Finite Group Databasebeta

← C2×#11722816 #13170501

Subgroups — C2×#11722816 #13170501

Order 1728.

1426 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D32C211yesmaximal, characteristic
2C2×#88808C211yesmaximal, characteristic, complemented
3C2×#88786C212yesmaximal, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×#3711C212yesmaximal, complemented
6#11722816C218yesmaximal, direct product factor
7C2×D3×#208—32nomaximal
8C2×#63791—31nomaximal
9C22×C3×D32C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×C4×D3C2212yes—
12C2×C3×#624C411yescharacteristic
13C2×C3×D3×Dic3C412yes—
14C3×C4×D32C414yes—
15C3×C4×D32C2214yes—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
17C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
18C2×C32×C3⋊C8C2211yescharacteristic, complemented
19#88808C2214yescomplemented
20#88786C2218yescomplemented
21C3⋊D3×C3⋊C8C2214yescomplemented
22C3×#3711C2218yescomplemented
23C22×C3×C4×D3—32no—
24C2×C4×D32D311yescharacteristic
25C2×#3748—32no—
26C2×C8×C3⋊D3—31no—
27C2×C3×#208—32no—
28C2×C3×ΓL(1,𝔽25)—32no—
29C2×#3711—32no—
30C2×#3711—32no—
31C2×#3711—62no—
32C2×#3711D312yescomplemented
33C2×#3710—31no—
34D3×#208—316no—
35C2×D3×C3⋊C8—32no—
36C2×D3×C3⋊C8—32no—
37C2×D3×C3⋊C8—61no—
38#63791—38no—
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C22×C32×D3C2×C412yes—
41C2×C3×D32—24no—
42C2×C3×D32C2×C414yes—
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4×C3⋊D3C2×C412yes—
46C3×C4×C3⋊D3C2312yes—
47C2×C32×Dic3C2×C412yes—
48C32×C4×D3C2×C414yes—
49C32×C4×D3C2314yes—
50C3×#624C2×C414yes—
51C3×D3×Dic3C2×C418yes—
52C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
53C3×C3⋊(C3⋊C8)C2314yescomplemented
54C32×C3⋊C8C2312yescomplemented
55C22×#208—91no—
56C2×D3×M16—92no—
57C23×C3×D3—32no—
58C22×D32Dic311yescharacteristic
59C22×C32×C4—32no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C22×C3×Dic3—32no—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—38no—
67C2×C3×C4×D3—38no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3C2×D312yes—
70C2×C3×C4×D3C2×D312yes—
71C2×#624Dic311yescharacteristic
72C2×D3×Dic3Dic312yes—
73C4×D32Dic314yes—
74C4×D32C2×D314yes—
75C2×C32×C8—31no—
76#3748—38no—
77C2×C3⋊(C3⋊C8)—32no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)—62no—
81C2×C3⋊(C3⋊C8)—121no—
82C2×C3⋊(C3⋊C8)C2×D312yescomplemented
83C8×C3⋊D3—34no—
84C3×#208—38no—
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—62no—
89C2×C3×C3⋊C8C2×D312yescomplemented
90C3×ΓL(1,𝔽25)—38no—
91#3711—38no—
92#3711—38no—
93#3711—68no—
94#3711C2×D318yescomplemented
95#3710—34no—
96D3×C3⋊C8—38no—
97D3×C3⋊C8—38no—
98D3×C3⋊C8—64no—
99C22×C33C22×C411yescharacteristic
100C2×C3×C3⋊D3M1614yescomplemented
101C2×C3×C3⋊D3C22×C412yes—
102C2×C32×D3—24no—
103C2×C32×D3C22×C414yes—
