Finite Group Databasebeta

← C2×#11721053 #13150585

Subgroups — C2×#11721053 #13150585

Order 1728.

1111 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#88538C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88430C212yesmaximal, complemented
5#11721053C218yesmaximal, direct product factor
6C2×((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
7C2×#63818—31nomaximal
8C2×D3×#208—31nomaximal
9C23×(C3⋊D3)⋊C3C411yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C411yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C414yes—
14C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2214yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18#88538C2214yescomplemented
19C2×#6180C2212yescomplemented
20#3748⋊C3C2214yescomplemented
21C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
22#88430C2218yescomplemented
23((C3⋊D3)⋊C3)⋊C8C2218yescomplemented
24C22×C4×C3⋊D3D311yescharacteristic
25C22×C3×C4×D3—31no—
26C2×#3748D311yescharacteristic, complemented
27C2×C3×#208—31no—
28C2×C3×#208—31no—
29C2×C3×#208—31no—
30C2×C3×#208—61no—
31C2×#3712—32no—
32C2×#3711—32no—
33#63818—38no—
34C2×#3710—32no—
35D3×#208—38no—
36C2×D3×C3⋊C8—32no—
37C23×H(3,𝔽3)C2×C411yescharacteristic
38C22×(C3⋊D3)⋊C3—24no—
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C22×(C3⋊D3)⋊C3C2×C414yes—
41C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
42C2×C4×H(3,𝔽3)C2×C412yes—
43C2×C4×H(3,𝔽3)C2×C411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2312yes—
46C2×(C3⋊Dic3)⋊C3C2×C414yes—
47C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
48C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
49C4×(C3⋊D3)⋊C3C2×C414yes—
50C4×(C3⋊D3)⋊C3C2×C412yes—
51C4×(C3⋊D3)⋊C3C2×C412yes—
52C4×(C3⋊D3)⋊C3C2×C414yes—
53C4×(C3⋊D3)⋊C3C2312yes—
54C4×(C3⋊D3)⋊C3C2312yes—
55#6180C2314yescomplemented
56(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
57C22×#208—91no—
58C2×D3×M16—91no—
59C2×D3×M16—91no—
60C23×C3⋊D3Dic311yescharacteristic
61C23×C3×D3—31no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C3⋊Dic3Dic311yescharacteristic
66C2×C4×C3⋊D3Dic311yescharacteristic
67C2×C4×C3⋊D3Dic314yes—
68C2×C4×C3⋊D3Dic311yescharacteristic
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C4×C3⋊D3C2×D314yes—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—31no—
79#3748C2×D314yescomplemented
80C2×C3⋊(C3⋊C8)C2×D312yescomplemented
81C3×#208—34no—
82C3×#208—34no—
83C3×#208—34no—
84C3×#208—64no—
85C2×C3×C3⋊C8—32no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—62no—
89#3712—38no—
90#3711—38no—
91#3710—38no—
92D3×C3⋊C8—38no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)C22×C412yes—
95C22×H(3,𝔽3)C22×C411yescharacteristic
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—28no—
98C2×(C3⋊D3)⋊C3M1614yescomplemented
99C2×(C3⋊D3)⋊C3C22×C412yes—
100C2×(C3⋊D3)⋊C3C22×C412yes—
101C4×H(3,𝔽3)C22×C411yescharacteristic
