Finite Group Databasebeta

← C2×#11722820 #13170508

Subgroups — C2×#11722820 #13170508

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×C4×#1113C211yesmaximal, characteristic
2C2×#88787C212yesmaximal, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×C3×#3712C212yesmaximal, complemented
5C2×C3×#3710C211yesmaximal, characteristic, complemented
6#11722820C218yesmaximal, direct product factor
7C2×#63791—31nomaximal
8C2×D3×ΓL(1,𝔽25)—32nomaximal
9C22×#1113C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2212yes—
12C2×#6294C412yes—
13C2×#6294C411yescharacteristic
14C4×#1113C414yes—
15C4×#1113C2214yes—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
17C2×C32×C3⋊C8C2211yescharacteristic, complemented
18C2×C32×C3⋊C8C2212yescomplemented
19#88787C2218yescomplemented
20C3⋊D3×C3⋊C8C2214yescomplemented
21C3×#3712C2218yescomplemented
22C3×#3710C2214yescomplemented
23C2×C4×D32—32no—
24C2×C4×D32—31no—
25C2×#3743—32no—
26C2×C8×C3⋊D3—31no—
27C2×C3×ΓL(1,𝔽25)—32no—
28C2×C3×ΓL(1,𝔽25)—32no—
29C2×C3×D3×C8—32no—
30C2×#3712D312yescomplemented
31C2×#3711—32no—
32C2×#3711—32no—
33C2×#3711—62no—
34C2×#3710D311yescharacteristic, complemented
35C2×D3×C3⋊C8—32no—
36C2×D3×C3⋊C8—61no—
37#63791—38no—
38D3×ΓL(1,𝔽25)—316no—
39C22×C3×C3⋊D3C2×C412yes—
40C22×C3×C3⋊D3C2×C411yescharacteristic
41C2×#1113—24no—
42C2×#1113C2×C414yes—
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C2×C3×C3⋊Dic3C2×C412yes—
46C3×C4×C3⋊D3C2×C414yes—
47C3×C4×C3⋊D3C2×C412yes—
48C3×C4×C3⋊D3C2312yes—
49C3×C4×C3⋊D3C2314yes—
50#6294C2×C418yes—
51#6294C2×C414yes—
52C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
53C32×C3⋊C8C2314yescomplemented
54C32×C3⋊C8C2312yescomplemented
55C2×D3×M16—92no—
56C22×ΓL(1,𝔽25)—91no—
57C22×D32—32no—
58C22×D32—31no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×C4×C3⋊D3C2×D312yes—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—62no—
66C2×#624—32no—
67C2×#624—31no—
68C2×D3×Dic3—32no—
69C2×D3×Dic3—32no—
70C2×D3×Dic3—32no—
71C4×D32—38no—
72C4×D32—34no—
73C4×D32—38no—
74C4×D32—34no—
75C2×C32×C8—32no—
76C2×C32×C8—31no—
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—
82#3743—38no—
83C8×C3⋊D3—34no—
84C2×C3×C3⋊C8—21no—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2×D312yescomplemented
87C2×C3×C3⋊C8C2×D312yescomplemented
88C2×C3×C3⋊C8C2×D312yescomplemented
89C3×ΓL(1,𝔽25)—38no—
90C3×ΓL(1,𝔽25)—38no—
91C3×D3×C8—38no—
92#3712C2×D318yescomplemented
93#3711—38no—
94#3711—38no—
95#3711—68no—
96#3710C2×D314yescomplemented
97D3×C3⋊C8—38no—
98D3×C3⋊C8—64no—
99C22×C33C22×C411yescharacteristic
100C2×C3×C3⋊D3—24no—
101C2×C3×C3⋊D3M1614yescomplemented
102C2×C3×C3⋊D3C22×C414yes—
103C2×C3×C3⋊D3C22×C412yes—
104#1113—28no—
105C33×C4C22×C411yescharacteristic
106C33×C4C2411yescharacteristic