104C3×D32—28no—
105C33×C4C22×C411yescharacteristic
106C33×C4C2411yescharacteristic
107C3×C3⋊Dic3C22×C412yes—
108C32×Dic3C22×C414yes—
109C23×C3×C4—91no—
110C22×C4×D3—32no—
111C2×C3×M16—92no—
112C2×#208—32no—
113C2×#208—92no—
114C2×#208—92no—
115C2×#208—98no—
116C2×#208—182no—
117C22×C3⋊C8—91no—
118C22×C3⋊C8—91no—
119D3×M16—916no—
120C2×ΓL(1,𝔽25)—32no—
121C2×ΓL(1,𝔽25)—92no—
122C2×D3×C8—92no—
123C2×D3×C8—92no—
124C2×D3×C8—181no—
125C23×C32—32no—
126C22×C3⋊D3C2×Dic311yescharacteristic
127C22×C3×D3—22no—
128C22×C3×D3—32no—
129C22×C3×D3—38no—
130C22×C3×D3—61no—
131C22×C3×D3—68no—
132C22×C3×D3C4×D312yes—
133C22×C3×D3C2×Dic312yes—
134C2×D32—24no—
135C2×D32C2×Dic314yes—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—32no—
139C2×C32×C4—34no—
140C2×C32×C4—34no—
141C2×C32×C4—41no—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3C2×Dic311yescharacteristic
145C4×C3⋊D3C2×Dic312yes—
146C4×C3⋊D3C22×D312yes—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—38no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3C4×D312yes—
152C2×C3×Dic3C2×Dic312yes—
153C3×C4×D3—24no—
154C3×C4×D3—24no—
155C3×C4×D3—38no—
156C3×C4×D3—34no—
157C3×C4×D3—34no—
158C3×C4×D3—38no—
159C3×C4×D3—62no—
160C3×C4×D3—62no—
161C3×C4×D3C4×D314yes—
162C3×C4×D3C2×Dic314yes—
163C3×C4×D3C22×D314yes—
164C3×C4×D3C22×D314yes—
165#624C2×Dic314yes—
166D3×Dic3C2×Dic318yes—
167C32×C8—32no—
168C3⋊(C3⋊C8)—34no—
169C3⋊(C3⋊C8)—32no—
170C3⋊(C3⋊C8)—62no—
171C3⋊(C3⋊C8)—64no—
172C3⋊(C3⋊C8)—122no—
173C3⋊(C3⋊C8)C22×D314yescomplemented
174C3×C3⋊C8—22no—
175C3×C3⋊C8—34no—
176C3×C3⋊C8—34no—
177C3×C3⋊C8—64no—
178C3×C3⋊C8C22×D314yescomplemented
179C22×M16—271noSylow
180C2×C33C2×M1612yescomplemented
181C2×C33C23×C411yescharacteristic
182C3×C3⋊D3C2×M1614yescomplemented
183C32×D3—24no—
184C24×C3—91no—
185C23×D3—32no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—62no—
189C22×C3×C4—94no—
190C22×C3×C4—91no—
191C22×C3×C4—94no—
192C22×C3×C4—92no—
193C22×C3×C4—91no—
194C22×Dic3—32no—
195C2×C4×D3—32no—
196C2×C4×D3—32no—
197C2×C4×D3—32no—
198C2×C4×D3—38no—
199C2×C4×D3—38no—
200C2×C4×D3—61no—
201C2×C4×D3D3212yes—
202C3×M16—98no—
203C2×C3×C8—32no—
204C2×C3×C8—61no—
205C2×C3×C8—92no—
206#208—38no—
207#208—98no—
208#208—98no—
209#208—188no—
210C2×C3⋊C8—32no—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—62no—
213C2×C3⋊C8—94no—
214C2×C3⋊C8—92no—
215C2×C3⋊C8—91no—
216C2×C3⋊C8—94no—
217C2×C3⋊C8—181no—
218C2×C3⋊C8—182no—
219C2×C3⋊C8—361no—
220C2×C3⋊C8D3211yescharacteristic, complemented
221ΓL(1,𝔽25)—38no—
222ΓL(1,𝔽25)—98no—
223D3×C8—98no—
224D3×C8—98no—
225D3×C8—184no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—34no—
229C22×C32—41no—
230C22×C32—64no—
231C22×C32C2×C4×D312yes—
232C22×C32C22×Dic311yescharacteristic
233C2×C3⋊D3#20814yescomplemented
