Finite Group Databasebeta

← #11860479

Subgroups — #11860479

Order 1152.

1518 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#10596298C211yesmaximal, characteristic, complemented
3#10596269C211yesmaximal, characteristic, complemented
4#10592932C211yesmaximal, characteristic, complemented
5C2×#63659C211yesmaximal, characteristic, complemented
6#10592863C212yesmaximal, complemented
7SD16×C3⋊Dic3C212yesmaximal, complemented
8#10592851C212yesmaximal, complemented
9D8×C3⋊Dic3C212yesmaximal, complemented
10#10592821C211yesmaximal, characteristic, complemented
11#10592820C211yesmaximal, characteristic, complemented
12#83930—34nomaximal
13C2×C32×#41C2211yescharacteristic
14C22×C32×D4C2211yescharacteristic
15C32×AGL(1,ℤ/8)C418yescomplemented
16C2×C32×SD16C2212yes—
17C2×C32×D8C2212yes—
18C2×C32×M16C2211yescharacteristic
19C2×#3757—22no—
20Q8×C3⋊Dic3C2212yescomplemented
21D4×C3⋊Dic3—22no—
22D4×C3⋊Dic3—21no—
23D4×C3⋊Dic3—21no—
24D4×C3⋊Dic3C2212yes—
25D4×C3⋊Dic3C2212yes—
26D4×C3⋊Dic3C2212yescomplemented
27#64135—21no—
28#64135C2211yescharacteristic
29#64135C2211yescharacteristic
30C2×C4⋊(C3⋊Dic3)—21no—
31C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C22×C3⋊(C3⋊C8)C2211yescharacteristic
33#63662C2211yescharacteristic, complemented
34#63662C2211yescharacteristic, complemented
35#63659C2212yes—
36#63659C2212yes—
37#63659C2211yescharacteristic, complemented
38#63659C2211yescharacteristic, complemented
39#63645C2212yescomplemented
40#63644C2212yescomplemented
41#63642C2212yescomplemented
42C8×C3⋊Dic3C2212yescomplemented
43C2×C3×AGL(1,ℤ/8)D314yes—
44#5827—34no—
45#5798—34no—
46#5233—34no—
47C2×#832—34no—
48#5164—38no—
49Dic3×SD16—38no—
50#5152—38no—
51Dic3×D8—38no—
52#5122—34no—
53#5121—34no—
54C24×C32—21no—
55C32×#41—22no—
56C32×#41C2×C414yescomplemented
57C2×C32×Q8C2311yescharacteristic
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—22no—
61C2×C32×D4C2×C414yescomplemented
62C2×C32×D4C2312yes—
63C2×C32×D4C2311yescharacteristic
64C22×C32×C4—21no—
65C22×C32×C4C2311yescharacteristic
66C22×C3⋊Dic3—21no—
67C22×C3⋊Dic3—22no—
68C22×C3⋊Dic3—22no—
69C22×C3⋊Dic3—22no—
70C32×SD16C2×C418yescomplemented
71C32×D8C2×C418yescomplemented
72C32×M16C2×C414yescomplemented
73C2×C32×C8C2312yes—
74#3757—21no—
75#3757—22no—
76#3757—22no—
77#3757—22no—
78#3757D412yes—
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)—22no—
81C4⋊(C3⋊Dic3)D411yescharacteristic
82C4⋊(C3⋊Dic3)D411yescharacteristic
83C4⋊(C3⋊Dic3)C2312yes—
84C4⋊(C3⋊Dic3)C2311yescharacteristic
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86C4×C3⋊Dic3—21no—
87C4×C3⋊Dic3—22no—
88C4×C3⋊Dic3C2312yes—
89C2×C3⋊(C3⋊C8)—22no—
