Finite Group Databasebeta

← #11868077

Subgroups — #11868077

Order 1152.

1398 conjugacy classes of subgroups.

The subgroup lattice has 444 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#10596293C211yesmaximal, characteristic, complemented
3#10596290C211yesmaximal, characteristic, complemented
4C2×#64141C211yesmaximal, characteristic, complemented
5C2×#63660C211yesmaximal, characteristic, complemented
6#10592899C214yesmaximal, complemented
7#10592898C214yesmaximal, complemented
8#10592831C211yesmaximal, characteristic, complemented
9#10592823C211yesmaximal, characteristic, complemented
10#83943—34nomaximal
11C2×C32×#41C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C2×#3828C2211yescharacteristic
14C32×AGL(1,ℤ/8)—22no—
15C32×AGL(1,ℤ/8)C2214yes—
16C2×C32×SD16—21no—
17C2×C32×D8—21no—
18C2×C32×M16C2211yescharacteristic
19#64160C2212yescomplemented
20#64159—22no—
21#64158C2212yescomplemented
22#64157—22no—
23#64157C2214yescomplemented
24#64151—21no—
25C2×#3754C2212yescomplemented
26#64141—21no—
27#64141—22no—
28#64141—22no—
29C2×#3753C2212yescomplemented
30C2×#3748—22no—
31C2×#3748C2211yescharacteristic
32C2×#3748C2211yescharacteristic
33C2×#3748C2211yescharacteristic
34#63660C2214yescomplemented
35#63651C2212yescomplemented
36#63650C2212yescomplemented
37#63649C2214yescomplemented
38C2×C3×AGL(1,ℤ/8)D314yescomplemented
39#5822—34no—
40#5819—34no—
41C2×#932—34no—
42C2×#833—34no—
43#5200—316no—
44#5199—316no—
45#5132—34no—
46#5124—34no—
47C24×C32—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41D412yes—
51C32×#41C2312yes—
52C2×C32×Q8D411yescharacteristic
53C2×C32×D4—21no—
54C2×C32×D4—21no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4D411yescharacteristic
60C2×C32×D4C2312yes—
61C22×C32×C4—21no—
62C22×C32×C4C2311yescharacteristic
63C2×#638—21no—
64#3828—21no—
65#3828—21no—
66#3828—21no—
67#3828—21no—
68C2×C4⋊(C3⋊D3)C2311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×#634C2311yescharacteristic
71C32×SD16—24no—
72C32×D8—24no—
73C32×M16—21no—
74C32×M16C2312yes—
75C2×C32×C8—21no—
76#3756—22no—
77#3755—22no—
78#3754—22no—
79#3754—24no—
80#3753—22no—
81#3753—24no—
82#3748—21no—
83#3748—21no—
84#3748—24no—
85#3748D412yes—
86#3748D412yes—
87#3748C2312yes—
88#3748C2312yes—
89C2×C3⋊(C3⋊C8)—21no—
90C2×C3⋊(C3⋊C8)—21no—
91C2×C3⋊(C3⋊C8)—24no—
92C2×C3⋊(C3⋊C8)C2312yes—
93#3068—91noSylow
94C2×C3×#41C2×D314yes—
95C22×C3×D4C2×D314yes—
96C2×#235—34no—
97C3×AGL(1,ℤ/8)—28no—
98C3×AGL(1,ℤ/8)C2×D3116yes—
99C2×C3×SD16—24no—
100C2×C3×D8—24no—
101C2×C3×M16C2×D314yes—
102#951—38no—
103#950—68no—
104#949—38no—
105#948—316no—
106#948—68no—
107#942—64no—
108C2×#214—38no—
109#932—68no—
110#932—68no—
111#932—64no—
112C2×#213—38no—
113C2×#208—34no—
114C2×#208—34no—
115C2×#208—34no—
116C2×#208—68no—
117#833—316no—
118#824—38no—
119#823—38no—
120#822—316no—
121C23×C32—21no—
122C23×C32—22no—
123C23×C32—41no—
124C23×C32—41no—
125C23×C32—82no—
126C23×C32#4111yescharacteristic
127C22×C3⋊D3—21no—
128C32×Q8—21no—
129C32×Q8C2×D412yes—
130C32×D4—22no—
131C32×D4—22no—
132C32×D4—21no—
133C32×D4—22no—
134C32×D4—22no—
135C32×D4—41no—
136C32×D4—41no—
137C32×D4C2×D412yes—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4#4111yescharacteristic
