Finite Group Databasebeta

← #11865835

Subgroups — #11865835

Order 1152.

1644 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#10596303C211yesmaximal, characteristic, complemented
3C2×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#64141C211yesmaximal, characteristic, complemented
5#10592931C211yesmaximal, characteristic, complemented
6#10592864C212yesmaximal, complemented
7#10592861C212yesmaximal, complemented
8#10592852C212yesmaximal, complemented
9#10592849C212yesmaximal, complemented
10#10592832C211yesmaximal, characteristic, complemented
11#10592817C211yesmaximal, characteristic, complemented
12#83931—34nomaximal
13C2×C32×#41C2211yescharacteristic
14C22×C32×D4C2211yescharacteristic
15C23×C3⋊Dic3—21no—
16C2×#3828C2211yescharacteristic
17C32×AGL(1,ℤ/8)—24no—
18C2×C32×SD16C2212yes—
19C2×C32×D8C2212yes—
20C2×C32×M16C2211yescharacteristic
21C2×#3757—21no—
22#64156C2211yescharacteristic, complemented
23#64153C2211yescharacteristic, complemented
24#64149C2211yescharacteristic, complemented
25#64148—21no—
26#64148—21no—
27#64145C2211yescharacteristic, complemented
28D4×C3⋊Dic3—21no—
29D4×C3⋊Dic3—21no—
30D4×C3⋊Dic3C2212yes—
31D4×C3⋊Dic3C2212yes—
32C2×#3754C2212yes—
33#64141—24no—
34C2×#3753C2212yes—
35C4×#638—21no—
36C4×#638—21no—
37C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
38C2×C4×C3⋊Dic3C2211yescharacteristic
39C2×#3748C2211yescharacteristic
40#63662C2212yescomplemented
41#63659C2212yescomplemented
42#63648C2212yescomplemented
43#63643C2212yescomplemented
44#63641C2212yescomplemented
45#63641C2212yescomplemented
46C2×C3×AGL(1,ℤ/8)D314yescomplemented
47#5832—34no—
48C2×D4×Dic3—34no—
49C2×#932—34no—
50#5232—34no—
51#5165—38no—
52#5162—38no—
53#5153—38no—
54#5150—38no—
55#5133—34no—
56#5118—34no—
57C24×C32—21no—
58C32×#41—22no—
59C32×#41—22no—
60C2×C32×Q8C2311yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—22no—
64C2×C32×D4D414yescomplemented
65C2×C32×D4C2311yescharacteristic
66C2×C32×D4C2312yes—
67C22×C32×C4—21no—
68C22×C32×C4C2311yescharacteristic
69C2×#638—21no—
70C2×#638—21no—
71C2×#638—21no—
72C22×C3⋊Dic3—21no—
73C22×C3⋊Dic3—22no—
74C22×C3⋊Dic3—22no—
75C22×C3⋊Dic3—44no—
76C22×C3⋊Dic3D411yescharacteristic
77C22×C3⋊Dic3D411yescharacteristic
78#3828—22no—
79#3828—22no—
80C2×C4⋊(C3⋊D3)C2311yescharacteristic
81C2×C4×C3⋊D3—21no—
82C2×#634C2311yescharacteristic
83C32×SD16—24no—
84C32×D8—24no—
85C32×M16—22no—
86C2×C32×C8C2312yes—
87#3757—21no—
88#3757—21no—
89#3757—22no—
90#3757—22no—
91#3754—24no—
92#3753—24no—
93#3752—21no—
94#3752—21no—
95C4⋊(C3⋊Dic3)—21no—
96C4⋊(C3⋊Dic3)C2312yes—
97#3750—21no—
98#3750—21no—
99C4×C3⋊Dic3D411yescharacteristic
100C4×C3⋊Dic3D411yescharacteristic
101C4×C3⋊Dic3C2311yescharacteristic
102C4×C3⋊Dic3C2311yescharacteristic
103#3748—22no—
