Finite Group Databasebeta

← C2×#11723546 #13177535

Subgroups — C2×#11723546 #13177535

Order 1728.

2092 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11723546C218yesmaximal, direct product factor
6C2×#88797C211yesmaximal, characteristic, complemented
7C2×#88792C211yesmaximal, characteristic, complemented
8C2×#88787C211yesmaximal, characteristic, complemented
9C2×#64120—31nomaximal
10C2×#63925—34nomaximal
11C22×D3×C3⋊D3—21no—
12C2×C33×D4C2211yescharacteristic, complemented
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C3×#638—21no—
15C22×C3×C3⋊Dic3—21no—
16C3×#3830—22no—
17C3×#3830C2214yescomplemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic, complemented
20C2×C32×D12C2211yescharacteristic
21C2×#6291—21no—
22C3⋊D3×D12—22no—
23C3⋊D3×D12C2214yescomplemented
24#88860C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26C32×#213C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic, complemented
28#88797C2214yescomplemented
29#88792C2214yescomplemented
30#88787C2214yescomplemented
31C2×#3830D311yescharacteristic, complemented
32C2×D4×C3⋊D3—31no—
33C2×C3×#237—34no—
34C2×D3×D12—34no—
35C2×C32×D8—31no—
36C2×#3754—31no—
37C2×#3753—31no—
38C2×#3753—34no—
39C2×#3753—64no—
40#64120—38no—
41C2×#3744—31no—
42C2×#3743—31no—
43C2×C3×#213D314yescomplemented
44#63925—332no—
45C2×#3717—34no—
46C2×#3715—34no—
47C2×#3711—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3×C3⋊D3D411yescharacteristic
51C22×C32×D3—21no—
52C2×D3×C3⋊D3—24no—
53C2×D3×C3⋊D3—44no—
54C33×D4—21no—
55C33×D4C2312yescomplemented
56C2×C33×C4C2311yescharacteristic
57C4⋊(C3⋊(C3⋊D3))—21no—
58C4⋊(C3⋊(C3⋊D3))C2312yes—
59C3×#638—24no—
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3—24no—
62C2×C3×C3⋊Dic3D411yescharacteristic
63C3×C4×C3⋊D3D412yes—
64C3×C4×C3⋊D3C2312yes—
65C3×#634—21no—
66C3×#634C2312yescomplemented
67C32×D12—21no—
68C32×D12C2312yes—
69#6291—24no—
70C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
71C32×C3⋊C8C2312yescomplemented
72C2×#949—91no—
73C2×#911—94no—
74C23×C3⋊D3—61no—
75C22×D32—64no—
76C2×C32×D4—24no—
77C2×C32×D4—31no—
78C2×C32×D4C2×D314yescomplemented
79C2×C32×D4C2×D311yescharacteristic, complemented
80C2×#638—21no—
81C2×#638—61no—
82C22×C3⋊Dic3—21no—
83#3830—22no—
84#3830C2×D314yescomplemented
85D4×C3⋊D3—34no—
86D4×C3⋊D3—62no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×#634C2×D311yescharacteristic, complemented
92C2×C3×#67—64no—
93C22×C3×Dic3—64no—
94C3×#237—316no—
95C3×#237—68no—
96C2×C3×D12C2×D314yes—
97C2×C3×C4×D3—34no—
98C2×C3×Dic6—34no—
99C2×#626—64no—
100D3×D12—316no—
101D3×D12—68no—
102C32×D8—34no—
103C2×C32×C8—31no—
104#3754—34no—
105#3753—34no—
106#3753—316no—
107#3753—616no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—34no—
110C2×C3⋊(C3⋊C8)—64no—
111#3744—34no—
112#3743—34no—
113C3×#213C2×D3116yescomplemented
114C2×C3×C3⋊C8C2×D314yescomplemented
115#3717—316no—
116#3715—316no—
117#3711—316no—
118C22×C33—22no—
119C22×C33—42no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)—42no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3C2×D412yes—