234C2×C3⋊D3C22×Dic312yes—
235C2×C3×D3—24no—
236C2×C3×D3—24no—
237C2×C3×D3—24no—
238C2×C3×D3—38no—
239C2×C3×D3—34no—
240C2×C3×D3—44no—
241C2×C3×D3—64no—
242C2×C3×D3—62no—
243C2×C3×D3—616no—
244C2×C3×D3C2×C4×D314yes—
245C2×C3×D3C22×Dic314yes—
246D32—28no—
247C32×C4—21no—
248C32×C4—22no—
249C32×C4—21no—
250C32×C4—22no—
251C32×C4—34no—
252C32×C4—41no—
253C32×C4—41no—
254C32×C4C2×C4×D312yes—
255C32×C4C22×Dic311yescharacteristic
256C32×C4C23×D312yes—
257C32×C4C23×D311yescharacteristic
258C3⋊Dic3C22×Dic312yes—
259C3×Dic3—24no—
260C3×Dic3—34no—
261C3×Dic3—62no—
262C3×Dic3C2×C4×D314yes—
263C3×Dic3C22×Dic314yes—
264C23×C4—91no—
265C2×M16—92no—
266C2×M16—272no—
267C2×M16—278no—
268C22×C8—271no—
269C22×C8—271no—
270C33C22×M1611yescharacteristic, complemented, Sylow
271C23×C3—32no—
272C23×C3—32no—
273C23×C3—62no—
274C23×C3—94no—
275C23×C3—91no—
276C23×C3—184no—
277C22×D3—32no—
278C22×D3—38no—
279C22×D3—61no—
280C22×D3—68no—
281C22×D3D3×Dic312yes—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—34no—
287C2×C3×C4—34no—
288C2×C3×C4—34no—
289C2×C3×C4—34no—
290C2×C3×C4—41no—
291C2×C3×C4—64no—
292C2×C3×C4—64no—
293C2×C3×C4—62no—
294C2×C3×C4—94no—
295C2×C3×C4—98no—
296C2×C3×C4—91no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4C2×D3211yescharacteristic
300C2×C3×C4C2×D3212yes—
301C2×Dic3—32no—
302C2×Dic3—38no—
303C2×Dic3—61no—
304C2×Dic3D3×Dic312yes—
305C4×D3—38no—
306C4×D3—34no—
307C4×D3—34no—
308C4×D3—38no—
309C4×D3—62no—
310C4×D3—62no—
311C4×D3D3×Dic314yes—
312C4×D3C2×D3214yes—
313C3×C8—34no—
314C3×C8—62no—
315C3×C8—94no—
316C3⋊C8—34no—
317C3⋊C8—34no—
318C3⋊C8—64no—
319C3⋊C8—94no—
320C3⋊C8—92no—
321C3⋊C8—182no—
322C3⋊C8—184no—
323C3⋊C8—362no—
324C3⋊C8C2×D3212yescomplemented
325C2×C32—21no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—24no—
329C2×C32—42no—
330C2×C32—41no—
331C2×C32—64no—
332C2×C32D3×M1614yescomplemented
333C2×C32C2×#20812yescomplemented
334C2×C32C22×C4×D312yes—
335C2×C32C23×Dic311yescharacteristic
336C3⋊D3C2×#20814yescomplemented
337C3×D3—24no—
338C3×D3—24no—
339C3×D3—38no—
340C3×D3—44no—
341C3×D3—64no—
342C24—91no—
343C22×C4—32no—
344C22×C4—94no—
345C22×C4—91no—
346C22×C4—94no—
347C22×C4—92no—
348C22×C4—91no—
349M16—98no—
350M16—278no—
351C2×C8—31no—
352C2×C8—92no—
353C2×C8—274no—
354C2×C8—274no—
355C2×C8—271no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—34no—
359C22×C3—34no—
360C22×C3—41no—
361C22×C3—64no—
362C22×C3—64no—
363C22×C3—64no—
364C22×C3—92no—
365C22×C3—94no—
366C22×C3—124no—
367C22×C3—188no—
368C22×C3C4×D3211yescharacteristic
369C22×C3C2×D3×Dic312yes—
370C2×D3—24no—
371C2×D3—34no—
372C2×D3—38no—
373C2×D3—62no—
374C2×D3—64no—
375C2×D3—616no—
376C2×D3C2×D3×Dic314yes—
377C3×C4—22no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4—34no—