146C2×C32×C4C2411yescharacteristic
147#638—41no—
148#638—41no—
149C2×C3⋊Dic3—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)—21no—
152C4×C3⋊D3—41no—
153C4×C3⋊D3—41no—
154#634—21no—
155#634—21no—
156C32×C8—22no—
157C3⋊(C3⋊C8)—22no—
158C3⋊(C3⋊C8)—22no—
159C3⋊(C3⋊C8)C2×D414yes—
160#575—91no—
161C2×AGL(1,ℤ/8)—91no—
162C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
163#568—91no—
164#472—94no—
165#471—94no—
166#430—91no—
167#413—91no—
168C2×#100—91no—
169C24×C3—24no—
170C3×#41—24no—
171C3×#41—24no—
172C3×#41#6718yes—
173C3×#41C22×D318yes—
174C2×C3×Q8#6714yes—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4#6714yes—
182C2×C3×D4C22×D318yes—
183C22×C3×C4—24no—
184C22×C3×C4C22×D314yes—
185C2×#67—64no—
186#235—64no—
187#235—64no—
188#235—64no—
189#235—64no—
190C2×D12—34no—
191C2×C4×D3—64no—
192C2×Dic6—34no—
193C3×SD16—216no—
194C3×D8—216no—
195C3×M16—24no—
196C3×M16C22×D318yes—
197C2×C3×C8—24no—
198#216—68no—
199#215—68no—
200#214—616no—
201#214—68no—
202#213—616no—
203#213—68no—
204#208—38no—
205#208—38no—
206#208—38no—
207#208—38no—
208#208—616no—
209#208—64no—
210#208—64no—
211C2×C3⋊C8—38no—
212C2×C3⋊C8—616no—
213C2×C3⋊C8—64no—
214C2×C3⋊C8—64no—
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—42no—
218C22×C32—81no—
219C22×C32—82no—
220C22×C32—82no—
221C22×C32—82no—
222C22×C32#12111yescharacteristic
223C22×C32#12111yescharacteristic
224C22×C32C2×#4111yescharacteristic
225C2×C3⋊D3—41no—
226C2×C3⋊D3—81no—
227C32×C4—22no—
228C32×C4#12111yescharacteristic
229C32×C4#12111yescharacteristic
230C32×C4C22×D411yescharacteristic
231C32×C4C22×D411yescharacteristic
232C3⋊Dic3—41no—
233C2×#41—91no—
234C2×#41C2×C3⋊D311yescharacteristic
235C22×D4C2×C3⋊D311yescharacteristic
236#137—92no—
237#137—181no—
238AGL(1,ℤ/8)—22no—
239AGL(1,ℤ/8)—92no—
240AGL(1,ℤ/8)—182no—
241AGL(1,ℤ/8)—182no—
242AGL(1,ℤ/8)—181no—
243AGL(1,ℤ/8)C2×C3⋊D314yes—
244#135—182no—
245C2×SD16—21no—
246C2×SD16—92no—
247C2×D8—21no—
248C2×D8—92no—
249#131—94no—
250#131—182no—
251C2×M16—91no—
252C2×M16—91no—
253C2×M16—91no—
254C2×M16—182no—
255C2×M16C2×C3⋊D311yescharacteristic
256#108—94no—
257#101—92no—
258#100—94no—
259#100—92no—
260C23×C3—24no—
261C23×C3—28no—
262C23×C3—44no—
263C23×C3—44no—
264C23×C3—88no—
265C23×C3#23714yes—
266C22×D3—64no—
267C3×Q8—24no—
268C3×Q8C2×#6718yes—
269C3×D4—28no—
270C3×D4—28no—
271C3×D4—24no—
272C3×D4—28no—
273C3×D4—28no—
274C3×D4—44no—
275C3×D4—44no—
276C3×D4C2×#6718yes—
277C2×C3×C4—24no—
278C2×C3×C4—28no—
279C2×C3×C4—28no—
280C2×C3×C4D3×D414yes—
281C2×C3×C4D3×D414yes—
282C2×C3×C4#23714yes—
283C2×C3×C4C2×#6714yes—
284C2×C3×C4C2×#6714yes—
285C2×C3×C4C23×D314yes—
286#67—124no—
287#67—124no—
288C2×Dic3—64no—
289D12—64no—
290D12—64no—
291C4×D3—124no—
292C4×D3—124no—
293Dic6—64no—
294Dic6—64no—
295C3×C8—28no—
296C3⋊C8—316no—
297C3⋊C8—68no—
298C3⋊C8—68no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—42no—
303C2×C32—82no—
304C2×C32C2×#12111yescharacteristic
305C3⋊D3—81no—
306C24—21no—
307#41—21no—
308#41—21no—
309#41—181no—
310#41—181no—
311#41—181no—
312#41—181no—
313#41#63812yes—
314#41C22×C3⋊D312yes—
315C2×Q8—91no—
316C2×Q8#63811yescharacteristic
