Finite Group Databasebeta

← C2×#11723561 #13177706

Subgroups — C2×#11723561 #13177706

Order 1728.

1900 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11723561C218yesmaximal, direct product factor
6C2×#88798C211yesmaximal, characteristic, complemented
7C2×#88797C211yesmaximal, characteristic, complemented
8C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×#64125—31nomaximal
10C2×#63935—34nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C3×#638—21no—
14C22×C3×C3⋊Dic3—21no—
15C3×#3830—22no—
16C3×#3830C2214yescomplemented
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6292—21no—
21C2×#6289—21no—
22#89018—22no—
23#89018C2214yescomplemented
24#88860C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26C32×#214C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic
28#88798C2214yescomplemented
29#88797C2214yescomplemented
30C3⋊D3×C3⋊C8C2214yescomplemented
31C2×#3832—31no—
32C2×#3830D311yescharacteristic, complemented
33C2×C3×#237—34no—
34C2×#3799—34no—
35C2×C32×SD16—31no—
36C2×#3756—31no—
37C2×#3753—31no—
38C2×#3753—34no—
39C2×#3753—64no—
40#64125—38no—
41C2×#3744—31no—
42C2×C8×C3⋊D3—31no—
43C2×C3×#214D314yescomplemented
44#63935—332no—
45C2×#3718—34no—
46C2×#3717—34no—
47C2×D3×C3⋊C8—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3×C3⋊D3D411yescharacteristic
51C33×D4—21no—
52C33×D4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C4⋊(C3⋊(C3⋊D3))—21no—
55C4⋊(C3⋊(C3⋊D3))C2312yes—
56C3×#638—24no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—24no—
59C2×C3×C3⋊Dic3D411yescharacteristic
60C3×C4×C3⋊D3D412yes—
61C3×C4×C3⋊D3C2312yes—
62C3×#634—21no—
63C3×#634C2312yes—
64C2×C32×Dic3—21no—
65C32×Dic6—21no—
66C32×Dic6C2312yes—
67#6292—24no—
68#6289—24no—
69C3⋊(C3⋊(C3⋊C8))C2312yes—
70C32×C3⋊C8C2312yes—
71C2×#950—91no—
72C2×#916—94no—
73C2×C32×Q8—31no—
74C2×C32×D4—24no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C2×D311yescharacteristic
77C2×#638—21no—
78C22×C3⋊Dic3—21no—
79#3832—34no—
80#3832—62no—
81#3830—22no—
82#3830C2×D314yescomplemented
83C2×C4⋊(C3⋊D3)—31no—
84C2×C4⋊(C3⋊D3)—34no—
85C2×C4⋊(C3⋊D3)—61no—
86C2×C4⋊(C3⋊D3)—64no—
87C2×C4×C3⋊D3—61no—
88C2×C4×C3⋊D3C2×D311yescharacteristic
89C2×#634C2×D311yescharacteristic
90C2×C3×#67—64no—
91C22×C3×Dic3—64no—
92C3×#237—316no—
93C3×#237—68no—
94C2×C3×C4×D3—34no—
95C2×C3×Dic6—34no—
96C2×C3×Dic6C2×D314yes—
97C2×#626—64no—
98C2×#624—64no—
99#3799—316no—
100#3799—68no—
101C32×SD16—34no—
102C2×C32×C8—31no—
103#3756—34no—
104#3753—34no—
105#3753—316no—
106#3753—616no—
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—34no—
109C2×C3⋊(C3⋊C8)—64no—
110#3744—34no—
111C8×C3⋊D3—34no—
112C3×#214C2×D3116yescomplemented
