complemented
5#11721316C218yesmaximal, direct product factor
6C2×#88450C212yesmaximal, complemented
7C2×#63981—31nomaximal
8C2×#63977—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3—24no—
11C23×(C3⋊D3)⋊C3—28no—
12C23×(C3⋊D3)⋊C3C2212yes—
13C23×(C3⋊D3)⋊C3C2214yescomplemented
14C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
15C22×#1084C2212yescomplemented
16C2×#638⋊C3—24no—
17C2×#638⋊C3C2214yescomplemented
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
19C2×#6198—28no—
20C2×#6198C2218yescomplemented
21#88552C2214yescomplemented
22#88450C2218yescomplemented
23C24×C3⋊D3D311yescharacteristic, complemented
24C24×C3×D3—31no—
25C22×#638D311yescharacteristic, complemented
26C22×C3×#67—31no—
27C22×#626—32no—
28C22×#626—32no—
29C2×C3×#217—31no—
30C2×C3×#217—31no—
31C2×C3×#217—61no—
32#63981—38no—
33#63977—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×(C3⋊D3)⋊C3—232no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—24no—
43C22×(C3⋊D3)⋊C3—28no—
44C22×(C3⋊D3)⋊C3—216no—
45C22×(C3⋊D3)⋊C3—24no—
46C22×(C3⋊D3)⋊C3—22no—
47C22×(C3⋊D3)⋊C3D418yescomplemented
48C22×(C3⋊D3)⋊C3C2314yes—
49C22×(C3⋊D3)⋊C3C2312yescomplemented
50C2×#1084—24no—
51C2×#1084C2314yescomplemented
52#638⋊C3—28no—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3C2312yescomplemented
55#6198—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⋊D3C2×D311yescharacteristic, complemented
67C23×C3×D3—32no—
68C23×C3×D3—31no—
69C23×C3×D3—34no—
70C23×C3×D3—68no—
71C23×C3×D3—64no—
72C2×#638—24no—
73C2×#638C2×D314yescomplemented
74C22×C3⋊Dic3C2×D311yescharacteristic, complemented
75C2×C3×#67—34no—
76C2×C3×#67—64no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—62no—
81C2×#626—38no—
82C2×#626—38no—
83C2×#626—68no—
84C2×#626—68no—
85C3×#217—34no—
86C3×#217—34no—
87C3×#217—64no—
88#3722—38no—
89(C2×D3)⋊Dic3—38no—
90C22×H(3,𝔽3)—28no—
91C22×H(3,𝔽3)—24no—
92C22×H(3,𝔽3)C2×D414yes—
93C22×H(3,𝔽3)C2×D412yes—
94C22×H(3,𝔽3)C2×D414yescomplemented
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—232no—
97C2×(C3⋊D3)⋊C3—216no—
98C2×(C3⋊D3)⋊C3—24no—
99C2×(C3⋊D3)⋊C3—24no—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—44no—
103C2×(C3⋊D3)⋊C3C2×D418yescomplemented
104#1084—24no—
105(C3⋊Dic3)⋊C3—22no—
106C25×C3—91no—
107C24×D3—31no—
108C24×D3—91no—
109C22×C3×D4—91no—
110C22×#67—31no—
111C22×#67—91no—
112C22×#67—92no—
113C22×#67—92no—
114C22×D12—92no—
115C2×C3×C22⋊C4—91no—
116#953—98no—
117C2×#217—31no—
118C2×#217—31no—
119C2×#217—61no—
120C2×#217—92no—
121#937—98no—
122C2×(C2×D3)⋊C4—92no—
123C2×(C2×D3)⋊C4—92no—
124#882—98no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—24no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—44no—
