Finite Group Databasebeta

← C2×#11721310 #13152060

Subgroups — C2×#11721310 #13152060

Order 1728.

2491 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
3C2×#88552C211yesmaximal, characteristic, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5#11721310C218yesmaximal, direct product factor
6C2×#88450C212yesmaximal, complemented
7C2×#63974—31nomaximal
8C2×D3×#217—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3—28no—
11C23×(C3⋊D3)⋊C3C418yescomplemented
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C23×(C3⋊D3)⋊C3C2214yes—
14C23×(C3⋊D3)⋊C3C2211yescharacteristic
15C22×#1084C2212yescomplemented
16C22×(C3⋊Dic3)⋊C3C2212yescomplemented
17C2×((C3⋊D3)⋊C3)⋊C4—28no—
18C2×((C3⋊D3)⋊C3)⋊C4C2218yescomplemented
19#88552C2214yescomplemented
20#3757⋊C3C2214yescomplemented
21#88450C2218yescomplemented
22C24×C3⋊D3D311yescharacteristic
23C24×C3×D3—31no—
24C22×#624—32no—
25C22×D3×Dic3—32no—
26C2×#3757D311yescharacteristic, complemented
27C2×C3×#217—31no—
28C2×C3×#217—31no—
29C2×C3×#217—31no—
30C2×C3×#217—61no—
31#63974—38no—
32D3×#217—38no—
33C2×#3722—32no—
34C2×(C2×D3)⋊Dic3—32no—
35C23×H(3,𝔽3)—24no—
36C23×H(3,𝔽3)C2×C414yescomplemented
37C23×H(3,𝔽3)C2312yes—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3—232no—
41C22×(C3⋊D3)⋊C3—216no—
42C22×(C3⋊D3)⋊C3C2×C414yes—
43C22×(C3⋊D3)⋊C3C2×C4116yescomplemented
44C22×(C3⋊D3)⋊C3C2×C414yescomplemented
45C22×(C3⋊D3)⋊C3D418yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47C22×(C3⋊D3)⋊C3C2312yes—
48C2×#1084—24no—
49C2×#1084C2314yescomplemented
50C2×(C3⋊Dic3)⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yescomplemented
52((C3⋊D3)⋊C3)⋊C4—216no—
53C22×#217—91no—
54C2×D3×C22⋊C4—91no—
55C2×D3×C22⋊C4—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C2×D311yescharacteristic
59C23×C3⋊D3—28no—
60C23×C3⋊D3Dic318yescomplemented
61C23×C3⋊D3C2×D311yescharacteristic
62C23×C3⋊D3C2×D314yes—
63C23×C3⋊D3C2×D311yescharacteristic
64C23×C3×D3—31no—
65C23×C3×D3—38no—
66C23×C3×D3—31no—
67C23×C3×D3—34no—
68C23×C3×D3—68no—
69C22×C3⋊Dic3C2×D312yescomplemented
70C22×C3×Dic3—32no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—62no—
74C2×#624—38no—
75C2×#624—68no—
76C2×D3×Dic3—38no—
77C2×D3×Dic3—68no—
78#3757C2×D314yescomplemented
79C3×#217—34no—
80C3×#217—34no—
81C3×#217—34no—
82C3×#217—64no—
83#3722—38no—
84(C2×D3)⋊Dic3—38no—
85C22×H(3,𝔽3)—28no—
86C22×H(3,𝔽3)—24no—
87C22×H(3,𝔽3)C22×C412yes—
88C22×H(3,𝔽3)C22×C414yescomplemented
89C22×H(3,𝔽3)C2×D414yes—
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×(C3⋊D3)⋊C3—28no—
92C2×(C3⋊D3)⋊C3—24no—
93C2×(C3⋊D3)⋊C3—232no—
94C2×(C3⋊D3)⋊C3C22⋊C4116yescomplemented
95C2×(C3⋊D3)⋊C3C22×C414yes—
96C2×(C3⋊D3)⋊C3C22×C414yescomplemented
97C2×(C3⋊D3)⋊C3C2×D418yes—
98#1084—24no—
99(C3⋊Dic3)⋊C3—24no—
100C25×C3—91no—
101C24×D3—31no—
102C24×D3—91no—
103C23×Dic3—92no—
104C22×C4×D3—92no—
105C22×C4×D3—92no—
106C2×C3×C22⋊C4—91no—
