Finite Group Databasebeta

← #11867125

Subgroups — #11867125

Order 1152.

1644 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic, complemented
2#10596262C211yesmaximal, characteristic, complemented
3C2×#64139C211yesmaximal, characteristic, complemented
4C2×#64117C211yesmaximal, characteristic, complemented
5#10592834C212yesmaximal, complemented
6#10592833C212yesmaximal, complemented
7#10592832C211yesmaximal, characteristic, complemented
8#10592830C211yesmaximal, characteristic, complemented
9#10592820C211yesmaximal, characteristic, complemented
10#10592814C212yesmaximal, complemented
11#10592813C212yesmaximal, complemented
12#83772—34nomaximal
13C23×C32×C4C2211yescharacteristic
14C22×#638—21no—
15C2×#3828C2211yescharacteristic
16C22×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C32×M16—24no—
18C2×C32×M16C2211yescharacteristic
19C2×C32×M16C2211yescharacteristic
20C2×C32×M16C2212yescomplemented
21C22×C32×C8C2212yescomplemented
22#64162—21no—
23#64139—21no—
24#64139—21no—
25#64139C2212yes—
26#64139C2212yes—
27#64139C2211yescharacteristic, complemented
28#64139C2211yescharacteristic, complemented
29#64138—21no—
30C4×#638—21no—
31C4×#638—21no—
32C2×#3752—21no—
33#64135C2211yescharacteristic
34C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
35#64133C2211yescharacteristic, complemented
36#64133C2211yescharacteristic, complemented
37#64117—24no—
38C2×#3745C2212yescomplemented
39C2×#3744C2212yescomplemented
40#63648C2212yescomplemented
41#63648C2212yescomplemented
42#63648C2212yescomplemented
43#63645C2212yescomplemented
44#63644C2212yescomplemented
45#63643C2212yescomplemented
46C22×C3×M16D314yescomplemented
47#5791—34no—
48C2×#930—34no—
49C2×#908—34no—
50#5135—38no—
51#5134—38no—
52#5133—34no—
53#5131—34no—
54#5121—34no—
55#5115—38no—
56#5114—38no—
57C24×C32D411yescharacteristic
58C23×C3⋊D3—21no—
59C22×C32×C4—22no—
60C22×C32×C4—22no—
61C22×C32×C4D411yescharacteristic
62C22×C32×C4D411yescharacteristic
63C22×C32×C4D411yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C2×#638—21no—
68C2×#638—22no—
69C2×#638—22no—
70C2×#638—21no—
71C2×#638—21no—
72C2×#638—44no—
73C22×C3⋊Dic3—21no—
74#3828—22no—
75#3828—22no—
76C2×C4⋊(C3⋊D3)—22no—
77C2×C4⋊(C3⋊D3)—22no—
78C2×C4⋊(C3⋊D3)—21no—
79C2×C4⋊(C3⋊D3)C2312yes—
80C2×C4⋊(C3⋊D3)C2311yescharacteristic
81C2×C4×C3⋊D3—21no—
82C2×#634C2311yescharacteristic
83C32×M16—22no—
84C32×M16—24no—
85C32×M16D414yescomplemented
86C2×C32×C8—24no—
87C2×C32×C8C2312yes—
88C2×C32×C8C2312yes—
89#3757—21no—
90#3757—21no—
91#3757—21no—
92#3752—22no—
93#3752—22no—
94#3752—21no—
95#3752—21no—
96C4⋊(C3⋊Dic3)—21no—
97C4⋊(C3⋊Dic3)C2311yescharacteristic
98C4⋊(C3⋊Dic3)C2311yescharacteristic
99C4⋊(C3⋊Dic3)C2312yes—
100#3750—21no—
101#3750—21no—
102C4×C3⋊Dic3—21no—
103#3745—24no—
104#3744—24no—
105#3055—91noSylow
106C23×C3×C4C2×D314yes—
107C22×#67—64no—
108C2×#235—34no—
109C22×D12—34no—
110C2×C3×M16—216no—
111C2×C3×M16C2×D314yes—
112C2×C3×M16C2×D314yes—