102C4×H(3,𝔽3)C22×C412yes—
103C4×H(3,𝔽3)C2411yescharacteristic
104(C3⋊Dic3)⋊C3C22×C412yes—
105(C3⋊Dic3)⋊C3C22×C412yes—
106C23×C3×C4—91no—
107C22×C4×D3—31no—
108C22×C4×D3—91no—
109C2×C3×M16—91no—
110C2×C3×M16—91no—
111C2×#208—31no—
112C2×#208—31no—
113C2×#208—31no—
114C2×#208—61no—
115C2×#208—91no—
116C2×#208—92no—
117C2×#208—98no—
118C22×C3⋊C8—92no—
119D3×M16—98no—
120D3×M16—98no—
121C2×ΓL(1,𝔽25)—92no—
122C2×ΓL(1,𝔽25)—92no—
123C2×D3×C8—92no—
124C2×D3×C8—92no—
125C23×C32—21no—
126C23×C32C4×D311yescharacteristic
127C23×C32C2×Dic311yescharacteristic
128C22×C3⋊D3—24no—
129C22×C3⋊D3C2×Dic311yescharacteristic
130C22×C3⋊D3C2×Dic314yes—
131C22×C3⋊D3C2×Dic311yescharacteristic
132C22×C3×D3—31no—
133C22×C3×D3—34no—
134C22×C3×D3—31no—
135C22×C3×D3—64no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4C4×D312yes—
141C2×C32×C4C4×D311yescharacteristic
142C2×C32×C4C2×Dic312yes—
143C2×C32×C4C2×Dic311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D312yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D312yes—
148C2×C3⋊Dic3C2×Dic314yes—
149C2×C3⋊Dic3C2×Dic311yescharacteristic
150C2×C3⋊Dic3C2×Dic311yescharacteristic
151C4×C3⋊D3C2×Dic314yes—
152C4×C3⋊D3C2×Dic312yes—
153C4×C3⋊D3C2×Dic312yes—
154C4×C3⋊D3C2×Dic314yes—
155C4×C3⋊D3C22×D312yes—
156C4×C3⋊D3C22×D312yes—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—31no—
159C2×C3×Dic3—31no—
160C3×C4×D3—32no—
161C3×C4×D3—34no—
162C3×C4×D3—34no—
163C3×C4×D3—32no—
164C3×C4×D3—32no—
165C3×C4×D3—32no—
166C3⋊(C3⋊C8)C22×D314yescomplemented
167C3×C3⋊C8—34no—
168C3×C3⋊C8—34no—
169C3×C3⋊C8—34no—
170C3×C3⋊C8—64no—
171C22×M16—271noSylow
172C2×H(3,𝔽3)—22no—
173C2×H(3,𝔽3)C2×M1612yescomplemented
174C2×H(3,𝔽3)C23×C411yescharacteristic
175(C3⋊D3)⋊C3—22no—
176(C3⋊D3)⋊C3C2×M1614yescomplemented
177C24×C3—91no—
178C23×D3—31no—
179C23×D3—91no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—61no—
183C22×C3×C4—91no—
184C22×C3×C4—94no—
185C22×C3×C4—91no—
186C22×C3×C4—91no—
187C22×C3×C4—91no—
188C22×C3×C4—94no—
189C22×C3×C4—91no—
190C22×C3×C4D3211yescharacteristic
191C22×Dic3—31no—
192C22×Dic3—91no—
193C2×C4×D3—31no—
194C2×C4×D3—34no—
195C2×C4×D3—31no—
196C2×C4×D3—31no—
197C2×C4×D3—31no—
198C2×C4×D3—34no—
199C2×C4×D3—91no—
200C2×C4×D3—91no—
201C2×C4×D3—94no—
202C2×C4×D3—91no—
203C2×C4×D3—94no—
204C2×C4×D3—91no—
205C3×M16—94no—
206C3×M16—94no—
207C2×C3×C8—92no—
208C2×C3×C8—92no—
209#208—34no—
210#208—34no—
211#208—34no—
212#208—64no—
213#208—94no—
214#208—98no—
215C2×C3⋊C8—32no—
216C2×C3⋊C8—32no—
217C2×C3⋊C8—32no—
218C2×C3⋊C8—62no—
219C2×C3⋊C8—92no—
220C2×C3⋊C8—98no—
221ΓL(1,𝔽25)—98no—
222ΓL(1,𝔽25)—98no—
223D3×C8—98no—
224D3×C8—98no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—42no—
230C22×C32C2×C4×D312yes—
231C22×C32C2×C4×D311yescharacteristic
232C22×C32C22×Dic312yes—
233C22×C32C22×Dic311yescharacteristic
234C2×C3⋊D3—22no—
235C2×C3⋊D3—28no—