113C2×C3×C3⋊C8C2×D314yes—
114#3718—316no—
115#3717—316no—
116D3×C3⋊C8—316no—
117C22×C33—22no—
118C22×C33—42no—
119C22×C33C2×D411yescharacteristic
120C2×C3⋊(C3⋊D3)—22no—
121C2×C3⋊(C3⋊D3)—42no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3C2×D412yes—
124C33×C4C2×D411yescharacteristic
125C33×C4C2411yescharacteristic
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3C2×D412yes—
128C32×Dic3—22no—
129C2×C3×#41—91no—
130C2×#239—94no—
131C2×#237—34no—
132C2×#235—91no—
133C2×C3×SD16—34no—
134#950—98no—
135C2×#216—94no—
136C2×#216—91no—
137C2×#215—91no—
138C2×#214C3⋊D311yescharacteristic, complemented
139C2×#213—94no—
140C2×#213—91no—
141C2×#213—184no—
142C22×C3⋊C8—91no—
143#916—932no—
144C2×#204—94no—
145C2×D3×C8—94no—
146C23×C32—21no—
147C23×C32—24no—
148C23×C32—44no—
149C22×C3⋊D3—61no—
150C22×C3⋊D3—64no—
151C22×C3⋊D3—124no—
152C22×C3⋊D3#6711yescharacteristic
153C22×C3×D3—34no—
154C32×Q8—32no—
155C32×Q8—61no—
156C32×D4—21no—
157C32×D4—28no—
158C32×D4—24no—
159C32×D4—44no—
160C32×D4C22×D318yes—
161C32×D4C22×D312yes—
162C2×C32×C4—24no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D314yes—
165C2×C32×C4C22×D311yescharacteristic
166#638—24no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—24no—
169C2×C3⋊Dic3#6711yescharacteristic
170C4⋊(C3⋊D3)—32no—
171C4⋊(C3⋊D3)—38no—
172C4⋊(C3⋊D3)—68no—
173C4⋊(C3⋊D3)—64no—
174C4⋊(C3⋊D3)—61no—
175C4⋊(C3⋊D3)—64no—
176C4⋊(C3⋊D3)—124no—
177C4×C3⋊D3—64no—
178C4×C3⋊D3#6712yes—
179C4×C3⋊D3C22×D312yes—
180#634—21no—
181#634C22×D312yes—
182C3×#67—616no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—616no—
186C2×C3×Dic3—64no—
187C3×C4×D3—38no—
188C3×C4×D3—38no—
189C3×Dic6—24no—
190C3×Dic6—38no—
191C3×Dic6—64no—
192C3×Dic6C22×D318yes—
193#626—616no—
194#624—616no—
195C32×C8—32no—
196C3⋊(C3⋊C8)—32no—
197C3⋊(C3⋊C8)—38no—
198C3⋊(C3⋊C8)—68no—
199C3×C3⋊C8C22×D318yes—
200C2×#135—271noSylow
201C2×C33—42no—
202C2×C33#13512yescomplemented
203C2×C33C22×D411yescharacteristic
204C3⋊(C3⋊D3)—42no—
205C3×C3⋊D3—22no—
206C3×#41—94no—
207C3×#41—182no—
208C2×C3×Q8—34no—
209C2×C3×Q8—91no—
210C2×C3×D4—24no—
211C2×C3×D4—181no—
212C2×C3×D4D3214yes—
213C2×C3×D4C2×C3⋊D311yescharacteristic
214C22×C3×C4—91no—
215C22×C3×C4—181no—
216C2×#67—64no—
217C2×#67—181no—
218C22×Dic3—64no—
219#239—916no—
220#239—188no—
221#237—316no—
222#237—68no—
223#235—94no—
224#235—182no—
225C2×D12—94no—
226C2×D12—91no—
227C2×D12—184no—
228C2×D12—184no—
229C2×C4×D3—34no—
230C2×C4×D3—181no—
231C2×C4×D3—184no—
232C2×Dic6—34no—
233C2×Dic6C2×C3⋊D311yescharacteristic
234C3×SD16—316no—
235C2×C3×C8—34no—
236#216—916no—
237#216—94no—
238#215—94no—
239#214C2×C3⋊D314yescomplemented