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—91no—
324C2×D4—181no—
325C2×D4#63811yescharacteristic
326C2×D4C22×C3⋊D312yes—
327C22×C4—21no—
328C22×C4—181no—
329C22×C4C22×C3⋊D311yescharacteristic
330Q16—182no—
331SD16—24no—
332SD16—184no—
333SD16—182no—
334SD16—182no—
335D8—24no—
336D8—184no—
337D8—182no—
338M16—21no—
339M16—92no—
340M16—92no—
341M16—92no—
342M16—92no—
343M16—184no—
344M16—181no—
345M16—181no—
346M16C22×C3⋊D312yes—
347C2×C8—21no—
348C2×C8—92no—
349C2×C8—184no—
350C2×C8—181no—
351C2×C8—181no—
352C22×C3—28no—
353C22×C3—44no—
354C22×C3—48no—
355C22×C3—84no—
356C22×C3—88no—
357C22×C3—88no—
358C22×C3—88no—
359C22×C3#93914yes—
360C22×C3#93914yes—
361C22×C3C2×#23714yes—
362C2×D3—124no—
363C2×D3—244no—
364C3×C4—28no—
365C3×C4#93914yes—
366C3×C4#93914yes—
367C3×C4C2×D3×D414yes—
368C3×C4C22×#6714yes—
369Dic3—124no—
370C32#306811yescharacteristic, complemented, Sylow
371C23—21no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—82no—
376C23—181no—
377C23#383011yescharacteristic
378Q8—21no—
379Q8—181no—
380Q8—181no—
381Q8C2×#63812yes—
382D4—22no—
383D4—22no—
384D4—22no—
385D4—22no—
386D4—21no—
387D4—41no—
388D4—41no—
389D4—181no—
390D4—181no—
391D4—361no—
392D4—361no—
393D4C2×#63812yes—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—181no—
398C2×C4—361no—
399C2×C4—361no—
400C2×C4D4×C3⋊D311yescharacteristic
401C2×C4D4×C3⋊D311yescharacteristic
402C2×C4#383011yescharacteristic
403C2×C4C2×#63811yescharacteristic
404C2×C4C2×#63811yescharacteristic
405C2×C4C23×C3⋊D311yescharacteristic
406C8—22no—
407C8—94no—
408C8—182no—
409C8—182no—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—48no—
414C2×C3—88no—
415C2×C3C2×#93914yes—
416D3—244no—
417C22—22no—
418C22—41no—
419C22—42no—
420C22—81no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—361no—
425C22—721no—
426C22#6414811yescharacteristic
427C22#6414811yescharacteristic
428C22C2×#383011yescharacteristic
429C4—22no—
430C4—361no—
431C4#6414811yescharacteristic
432C4#6414811yescharacteristic
433C4C2×D4×C3⋊D311yescharacteristic
434C4C22×#63811yescharacteristic
435C3#8394314yescomplemented
436C2—21no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—82no—
441C2—721no—
442C2C2×#6414811yescharacteristic
subgroupG/H
centerC2C2×#64148
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC2×C32C2×#121
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#3068
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ 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. #10596293C2semidirect
3. #10596290C2semidirect
4. C2×#64141C2semidirect
5. C2×#63660C2semidirect
6. #10592899C2semidirect
7. #10592898C2semidirect
8. #10592831C2semidirect
9. #10592823C2semidirect
19. #64160C22semidirect
21. #64158C22semidirect
23. #64157C22semidirect
25. C2×#3754C22semidirect
29. C2×#3753C22semidirect
34. #63660C22semidirect
35. #63651C22semidirect
36. #63650C22semidirect
37. #63649C22semidirect
38. C2×C3×AGL(1,ℤ/8)D3semidirect
162. C2×AGL(1,ℤ/8)C3⋊D3semidirect
370. C32#3068semidirect
435. C3#83943semidirect

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
#105962932
#105962902
C2×#641412
C2×#636602
#105928992
#105928982
#105928312
#105928232
#839433