236C2×C3⋊D3#20814yescomplemented
237C2×C3⋊D3C22×Dic312yes—
238C2×C3⋊D3C22×Dic312yes—
239C2×C3×D3—34no—
240C2×C3×D3—32no—
241C2×C3×D3—32no—
242C2×C3×D3—68no—
243C2×C3×D3—62no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—22no—
247C32×C4C2×C4×D311yescharacteristic
248C32×C4C2×C4×D312yes—
249C32×C4C22×Dic311yescharacteristic
250C32×C4C22×Dic312yes—
251C32×C4C23×D311yescharacteristic
252C32×C4C23×D311yescharacteristic
253C3⋊Dic3C22×Dic312yes—
254C3⋊Dic3C22×Dic312yes—
255C3×Dic3—32no—
256C3×Dic3—32no—
257C23×C4—91no—
258C2×M16—91no—
259C2×M16—91no—
260C2×M16—272no—
261C2×M16—278no—
262C22×C8—272no—
263H(3,𝔽3)C22×M1611yescharacteristic, complemented, Sylow
264C23×C3—31no—
265C23×C3—31no—
266C23×C3—61no—
267C23×C3—91no—
268C23×C3—91no—
269C23×C3—94no—
270C23×C3—184no—
271C23×C3D3×Dic311yescharacteristic
272C22×D3—31no—
273C22×D3—34no—
274C22×D3—31no—
275C22×D3—64no—
276C22×D3—91no—
277C22×D3—94no—
278C22×D3—91no—
279C22×D3—184no—
280C2×C3×C4—31no—
281C2×C3×C4—32no—
282C2×C3×C4—31no—
283C2×C3×C4—32no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—61no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—61no—
292C2×C3×C4—92no—
293C2×C3×C4—92no—
294C2×C3×C4—92no—
295C2×C3×C4—94no—
296C2×C3×C4—94no—
297C2×C3×C4—92no—
298C2×C3×C4—91no—
299C2×C3×C4—91no—
300C2×C3×C4—94no—
301C2×C3×C4D3×Dic312yes—
302C2×C3×C4D3×Dic311yescharacteristic
303C2×C3×C4C2×D3212yes—
304C2×C3×C4C2×D3211yescharacteristic
305C2×Dic3—34no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—91no—
309C2×Dic3—91no—
310C2×Dic3—94no—
311C4×D3—32no—
312C4×D3—32no—
313C4×D3—32no—
314C4×D3—34no—
315C4×D3—32no—
316C4×D3—34no—
317C4×D3—92no—
318C4×D3—94no—
319C4×D3—92no—
320C4×D3—94no—
321C4×D3—92no—
322C4×D3—92no—
323C3×C8—94no—
324C3×C8—94no—
325C3⋊C8—34no—
326C3⋊C8—34no—
327C3⋊C8—34no—
328C3⋊C8—64no—
329C3⋊C8—94no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—21no—
333C2×C32—22no—
334C2×C32—42no—
335C2×C32D3×M1612yescomplemented
336C2×C32C2×#20812yescomplemented
337C2×C32C22×C4×D311yescharacteristic
338C2×C32C23×Dic311yescharacteristic
339C3⋊D3—22no—
340C3⋊D3C2×#20814yescomplemented
341C3×D3—34no—
342C3×D3—62no—
343C24—91no—
344C22×C4—91no—
345C22×C4—91no—
346C22×C4—94no—
347C22×C4—91no—
348C22×C4—94no—
349C22×C4—91no—
350C22×C4—91no—
351C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
352M16—94no—
353M16—94no—
354M16—278no—
355C2×C8—92no—
356C2×C8—92no—
357C2×C8—278no—
358C22×C3—22no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—32no—
362C22×C3—32no—
363C22×C3—62no—
364C22×C3—62no—
365C22×C3—61no—
366C22×C3—62no—
367C22×C3—92no—
368C22×C3—94no—
369C22×C3—92no—
370C22×C3—122no—
371C22×C3—182no—
372C22×C3—188no—
373C22×C3C2×D3×Dic312yes—
374C22×C3C2×D3×Dic311yescharacteristic
375C2×D3—32no—
376C2×D3—34no—
377C2×D3—32no—
378C2×D3—68no—
379C2×D3—62no—
380C2×D3—92no—
381C2×D3—94no—