104C2×C3⋊(C3⋊C8)C2312yes—
105#3036—91noSylow
106C2×C3×#41C2×D314yes—
107C22×C3×D4C2×D314yes—
108C23×Dic3—64no—
109C2×#235—34no—
110C3×AGL(1,ℤ/8)—216no—
111C2×C3×SD16C2×D318yes—
112C2×C3×D8C2×D318yes—
113C2×C3×M16C2×D314yes—
114C2×#217—64no—
115#947—34no—
116#944—34no—
117#940—34no—
118#939—64no—
119#939—64no—
120#936—34no—
121D4×Dic3—38no—
122D4×Dic3—38no—
123D4×Dic3—64no—
124D4×Dic3—64no—
125C2×#214—38no—
126#932—616no—
127C2×#213—38no—
128C4×#67—64no—
129C4×#67—64no—
130C2×C4⋊Dic3—34no—
131C2×C4×Dic3—34no—
132C2×#208—34no—
133#835—38no—
134#832—38no—
135#821—38no—
136#816—38no—
137Dic3⋊C8—38no—
138Dic3⋊C8—38no—
139C23×C32—24no—
140C23×C32—22no—
141C23×C32—21no—
142C23×C32—42no—
143C23×C32—42no—
144C23×C32C2×D411yescharacteristic
145C22×C3⋊D3—21no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×D4—24no—
149C32×D4—21no—
150C32×D4—22no—
151C32×D4—21no—
152C32×D4—42no—
153C32×D4C2×D414yes—
154C2×C32×C4—21no—
155C2×C32×C4—42no—
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D412yes—
158C2×C32×C4#4112yes—
159C2×C32×C4C2411yescharacteristic
160#638—42no—
161#638—42no—
162#638—42no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—41no—
167C2×C3⋊Dic3—44no—
168C2×C3⋊Dic3—42no—
169C2×C3⋊Dic3—44no—
170C2×C3⋊Dic3C2×D411yescharacteristic
171C2×C3⋊Dic3C2×D411yescharacteristic
172C2×C3⋊Dic3C2×D411yescharacteristic
173C2×C3⋊Dic3C2×D411yescharacteristic
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176C4×C3⋊D3—42no—
177#634—21no—
178#634—21no—
179C32×C8—22no—
180C3⋊(C3⋊C8)—22no—
181C2×AGL(1,ℤ/8)—91no—
182C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
183#532—91no—
184C2×C4×D4—91no—
185#463—92no—
186#463—92no—
187#461—92no—
188#459—92no—
189#423—91no—
190#417—91no—
191#417—91no—
192C24×C3—24no—
193C3×#41—28no—
194C3×#41—28no—
195C2×C3×Q8C22×D314yes—
196C2×C3×D4—24no—
197C2×C3×D4—24no—
198C2×C3×D4—28no—
199C2×C3×D4#67116yescomplemented
200C2×C3×D4C22×D314yes—
201C2×C3×D4C22×D318yes—
202C22×C3×C4—24no—
203C22×C3×C4C22×D314yes—
204C2×#67—64no—
205C2×#67—64no—
206C2×#67—64no—
207C22×Dic3—34no—
208C22×Dic3—34no—
209C22×Dic3—68no—
210C22×Dic3—68no—
211C22×Dic3—64no—
212C22×Dic3—1216no—
213#235—68no—
214#235—68no—
215C2×D12—34no—
216C2×C4×D3—64no—
217C2×Dic6—34no—
218C3×SD16—216no—
219C3×D8—216no—
220C3×M16—28no—
221C2×C3×C8C22×D318yes—
222#217—64no—
223#217—64no—
224#217—68no—
225#217—68no—
226#214—616no—
227#213—616no—
228(C2×D3)⋊C4—64no—
229(C2×D3)⋊C4—64no—
230C4⋊Dic3—38no—
231C4⋊Dic3—64no—
232Dic3⋊C4—64no—
233Dic3⋊C4—64no—
234C4×Dic3—34no—
235C4×Dic3—34no—
236C4×Dic3—34no—
237C4×Dic3—34no—
238#208—68no—
239C2×C3⋊C8—38no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—24no—
243C22×C32—48no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32#12112yes—