240#213—916no—
241#213—94no—
242#213—1816no—
243C2×C3⋊C8—94no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—94no—
246C2×C3⋊C8—184no—
247C2×C3⋊C8C2×C3⋊D311yescharacteristic
248#204—916no—
249D3×C8—916no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—48no—
254C22×C32—42no—
255C22×C32—48no—
256C22×C32—88no—
257C22×C32D3×D414yes—
258C22×C32C2×#6711yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—68no—
262C2×C3⋊D3—128no—
263C2×C3⋊D3—122no—
264C2×C3⋊D3—128no—
265C2×C3⋊D3—248no—
266C2×C3⋊D3C2×#6712yes—
267C2×C3×D3—38no—
268C2×C3×D3—68no—
269C32×C4—24no—
270C32×C4—24no—
271C32×C4—62no—
272C32×C4D3×D414yes—
273C32×C4C2×#6711yescharacteristic
274C32×C4C23×D314yes—
275C32×C4C23×D311yescharacteristic
276C3⋊Dic3—22no—
277C3⋊Dic3C2×#6712yes—
278C3×Dic3—28no—
279C3×Dic3—38no—
280C3×Dic3—68no—
281C2×#41—91no—
282C2×#41—271no—
283#135—278no—
284C2×Q16—271no—
285C2×SD16—31no—
286C2×SD16—271no—
287C2×D8—271no—
288C22×C8—271no—
289C33C2×#13511yescharacteristic, complemented, Sylow
290C23×C3—24no—
291C23×C3—21no—
292C23×C3—44no—
293C23×C3—91no—
294C22×D3—34no—
295C22×D3—184no—
296C22×D3—181no—
297C22×D3—364no—
298C3×Q8—38no—
299C3×Q8—64no—
300C3×Q8—92no—
301C3×Q8—181no—
302C3×D4—24no—
303C3×D4—21no—
304C3×D4—28no—
305C3×D4—44no—
306C3×D4—184no—
307C3×D4C2×D3218yes—
308C3×D4C22×C3⋊D312yes—
309C2×C3×C4—24no—
310C2×C3×C4—64no—
311C2×C3×C4—91no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—184no—
315C2×C3×C4—181no—
316C2×C3×C4C2×D3214yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318#67—616no—
319#67—184no—
320C2×Dic3—21no—
321C2×Dic3—34no—
322C2×Dic3—616no—
323C2×Dic3—64no—
324D12—98no—
325D12—92no—
326D12—188no—
327D12—1816no—
328D12—184no—
329D12—181no—
330D12—364no—
331C4×D3—38no—
332C4×D3—38no—
333C4×D3—184no—
334C4×D3—1816no—
335Dic6—21no—
336Dic6—38no—
337Dic6—64no—
338Dic6C22×C3⋊D312yes—
339C3×C8—38no—
340C3⋊C8—98no—
341C3⋊C8—92no—
342C3⋊C8—188no—
343C3⋊C8C22×C3⋊D312yes—
344C2×C32—24no—
345C2×C32—28no—
346C2×C32—42no—
347C2×C32—48no—
348C2×C32—88no—
349C2×C32#91618yescomplemented
350C2×C32#95012yescomplemented
351C2×C32C2×D3×D414yes—
352C2×C32C22×#6711yescharacteristic
353C3⋊D3—22no—
354C3⋊D3—122no—
355C3⋊D3—128no—
356C3⋊D3—248no—
357C3×D3—68no—
358#41—94no—
359#41—182no—
360#41—274no—
361#41—542no—
362C2×Q8—31no—
363C2×Q8—91no—
364C2×D4—181no—
365C2×D4—271no—
366C2×D4—541no—
367C2×D4D3×C3⋊D311yescharacteristic
368C22×C4—91no—
369C22×C4—181no—
370C22×C4—541no—
371Q16—274no—
372SD16—34no—
373SD16—274no—
374D8—274no—
375C2×C8—31no—
376C2×C8—274no—
377C2×C8—271no—
378C22×C3—28no—
379C22×C3—22no—
380C22×C3—24no—
381C22×C3—48no—
382C22×C3—42no—