125C2×C32×D3—22no—
126C2×C32×D3—42no—
127D3×C3⋊D3—48no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2411yescharacteristic
130C3×C3⋊Dic3—22no—
131C3×C3⋊Dic3C2×D412yes—
132C2×C3×#41—91no—
133C2×#237—34no—
134C2×D3×D4—94no—
135C22×D12—91no—
136C2×C3×D8—34no—
137#949—98no—
138C2×#215—91no—
139C2×#215—91no—
140C2×#214—94no—
141C2×#213—94no—
142C2×#213—91no—
143C2×#213—184no—
144C2×#213C3⋊D311yescharacteristic, complemented
145C2×#208—91no—
146#911—932no—
147C2×#204—94no—
148C2×ΓL(1,𝔽25)—94no—
149C23×C32—21no—
150C23×C32—24no—
151C23×C32—44no—
152C23×C32—61no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—64no—
155C22×C3⋊D3—64no—
156C22×C3⋊D3—124no—
157C22×C3⋊D3—124no—
158C22×C3⋊D3#6711yescharacteristic
159C22×C3×D3—24no—
160C22×C3×D3—34no—
161C2×D32—616no—
162C2×D32—1216no—
163C32×D4—21no—
164C32×D4—28no—
165C32×D4—24no—
166C32×D4—32no—
167C32×D4—44no—
168C32×D4—61no—
169C32×D4C22×D318yescomplemented
170C32×D4C22×D312yescomplemented
171C2×C32×C4—24no—
172C2×C32×C4C22×D314yes—
173C2×C32×C4C22×D311yescharacteristic
174#638—24no—
175#638—64no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—24no—
178C2×C3⋊Dic3#6711yescharacteristic
179C4⋊(C3⋊D3)—32no—
180C4⋊(C3⋊D3)—38no—
181C4⋊(C3⋊D3)—68no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—64no—
184C4⋊(C3⋊D3)—124no—
185C4×C3⋊D3#6712yes—
186C4×C3⋊D3C22×D312yes—
187#634—21no—
188#634C22×D312yescomplemented
189C3×#67—616no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—616no—
192C2×C3×Dic3—64no—
193C3×D12—24no—
194C3×D12C22×D318yes—
195C3×C4×D3—38no—
196C3×C4×D3—38no—
197C3×Dic6—38no—
198C3×Dic6—64no—
199#626—616no—
200C32×C8—32no—
201C3⋊(C3⋊C8)—32no—
202C3⋊(C3⋊C8)—38no—
203C3⋊(C3⋊C8)—68no—
204C3×C3⋊C8C22×D318yescomplemented
205C2×AGL(1,ℤ/8)—271noSylow
206C2×C33—42no—
207C2×C33AGL(1,ℤ/8)12yescomplemented
208C2×C33C22×D411yescharacteristic
209C3⋊(C3⋊D3)—42no—
210C3×C3⋊D3—22no—
211C32×D3—42no—
212C23×D3—181no—
213C23×D3—184no—
214C3×#41—94no—
215C3×#41—182no—
216C2×C3×Q8—91no—
217C2×C3×D4—24no—
218C2×C3×D4—34no—
219C2×C3×D4—181no—
220C2×C3×D4D3214yescomplemented
221C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
222C22×C3×C4—91no—
223C22×C3×C4—181no—
224C2×#67—64no—
225C2×#67—184no—
226C22×Dic3—64no—
227#237—316no—
228#237—68no—
229D3×D4—916no—
230D3×D4—188no—
231C2×D12—94no—
232C2×D12—91no—
233C2×D12—94no—
234C2×D12—184no—
235C2×D12—182no—
236C2×D12—181no—
237C2×D12C2×C3⋊D311yescharacteristic
238C2×C4×D3—34no—
239C2×Dic6—34no—
240C3×D8—316no—
241C2×C3×C8—34no—
242#215—94no—
243#215—94no—
244#214—916no—
245#213—916no—
246#213—94no—
247#213—1816no—
248#213C2×C3⋊D314yescomplemented
249#208—94no—
250C2×C3⋊C8—94no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—184no—
253C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
254#204—916no—
255ΓL(1,𝔽25)—916no—
256C22×C32—22no—
257C22×C32—24no—
258C22×C32—28no—
259C22×C32—48no—
260C22×C32—42no—
261C22×C32—48no—
262C22×C32—62no—
263C22×C32—88no—
264C22×C32—122no—
265C22×C32D3×D414yes—