107C3×C3⋊Dic3C22×C412yes—
108C3×C3⋊Dic3C22×C414yes—
109C22×C4×D3—91no—
110C22×C4×D3—92no—
111C2×C3×M16—92no—
112C22×C3×C8—91no—
113C2×#208—92no—
114C22×C3⋊C8—91no—
115D3×M16—916no—
116C2×ΓL(1,𝔽25)—32no—
117C2×ΓL(1,𝔽25)—32no—
118C2×ΓL(1,𝔽25)—98no—
119C2×ΓL(1,𝔽25)—92no—
120C2×ΓL(1,𝔽25)—92no—
121C2×ΓL(1,𝔽25)—182no—
122C2×D3×C8—32no—
123C2×D3×C8—92no—
124C2×D3×C8—181no—
125C22×C3⋊D3C4×D312yes—
126C22×C3⋊D3C4×D311yescharacteristic
127C22×C3×D3—32no—
128C22×C3×D3—32no—
129C22×C3×D3—32no—
130C22×C3×D3—61no—
131C22×C3×D3—62no—
132C2×D32—38no—
133C2×D32—34no—
134C2×D32—68no—
135C2×D32—64no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D312yes—
141C2×C3⋊Dic3C4×D311yescharacteristic
142C2×C3⋊Dic3C4×D312yes—
143C4×C3⋊D3C4×D314yes—
144C4×C3⋊D3C4×D312yes—
145C4×C3⋊D3C22×D312yes—
146C4×C3⋊D3C22×D314yes—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—62no—
152C3×C4×D3—34no—
153C3×C4×D3—34no—
154C3×C4×D3—34no—
155C3×C4×D3—34no—
156C3×C4×D3—34no—
157C3×C4×D3—34no—
158C3×C4×D3—62no—
159C3×C4×D3—64no—
160C3×C4×D3—62no—
161C3×C4×D3—64no—
162#624—38no—
163#624—34no—
164D3×Dic3—38no—
165D3×Dic3—38no—
166D3×Dic3—38no—
167C32×C8—34no—
168C32×C8—32no—
169C3⋊(C3⋊C8)—34no—
170C3⋊(C3⋊C8)—32no—
171C3⋊(C3⋊C8)—62no—
172C3⋊(C3⋊C8)—64no—
173C3⋊(C3⋊C8)—122no—
174C3×C3⋊C8—22no—
175C3×C3⋊C8—24no—
176C3×C3⋊C8C22×D314yescomplemented
177C3×C3⋊C8C22×D314yescomplemented
178C3×C3⋊C8C22×D314yescomplemented
179C22×M16—271noSylow
180C2×C33C2×M1612yescomplemented
181C2×C33C23×C411yescharacteristic
182C3×C3⋊D3—24no—
183C3×C3⋊D3C2×M1614yescomplemented
184C23×D3—92no—
185C23×D3—91no—
186C22×C3×C4—91no—
187C22×C3×C4—92no—
188C22×Dic3—91no—
189C22×Dic3—92no—
190C2×C4×D3—32no—
191C2×C4×D3—32no—
192C2×C4×D3—32no—
193C2×C4×D3—61no—
194C2×C4×D3—62no—
195C2×C4×D3—94no—
196C2×C4×D3—98no—
197C2×C4×D3—94no—
198C2×C4×D3—98no—
199C2×C4×D3—92no—
200C2×C4×D3—92no—
201C2×C4×D3—92no—
202C3×M16—98no—
203C2×C3×C8—32no—
204C2×C3×C8—32no—
205C2×C3×C8—32no—
206C2×C3×C8—61no—
207C2×C3×C8—62no—
208C2×C3×C8—94no—
209#208—98no—
210C2×C3⋊C8—92no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—94no—
213C2×C3⋊C8—181no—
214C2×C3⋊C8—182no—
215C2×C3⋊C8—361no—
216C2×C3⋊C8D3212yescomplemented
217C2×C3⋊C8D3211yescharacteristic, complemented
218ΓL(1,𝔽25)—38no—
219ΓL(1,𝔽25)—38no—
220ΓL(1,𝔽25)—98no—
221ΓL(1,𝔽25)—98no—
222ΓL(1,𝔽25)—188no—
223D3×C8—38no—
224D3×C8—98no—
225D3×C8—184no—
226C22×C32—22no—
227C22×C32—21no—
228C22×C32—41no—
229C22×C32C2×C4×D311yescharacteristic
230C22×C32C2×C4×D312yes—
231C2×C3⋊D3—24no—
232C2×C3⋊D3ΓL(1,𝔽25)14yescomplemented
233C2×C3⋊D3C2×C4×D314yes—
234C2×C3⋊D3C2×C4×D312yes—
235C2×C3×D3—38no—
236C2×C3×D3—34no—
237C2×C3×D3—34no—
238C2×C3×D3—34no—
239C2×C3×D3—64no—
240C2×C3×D3—62no—