383C22×C3—48no—
384C22×C3—88no—
385C22×C3—92no—
386C22×C3—182no—
387C22×C3D3×#6714yes—
388C22×C3D4×C3⋊D311yescharacteristic
389C2×D3—38no—
390C2×D3—68no—
391C2×D3—188no—
392C2×D3—182no—
393C2×D3—368no—
394C2×D3—362no—
395C2×D3—368no—
396C2×D3—728no—
397C3×C4—24no—
398C3×C4—24no—
399C3×C4—68no—
400C3×C4—92no—
401C3×C4—182no—
402C3×C4D3×#6714yes—
403C3×C4D4×C3⋊D311yescharacteristic
404C3×C4C22×D3214yes—
405C3×C4C23×C3⋊D311yescharacteristic
406Dic3—22no—
407Dic3—38no—
408Dic3—68no—
409C32—24no—
410C32C2×#91614yescomplemented
411C32C2×#95011yescharacteristic, complemented
412C23—21no—
413C23—91no—
414C23—541no—
415Q8—32no—
416Q8—61no—
417Q8—92no—
418Q8—181no—
419D4—21no—
420D4—184no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4C2×D3×C3⋊D312yes—
425C2×C4—61no—
426C2×C4—91no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—184no—
430C2×C4—181no—
431C2×C4—544no—
432C2×C4C2×D3×C3⋊D311yescharacteristic
433C8—32no—
434C8—272no—
435C2×C3—28no—
436C2×C3—24no—
437C2×C3—48no—
438C2×C3—42no—
439C2×C3—88no—
440C2×C3—182no—
441C2×C3#6393518yescomplemented
442C2×C3#6412512yescomplemented
443C2×C3C2×D3×#6714yes—
444C2×C3C2×D4×C3⋊D311yescharacteristic
445D3—68no—
446D3—368no—
447D3—362no—
448D3—728no—
449C22—22no—
450C22—42no—
451C22—92no—
452C22—182no—
453C22—542no—
454C22—1082no—
455C22C3⋊D3×#6711yescharacteristic
456C4—62no—
457C4—92no—
458C4—182no—
459C4C3⋊D3×#6711yescharacteristic
460C4C22×D3×C3⋊D311yescharacteristic
461C3—24no—
462C3C2×#6393514yescomplemented
463C3C2×#6412511yescharacteristic, complemented
464C2—42no—
465C2—182no—
466C2—1082no—
467C2#1172356112yesdirect product factor
468C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×#89018C2semidirect
3. C2×#88860C2semidirect
4. C2×C32×#214C2semidirect
5. #11723561C2direct (N × H)
6. C2×#88798C2semidirect
7. C2×#88797C2semidirect
8. C2×C3⋊D3×C3⋊C8C2semidirect
16. C3×#3830C22semidirect
23. #89018C22semidirect
24. #88860C22semidirect
26. C32×#214C22semidirect
28. #88798C22semidirect
29. #88797C22semidirect
30. C3⋊D3×C3⋊C8C22semidirect
32. C2×#3830D3semidirect
43. C2×C3×#214D3semidirect
82. #3830C2×D3semidirect
112. C3×#214C2×D3semidirect
138. C2×#214C3⋊D3semidirect
202. C2×C33#135semidirect
239. #214C2×C3⋊D3semidirect
289. C33C2×#135semidirect
349. C2×C32#916semidirect
350. C2×C32#950semidirect
410. C32C2×#916semidirect
411. C32C2×#950semidirect
441. C2×C3#63935semidirect
442. C2×C3#64125semidirect
462. C3C2×#63935semidirect
463. C3C2×#64125semidirect
467. C2#11723561direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890182
C2×#888602
C2×C32×#2142
#117235612
C2×#887982
C2×#887972
C2×C3⋊D3×C3⋊C82
C2×#641253
C2×#639353