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—22no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—68no—
270C2×C3⋊D3—128no—
271C2×C3⋊D3—128no—
272C2×C3⋊D3—122no—
273C2×C3⋊D3—128no—
274C2×C3⋊D3—248no—
275C2×C3⋊D3C2×#6712yes—
276C2×C3×D3—28no—
277C2×C3×D3—38no—
278C2×C3×D3—48no—
279C2×C3×D3—68no—
280D32—1232no—
281C32×C4—24no—
282C32×C4—24no—
283C32×C4D3×D414yes—
284C32×C4C2×#6711yescharacteristic
285C32×C4C23×D314yes—
286C32×C4C23×D311yescharacteristic
287C3⋊Dic3—22no—
288C3⋊Dic3C2×#6712yes—
289C3×Dic3—38no—
290C3×Dic3—68no—
291C2×#41—91no—
292C22×D4—271no—
293AGL(1,ℤ/8)—278no—
294C2×SD16—271no—
295C2×SD16—271no—
296C2×D8—31no—
297C2×D8—271no—
298C2×M16—271no—
299C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
300C23×C3—24no—
301C23×C3—21no—
302C23×C3—44no—
303C23×C3—64no—
304C23×C3—91no—
305C22×D3—21no—
306C22×D3—34no—
307C22×D3—184no—
308C22×D3—1816no—
309C22×D3—184no—
310C22×D3—181no—
311C22×D3—364no—
312C22×D3—364no—
313C22×D3—3616no—
314C3×Q8—92no—
315C3×Q8—181no—
316C3×D4—24no—
317C3×D4—21no—
318C3×D4—28no—
319C3×D4—38no—
320C3×D4—44no—
321C3×D4—64no—
322C3×D4—184no—
323C3×D4C2×D3218yescomplemented
324C3×D4C22×C3⋊D312yescomplemented
325C2×C3×C4—24no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—184no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C22×C3⋊D311yescharacteristic
333#67—616no—
334#67—1816no—
335C2×Dic3—34no—
336C2×Dic3—616no—
337C2×Dic3—64no—
338D12—21no—
339D12—98no—
340D12—92no—
341D12—188no—
342D12—184no—
343D12—184no—
344D12—181no—
345D12—364no—
346D12C22×C3⋊D312yes—
347C4×D3—38no—
348C4×D3—38no—
349Dic6—38no—
350Dic6—64no—
351C3×C8—38no—
352C3⋊C8—98no—
353C3⋊C8—92no—
354C3⋊C8—188no—
355C3⋊C8C22×C3⋊D312yescomplemented
356C2×C32—24no—
357C2×C32—28no—
358C2×C32—42no—
359C2×C32—48no—
360C2×C32—88no—
361C2×C32—122no—
362C2×C32#91118yescomplemented
363C2×C32#94912yescomplemented
364C2×C32C2×D3×D414yes—
365C2×C32C22×#6711yescharacteristic
366C3⋊D3—22no—
367C3⋊D3—122no—
368C3⋊D3—128no—
369C3⋊D3—248no—
370C3×D3—48no—
371C3×D3—68no—
372C24—541no—
373#41—94no—
374#41—182no—
375C2×Q8—91no—
376C2×D4—31no—
377C2×D4—181no—
378C2×D4—274no—
379C2×D4—271no—
380C2×D4—542no—
381C2×D4—541no—
382C2×D4D3×C3⋊D311yescharacteristic, complemented
383C22×C4—91no—
384C22×C4—181no—
385SD16—274no—
386SD16—274no—
387D8—34no—
388D8—274no—
389M16—274no—
390C2×C8—31no—
391C2×C8—271no—
392C22×C3—28no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—48no—
396C22×C3—42no—
397C22×C3—48no—
398C22×C3—68no—
399C22×C3—88no—
400C22×C3—92no—
401C22×C3—128no—
402C22×C3—182no—
403C22×C3D3×#6714yes—
404C22×C3D4×C3⋊D311yescharacteristic
405C2×D3—22no—
406C2×D3—38no—
407C2×D3—42no—
408C2×D3—68no—
409C2×D3—188no—
410C2×D3—182no—
411C2×D3—368no—
412C2×D3—368no—
413C2×D3—3632no—
414C2×D3—362no—
415C2×D3—368no—
416C2×D3—728no—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—92no—
420C3×C4—182no—
421C3×C4D3×#6714yes—
422C3×C4D4×C3⋊D311yescharacteristic
423C3×C4C22×D3214yes—
424C3×C4C23×C3⋊D311yescharacteristic
425Dic3—38no—
426Dic3—68no—
427C32—24no—
