Finite Group Databasebeta

← #11863956

Subgroups — #11863956

Order 1152.

2153 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D12C211yesmaximal, characteristic, complemented
2#10596262C211yesmaximal, characteristic, complemented
3C2×#63830C212yesmaximal, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5#10591172C214yesmaximal, complemented
6#10591162C214yesmaximal, complemented
7#10591086C212yesmaximal, complemented
8#84038—31nomaximal
9#78637—31nomaximal
10C24×C3×D3—21no—
11C23×C32×C4C2211yescharacteristic
12C2×#3828C2211yescharacteristic, complemented
13C22×C3×D12—22no—
14C22×C3×D12—21no—
15C22×C3×D12—28no—
16C22×C3×D12C2214yes—
17C22×C3×D12C2212yes—
18#64162—21no—
19#64139C2212yescomplemented
20#64138—21no—
21C4×#638—22no—
22#64135C2211yescharacteristic, complemented
23#64133C2212yescomplemented
24C2×C3×#208C2212yescomplemented
25#63830—28no—
26C2×#3716C2214yescomplemented
27C2×#3714C2214yescomplemented
28C3×#830C2214yescomplemented
29#63563C2218yescomplemented
30C23×C3×D4—31no—
31C23×D12D311yescharacteristic, complemented
32C2×#932—32no—
33#5791—31no—
34#5791—31no—
35#5791—61no—
36C2×#908—32no—
37C3×#407—31no—
38#5218—34no—
39#5217—34no—
40#5215—32no—
41#5207D311yescharacteristic, complemented
42#5131—32no—
43#4732—34no—
44#4731—34no—
45C24×C32D411yescharacteristic
46C23×C3×D3—28no—
47C23×C3×D3—24no—
48C23×C3×D3—22no—
49C23×C3×D3—48no—
50C22×C32×C4—24no—
51C22×C32×C4D412yes—
52C22×C32×C4D411yescharacteristic
53C22×C32×C4C2312yes—
54C22×C32×C4C2311yescharacteristic
55C2×#638—21no—
56C2×#638—22no—
57#3828—24no—
58C2×C4⋊(C3⋊D3)C2311yescharacteristic, complemented
59C2×C4×C3⋊D3—21no—
60C2×#634C2311yescharacteristic, complemented
61C2×C3×D12—28no—
62C2×C3×D12—216no—
63C2×C3×D12—24no—
64C2×C3×D12—22no—
65C2×C3×D12—24no—
66C2×C3×D12—216no—
67C2×C3×D12D418yescomplemented
68C2×C3×D12C2314yes—
69#3757—22no—
70#3757—21no—
71#3752—22no—
72C4⋊(C3⋊Dic3)C2312yescomplemented
73#3750—22no—
74C4×C3⋊Dic3—21no—
75C3×#208—24no—
76C2×C3×C3⋊C8C2314yescomplemented
77#3716—28no—
78#3714—28no—
79#3003—91noSylow
80C25×C3—61no—
81C24×D3—21no—
82C22×C3×D4—32no—
83C22×C3×D4—34no—
84C22×C3×D4—68no—
85C22×C3×D4—62no—
86C22×C3×D4—61no—
87C23×C3×C4—21no—
88C23×C3×C4C2×D311yescharacteristic
89C23×C3×C4C2×D311yescharacteristic
90C2×#235—31no—
91C2×#235—31no—
92C2×#235—61no—
93C22×D12—22no—
94C22×D12—21no—
95C22×D12—28no—
96C22×D12C2×D314yes—
97C22×D12C2×D312yes—
98C2×C3×M16—32no—
99#953—62no—
100#953—61no—
101#953—61no—
102C2×#214—34no—
103#932—68no—
104C2×#213—34no—
105#930—32no—
106#930—32no—
107#930—62no—
108#929—62no—
109#929—61no—
110#929—61no—
111C4×#67—64no—
112C4×#67—62no—
113C4×#67—62no—
114#926—31no—
115#926—31no—
116#926—61no—
117#924—32no—
118#924—32no—
119#924—62no—
120C2×#208C2×D312yescomplemented