382C2×D3—92no—
383C2×D3—188no—
384C2×D3—182no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—32no—
388C3×C4—31no—
389C3×C4—32no—
390C3×C4—31no—
391C3×C4—61no—
392C3×C4—62no—
393C3×C4—61no—
394C3×C4—92no—
395C3×C4—92no—
396C3×C4C2×D3×Dic312yes—
397C3×C4C2×D3×Dic311yescharacteristic
398C3×C4C22×D3211yescharacteristic
399Dic3—32no—
400Dic3—32no—
401Dic3—92no—
402Dic3—92no—
403C32—21no—
404C32C2×D3×M1611yescharacteristic, complemented
405C32C22×#20811yescharacteristic, complemented
406C23—91no—
407C23—94no—
408C23—91no—
409C23—184no—
410C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
411C2×C4—94no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—94no—
417C2×C4—94no—
418C2×C4—92no—
419C2×C4—92no—
420C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
421C2×C4((C3⋊D3)⋊C3)⋊C412yes—
422C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
423C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C8—94no—
425C8—94no—
426C2×C3—22no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—32no—
430C2×C3—32no—
431C2×C3—62no—
432C2×C3—62no—
433C2×C3—61no—
434C2×C3—62no—
435C2×C3—94no—
436C2×C3—122no—
437C2×C3—182no—
438C2×C3D3×#20812yes—
439C2×C3C22×D3×Dic311yescharacteristic
440D3—34no—
441D3—62no—
442D3—94no—
443D3—182no—
444C22—22no—
445C22—92no—
446C22—94no—
447C22—92no—
448C22—188no—
449C22—182no—
450C22C2×((C3⋊D3)⋊C3)⋊C412yes—
451C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
452C4—92no—
453C4—92no—
454C4C2×((C3⋊D3)⋊C3)⋊C412yes—
455C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
456C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
457C3—31no—
458C3—31no—
459C3—61no—
460C3C2×D3×#20811yescharacteristic
461C2—22no—
462C2—94no—
463C2—182no—
464C2#1172105312yesdirect product factor
465C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC23×(C3⋊D3)⋊C3C4
cosocleC4×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88538C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88430C2semidirect
5. #11721053C2direct (N × H)
6. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
18. #88538C22semidirect
19. C2×#6180C22semidirect
20. #3748⋊C3C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. #88430C22semidirect
23. ((C3⋊D3)⋊C3)⋊C8C22semidirect
26. C2×#3748D3semidirect
55. #6180C23semidirect
56. (C3⋊(C3⋊C8))⋊C3C23semidirect
79. #3748C2×D3semidirect
80. C2×C3⋊(C3⋊C8)C2×D3semidirect
98. C2×(C3⋊D3)⋊C3M16semidirect
166. C3⋊(C3⋊C8)C22×D3semidirect
173. C2×H(3,𝔽3)C2×M16semidirect
176. (C3⋊D3)⋊C3C2×M16semidirect
236. C2×C3⋊D3#208semidirect
263. H(3,𝔽3)C22×M16semidirect
335. C2×C32D3×M16semidirect
336. C2×C32C2×#208semidirect
340. C3⋊D3C2×#208semidirect
404. C32C2×D3×M16semidirect
405. C32C22×#208semidirect
464. C2#11721053direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×#885382
C2×#3748⋊C32
C2×#884302
#117210532
C2×((C3⋊D3)⋊C3)⋊C82
C2×#638183
C2×D3×#2083