131C23×C32#6714yescomplemented
132C23×C32#6714yescomplemented
133C23×C32C22×D312yes—
134C23×C32C22×D312yes—
135C23×C32C22×D311yescharacteristic
136C23×C32C22×D311yescharacteristic
137C22×C3⋊D3—232no—
138C22×C3⋊D3—24no—
139C22×C3⋊D3—24no—
140C22×C3⋊D3—28no—
141C22×C3⋊D3—216no—
142C22×C3⋊D3—24no—
143C22×C3⋊D3—22no—
144C22×C3⋊D3#6718yescomplemented
145C22×C3⋊D3C22×D314yes—
146C22×C3⋊D3C22×D312yescomplemented
147C22×C3×D3—38no—
148C22×C3×D3—34no—
149C22×C3×D3—32no—
150C22×C3×D3—616no—
151C22×C3×D3—632no—
152C22×C3×D3—64no—
153C22×C3×D3—64no—
154C22×C3×D3—68no—
155C22×C3×D3—64no—
156C22×C3×D3—62no—
157#638—28no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3C22×D312yescomplemented
160C3×#67—68no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3—124no—
169#626—616no—
170#626—616no—
171C2×#120—271noSylow
172C2×H(3,𝔽3)—24no—
173C2×H(3,𝔽3)#12014yescomplemented
174C2×H(3,𝔽3)C22×D412yes—
175C2×H(3,𝔽3)C22×D411yescharacteristic
176(C3⋊D3)⋊C3—28no—
177(C3⋊D3)⋊C3—42no—
178C24×C3—31no—
179C24×C3—31no—
180C24×C3—61no—
181C24×C3—92no—
182C24×C3—94no—
183C24×C3—91no—
184C24×C3—184no—
185C24×C3—188no—
186C24×C3D3211yescharacteristic
187C23×D3—32no—
188C23×D3—34no—
189C23×D3—31no—
190C23×D3—68no—
191C23×D3—64no—
192C23×D3—92no—
193C23×D3—94no—
194C23×D3—91no—
195C23×D3—188no—
196C23×D3—184no—
197C2×C3×D4—94no—
198C2×C3×D4—184no—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C2×#67—34no—
202C2×#67—64no—
203C2×#67—94no—
204C2×#67—98no—
205C2×#67—98no—
206C2×#67—184no—
207C2×#67—188no—
208C2×#67—188no—
209C22×Dic3—32no—
210C22×Dic3—32no—
211C22×Dic3—31no—
212C22×Dic3—62no—
213C22×Dic3—91no—
214C2×D12—98no—
215C2×D12—188no—
216C3×C22⋊C4—94no—
217#217—34no—
218#217—34no—
219#217—64no—
220#217—98no—
221(C2×D3)⋊C4—98no—
222(C2×D3)⋊C4—98no—
223C22×C32—28no—
224C22×C32—28no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—44no—
232C22×C32—48no—
233C22×C32D3×D414yes—
234C22×C32C2×#6712yes—
235C22×C32C2×#6712yes—
236C22×C32C2×#6714yes—
237C22×C32C2×#6714yescomplemented
238C22×C32C2×#6714yescomplemented
239C22×C32C23×D311yescharacteristic
240C22×C32C23×D311yescharacteristic
241C2×C3⋊D3—232no—
242C2×C3⋊D3—216no—
243C2×C3⋊D3—24no—
244C2×C3⋊D3—24no—
245C2×C3⋊D3—24no—
246C2×C3⋊D3—22no—
247C2×C3⋊D3—44no—
248C2×C3⋊D3C2×#6718yescomplemented
249C2×C3×D3—38no—
250C2×C3×D3—632no—
251C2×C3×D3—616no—
252C2×C3×D3—64no—
253C2×C3×D3—64no—
254C2×C3×D3—64no—
255C2×C3×D3—62no—
256C2×C3×D3—124no—
257C3⋊Dic3—22no—
258C3×Dic3—64no—
259C3×Dic3—62no—
260C3×Dic3—64no—
261C3×Dic3—124no—
262C25—91no—
263C22×D4—91no—
264C22×D4—272no—
265#120—278no—
266C2×C22⋊C4—91no—
267C2×C22⋊C4—272no—
268H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
269C23×C3—24no—
270C23×C3—34no—