121#908—68no—
122C2×D24—34no—
123C2×#204—34no—
124C3×C22⋊C8—34no—
125#832—38no—
126#830C2×D314yescomplemented
127#821—38no—
128C23×C32—22no—
129C23×C32—24no—
130C23×C32C2×D411yescharacteristic
131C23×C32C2×D412yes—
132C22×C3⋊D3—21no—
133C22×C3×D3—216no—
134C22×C3×D3—28no—
135C22×C3×D3—24no—
136C22×C3×D3—432no—
137C22×C3×D3—48no—
138C22×C3×D3—48no—
139C22×C3×D3—48no—
140C2×C32×C4—22no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4C2×D412yes—
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D414yes—
146C2×C32×C4C2411yescharacteristic
147#638—42no—
148#638—44no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—21no—
151C4⋊(C3⋊D3)—22no—
152C4×C3⋊D3—42no—
153#634—22no—
154C3×D12—216no—
155C3×D12—28no—
156C3×D12—24no—
157C3×D12C2×D418yes—
158C3×C3⋊C8—24no—
159C23×D4—31no—
160C2×AGL(1,ℤ/8)—92no—
161#525—91no—
162#450—94no—
163#447—94no—
164#418—92no—
165#407—31no—
166C24×C3—21no—
167C24×C3—68no—
168C24×C3—64no—
169C24×C3—62no—
170C24×C3—128no—
171C24×C3D1211yescharacteristic
172C24×C3#6711yescharacteristic
173C23×D3—28no—
174C23×D3—24no—
175C23×D3—22no—
176C23×D3—48no—
177C2×C3×D4—38no—
178C2×C3×D4—34no—
179C2×C3×D4—616no—
180C2×C3×D4—68no—
181C2×C3×D4—616no—
182C2×C3×D4—64no—
183C2×C3×D4—62no—
184C2×C3×D4—64no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—44no—
192C22×C3×C4D1212yes—
193C22×C3×C4D1211yescharacteristic
194C22×C3×C4#6712yes—
195C22×C3×C4#6711yescharacteristic
196C22×C3×C4C22×D312yes—
197C22×C3×C4C22×D312yes—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D311yescharacteristic
200C2×#67—64no—
201C2×#67—62no—
202C2×#67—62no—
203C2×#67—61no—
204C2×#67—62no—
205C2×#67—61no—
206#235—64no—
207#235—64no—
208#235—124no—
209C2×D12—28no—
210C2×D12—216no—
211C2×D12—24no—
212C2×D12—22no—
213C2×D12—24no—
214C2×D12—216no—
215C2×D12—31no—
216C2×D12—31no—
217C2×D12—61no—
218C2×D12#6718yescomplemented
219C2×D12C22×D314yes—
220C2×C4×D3—62no—
221C2×C4×D3—61no—
222C2×C4×D3—61no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C3×M16—64no—
227C2×C3×C8—34no—
228#217—64no—
229#217—62no—
230#217—62no—
231#217—61no—
232#217—62no—
233#217—61no—
234#214—68no—
235#213—68no—
236(C2×D3)⋊C4—64no—
237(C2×D3)⋊C4—62no—
238(C2×D3)⋊C4—62no—
239C4⋊Dic3—32no—
240C4⋊Dic3—32no—
241C4⋊Dic3—62no—
242Dic3⋊C4—64no—
243Dic3⋊C4—62no—
244Dic3⋊C4—62no—
245C4×Dic3—62no—
246C4×Dic3—61no—
247C4×Dic3—61no—
248#208—24no—
249C2×C3⋊C8C22×D314yescomplemented
250D24—68no—
251#204—68no—
252C22×C32—28no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32#12012yes—
257C22×C32AGL(1,ℤ/8)14yescomplemented
258C22×C32C22×D411yescharacteristic
259C2×C3⋊D3—41no—
260C2×C3⋊D3—42no—
261C2×C3×D3—28no—
262C2×C3×D3—416no—
263C2×C3×D3—432no—
264C2×C3×D3—48no—
265C32×C4—22no—
266C32×C4#12012yes—
267C32×C4C22×D412yes—
268C3⋊Dic3—41no—