241C2×C3×D3—64no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244C2×C3×D3—124no—
245D32—616no—
246D32—68no—
247C32×C4—21no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—21no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4C2×C4×D311yescharacteristic
254C32×C4C2×C4×D312yes—
255C32×C4C23×D311yescharacteristic
256C32×C4C23×D312yes—
257C3⋊Dic3C2×C4×D312yes—
258C3⋊Dic3C2×C4×D314yes—
259C3×Dic3—34no—
260C3×Dic3—34no—
261C3×Dic3—34no—
262C3×Dic3—62no—
263C3×Dic3—64no—
264C23×C4—271no—
265C2×M16—92no—
266C2×M16—272no—
267C2×M16—278no—
268C22×C8—91no—
269C22×C8—271no—
270C33C22×M1611yescharacteristic, complemented, Sylow
271C23×C3—91no—
272C23×C3—92no—
273C22×D3—32no—
274C22×D3—32no—
275C22×D3—32no—
276C22×D3—61no—
277C22×D3—62no—
278C22×D3—98no—
279C22×D3—94no—
280C22×D3—188no—
281C22×D3—184no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—92no—
286C2×C3×C4—91no—
287C2×C3×C4—92no—
288C2×C3×C4—94no—
289C2×C3×C4—92no—
290C2×C3×C4—94no—
291C2×C3×C4C2×D3212yes—
292C2×C3×C4C2×D3211yescharacteristic
293C2×Dic3—32no—
294C2×Dic3—32no—
295C2×Dic3—32no—
296C2×Dic3—61no—
297C2×Dic3—62no—
298C2×Dic3—94no—
299C2×Dic3—98no—
300C4×D3—34no—
301C4×D3—34no—
302C4×D3—34no—
303C4×D3—34no—
304C4×D3—34no—
305C4×D3—34no—
306C4×D3—62no—
307C4×D3—64no—
308C4×D3—62no—
309C4×D3—64no—
310C4×D3—98no—
311C4×D3—98no—
312C4×D3—98no—
313C3×C8—34no—
314C3×C8—34no—
315C3×C8—34no—
316C3×C8—62no—
317C3×C8—64no—
318C3⋊C8—94no—
319C3⋊C8—92no—
320C3⋊C8—182no—
321C3⋊C8—184no—
322C3⋊C8—362no—
323C3⋊C8C2×D3214yescomplemented
324C3⋊C8C2×D3212yescomplemented
325C2×C32—24no—
326C2×C32—22no—
327C2×C32—21no—
328C2×C32—22no—
329C2×C32—42no—
330C2×C32—41no—
331C2×C32C2×ΓL(1,𝔽25)12yescomplemented
332C2×C32D3×M1614yescomplemented
333C2×C32C22×C4×D311yescharacteristic
334C2×C32C22×C4×D312yes—
335C3⋊D3—24no—
336C3⋊D3C2×ΓL(1,𝔽25)14yescomplemented
337C3×D3—38no—
338C3×D3—64no—
339C3×D3—64no—
340C3×D3—64no—
341C3×D3—124no—
342C24—271no—
343C22×C4—91no—
344C22×C4—92no—
345C22×C4—274no—
346C22×C4—274no—
347C22×C4—271no—
348C22×C4—272no—
349M16—98no—
350M16—278no—
351C2×C8—32no—
352C2×C8—31no—
353C2×C8—94no—
354C2×C8—274no—
355C2×C8—271no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3—92no—
360C22×C3—94no—
361C22×C3—94no—
362C22×C3—184no—
363C22×C3C4×D3212yes—
364C22×C3C4×D3211yescharacteristic
365C2×D3—38no—
366C2×D3—34no—
367C2×D3—34no—
368C2×D3—34no—
369C2×D3—62no—
370C2×D3—64no—
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—64no—
374C2×D3—124no—
375C2×D3—1816no—
376C2×D3—188no—
377C3×C4—21no—
378C3×C4—22no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—94no—
384C3×C4—92no—
385C3×C4C4×D3212yes—
386C3×C4C4×D3211yescharacteristic
387C3×C4C22×D3212yes—
388C3×C4C22×D3211yescharacteristic
389Dic3—34no—
390Dic3—34no—
391Dic3—34no—