113C2×C3×M16C2×D318yescomplemented
114C22×C3×C8C2×D318yescomplemented
115#953—64no—
116#930—38no—
117#930—34no—
118#930—38no—
119#930—34no—
120#930—64no—
121#930—64no—
122#929—64no—
123C4×#67—64no—
124C4×#67—64no—
125C2×(C2×D3)⋊C4—64no—
126#926—34no—
127C2×C4⋊Dic3—34no—
128#924—34no—
129#924—34no—
130#908—616no—
131C2×D24—38no—
132C2×#204—38no—
133#821—38no—
134#821—38no—
135#821—38no—
136#818—38no—
137#817—38no—
138#816—38no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—41no—
142C23×C32C2×D411yescharacteristic
143C23×C32C2×D411yescharacteristic
144C23×C32C2×D411yescharacteristic
145C22×C3⋊D3—22no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3—42no—
148C22×C3⋊D3—42no—
149C22×C3⋊D3—42no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D412yes—
161C2×C32×C4#4112yes—
162C2×C32×C4C2411yescharacteristic
163#638—42no—
164#638—44no—
165#638—42no—
166#638—42no—
167#638—44no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—42no—
173C4⋊(C3⋊D3)—22no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—22no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—42no—
178C4×C3⋊D3—42no—
179#634—21no—
180#634—21no—
181C32×C8—22no—
182C32×C8C2×D414yescomplemented
183C2×AGL(1,ℤ/8)—91no—
184C22×M16C3⋊D311yescharacteristic, complemented
185#525—91no—
186C2×#121—91no—
187#469—92no—
188#468—92no—
189#466—92no—
190#465—92no—
191#428—91no—
192#418—91no—
193#417—91no—
194C24×C3D1214yes—
195C23×D3—64no—
196C22×C3×C4—28no—
197C22×C3×C4—28no—
198C22×C3×C4D1214yes—
199C22×C3×C4D1214yes—
200C22×C3×C4D1214yes—
201C22×C3×C4C22×D314yes—
202C22×C3×C4C22×D314yes—
203C22×C3×C4C22×D314yes—
204C2×#67—64no—
205C2×#67—64no—
206C2×#67—64no—
207C2×#67—68no—
208C2×#67—68no—
209C2×#67—1216no—
210C22×Dic3—64no—
211#235—68no—
212#235—68no—
213C2×D12—38no—
214C2×D12—34no—
215C2×D12—68no—
216C2×D12—68no—
217C2×D12—64no—
218C2×C4×D3—64no—
219C2×Dic6—34no—
220C3×M16—28no—
221C3×M16—216no—
222C3×M16#67116yescomplemented
223C2×C3×C8—216no—
224C2×C3×C8C22×D318yes—
225C2×C3×C8C22×D318yes—
226#217—64no—
227#217—64no—
228#217—64no—
229(C2×D3)⋊C4—68no—
230(C2×D3)⋊C4—68no—
231(C2×D3)⋊C4—64no—
232(C2×D3)⋊C4—64no—
233C4⋊Dic3—38no—
234C4⋊Dic3—34no—
235C4⋊Dic3—34no—
236C4⋊Dic3—64no—
237Dic3⋊C4—64no—
238Dic3⋊C4—64no—
239C4×Dic3—64no—
240D24—616no—
241#204—616no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—44no—
250C22×C32#12112yes—
251C22×C32AGL(1,ℤ/8)12yescomplemented
252C22×C32C22×D411yescharacteristic
253C2×C3⋊D3—41no—
254C2×C3⋊D3—42no—
255C2×C3⋊D3—42no—
256C2×C3⋊D3—41no—
257C2×C3⋊D3—41no—
258C2×C3⋊D3—42no—
259C2×C3⋊D3—84no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4#12112yes—
263C32×C4C22×D411yescharacteristic
264C32×C4C2×#4111yescharacteristic
265C3⋊Dic3—41no—
266C3⋊Dic3—42no—
267C3⋊Dic3—41no—
268C3⋊Dic3—41no—
269C2×#41—91no—
270C22×D4—91no—
271C22×D4—181no—
272C23×C4C2×C3⋊D311yescharacteristic
273AGL(1,ℤ/8)—184no—
274C2×SD16—92no—
275C2×D8—92no—
276C2×M16—24no—