269C3⋊Dic3—42no—
270C25—61no—
271C2×#41—91no—
272C22×D4—32no—
273C22×D4—34no—
274C22×D4—68no—
275C22×D4—62no—
276C22×D4—61no—
277C23×C4D3211yescharacteristic
278AGL(1,ℤ/8)—188no—
279C2×SD16—94no—
280C2×D8—94no—
281C2×M16—32no—
282#123—181no—
283C22⋊Q8—92no—
284#121—92no—
285#120—181no—
286C4×D4—182no—
287#117—91no—
288#102—98no—
289C22⋊C8—34no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3—616no—
299C23×C3—68no—
300C23×C3—64no—
301C23×C3—1232no—
302C23×C3—128no—
303C23×C3—128no—
304C23×C3—128no—
305C23×C3C2×D1211yescharacteristic
306C23×C3C2×D1212yes—
307C23×C3C2×#6712yes—
308C23×C3C2×#6711yescharacteristic
309C22×D3—216no—
310C22×D3—28no—
311C22×D3—24no—
312C22×D3—432no—
313C22×D3—48no—
314C22×D3—48no—
315C22×D3—48no—
316C22×D3—62no—
317C22×D3—61no—
318C22×D3—61no—
319C3×D4—38no—
320C3×D4—616no—
321C3×D4—68no—
322C3×D4—64no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4C2×D1212yes—
337C2×C3×C4C2×D1211yescharacteristic
338C2×C3×C4D3×D414yes—
339C2×C3×C4C2×#6712yes—
340C2×C3×C4C2×#6711yescharacteristic
341C2×C3×C4C2×#6714yes—
342C2×C3×C4C23×D311yescharacteristic
343C2×C3×C4C23×D311yescharacteristic
344#67—124no—
345#67—128no—
346#67—122no—
347#67—124no—
348#67—124no—
349#67—122no—
350C2×Dic3—64no—
351C2×Dic3—62no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—62no—
355C2×Dic3—61no—
356D12—216no—
357D12—28no—
358D12—24no—
359D12—62no—
360D12—62no—
361D12—122no—
362D12C2×#6718yes—
363C4×D3—124no—
364C4×D3—122no—
365C4×D3—122no—
366Dic6—62no—
367Dic6—62no—
368Dic6—122no—
369C3×C8—64no—
370C3⋊C8—24no—
371C2×C32—22no—
372C2×C32—24no—
373C2×C32C2×#12011yescharacteristic
374C2×C32C2×AGL(1,ℤ/8)12yes—
375C3⋊D3—81no—
376C3×D3—48no—
377C24—68no—
378C24—64no—
379C24—62no—
380C24—128no—
381C24#62611yescharacteristic
382#41—184no—
383C2×Q8—91no—
384C2×D4—34no—
385C2×D4—38no—
386C2×D4—616no—
387C2×D4—68no—
388C2×D4—616no—
389C2×D4—64no—
390C2×D4—62no—
391C2×D4—64no—
392C2×D4—91no—
393C2×D4—182no—
394C2×D4—181no—
395C22×C4—24no—
396C22×C4—181no—
397C22×C4#62612yes—
398C22×C4#62611yescharacteristic
399C22×C4C2×D3212yes—
400C22×C4C2×D3211yescharacteristic
401SD16—188no—
402D8—188no—
403M16—64no—
404C2×C8—34no—
405C4⋊C4—92no—
406C4⋊C4—182no—
407C22⋊C4—181no—
408C22⋊C4—182no—
409C22⋊C4—182no—
410C42—181no—
411C22×C3—28no—
412C22×C3—28no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—48no—
426C22×C3—68no—
427C22×C3—1216no—
428C22×C3—1232no—
429C22×C3—128no—
430C22×C3#88212yes—
431C22×C3#90814yescomplemented
432C22×C3#93214yescomplemented
433C22×C3#95312yes—
434C22×C3C22×D1211yescharacteristic
435C22×C3C22×#6711yescharacteristic
436C2×D3—28no—
437C2×D3—432no—
438C2×D3—416no—
439C2×D3—48no—
440C2×D3—122no—
441C2×D3—124no—
442C2×D3—121no—
443C2×D3—122no—
444C2×D3—122no—