90C2×C3⋊(C3⋊C8)C2311yescharacteristic
91C2×C3⋊(C3⋊C8)C2311yescharacteristic
92#2946—91noSylow
93C2×C3×#41C2×D314yes—
94C22×C3×D4C2×D314yes—
95C3×AGL(1,ℤ/8)Dic3132yescomplemented
96C2×C3×SD16C2×D318yes—
97C2×C3×D8C2×D318yes—
98C2×C3×M16C2×D314yes—
99C2×#217—68no—
100Q8×Dic3—38no—
101D4×Dic3—38no—
102D4×Dic3—38no—
103D4×Dic3—38no—
104D4×Dic3—64no—
105D4×Dic3—64no—
106D4×Dic3—68no—
107#926—34no—
108#926—34no—
109#926—64no—
110C2×C4⋊Dic3—34no—
111C2×C4⋊Dic3—64no—
112C22×C3⋊C8—34no—
113#835—34no—
114#835—34no—
115#832—34no—
116#832—38no—
117#832—34no—
118#832—38no—
119#818—38no—
120#817—38no—
121#815—38no—
122C8×Dic3—38no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—24no—
126C23×C32—81no—
127C23×C32—81no—
128C23×C32C2×D411yescharacteristic
129C32×Q8—21no—
130C32×Q8C22×C412yes—
131C32×D4—24no—
132C32×D4—24no—
133C32×D4—22no—
134C32×D4—21no—
135C32×D4C22×C412yes—
136C32×D4C22×C414yes—
137C2×C32×C4—21no—
138C2×C32×C4—24no—
139C2×C32×C4C22×C412yes—
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4112yes—
142C2×C32×C4C2411yescharacteristic
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—82no—
147C2×C3⋊Dic3—82no—
148C2×C3⋊Dic3—82no—
149C2×C3⋊Dic3—81no—
150C2×C3⋊Dic3C2×D412yes—
151C2×C3⋊Dic3C2×D412yes—
152C32×C8C22×C414yes—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic
156#520—91no—
157#518—91no—
158#442—92no—
159#440—92no—
160C4×SD16—92no—
161C4×D8—92no—
162#428—91no—
163#416—91no—
164C2×#102—91no—
165#405—91no—
166C24×C3—24no—
167C3×#41—28no—
168C3×#41C2×Dic3116yescomplemented
169C2×C3×Q8C22×D314yes—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—28no—
173C2×C3×D4C2×Dic3116yescomplemented
174C2×C3×D4C22×D314yes—
175C2×C3×D4C22×D318yes—
176C22×C3×C4—24no—
177C22×C3×C4C22×D314yes—
178C22×Dic3—68no—
179C22×Dic3—68no—
180C22×Dic3—68no—
181C22×Dic3—64no—
182C3×SD16C2×Dic3132yescomplemented
183C3×D8C2×Dic3132yescomplemented
184C3×M16C2×Dic3116yescomplemented
185C2×C3×C8C22×D318yes—
186#217—38no—
187#217—68no—
188#217—68no—
189#217—68no—
190#217—64no—
191C4⋊Dic3—34no—
192C4⋊Dic3—34no—
193C4⋊Dic3—34no—
194C4⋊Dic3—38no—
195C4⋊Dic3—34no—
196C4⋊Dic3—68no—
197C4⋊Dic3—64no—
198C4×Dic3—38no—
199C4×Dic3—68no—
200C4×Dic3—64no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8—34no—
203C2×C3⋊C8—68no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—22no—
207C22×C32—24no—
208C22×C32—41no—
209C22×C32—42no—
210C22×C32—42no—
211C22×C32—41no—
212C22×C32—84no—
213C22×C32C4×D412yes—
214C22×C32C22×D411yescharacteristic
215C32×C4—22no—
216C32×C4C4×D412yes—
217C32×C4C23×C411yescharacteristic
218C32×C4C2×#4111yescharacteristic
219C3⋊Dic3—22no—
220C3⋊Dic3—22no—
221C3⋊Dic3—42no—
222C3⋊Dic3—41no—