271C23×C3—34no—
272C23×C3—32no—
273C23×C3—32no—
274C23×C3—31no—
275C23×C3—31no—
276C23×C3—64no—
277C23×C3—64no—
278C23×C3—64no—
279C23×C3—62no—
280C23×C3—61no—
281C23×C3—94no—
282C23×C3—98no—
283C23×C3—92no—
284C23×C3—124no—
285C23×C3—184no—
286C23×C3—188no—
287C23×C3—184no—
288C23×C3—184no—
289C23×C3—182no—
290C23×C3—1816no—
291C23×C3—1832no—
292C23×C3#62514yes—
293C23×C3C2×D3212yes—
294C23×C3C2×D3211yescharacteristic
295C22×D3—34no—
296C22×D3—38no—
297C22×D3—32no—
298C22×D3—616no—
299C22×D3—632no—
300C22×D3—64no—
301C22×D3—68no—
302C22×D3—64no—
303C22×D3—62no—
304C22×D3—64no—
305C22×D3—98no—
306C22×D3—94no—
307C22×D3—92no—
308C22×D3—1816no—
309C22×D3—184no—
310C22×D3—188no—
311C22×D3—184no—
312C22×D3—1832no—
313C22×D3—182no—
314C22×D3—184no—
315C3×D4—188no—
316C2×C3×C4—92no—
317C2×C3×C4—94no—
318C2×C3×C4—182no—
319C2×C3×C4—184no—
320#67—68no—
321#67—188no—
322#67—1816no—
323#67—1816no—
324C2×Dic3—34no—
325C2×Dic3—34no—
326C2×Dic3—32no—
327C2×Dic3—64no—
328C2×Dic3—64no—
329C2×Dic3—64no—
330C2×Dic3—62no—
331C2×Dic3—92no—
332C2×Dic3—124no—
333C2×Dic3—182no—
334D12—1816no—
335C2×C32—24no—
336C2×C32—24no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—24no—
340C2×C32—44no—
341C2×C32#93714yescomplemented
342C2×C32#95314yescomplemented
343C2×C32C2×D3×D412yes—
344C2×C32C22×#6711yescharacteristic
345C2×C32C22×#6711yescharacteristic
346C2×C32C22×#6712yes—
347C3⋊D3—28no—
348C3⋊D3—42no—
349C3×D3—68no—
350C3×D3—122no—
351C24—92no—
352C24—94no—
353C24—91no—
354C24—188no—
355C24—184no—
356C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C2×D4—94no—
358C2×D4—184no—
359C2×D4—278no—
360C2×D4—548no—
361C22×C4—92no—
362C22×C4—91no—
363C22⋊C4—94no—
364C22⋊C4—278no—
365C22×C3—24no—
366C22×C3—28no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—32no—
370C22×C3—32no—
371C22×C3—34no—
372C22×C3—34no—
373C22×C3—34no—
374C22×C3—34no—
375C22×C3—64no—
376C22×C3—68no—
377C22×C3—64no—
378C22×C3—68no—
379C22×C3—61no—
380C22×C3—62no—
381C22×C3—64no—
382C22×C3—64no—
383C22×C3—98no—
384C22×C3—124no—
385C22×C3—128no—
386C22×C3—184no—
387C22×C3—184no—
388C22×C3—1816no—
389C22×C3—184no—
390C22×C3—182no—
391C22×C3—1832no—
392C22×C3—364no—
393C22×C3C2×#62512yes—
394C22×C3C2×#62514yes—
395C22×C3D3×#6714yes—
396C22×C3C22×D3211yescharacteristic
397C2×D3—38no—
398C2×D3—632no—
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—616no—
402C2×D3—62no—
403C2×D3—64no—
404C2×D3—98no—
405C2×D3—124no—
406C2×D3—184no—
407C2×D3—184no—
408C2×D3—1816no—
409C2×D3—1832no—
410C2×D3—182no—
411C2×D3—184no—
412C2×D3—364no—
413C3×C4—182no—
414C3×C4—184no—
415Dic3—64no—
416Dic3—64no—
417Dic3—62no—
418Dic3—124no—
419Dic3—182no—
420C32—21no—
421C32C2×#93711yescharacteristic, complemented
