Finite Group Databasebeta

← C2×#11723548 #13177563

Subgroups — C2×#11723548 #13177563

Order 1728.

2444 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic, complemented
2C2×#89026C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11723548C218yesmaximal, direct product factor
6C2×#88796C211yesmaximal, characteristic, complemented
7C2×#88794C211yesmaximal, characteristic, complemented
8C2×#88788C211yesmaximal, characteristic, complemented
9C2×#64120—31nomaximal
10C2×#63927—34nomaximal
11C22×D3×C3⋊D3—21no—
12C2×C33×D4C2211yescharacteristic, complemented
13C2×#6312—21no—
14C22×C3⋊(C3⋊Dic3)—21no—
15#89078—22no—
16#89078C2214yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×#6308C2211yescharacteristic, complemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C32×D12C2211yescharacteristic
21C2×#6290—21no—
22#89026—22no—
23#89026C2214yescomplemented
24C3×#3753C2214yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
26C32×#213C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic, complemented
28#88796C2214yescomplemented
29#88794C2214yescomplemented
30#88788C2214yescomplemented
31C2×#3830—31no—
32C2×#3830—34no—
33C2×#3830—64no—
34C2×D4×C3⋊D3—31no—
35C2×#3802—34no—
36C2×C32×D8—31no—
37C2×#3754—31no—
38C2×#3753D311yescharacteristic, complemented
39#64120—38no—
40C2×#3744—31no—
41C2×#3743—31no—
42C2×C3×#213—34no—
43C2×C3×#213D314yescomplemented
44#63927—332no—
45C2×#3716—34no—
46C2×#3716—34no—
47C2×#3712—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)D411yescharacteristic
50C22×C3×C3⋊D3—21no—
51C22×C32×D3—21no—
52C2×D3×C3⋊D3—24no—
53C2×D3×C3⋊D3—44no—
54C33×D4—21no—
55C33×D4C2312yescomplemented
56C2×C33×C4C2311yescharacteristic
57#6312—24no—
58C2×C3⋊(C3⋊Dic3)—21no—
59C2×C3⋊(C3⋊Dic3)—24no—
60C2×C3⋊(C3⋊Dic3)D411yescharacteristic
61C4×C3⋊(C3⋊D3)D412yes—
62C4×C3⋊(C3⋊D3)C2312yes—
63#6308—21no—
64#6308C2312yescomplemented
65C3×C4⋊(C3⋊D3)—21no—
66C3×C4⋊(C3⋊D3)C2312yes—
67C32×D12—21no—
68C32×D12C2312yes—
69#6290—24no—
70C3×C3⋊(C3⋊C8)C2312yescomplemented
71C32×C3⋊C8C2312yescomplemented
72C2×#911—94no—
73C2×#911—91no—
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—61no—
81C2×#638—61no—
82C2×#638—64no—
83C2×#638—68no—
84C22×C3⋊Dic3—61no—
85C22×C3⋊Dic3—64no—
86C22×C3⋊Dic3—68no—
87#3830—34no—
88#3830—316no—
89#3830—62no—
90#3830—68no—
91#3830—616no—
92#3830—616no—
93D4×C3⋊D3—34no—
94D4×C3⋊D3—62no—
95C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
96C2×C4×C3⋊D3—31no—
97C2×C4×C3⋊D3—34no—
98C2×C4×C3⋊D3—64no—
99C2×#634—31no—
100C2×#634—34no—
101C2×#634—64no—
102C2×C3×D12—34no—
103C2×C3×D12C2×D314yes—
104C2×#625—64no—
105#3802—316no—
106#3802—68no—
107C32×D8—34no—
108C2×C32×C8—31no—
109#3754—34no—
110#3753C2×D314yescomplemented
111C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
112#3744—34no—
113#3743—34no—
114C3×#213—316no—
115C3×#213C2×D3116yescomplemented
116C2×C3×C3⋊C8—34no—
117C2×C3×C3⋊C8C2×D314yescomplemented
118#3716—316no—
119#3716—316no—
120#3712—316no—
121C22×C33—22no—
122C22×C33—42no—
123C22×C33C2×D411yescharacteristic