223C3⋊Dic3—41no—
224C2×#41C2×C3⋊D311yescharacteristic
225C22×D4C2×C3⋊D311yescharacteristic
226AGL(1,ℤ/8)C3⋊Dic318yescomplemented
227C2×SD16C2×C3⋊D312yes—
228C2×D8C2×C3⋊D312yes—
229C2×M16C2×C3⋊D311yescharacteristic
230C22×C8—91no—
231C4×Q8—92no—
232C4×D4—92no—
233C4×D4—92no—
234C4×D4—92no—
235C4×D4—182no—
236C4×D4—181no—
237C4×D4—181no—
238#117—91no—
239#117—91no—
240#117—181no—
241C2×C4⋊C4—91no—
242C2×C4⋊C4—181no—
243C2×C22⋊C4—182no—
244#107—92no—
245#106—92no—
246#103—91no—
247#103—91no—
248#102—92no—
249#102—92no—
250#102—91no—
251#102—91no—
252#97—92no—
253C4×C8—92no—
254C23×C3—24no—
255C23×C3—28no—
256C23×C3—216no—
257C23×C3—84no—
258C23×C3—84no—
259C23×C3D3×D414yes—
260C3×Q8—24no—
261C3×Q8C22×Dic318yes—
262C3×D4—216no—
263C3×D4—216no—
264C3×D4—28no—
265C3×D4—24no—
266C3×D4C22×Dic318yes—
267C3×D4C22×Dic3116yes—
268C2×C3×C4—24no—
269C2×C3×C4—216no—
270C2×C3×C4D3×D414yes—
271C2×C3×C4#23718yes—
272C2×C3×C4C22×Dic318yes—
273C2×C3×C4C23×D314yes—
274C2×Dic3—38no—
275C2×Dic3—38no—
276C2×Dic3—68no—
277C2×Dic3—64no—
278C2×Dic3—64no—
279C2×Dic3—248no—
280C2×Dic3—248no—
281C2×Dic3—248no—
282C2×Dic3—244no—
283C3×C8C22×Dic3116yes—
284C3⋊C8—64no—
285C3⋊C8—64no—
286C2×C32—22no—
287C2×C32—42no—
288C2×C32—44no—
289C2×C32C2×C4×D411yescharacteristic
290C2×C32#57611yescharacteristic
291C2×C32#57711yescharacteristic
292C24—21no—
293#41—22no—
294#41C2×C3⋊Dic314yescomplemented
295C2×Q8C22×C3⋊D311yescharacteristic
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—22no—
299C2×D4C2×C3⋊Dic314yescomplemented
300C2×D4C22×C3⋊D311yescharacteristic
301C2×D4C22×C3⋊D312yes—
302C22×C4—21no—
303C22×C4—182no—
304C22×C4—182no—
305C22×C4—182no—
306C22×C4—181no—
307C22×C4C22×C3⋊D311yescharacteristic
308SD16C2×C3⋊Dic318yescomplemented
309D8C2×C3⋊Dic318yescomplemented
310M16C2×C3⋊Dic314yescomplemented
311C2×C8—91no—
312C2×C8—91no—
313C2×C8—182no—
314C2×C8C22×C3⋊D312yes—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—92no—
318C4⋊C4—91no—
319C4⋊C4—91no—
320C4⋊C4—182no—
321C4⋊C4—181no—
322C22⋊C4—92no—
323C22⋊C4—182no—
324C22⋊C4—182no—
325C22⋊C4—182no—
326C22⋊C4—181no—
327C42—92no—
328C42—182no—
329C42—181no—
330C22×C3—24no—
331C22×C3—24no—
332C22×C3—28no—
333C22×C3—216no—
334C22×C3—44no—
335C22×C3—48no—
336C22×C3—48no—
337C22×C3—44no—
338C22×C3—816no—
339C22×C3D4×Dic318yes—
340C22×C3C2×D3×D414yes—
341C3×C4—28no—
342C3×C4D4×Dic318yes—
343C3×C4C2×#23714yes—
344C3×C4C23×Dic314yes—
345Dic3—68no—
346Dic3—68no—
347Dic3—128no—
348Dic3—124no—
349Dic3—124no—
350C32#294611yescharacteristic, complemented, Sylow
351C23—21no—
352C23—22no—
353C23—24no—
354C23—81no—
355C23—81no—
356C23D4×C3⋊D311yescharacteristic
357Q8—21no—