277C2×M16C2×C3⋊D311yescharacteristic
278C2×M16C2×C3⋊D311yescharacteristic
279C2×M16C2×C3⋊D312yescomplemented
280C22×C8C2×C3⋊D312yescomplemented
281#123—181no—
282C22⋊Q8—91no—
283C22⋊Q8—91no—
284#121—92no—
285#121—92no—
286#121—91no—
287#121—91no—
288#121—181no—
289#121—181no—
290#120—181no—
291C4×D4—181no—
292C4×D4—181no—
293#117—91no—
294C2×C4⋊C4—91no—
295C2×C22⋊C4—181no—
296#107—92no—
297#106—92no—
298#103—92no—
299#102—92no—
300#102—92no—
301#102—92no—
302C23×C3—28no—
303C23×C3—28no—
304C23×C3—44no—
305C23×C3C2×D1214yes—
306C23×C3C2×D1214yes—
307C23×C3C2×D1214yes—
308C22×D3—64no—
309C22×D3—68no—
310C22×D3—128no—
311C22×D3—128no—
312C22×D3—128no—
313C2×C3×C4—28no—
314C2×C3×C4—28no—
315C2×C3×C4—28no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4C2×D1214yes—
321C2×C3×C4C2×D1214yes—
322C2×C3×C4C2×D1214yes—
323C2×C3×C4#23518yes—
324C2×C3×C4C2×#6718yes—
325C2×C3×C4C23×D314yes—
326#67—128no—
327#67—1216no—
328#67—128no—
329#67—1216no—
330#67—128no—
331C2×Dic3—64no—
332C2×Dic3—64no—
333C2×Dic3—64no—
334C2×Dic3—68no—
335C2×Dic3—128no—
336D12—68no—
337D12—64no—
338D12—68no—
339D12—64no—
340D12—128no—
341C4×D3—128no—
342Dic6—64no—
343Dic6—64no—
344C3×C8—28no—
345C3×C8C2×#67116yescomplemented
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—41no—
349C2×C32C2×#12111yescharacteristic
350C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
351C2×C32#57511yescharacteristic
352C3⋊D3—81no—
353C3⋊D3—82no—
354C24—181no—
355C24C4⋊(C3⋊D3)11yescharacteristic
356#41—182no—
357#41—182no—
358C2×Q8—91no—
359C2×D4—91no—
360C2×D4—92no—
361C2×D4—181no—
362C2×D4—181no—
363C2×D4—181no—
364C2×D4—182no—
365C2×D4—182no—
366C2×D4—181no—
367C2×D4—182no—
368C2×D4—182no—
369C2×D4—364no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—181no—
373C22×C4—181no—
374C22×C4C4⋊(C3⋊D3)11yescharacteristic
375C22×C4C4⋊(C3⋊D3)11yescharacteristic
376C22×C4C4⋊(C3⋊D3)11yescharacteristic
377C22×C4C22×C3⋊D311yescharacteristic
378C22×C4C22×C3⋊D311yescharacteristic
379C22×C4C22×C3⋊D311yescharacteristic
380SD16—184no—
381D8—184no—
382M16—22no—
383M16—24no—
384M16#63814yescomplemented
385C2×C8—24no—
386C2×C8C22×C3⋊D312yes—
387C2×C8C22×C3⋊D312yes—
388C4⋊C4—91no—
389C4⋊C4—91no—
390C4⋊C4—92no—
391C4⋊C4—181no—
392C4⋊C4—181no—
393C4⋊C4—181no—
394C22⋊C4—181no—
395C22⋊C4—181no—
396C22⋊C4—181no—
397C22⋊C4—182no—
398C22⋊C4—182no—
399C22⋊C4—181no—
400C22⋊C4—181no—
401C42—181no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—416no—
410C22×C3#90818yescomplemented
411C22×C3#93018yes—
412C22×C3C22×D1214yes—
413C2×D3—128no—
414C2×D3—124no—
415C2×D3—124no—
416C2×D3—128no—
417C2×D3—124no—
418C2×D3—128no—
419C2×D3—2416no—
420C3×C4—24no—
421C3×C4—24no—
422C3×C4#93018yes—
423C3×C4C2×#23514yes—
424C3×C4C22×#6714yes—
425Dic3—128no—
426Dic3—124no—
427Dic3—124no—
428Dic3—124no—
429C32#305511yescharacteristic, complemented, Sylow
430C23—22no—
431C23—22no—
432C23—41no—
433C23—181no—
434C23—182no—
435C23—362no—
436C23—362no—
437C23—362no—