382C3×C4—34no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—64no—
386C3×C4—92no—
387C3×C4C4×D3211yescharacteristic
388C3×C4C2×D3×Dic312yes—
389C3×C4C22×D3211yescharacteristic
390C3×C4C22×D3212yes—
391Dic3—34no—
392Dic3—62no—
393Dic3C2×D3×Dic314yes—
394C32—21no—
395C32—22no—
396C32—41no—
397C32C2×D3×M1612yescomplemented
398C32C22×#20811yescharacteristic, complemented
399C23—32no—
400C23—94no—
401C23—91no—
402C23—184no—
403C2×C4—32no—
404C2×C4—34no—
405C2×C4—34no—
406C2×C4—94no—
407C2×C4—98no—
408C2×C4—91no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4D3311yescharacteristic
412C8—32no—
413C8—94no—
414C8—272no—
415C2×C3—22no—
416C2×C3—24no—
417C2×C3—21no—
418C2×C3—22no—
419C2×C3—42no—
420C2×C3—41no—
421C2×C3—64no—
422C2×C3—64no—
423C2×C3—94no—
424C2×C3—124no—
425C2×C3#6379112yescomplemented
426C2×C3D3×#20814yescomplemented
427C2×C3C2×C4×D3211yescharacteristic
428C2×C3C22×D3×Dic312yes—
429D3—24no—
430D3—38no—
431D3—64no—
432C22—34no—
433C22—64no—
434C22—94no—
435C22—92no—
436C22—188no—
437C22D32×Dic311yescharacteristic
438C4—34no—
439C4—92no—
440C4D32×Dic311yescharacteristic
441C4C2×D3311yescharacteristic
442C3—21no—
443C3—22no—
444C3—41no—
445C3C2×#6379111yescharacteristic, complemented
446C3C2×D3×#20812yescomplemented
447C2—64no—
448C2—94no—
449C2#1172281612yesdirect product factor
450C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutatorC2×C33C23×C4
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C3×D32C4
cosocleC33×C4C24
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88808C2semidirect
3. C2×#88786C2semidirect
4. C2×C3⋊D3×C3⋊C8C2semidirect
5. C2×C3×#3711C2semidirect
6. #11722816C2direct (N × H)
16. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
17. C2×C3×C3⋊(C3⋊C8)C22semidirect
18. C2×C32×C3⋊C8C22semidirect
19. #88808C22semidirect
20. #88786C22semidirect
21. C3⋊D3×C3⋊C8C22semidirect
22. C3×#3711C22semidirect
32. C2×#3711D3semidirect
52. C3⋊(C3⋊(C3⋊C8))C23semidirect
53. C3×C3⋊(C3⋊C8)C23semidirect
54. C32×C3⋊C8C23semidirect
82. C2×C3⋊(C3⋊C8)C2×D3semidirect
89. C2×C3×C3⋊C8C2×D3semidirect
94. #3711C2×D3semidirect
100. C2×C3×C3⋊D3M16semidirect
173. C3⋊(C3⋊C8)C22×D3semidirect
178. C3×C3⋊C8C22×D3semidirect
180. C2×C33C2×M16semidirect
182. C3×C3⋊D3C2×M16semidirect
220. C2×C3⋊C8D32semidirect
233. C2×C3⋊D3#208semidirect
270. C33C22×M16semidirect
324. C3⋊C8C2×D32semidirect
332. C2×C32D3×M16semidirect
333. C2×C32C2×#208semidirect
336. C3⋊D3C2×#208semidirect
397. C32C2×D3×M16semidirect
398. C32C22×#208semidirect
425. C2×C3#63791semidirect
426. C2×C3D3×#208semidirect
445. C3C2×#63791semidirect
446. C3C2×D3×#208semidirect
449. C2#11722816direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D322
C2×#888082
C2×#887862
C2×C3⋊D3×C3⋊C82
C2×C3×#37112
#117228162
C2×D3×#2083
C2×#637913