445C2×D3—121no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4#88212yes—
452C3×C4#95312yes—
453C3×C4C2×D3×D412yes—
454C3×C4C22×#6712yes—
455Dic3—122no—
456Dic3—124no—
457Dic3—121no—
458Dic3—122no—
459Dic3—122no—
460Dic3—121no—
461C32#300311yescharacteristic, complemented, Sylow
462C23—24no—
463C23—22no—
464C23—616no—
465C23—64no—
466C23—68no—
467C23—1232no—
468C23—128no—
469C23—128no—
470C23—128no—
471C23—181no—
472C23C2×#62612yes—
473C23C2×#62611yescharacteristic
474Q8—182no—
475D4—38no—
476D4—616no—
477D4—64no—
478D4—68no—
479D4—182no—
480D4—364no—
481D4—362no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—182no—
486C2×C4—181no—
487C2×C4—362no—
488C2×C4C2×#62612yes—
489C2×C4C2×#62611yescharacteristic
490C2×C4D3×#6714yes—
491C2×C4C22×D3211yescharacteristic
492C8—64no—
493C2×C3—22no—
494C2×C3—24no—
495C2×C3—24no—
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—44no—
500C2×C3—42no—
501C2×C3—128no—
502C2×C3C2×#88211yescharacteristic
503C2×C3C2×#90812yes—
504C2×C3C2×#93212yes—
505C2×C3C2×#95311yescharacteristic
506D3—48no—
507D3—242no—
508D3—241no—
509D3—241no—
510C22—28no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—68no—
515C22—1232no—
516C22—1216no—
517C22—128no—
518C22—361no—
519C22—362no—
520C22#6383014yescomplemented
521C22#6397712yes—
522C22C22×#62611yescharacteristic
523C4—22no—
524C4—361no—
525C4—362no—
526C4#6397712yes—
527C4C2×D3×#6712yes—
528C3—21no—
529C3#7863711yescharacteristic, complemented
530C3#8403811yescharacteristic, complemented
531C2—24no—
532C2—22no—
533C2—128no—
534C2—721no—
535C2C2×#6383012yes—
536C2C2×#6397711yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC23×C32×C4C22
socleC22×C32C22×D4
hypercenterC23×C4D32
hypocenterC32#3003
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C3×D12C2semidirect
2. #10596262C2semidirect
3. C2×#63830C2semidirect
4. C3×#5207C2semidirect
5. #10591172C2semidirect
6. #10591162C2semidirect
7. #10591086C2semidirect
12. C2×#3828C22semidirect
19. #64139C22semidirect
22. #64135C22semidirect
23. #64133C22semidirect
24. C2×C3×#208C22semidirect
26. C2×#3716C22semidirect
27. C2×#3714C22semidirect
28. C3×#830C22semidirect
29. #63563C22semidirect
31. C23×D12D3semidirect
41. #5207D3semidirect
58. C2×C4⋊(C3⋊D3)C23semidirect
60. C2×#634C23semidirect
67. C2×C3×D12D4semidirect
72. C4⋊(C3⋊Dic3)C23semidirect
76. C2×C3×C3⋊C8C23semidirect
120. C2×#208C2×D3semidirect
126. #830C2×D3semidirect
218. C2×D12#67semidirect
249. C2×C3⋊C8C22×D3semidirect
257. C22×C32AGL(1,ℤ/8)semidirect
431. C22×C3#908semidirect
432. C22×C3#932semidirect
461. C32#3003semidirect
520. C22#63830semidirect
529. C3#78637semidirect
530. C3#84038semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D122
#105962622
C2×#638302
C3×#52072
#105911722
#105911622
#105910862
#840383
#786373