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)C2×D412yes—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—42no—
128C2×C32×D3—22no—
129C2×C32×D3—42no—
130D3×C3⋊D3—48no—
131C33×C4C2×D411yescharacteristic
132C33×C4C2411yescharacteristic
133C3⋊(C3⋊Dic3)—22no—
134C3⋊(C3⋊Dic3)C2×D412yes—
135C2×#237—94no—
136C2×#237—91no—
137C2×#237—184no—
138C2×D3×D4—94no—
139C2×D3×D4—91no—
140C2×C3×D8—34no—
141C2×C3×D8—91no—
142C2×#214—94no—
143C2×#214—91no—
144C2×#213—34no—
145C2×#213C3⋊D311yescharacteristic, complemented
146#911—932no—
147#911—98no—
148C2×#204—94no—
149C2×#204—91no—
150C2×ΓL(1,𝔽25)—94no—
151C2×ΓL(1,𝔽25)—91no—
152C23×C32—28no—
153C23×C32—24no—
154C23×C32—21no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3⋊D3—31no—
158C22×C3⋊D3—34no—
159C22×C3⋊D3—64no—
160C22×C3⋊D3—64no—
161C22×C3⋊D3—124no—
162C22×C3×D3—24no—
163C22×C3×D3—64no—
164C2×D32—616no—
165C2×D32—1216no—
166C32×D4—28no—
167C32×D4—24no—
168C32×D4—21no—
169C32×D4—28no—
170C32×D4—32no—
171C32×D4—61no—
172C32×D4C22×D318yescomplemented
173C32×D4C22×D312yescomplemented
174C2×C32×C4—24no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D311yescharacteristic
177#638—64no—
178#638—64no—
179#638—616no—
180#638—632no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—616no—
186C2×C3⋊Dic3—68no—
187C2×C3⋊Dic3—632no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—61no—
190C4⋊(C3⋊D3)—21no—
191C4⋊(C3⋊D3)C22×D312yes—
192C4×C3⋊D3—32no—
193C4×C3⋊D3—32no—
194C4×C3⋊D3—38no—
195C4×C3⋊D3—38no—
196C4×C3⋊D3—68no—
197C4×C3⋊D3—68no—
198#634—32no—
199#634—38no—
200#634—68no—
201#634—68no—
202#634—64no—
203#634—61no—
204C3×D12—24no—
205C3×D12—38no—
206C3×D12—64no—
207C3×D12C22×D318yes—
208#625—616no—
209C32×C8—32no—
210C3⋊(C3⋊C8)C22×D312yescomplemented
211C3×C3⋊C8—38no—
212C3×C3⋊C8C22×D318yescomplemented
213C2×AGL(1,ℤ/8)—271noSylow
214C2×C33—42no—
215C2×C33AGL(1,ℤ/8)12yescomplemented
216C2×C33C22×D411yescharacteristic
217C3⋊(C3⋊D3)—22no—
218C3×C3⋊D3—42no—
219C32×D3—42no—
220C23×D3—181no—
221C23×D3—184no—
222C2×C3×D4—24no—
223C2×C3×D4—34no—
224C2×C3×D4—91no—
225C2×C3×D4D3214yescomplemented
226C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
227C2×#67—181no—
228C2×#67—184no—
229C2×#67—184no—
230C2×#67—181no—
231C2×#67—188no—
232C22×Dic3—184no—
233C22×Dic3—181no—
234C22×Dic3—188no—
235#237—916no—
236#237—94no—
237#237—1816no—
238#237—188no—
239#237—182no—
240#237—1816no—
241D3×D4—916no—
242D3×D4—94no—
243D3×D4—182no—
244D3×D4—188no—
245C2×D12—34no—
246C2×D12C2×C3⋊D311yescharacteristic
247C2×C4×D3—94no—
248C2×C4×D3—91no—
249C2×C4×D3—184no—
250C2×Dic6—94no—
251C2×Dic6—91no—
252C2×Dic6—184no—
253C3×D8—316no—
254C3×D8—94no—
255C2×C3×C8—34no—
256C2×C3×C8—91no—
257#214—916no—
258#214—94no—
259#213—316no—
260#213C2×C3⋊D314yescomplemented
261C2×C3⋊C8—34no—
262C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
263#204—916no—
264#204—94no—
265ΓL(1,𝔽25)—916no—
266ΓL(1,𝔽25)—94no—
267C22×C32—216no—
268C22×C32—28no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—416no—
272C22×C32—48no—
273C22×C32—42no—
274C22×C32—62no—
275C22×C32—122no—
276C22×C32D3×D414yes—