438C23C2×C4⋊(C3⋊D3)11yescharacteristic
439C23C2×C4⋊(C3⋊D3)11yescharacteristic
440C23C2×C4⋊(C3⋊D3)11yescharacteristic
441Q8—181no—
442Q8—181no—
443D4—182no—
444D4—181no—
445D4—181no—
446D4—182no—
447D4—364no—
448D4—362no—
449D4—364no—
450D4—362no—
451D4—362no—
452D4—362no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—182no—
463C2×C4—181no—
464C2×C4—362no—
465C2×C4—362no—
466C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
467C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
468C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
469C2×C4#382812yes—
470C2×C4C2×#63812yes—
471C2×C4C23×C3⋊D311yescharacteristic
472C8—22no—
473C8C2×#63814yescomplemented
474C2×C3—28no—
475C2×C3—28no—
476C2×C3—44no—
477C2×C3C2×#90814yes—
478C2×C3#574814yes—
479C2×C3C2×#93014yes—
480D3—248no—
481D3—244no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—44no—
490C22—362no—
491C22—361no—
492C22—361no—
493C22—362no—
494C22—361no—
495C22—362no—
496C22—724no—
497C22#6411712yescomplemented
498C22#6413912yes—
499C22C22×C4⋊(C3⋊D3)11yescharacteristic
500C4—21no—
501C4—21no—
502C4—362no—
503C4—361no—
504C4—361no—
505C4—361no—
506C4#6413912yes—
507C4C2×#382811yescharacteristic
508C4C22×#63811yescharacteristic
509C3#8377214yescomplemented
510C2—22no—
511C2—22no—
512C2—41no—
513C2—722no—
514C2—721no—
515C2C2×#6411711yescharacteristic
516C2#1059621911yescharacteristic
517C2C2×#6413911yescharacteristic
subgroupG/H
centerC22C22×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC22×C32×M16C2
socleC22×C32C22×D4
hypercenterC22×M16C3⋊D3
hypocenterC32#3055
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×C32×M16C2semidirect
2. #10596262C2semidirect
3. C2×#64139C2semidirect
4. C2×#64117C2semidirect
5. #10592834C2semidirect
6. #10592833C2semidirect
7. #10592832C2semidirect
8. #10592830C2semidirect
9. #10592820C2semidirect
10. #10592814C2semidirect
11. #10592813C2semidirect
20. C2×C32×M16C22semidirect
21. C22×C32×C8C22semidirect
27. #64139C22semidirect
28. #64139C22semidirect
35. #64133C22semidirect
36. #64133C22semidirect
38. C2×#3745C22semidirect
39. C2×#3744C22semidirect
40. #63648C22semidirect
41. #63648C22semidirect
42. #63648C22semidirect
43. #63645C22semidirect
44. #63644C22semidirect
45. #63643C22semidirect
46. C22×C3×M16D3semidirect
85. C32×M16D4semidirect
113. C2×C3×M16C2×D3semidirect
114. C22×C3×C8C2×D3semidirect
182. C32×C8C2×D4semidirect
184. C22×M16C3⋊D3semidirect
222. C3×M16#67semidirect
251. C22×C32AGL(1,ℤ/8)semidirect
279. C2×M16C2×C3⋊D3semidirect
280. C22×C8C2×C3⋊D3semidirect
345. C3×C8C2×#67semidirect
384. M16#638semidirect
410. C22×C3#908semidirect
429. C32#3055semidirect
473. C8C2×#638semidirect
497. C22#64117semidirect
509. C3#83772semidirect

Maximal subgroups

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

subgroupindex
C22×C32×M162
#105962622
C2×#641392
C2×#641172
#105928342
#105928332
#105928322
#105928302
#105928202
#105928142
#105928132
#837723