422C32C2×#95311yescharacteristic, complemented
423C23—24no—
424C23—94no—
425C23—98no—
426C23—92no—
427C23—1816no—
428C23—1832no—
429C23—184no—
430C23—188no—
431C23—184no—
432C23—182no—
433C23—184no—
434C23#619814yescomplemented
435C23C2×((C3⋊D3)⋊C3)⋊C212yes—
436C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
437D4—188no—
438D4—5416no—
439C2×C4—94no—
440C2×C4—92no—
441C2×C4—182no—
442C2×C4—184no—
443C2×C3—24no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—32no—
447C2×C3—32no—
448C2×C3—34no—
449C2×C3—34no—
450C2×C3—64no—
451C2×C3—64no—
452C2×C3—61no—
453C2×C3—62no—
454C2×C3—64no—
455C2×C3—124no—
456C2×C3—188no—
457C2×C3—362no—
458C2×C3#6397614yes—
459C2×C3C22×#62511yescharacteristic
460C2×C3C2×D3×#6712yes—
461D3—68no—
462D3—122no—
463D3—188no—
464D3—362no—
465C22—24no—
466C22—28no—
467C22—98no—
468C22—1832no—
469C22—184no—
470C22—184no—
471C22—1816no—
472C22—182no—
473C22—184no—
474C22—364no—
475C22C2×#619812yes—
476C22C2×#619814yescomplemented
477C22#8867614yes—
478C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
479C4—182no—
480C4—184no—
481C3—31no—
482C3—31no—
483C3—61no—
484C3C2×#6397611yescharacteristic
485C2—24no—
486C2—188no—
487C2—362no—
488C2#1172131614yesdirect product factor
489C2C22×#619811yescharacteristic
490C2C2×#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×#638⋊C3C2semidirect
3. C22×#6198C2semidirect
4. C2×#88552C2semidirect
5. #11721316C2direct (N × H)
6. C2×#88450C2semidirect
13. C23×(C3⋊D3)⋊C3C22semidirect
14. C23×(C3⋊D3)⋊C3C22semidirect
15. C22×#1084C22semidirect
17. C2×#638⋊C3C22semidirect
18. C22×(C3⋊Dic3)⋊C3C22semidirect
20. C2×#6198C22semidirect
21. #88552C22semidirect
22. #88450C22semidirect
23. C24×C3⋊D3D3semidirect
25. C22×#638D3semidirect
37. C23×H(3,𝔽3)D4semidirect
47. C22×(C3⋊D3)⋊C3D4semidirect
49. C22×(C3⋊D3)⋊C3C23semidirect
51. C2×#1084C23semidirect
54. C2×(C3⋊Dic3)⋊C3C23semidirect
65. C23×C3⋊D3C2×D3semidirect
66. C23×C3⋊D3C2×D3semidirect
73. C2×#638C2×D3semidirect
74. C22×C3⋊Dic3C2×D3semidirect
94. C22×H(3,𝔽3)C2×D4semidirect
103. C2×(C3⋊D3)⋊C3C2×D4semidirect
131. C23×C32#67semidirect
132. C23×C32#67semidirect
144. C22×C3⋊D3#67semidirect
146. C22×C3⋊D3C22×D3semidirect
159. C2×C3⋊Dic3C22×D3semidirect
173. C2×H(3,𝔽3)#120semidirect
237. C22×C32C2×#67semidirect
238. C22×C32C2×#67semidirect
248. C2×C3⋊D3C2×#67semidirect
268. H(3,𝔽3)C2×#120semidirect
341. C2×C32#937semidirect
342. C2×C32#953semidirect
421. C32C2×#937semidirect
422. C32C2×#953semidirect
434. C23#6198semidirect
476. C22C2×#6198semidirect
488. C2#11721316direct (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×#638⋊C32
C22×#61982
C2×#885522
#117213162
C2×#884502
C2×#639813
C2×#639773