392Dic3—62no—
393Dic3—64no—
394C32—22no—
395C32—21no—
396C32—41no—
397C32C22×ΓL(1,𝔽25)11yescharacteristic, complemented
398C32C2×D3×M1612yescomplemented
399C23—91no—
400C23—92no—
401C23—274no—
402C23—544no—
403C2×C4—91no—
404C2×C4—92no—
405C2×C4—92no—
406C2×C4—94no—
407C2×C4—92no—
408C2×C4—94no—
409C2×C4—274no—
410C2×C4—278no—
411C2×C4D3311yescharacteristic
412C8—34no—
413C8—32no—
414C8—272no—
415C2×C3—22no—
416C2×C3—24no—
417C2×C3—21no—
418C2×C3—22no—
419C2×C3—42no—
420C2×C3—41no—
421C2×C3—94no—
422C2×C3—184no—
423C2×C3D3×ΓL(1,𝔽25)14yescomplemented
424C2×C3#6379112yescomplemented
425C2×C3C2×C4×D3212yes—
426C2×C3C2×C4×D3211yescharacteristic
427D3—38no—
428D3—64no—
429D3—64no—
430D3—64no—
431D3—124no—
432C22—94no—
433C22—92no—
434C22—94no—
435C22—184no—
436C22—548no—
437C22#8894911yescharacteristic
438C4—92no—
439C4—94no—
440C4#8894911yescharacteristic
441C4C2×D3311yescharacteristic
442C3—21no—
443C3—22no—
444C3—41no—
445C3C2×D3×ΓL(1,𝔽25)12yescomplemented
446C3C2×#6379111yescharacteristic, complemented
447C2—94no—
448C2—184no—
449C2#1172282012yesdirect product factor
450C2C2×#8894911yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutatorC2×C33C23×C4
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×#1113C4
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×#88787C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. C2×C3×#3712C2semidirect
5. C2×C3×#3710C2semidirect
6. #11722820C2direct (N × H)
16. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
17. C2×C32×C3⋊C8C22semidirect
18. C2×C32×C3⋊C8C22semidirect
19. #88787C22semidirect
20. C3⋊D3×C3⋊C8C22semidirect
21. C3×#3712C22semidirect
22. C3×#3710C22semidirect
30. C2×#3712D3semidirect
34. C2×#3710D3semidirect
52. C3⋊(C3⋊(C3⋊C8))C23semidirect
53. C32×C3⋊C8C23semidirect
54. C32×C3⋊C8C23semidirect
86. C2×C3×C3⋊C8C2×D3semidirect
87. C2×C3×C3⋊C8C2×D3semidirect
88. C2×C3×C3⋊C8C2×D3semidirect
92. #3712C2×D3semidirect
96. #3710C2×D3semidirect
101. C2×C3×C3⋊D3M16semidirect
176. C3×C3⋊C8C22×D3semidirect
177. C3×C3⋊C8C22×D3semidirect
178. C3×C3⋊C8C22×D3semidirect
180. C2×C33C2×M16semidirect
183. C3×C3⋊D3C2×M16semidirect
216. C2×C3⋊C8D32semidirect
217. C2×C3⋊C8D32semidirect
232. C2×C3⋊D3ΓL(1,𝔽25)semidirect
270. C33C22×M16semidirect
323. C3⋊C8C2×D32semidirect
324. C3⋊C8C2×D32semidirect
331. C2×C32C2×ΓL(1,𝔽25)semidirect
332. C2×C32D3×M16semidirect
336. C3⋊D3C2×ΓL(1,𝔽25)semidirect
397. C32C22×ΓL(1,𝔽25)semidirect
398. C32C2×D3×M16semidirect
423. C2×C3D3×ΓL(1,𝔽25)semidirect
424. C2×C3#63791semidirect
445. C3C2×D3×ΓL(1,𝔽25)semidirect
446. C3C2×#63791semidirect
449. C2#11722820direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×#887872
C2×C3⋊D3×C3⋊C82
C2×C3×#37122
C2×C3×#37102
#117228202
C2×#637913
C2×D3×ΓL(1,𝔽25)3