358Q8C22×C3⋊Dic312yes—
359D4—24no—
360D4—22no—
361D4—21no—
362D4—24no—
363D4C22×C3⋊Dic312yes—
364D4C22×C3⋊Dic314yes—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—92no—
368C2×C4—92no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—182no—
372C2×C4—722no—
373C2×C4—722no—
374C2×C4—722no—
375C2×C4—721no—
376C2×C4D4×C3⋊D311yescharacteristic
377C2×C4#383012yes—
378C2×C4C22×C3⋊Dic312yes—
379C2×C4C23×C3⋊D311yescharacteristic
380C8—181no—
381C8—181no—
382C8C22×C3⋊Dic314yes—
383C2×C3—28no—
384C2×C3—416no—
385C2×C3—48no—
386C2×C3#577414yes—
387C2×C3#577514yes—
388C2×C3C2×D4×Dic314yes—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—24no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—84no—
398C22D4×C3⋊Dic312yes—
399C22C2×D4×C3⋊D311yescharacteristic
400C4—22no—
401C4—182no—
402C4—182no—
403C4—361no—
404C4—362no—
405C4—361no—
406C4D4×C3⋊Dic312yes—
407C4C2×#383011yescharacteristic
408C4C23×C3⋊Dic311yescharacteristic
409C3#8393014yescomplemented
410C2—22no—
411C2—44no—
412C2—42no—
413C2#1059624511yescharacteristic
414C2#1059624611yescharacteristic
415C2C2×D4×C3⋊Dic311yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutatorC32×C4C23×C4
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC22×C32C22×D4
cosocleC2×C32×C4C24
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#2946
LCSG ▷ 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
2. #10596298C2semidirect
3. #10596269C2semidirect
4. #10592932C2semidirect
5. C2×#63659C2semidirect
6. #10592863C2semidirect
7. SD16×C3⋊Dic3C2semidirect
8. #10592851C2semidirect
9. D8×C3⋊Dic3C2semidirect
10. #10592821C2semidirect
11. #10592820C2semidirect
15. C32×AGL(1,ℤ/8)C4semidirect
20. Q8×C3⋊Dic3C22semidirect
26. D4×C3⋊Dic3C22semidirect
33. #63662C22semidirect
34. #63662C22semidirect
37. #63659C22semidirect
38. #63659C22semidirect
39. #63645C22semidirect
40. #63644C22semidirect
41. #63642C22semidirect
42. C8×C3⋊Dic3C22semidirect
56. C32×#41C2×C4semidirect
61. C2×C32×D4C2×C4semidirect
70. C32×SD16C2×C4semidirect
71. C32×D8C2×C4semidirect
72. C32×M16C2×C4semidirect
95. C3×AGL(1,ℤ/8)Dic3semidirect
168. C3×#41C2×Dic3semidirect
173. C2×C3×D4C2×Dic3semidirect
182. C3×SD16C2×Dic3semidirect
183. C3×D8C2×Dic3semidirect
184. C3×M16C2×Dic3semidirect
226. AGL(1,ℤ/8)C3⋊Dic3semidirect
294. #41C2×C3⋊Dic3semidirect
299. C2×D4C2×C3⋊Dic3semidirect
308. SD16C2×C3⋊Dic3semidirect
309. D8C2×C3⋊Dic3semidirect
310. M16C2×C3⋊Dic3semidirect
350. C32#2946semidirect
409. C3#83930semidirect

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
#105962982
#105962692
#105929322
C2×#636592
#105928632
SD16×C3⋊Dic32
#105928512
D8×C3⋊Dic32
#105928212
#105928202
#839303