277C22×C32D3×D411yescharacteristic
278C2×C3⋊D3—22no—
279C2×C3⋊D3—32no—
280C2×C3⋊D3—38no—
281C2×C3⋊D3—42no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—128no—
286C2×C3⋊D3—128no—
287C2×C3×D3—28no—
288C2×C3×D3—48no—
289C2×C3×D3—68no—
290C2×C3×D3—128no—
291D32—1232no—
292C32×C4—24no—
293C32×C4—24no—
294C32×C4D3×D414yes—
295C32×C4D3×D411yescharacteristic
296C32×C4C23×D314yes—
297C32×C4C23×D311yescharacteristic
298C3⋊Dic3—32no—
299C3⋊Dic3—38no—
300C3⋊Dic3—68no—
301C3⋊Dic3—616no—
302C3⋊Dic3—68no—
303C3⋊Dic3—62no—
304C2×#41—271no—
305C22×D4—271no—
306AGL(1,ℤ/8)—278no—
307C2×SD16—271no—
308C2×SD16—271no—
309C2×D8—31no—
310C2×D8—91no—
311C2×M16—271no—
312C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
313C23×C3—28no—
314C23×C3—24no—
315C23×C3—21no—
316C23×C3—64no—
317C23×C3—181no—
318C22×D3—21no—
319C22×D3—64no—
320C22×D3—94no—
321C22×D3—91no—
322C22×D3—184no—
323C22×D3—184no—
324C22×D3—1816no—
325C22×D3—364no—
326C22×D3—3616no—
327C3×D4—28no—
328C3×D4—24no—
329C3×D4—21no—
330C3×D4—28no—
331C3×D4—38no—
332C3×D4—64no—
333C3×D4—92no—
334C3×D4—181no—
335C3×D4C2×D3218yescomplemented
336C3×D4C22×C3⋊D312yescomplemented
337C2×C3×C4—24no—
338C2×C3×C4C2×D3214yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—184no—
341#67—1816no—
342#67—1816no—
343#67—184no—
344#67—1832no—
345C2×Dic3—94no—
346C2×Dic3—91no—
347C2×Dic3—184no—
348C2×Dic3—1816no—
349C2×Dic3—184no—
350C2×Dic3—188no—
351C2×Dic3—1832no—
352C2×Dic3—184no—
353C2×Dic3—181no—
354D12—21no—
355D12—38no—
356D12—64no—
357D12C22×C3⋊D312yes—
358C4×D3—98no—
359C4×D3—92no—
360C4×D3—98no—
361C4×D3—92no—
362C4×D3—188no—
363C4×D3—188no—
364Dic6—98no—
365Dic6—92no—
366Dic6—188no—
367Dic6—188no—
368Dic6—184no—
369Dic6—181no—
370C3×C8—38no—
371C3×C8—92no—
372C3⋊C8—38no—
373C3⋊C8C22×C3⋊D312yescomplemented
374C2×C32—24no—
375C2×C32—28no—
376C2×C32—416no—
377C2×C32—48no—
378C2×C32—42no—
379C2×C32—122no—
380C2×C32#91118yescomplemented
381C2×C32#91112yescomplemented
382C2×C32C2×D3×D414yes—
383C2×C32C2×D3×D411yescharacteristic
384C3⋊D3—42no—
385C3⋊D3—62no—
386C3⋊D3—68no—
387C3⋊D3—128no—
388C3×D3—48no—
389C3×D3—128no—
390C24—541no—
391#41—274no—
392#41—542no—
393C2×Q8—271no—
394C2×D4—31no—
395C2×D4—91no—
396C2×D4—274no—
397C2×D4—542no—
398C2×D4—541no—
399C2×D4—541no—
400C2×D4D3×C3⋊D311yescharacteristic, complemented
401C22×C4—271no—
402C22×C4—541no—
403SD16—274no—
404SD16—274no—
405D8—34no—
406D8—94no—
407M16—274no—
408C2×C8—31no—
409C2×C8—91no—
410C22×C3—216no—
411C22×C3—28no—
412C22×C3—22no—
413C22×C3—24no—
414C22×C3—416no—
415C22×C3—42no—
416C22×C3—48no—
417C22×C3—68no—
418C22×C3—128no—
419C22×C3—182no—
420C22×C3—362no—
421C22×C3#381114yes—
422C22×C3D4×C3⋊D311yescharacteristic
423C2×D3—22no—
424C2×D3—42no—
425C2×D3—68no—
426C2×D3—98no—
427C2×D3—92no—
428C2×D3—128no—
429C2×D3—188no—
430C2×D3—182no—
431C2×D3—188no—
432C2×D3—368no—
433C2×D3—368no—
434C2×D3—3632no—
435C3×C4—24no—
436C3×C4—24no—
437C3×C4#381114yes—
438C3×C4D4×C3⋊D311yescharacteristic
439C3×C4C22×D3214yes—
440C3×C4C23×C3⋊D311yescharacteristic
441Dic3—98no—
442Dic3—92no—
443Dic3—188no—
444Dic3—1816no—