107C2×C3×C22⋊C4—91no—
108C2×#217—31no—
109C2×#217—31no—
110C2×#217—31no—
111C2×#217—61no—
112C2×#217—91no—
113C2×#217—98no—
114C2×#217—92no—
115C2×(C2×D3)⋊C4—92no—
116C2×(C2×D3)⋊C4—92no—
117D3×C22⋊C4—98no—
118D3×C22⋊C4—98no—
119C23×C32—24no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—21no—
123C23×C32—22no—
124C23×C32—44no—
125C23×C32C4×D314yescomplemented
126C23×C32C2×Dic314yescomplemented
127C23×C32C22×D312yes—
128C23×C32C22×D312yes—
129C23×C32C22×D311yescharacteristic
130C23×C32C22×D311yescharacteristic
131C22×C3⋊D3—24no—
132C22×C3⋊D3—232no—
133C22×C3⋊D3—216no—
134C22×C3⋊D3C2×Dic314yes—
135C22×C3⋊D3C2×Dic3116yescomplemented
136C22×C3⋊D3C2×Dic314yescomplemented
137C22×C3⋊D3#6718yes—
138C22×C3⋊D3C22×D312yes—
139C22×C3⋊D3C22×D312yes—
140C22×C3×D3—34no—
141C22×C3×D3—38no—
142C22×C3×D3—32no—
143C22×C3×D3—34no—
144C22×C3×D3—316no—
145C22×C3×D3—32no—
146C22×C3×D3—632no—
147C22×C3×D3—616no—
148C22×C3×D3—64no—
149C2×C3⋊Dic3—24no—
150C2×C3⋊Dic3C22×D314yescomplemented
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—64no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—124no—
159#624—616no—
160D3×Dic3—616no—
161C22×C22⋊C4—271noSylow
162C2×H(3,𝔽3)—24no—
163C2×H(3,𝔽3)C2×C22⋊C414yescomplemented
164C2×H(3,𝔽3)C23×C411yescharacteristic
165C2×H(3,𝔽3)C22×D412yes—
166(C3⋊D3)⋊C3—24no—
167(C3⋊D3)⋊C3C2×C22⋊C418yescomplemented
168C24×C3—31no—
169C24×C3—31no—
170C24×C3—61no—
171C24×C3—91no—
172C24×C3—98no—
173C24×C3—94no—
174C24×C3—91no—
175C24×C3—188no—
176C24×C3D3211yescharacteristic
177C23×D3—31no—
178C23×D3—38no—
179C23×D3—34no—
180C23×D3—31no—
181C23×D3—68no—
182C23×D3—91no—
183C23×D3—98no—
184C23×D3—94no—
185C23×D3—91no—
186C23×D3—188no—
187C22×C3×C4—92no—
188C22×C3×C4—92no—
189C22×Dic3—32no—
190C22×Dic3—32no—
191C22×Dic3—32no—
192C22×Dic3—62no—
193C22×Dic3—92no—
194C22×Dic3—98no—
195C22×Dic3—188no—
196C2×C4×D3—98no—
197C2×C4×D3—98no—
198C2×C4×D3—188no—
199C2×C4×D3—188no—
200C3×C22⋊C4—94no—
201C3×C22⋊C4—94no—
202#217—34no—
203#217—34no—
204#217—34no—
205#217—64no—
206#217—94no—
207#217—98no—
208(C2×D3)⋊C4—98no—
209(C2×D3)⋊C4—98no—
210C22×C32—28no—
211C22×C32—28no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—44no—
219C22×C32—48no—
220C22×C32C2×C4×D312yes—
221C22×C32C2×C4×D314yescomplemented
222C22×C32D3×D414yes—
223C22×C32C22×Dic312yes—
224C22×C32C22×Dic314yescomplemented
225C22×C32C2×#6714yes—
226C22×C32C23×D311yescharacteristic
227C22×C32C23×D311yescharacteristic
228C2×C3⋊D3—28no—
229C2×C3⋊D3—24no—
230C2×C3⋊D3—232no—
231C2×C3⋊D3#217116yescomplemented
232C2×C3⋊D3C22×Dic314yes—
233C2×C3⋊D3C22×Dic314yescomplemented
234C2×C3⋊D3C2×#6718yes—
235C2×C3×D3—316no—
236C2×C3×D3—34no—
237C2×C3×D3—38no—
238C2×C3×D3—34no—
239C2×C3×D3—632no—
240C2×C3×D3—68no—
241C2×C3×D3—64no—
242C3⋊Dic3—24no—
243C3×Dic3—64no—
244C3×Dic3—64no—
245C3×Dic3—64no—
246C3×Dic3—124no—
247C25—91no—
248C23×C4—272no—