428C32C2×#91114yescomplemented
429C32C2×#94911yescharacteristic, complemented
430C23—21no—
431C23—61no—
432C23—91no—
433C23—544no—
434C23—541no—
435C23—1084no—
436Q8—92no—
437Q8—181no—
438D4—21no—
439D4—32no—
440D4—61no—
441D4—184no—
442D4—272no—
443D4—544no—
444D4—541no—
445D4C2×D3×C3⋊D312yescomplemented
446C2×C4—91no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—184no—
450C2×C4—181no—
451C2×C4C2×D3×C3⋊D311yescharacteristic
452C8—32no—
453C8—272no—
454C2×C3—28no—
455C2×C3—24no—
456C2×C3—48no—
457C2×C3—42no—
458C2×C3—88no—
459C2×C3—128no—
460C2×C3—182no—
461C2×C3#6392518yescomplemented
462C2×C3#6412012yescomplemented
463C2×C3C2×D3×#6714yes—
464C2×C3C2×D4×C3⋊D311yescharacteristic
465D3—42no—
466D3—68no—
467D3—368no—
468D3—362no—
469D3—728no—
470C22—22no—
471C22—42no—
472C22—62no—
473C22—92no—
474C22—122no—
475C22—182no—
476C22—542no—
477C22—1088no—
478C22—1082no—
479C22C3⋊D3×#6711yescharacteristic
480C4—92no—
481C4—182no—
482C4C3⋊D3×#6711yescharacteristic
483C4C22×D3×C3⋊D311yescharacteristic
484C3—24no—
485C3C2×#6392514yescomplemented
486C3C2×#6412011yescharacteristic, complemented
487C2—42no—
488C2—122no—
489C2—182no—
490C2—1082no—
491C2#1172354612yesdirect product factor
492C2C2×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×AGL(1,ℤ/8)
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×C3⋊D3×D12C2semidirect
3. C2×#88860C2semidirect
4. C2×C32×#213C2semidirect
5. #11723546C2direct (N × H)
6. C2×#88797C2semidirect
7. C2×#88792C2semidirect
8. C2×#88787C2semidirect
12. C2×C33×D4C22semidirect
17. C3×#3830C22semidirect
19. C2×C3×#634C22semidirect
23. C3⋊D3×D12C22semidirect
24. #88860C22semidirect
25. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C32×#213C22semidirect
27. C2×C32×C3⋊C8C22semidirect
28. #88797C22semidirect
29. #88792C22semidirect
30. #88787C22semidirect
31. C2×#3830D3semidirect
43. C2×C3×#213D3semidirect
55. C33×D4C23semidirect
66. C3×#634C23semidirect
70. C3⋊(C3⋊(C3⋊C8))C23semidirect
71. C32×C3⋊C8C23semidirect
78. C2×C32×D4C2×D3semidirect
79. C2×C32×D4C2×D3semidirect
84. #3830C2×D3semidirect
91. C2×#634C2×D3semidirect
113. C3×#213C2×D3semidirect
114. C2×C3×C3⋊C8C2×D3semidirect
144. C2×#213C3⋊D3semidirect
169. C32×D4C22×D3semidirect
170. C32×D4C22×D3semidirect
188. #634C22×D3semidirect
204. C3×C3⋊C8C22×D3semidirect
207. C2×C33AGL(1,ℤ/8)semidirect
220. C2×C3×D4D32semidirect
221. C2×C3×D4C2×C3⋊D3semidirect
248. #213C2×C3⋊D3semidirect
253. C2×C3⋊C8C2×C3⋊D3semidirect
299. C33C2×AGL(1,ℤ/8)semidirect
323. C3×D4C2×D32semidirect
324. C3×D4C22×C3⋊D3semidirect
355. C3⋊C8C22×C3⋊D3semidirect
362. C2×C32#911semidirect
363. C2×C32#949semidirect
382. C2×D4D3×C3⋊D3semidirect
428. C32C2×#911semidirect
429. C32C2×#949semidirect
445. D4C2×D3×C3⋊D3semidirect
461. C2×C3#63925semidirect
462. C2×C3#64120semidirect
485. C3C2×#63925semidirect
486. C3C2×#64120semidirect
491. C2#11723546direct (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×C3⋊D3×D122
C2×#888602
C2×C32×#2132
#117235462
C2×#887972
C2×#887922
C2×#887872
C2×#641203
C2×#639253