250C22×C32C22×D411yescharacteristic
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C32×C4—41no—
255C32×C4#12112yes—
256C32×C4C22×D411yescharacteristic
257C32×C4C2×#4111yescharacteristic
258C3⋊Dic3—21no—
259C3⋊Dic3—21no—
260C3⋊Dic3—21no—
261C3⋊Dic3—41no—
262C3⋊Dic3—42no—
263C3⋊Dic3AGL(1,ℤ/8)12yescomplemented
264C2×#41—91no—
265C2×#41C2×C3⋊D311yescharacteristic
266C22×D4C2×C3⋊D311yescharacteristic
267C23×C4—181no—
268AGL(1,ℤ/8)—24no—
269AGL(1,ℤ/8)—184no—
270C2×SD16—92no—
271C2×SD16C2×C3⋊D312yes—
272C2×D8—92no—
273C2×D8C2×C3⋊D312yes—
274C2×M16—91no—
275C2×M16C2×C3⋊D311yescharacteristic
276C4⋊Q8—91no—
277#127—91no—
278#124—91no—
279#124—91no—
280#121—181no—
281#121—181no—
282C4×D4—92no—
283C4×D4—92no—
284C4×D4—181no—
285C4×D4—181no—
286C4×D4—181no—
287C4×D4—181no—
288C2×C4⋊C4—91no—
289C2×C22⋊C4—181no—
290C2×C42—91no—
291C4⋊C8—92no—
292C4⋊C8—92no—
293#103—92no—
294#103—92no—
295#102—92no—
296#102—92no—
297C23×C3—216no—
298C23×C3—28no—
299C23×C3—24no—
300C23×C3—48no—
301C23×C3—48no—
302C23×C3D3×D414yes—
303C22×D3—64no—
304C3×Q8—24no—
305C3×Q8—24no—
306C3×D4—216no—
307C3×D4—28no—
308C3×D4—24no—
309C3×D4—24no—
310C3×D4—48no—
311C3×D4C2×#67116yes—
312C2×C3×C4—24no—
313C2×C3×C4—48no—
314C2×C3×C4D3×D414yes—
315C2×C3×C4#23718yes—
316C2×C3×C4C2×#6718yes—
317C2×C3×C4C23×D314yes—
318#67—128no—
319#67—128no—
320#67—128no—
321C2×Dic3—34no—
322C2×Dic3—34no—
323C2×Dic3—34no—
324C2×Dic3—34no—
325C2×Dic3—68no—
326C2×Dic3—64no—
327C2×Dic3—68no—
328C2×Dic3—1216no—
329C2×Dic3—124no—
330C2×Dic3—1216no—
331C2×Dic3—128no—
332D12—64no—
333D12—64no—
334C4×D3—128no—
335Dic6—64no—
336Dic6—64no—
337C3×C8—28no—
338C3⋊C8—68no—
339C2×C32—22no—
340C2×C32—44no—
341C2×C32—81no—
342C2×C32C2×#12111yescharacteristic
343C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
344C2×C32#57511yescharacteristic
345C3⋊D3—81no—
346C24—21no—
347#41—22no—
348#41—22no—
349#41—182no—
350#41—182no—
351C2×Q8—91no—
352C2×Q8C22×C3⋊D311yescharacteristic
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—91no—
357C2×D4—181no—
358C2×D4—181no—
359C2×D4—181no—
360C2×D4#63814yescomplemented
361C2×D4C22×C3⋊D311yescharacteristic
362C2×D4C22×C3⋊D312yes—
363C22×C4—21no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C4—182no—
367C22×C4—182no—
368C22×C4—181no—
369C22×C4—181no—
370C22×C4—364no—
371C22×C4C22×C3⋊D311yescharacteristic
372SD16—24no—
373SD16—184no—
374D8—24no—
375D8—184no—
376M16—22no—
377M16—182no—
378C2×C8—92no—
379C2×C8C22×C3⋊D312yes—
380C4⋊C4—92no—
381C4⋊C4—181no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C22⋊C4—181no—
385C22⋊C4—181no—
386C22⋊C4—182no—
387C22⋊C4—182no—
388C22⋊C4—181no—
389C22⋊C4—181no—
390C42—91no—
391C42—91no—
392C42—91no—
393C42—91no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—216no—
397C22×C3—432no—