249C2×C22⋊C4—91no—
250C2×C22⋊C4—91no—
251C2×C22⋊C4—278no—
252C2×C22⋊C4—272no—
253H(3,𝔽3)C22×C22⋊C411yescharacteristic, complemented, Sylow
254C23×C3—24no—
255C23×C3—34no—
256C23×C3—34no—
257C23×C3—32no—
258C23×C3—32no—
259C23×C3—31no—
260C23×C3—31no—
261C23×C3—64no—
262C23×C3—64no—
263C23×C3—64no—
264C23×C3—62no—
265C23×C3—61no—
266C23×C3—92no—
267C23×C3—98no—
268C23×C3—94no—
269C23×C3—92no—
270C23×C3—94no—
271C23×C3—916no—
272C23×C3—124no—
273C23×C3—184no—
274C23×C3—1816no—
275C23×C3—1832no—
276C23×C3D3×Dic314yes—
277C23×C3C2×D3212yes—
278C23×C3C2×D3211yescharacteristic
279C22×D3—32no—
280C22×D3—34no—
281C22×D3—38no—
282C22×D3—32no—
283C22×D3—34no—
284C22×D3—316no—
285C22×D3—616no—
286C22×D3—632no—
287C22×D3—64no—
288C22×D3—94no—
289C22×D3—98no—
290C22×D3—92no—
291C22×D3—94no—
292C22×D3—916no—
293C22×D3—92no—
294C22×D3—1816no—
295C22×D3—1832no—
296C22×D3—184no—
297C2×C3×C4—94no—
298C2×C3×C4—94no—
299C2×C3×C4—184no—
300C2×C3×C4—184no—
301C2×Dic3—34no—
302C2×Dic3—34no—
303C2×Dic3—34no—
304C2×Dic3—64no—
305C2×Dic3—64no—
306C2×Dic3—64no—
307C2×Dic3—64no—
308C2×Dic3—94no—
309C2×Dic3—124no—
310C2×Dic3—184no—
311C2×Dic3—1816no—
312C4×D3—1816no—
313C4×D3—1816no—
314C2×C32—24no—
315C2×C32—24no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—24no—
319C2×C32—44no—
320C2×C32D3×C22⋊C414yescomplemented
321C2×C32C2×#21714yescomplemented
322C2×C32C22×C4×D311yescharacteristic
323C2×C32C2×D3×D412yes—
324C2×C32C23×Dic311yescharacteristic
325C2×C32C22×#6712yes—
326C3⋊D3—24no—
327C3⋊D3C2×#21718yescomplemented
328C3×D3—38no—
329C3×D3—64no—
330C24—91no—
331C24—98no—
332C24—94no—
333C24—91no—
334C24—188no—
335C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C22×C4—92no—
337C22×C4—92no—
338C22×C4—278no—
339C22×C4—548no—
340C22⋊C4—94no—
341C22⋊C4—94no—
342C22⋊C4—278no—
343C22×C3—24no—
344C22×C3—28no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—32no—
348C22×C3—32no—
349C22×C3—34no—
350C22×C3—34no—
351C22×C3—34no—
352C22×C3—34no—
353C22×C3—64no—
354C22×C3—68no—
355C22×C3—64no—
356C22×C3—68no—
357C22×C3—61no—
358C22×C3—62no—
359C22×C3—64no—
360C22×C3—64no—
361C22×C3—98no—
362C22×C3—94no—
363C22×C3—916no—
364C22×C3—94no—
365C22×C3—124no—
366C22×C3—128no—
367C22×C3—184no—
368C22×C3—188no—
369C22×C3—1832no—
370C22×C3C2×D3×Dic312yes—
371C22×C3C2×D3×Dic314yes—
372C22×C3D3×#6714yes—
373C22×C3C22×D3211yescharacteristic
374C2×D3—34no—
375C2×D3—38no—
376C2×D3—316no—
377C2×D3—34no—
378C2×D3—632no—
379C2×D3—64no—
380C2×D3—68no—
381C2×D3—98no—
382C2×D3—94no—
383C2×D3—916no—
384C2×D3—94no—
385C2×D3—1832no—
386C2×D3—184no—
387C2×D3—188no—
388C3×C4—184no—
389C3×C4—184no—
390Dic3—64no—
391Dic3—64no—
392Dic3—64no—
393Dic3—124no—
394Dic3—184no—
395C32—21no—
396C32C2×D3×C22⋊C411yescharacteristic, complemented
397C32C22×#21711yescharacteristic, complemented
398C23—24no—
399C23—92no—
400C23—94no—
401C23—98no—
402C23—92no—