445Dic3—188no—
446Dic3—182no—
447C32—24no—
448C32C2×#91114yescomplemented
449C32C2×#91111yescharacteristic, complemented
450C23—21no—
451C23—61no—
452C23—181no—
453C23—271no—
454C23—544no—
455C23—1084no—
456Q8—272no—
457Q8—541no—
458D4—21no—
459D4—32no—
460D4—61no—
461D4—92no—
462D4—181no—
463D4—544no—
464D4—544no—
465D4C2×D3×C3⋊D312yescomplemented
466C2×C4—271no—
467C2×C4—272no—
468C2×C4—272no—
469C2×C4—544no—
470C2×C4—541no—
471C2×C4C2×D3×C3⋊D311yescharacteristic
472C8—32no—
473C8—92no—
474C2×C3—28no—
475C2×C3—24no—
476C2×C3—416no—
477C2×C3—48no—
478C2×C3—42no—
479C2×C3—128no—
480C2×C3—362no—
481C2×C3#6392718yescomplemented
482C2×C3#6412012yescomplemented
483C2×C3C2×#381114yes—
484C2×C3C2×D4×C3⋊D311yescharacteristic
485D3—42no—
486D3—128no—
487D3—188no—
488D3—182no—
489D3—368no—
490C22—22no—
491C22—42no—
492C22—62no—
493C22—122no—
494C22—182no—
495C22—272no—
496C22—362no—
497C22—542no—
498C22—1088no—
499C22#8903911yescharacteristic
500C4—272no—
501C4—542no—
502C4#8903911yescharacteristic
503C4C22×D3×C3⋊D311yescharacteristic
504C3—24no—
505C3C2×#6392714yescomplemented
506C3C2×#6412011yescharacteristic, complemented
507C2—42no—
508C2—122no—
509C2—362no—
510C2—542no—
511C2#1172354812yesdirect product factor
512C2C2×#8903911yescharacteristic
subgroupG/H
centerC22#89039
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×#89078C2semidirect
2. C2×#89026C2semidirect
3. C2×C3×#3753C2semidirect
4. C2×C32×#213C2semidirect
5. #11723548C2direct (N × H)
6. C2×#88796C2semidirect
7. C2×#88794C2semidirect
8. C2×#88788C2semidirect
12. C2×C33×D4C22semidirect
16. #89078C22semidirect
18. C2×#6308C22semidirect
23. #89026C22semidirect
24. C3×#3753C22semidirect
25. C2×C3×C3⋊(C3⋊C8)C22semidirect
26. C32×#213C22semidirect
27. C2×C32×C3⋊C8C22semidirect
28. #88796C22semidirect
29. #88794C22semidirect
30. #88788C22semidirect
38. C2×#3753D3semidirect
43. C2×C3×#213D3semidirect
55. C33×D4C23semidirect
64. #6308C23semidirect
70. C3×C3⋊(C3⋊C8)C23semidirect
71. C32×C3⋊C8C23semidirect
78. C2×C32×D4C2×D3semidirect
79. C2×C32×D4C2×D3semidirect
110. #3753C2×D3semidirect
111. C2×C3⋊(C3⋊C8)C2×D3semidirect
115. C3×#213C2×D3semidirect
117. C2×C3×C3⋊C8C2×D3semidirect
145. C2×#213C3⋊D3semidirect
172. C32×D4C22×D3semidirect
173. C32×D4C22×D3semidirect
210. C3⋊(C3⋊C8)C22×D3semidirect
212. C3×C3⋊C8C22×D3semidirect
215. C2×C33AGL(1,ℤ/8)semidirect
225. C2×C3×D4D32semidirect
226. C2×C3×D4C2×C3⋊D3semidirect
260. #213C2×C3⋊D3semidirect
262. C2×C3⋊C8C2×C3⋊D3semidirect
312. C33C2×AGL(1,ℤ/8)semidirect
335. C3×D4C2×D32semidirect
336. C3×D4C22×C3⋊D3semidirect
373. C3⋊C8C22×C3⋊D3semidirect
380. C2×C32#911semidirect
381. C2×C32#911semidirect
400. C2×D4D3×C3⋊D3semidirect
448. C32C2×#911semidirect
449. C32C2×#911semidirect
465. D4C2×D3×C3⋊D3semidirect
481. C2×C3#63927semidirect
482. C2×C3#64120semidirect
505. C3C2×#63927semidirect
506. C3C2×#64120semidirect
511. C2#11723548direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890782
C2×#890262
C2×C3×#37532
C2×C32×#2132
#117235482
C2×#887962
C2×#887942
C2×#887882
C2×#641203
C2×#639273