398C22×C3—48no—
399C22×C3—48no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3#93918yes—
404C22×C3C2×D3×D414yes—
405C2×D3—124no—
406C2×D3—124no—
407C2×D3—124no—
408C3×C4—44no—
409C3×C4#93918yes—
410C3×C4C2×#23714yes—
411C3×C4C22×#6714yes—
412Dic3—38no—
413Dic3—64no—
414Dic3—64no—
415Dic3—64no—
416Dic3—128no—
417Dic3—124no—
418C32#303611yescharacteristic, complemented, Sylow
419C23—22no—
420C23—24no—
421C23—21no—
422C23—42no—
423C23—42no—
424C23—181no—
425C23D4×C3⋊D311yescharacteristic
426Q8—21no—
427Q8—21no—
428Q8—181no—
429Q8—181no—
430D4—22no—
431D4—21no—
432D4—21no—
433D4—24no—
434D4—42no—
435D4—181no—
436D4—181no—
437D4—362no—
438D4—362no—
439D4—362no—
440D4C2×#63814yes—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—181no—
450C2×C4—364no—
451C2×C4—361no—
452C2×C4—364no—
453C2×C4—362no—
454C2×C4—362no—
455C2×C4D4×C3⋊D311yescharacteristic
456C2×C4#383012yes—
457C2×C4C2×#63812yes—
458C2×C4C23×C3⋊D311yescharacteristic
459C8—22no—
460C8—182no—
461C2×C3—28no—
462C2×C3—416no—
463C2×C3—84no—
464C2×C3D3×AGL(1,ℤ/8)14yes—
465C2×C3#577314yes—
466C2×C3C2×#93914yes—
467D3—244no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—48no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—361no—
478C22—361no—
479C22—361no—
480C22#6414812yes—
481C22C2×D4×C3⋊D311yescharacteristic
482C4—41no—
483C4—92no—
484C4—181no—
485C4—181no—
486C4—181no—
487C4—362no—
488C4—361no—
489C4#6414812yes—
490C4C2×#383011yescharacteristic
491C4C22×#63811yescharacteristic
492C3#8393114yescomplemented
493C2—22no—
494C2—44no—
495C2—81no—
496C2—721no—
497C2C3⋊D3×AGL(1,ℤ/8)11yescharacteristic
498C2#1059624411yescharacteristic
499C2C2×#6414811yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC22×C32C22×D4
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#3036
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C32×AGL(1,ℤ/8)C2semidirect
2. #10596303C2semidirect
3. C2×D4×C3⋊Dic3C2semidirect
4. C2×#64141C2semidirect
5. #10592931C2semidirect
6. #10592864C2semidirect
7. #10592861C2semidirect
8. #10592852C2semidirect
9. #10592849C2semidirect
10. #10592832C2semidirect
11. #10592817C2semidirect
22. #64156C22semidirect
23. #64153C22semidirect
24. #64149C22semidirect
27. #64145C22semidirect
40. #63662C22semidirect
41. #63659C22semidirect
42. #63648C22semidirect
43. #63643C22semidirect
44. #63641C22semidirect
45. #63641C22semidirect
46. C2×C3×AGL(1,ℤ/8)D3semidirect
64. C2×C32×D4D4semidirect
182. C2×AGL(1,ℤ/8)C3⋊D3semidirect
199. C2×C3×D4#67semidirect
263. C3⋊Dic3AGL(1,ℤ/8)semidirect
360. C2×D4#638semidirect
418. C32#3036semidirect
492. C3#83931semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
#105963032
C2×D4×C3⋊Dic32
C2×#641412
#105929312
#105928642
#105928612
#105928522
#105928492
#105928322
#105928172
#839313