403C23—94no—
404C23—916no—
405C23—1816no—
406C23—1832no—
407C23—184no—
408C23((C3⋊D3)⋊C3)⋊C414yescomplemented
409C23C2×((C3⋊D3)⋊C3)⋊C212yes—
410C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
411C2×C4—94no—
412C2×C4—94no—
413C2×C4—184no—
414C2×C4—184no—
415C2×C4—5416no—
416C2×C3—24no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—32no—
420C2×C3—32no—
421C2×C3—34no—
422C2×C3—34no—
423C2×C3—64no—
424C2×C3—64no—
425C2×C3—61no—
426C2×C3—62no—
427C2×C3—64no—
428C2×C3—98no—
429C2×C3—124no—
430C2×C3—184no—
431C2×C3D3×#21714yes—
432C2×C3C22×D3×Dic311yescharacteristic
433C2×C3C2×D3×#6712yes—
434D3—38no—
435D3—64no—
436D3—98no—
437D3—184no—
438C22—24no—
439C22—28no—
440C22—94no—
441C22—98no—
442C22—916no—
443C22—94no—
444C22—1832no—
445C22—184no—
446C22—188no—
447C22C2×((C3⋊D3)⋊C3)⋊C412yes—
448C22C2×((C3⋊D3)⋊C3)⋊C414yescomplemented
449C22#8867614yes—
450C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
451C4—184no—
452C4—184no—
453C3—31no—
454C3—31no—
455C3—61no—
456C3C2×D3×#21711yescharacteristic
457C2—24no—
458C2—98no—
459C2—184no—
460C2#1172131014yesdirect product factor
461C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
462C2C2×#8867612yes—
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC24×(C3⋊D3)⋊C3C2
cosocleC22×H(3,𝔽3)C24
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×C22⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. C2×#88552C2semidirect
4. C2×#3757⋊C3C2semidirect
5. #11721310C2direct (N × H)
6. C2×#88450C2semidirect
11. C23×(C3⋊D3)⋊C3C4semidirect
15. C22×#1084C22semidirect
16. C22×(C3⋊Dic3)⋊C3C22semidirect
18. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
19. #88552C22semidirect
20. #3757⋊C3C22semidirect
21. #88450C22semidirect
26. C2×#3757D3semidirect
36. C23×H(3,𝔽3)C2×C4semidirect
43. C22×(C3⋊D3)⋊C3C2×C4semidirect
44. C22×(C3⋊D3)⋊C3C2×C4semidirect
49. C2×#1084C23semidirect
51. C2×(C3⋊Dic3)⋊C3C23semidirect
60. C23×C3⋊D3Dic3semidirect
69. C22×C3⋊Dic3C2×D3semidirect
78. #3757C2×D3semidirect
88. C22×H(3,𝔽3)C22×C4semidirect
94. C2×(C3⋊D3)⋊C3C22⋊C4semidirect
96. C2×(C3⋊D3)⋊C3C22×C4semidirect
125. C23×C32C4×D3semidirect
126. C23×C32C2×Dic3semidirect
135. C22×C3⋊D3C2×Dic3semidirect
136. C22×C3⋊D3C2×Dic3semidirect
150. C2×C3⋊Dic3C22×D3semidirect
163. C2×H(3,𝔽3)C2×C22⋊C4semidirect
167. (C3⋊D3)⋊C3C2×C22⋊C4semidirect
221. C22×C32C2×C4×D3semidirect
224. C22×C32C22×Dic3semidirect
231. C2×C3⋊D3#217semidirect
233. C2×C3⋊D3C22×Dic3semidirect
253. H(3,𝔽3)C22×C22⋊C4semidirect
320. C2×C32D3×C22⋊C4semidirect
321. C2×C32C2×#217semidirect
327. C3⋊D3C2×#217semidirect
396. C32C2×D3×C22⋊C4semidirect
397. C32C22×#217semidirect
408. C23((C3⋊D3)⋊C3)⋊C4semidirect
448. C22C2×((C3⋊D3)⋊C3)⋊C4semidirect
460. C2#11721310direct (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×((C3⋊D3)⋊C3)⋊C42
C2×#885522
C2×#3757⋊C32
#117213102
C2×#884502
C2×#639743
C2×D3×#2173