Finite Group Databasebeta

← #16855079

Subgroups — #16855079

Order 3968.

736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521884C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D31C212yesmaximal, complemented
3#14521612C211yesmaximal, characteristic, complemented
4#14521223C212yesmaximal, complemented
5#14521220C212yesmaximal, complemented
6#14521219C211yesmaximal, characteristic, complemented
7#14521217C211yesmaximal, characteristic, complemented
8#14521216C211yesmaximal, characteristic, complemented
9#14521064C212yesmaximal, complemented
10#14521030C211yesmaximal, characteristic, complemented
11C31×#453C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2211yescharacteristic, complemented
13#11758567—22no—
14#11758567C2212yescomplemented
15#41×D31—22no—
16#41×D31C2212yes—
17#11758561—22no—
18#11758561—22no—
19#11758561C2211yescharacteristic, complemented
20C2×#92497—22no—
21C2×#92497—22no—
22C2×#92497C2211yescharacteristic
23C2×D4×D31C2211yescharacteristic, complemented
24C31×AGL(1,ℤ/8)C2212yescomplemented
25C31×#127C2211yescharacteristic, complemented
26#11758526C2212yescomplemented
27#11758517C2211yescharacteristic, complemented
28#11758516—22no—
29#11758514—22no—
30#11758513C2211yescharacteristic, complemented
31#11758509C2212yescomplemented
32#11758491—22no—
33SD16×D31—22no—
34#11758488—22no—
35D8×D31—22no—
36#11758485C2212yescomplemented
37M16×D31C2212yescomplemented
38#11758452C2211yescharacteristic, complemented
39#11758449C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#100C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758421C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758412C2212yescomplemented
46#11758393C2212yescomplemented
47C23×D31—21no—
48#41×C31—22no—
49#41×C31C2312yes—
50C2×D4×C31—22no—
51C2×D4×C31—22no—
52C2×D4×C31—22no—
53C2×D4×C31C2311yescharacteristic
54C2×D4×C31C2311yescharacteristic
55C2×C22⋊D31—24no—
56C2×C22⋊D31—22no—
57C2×C22⋊D31—22no—
58C2×C22⋊D31—22no—
59C2×C22⋊D31—21no—
60C2×C22⋊D31D412yes—
61C22×Dic31—22no—
62C22×Dic31D412yes—
63#92499D412yescomplemented
64Q8×D31—22no—
65Q8×D31D412yescomplemented
66#92497—24no—
67#92497—24no—
68#92497—24no—
69#92497—22no—
70#92497—22no—
71#92497—24no—
72#92497—22no—
73#92497—22no—
74#92497—22no—
75#92497D412yescomplemented
76D4×D31—24no—
77D4×D31—22no—
78D4×D31—22no—
79D4×D31—22no—
80D4×D31—22no—
81D4×D31D412yescomplemented
82#92495—24no—
83#92495—22no—
84#92495—22no—
85#92495C2312yes—
86C2×D124C2311yescharacteristic
87C2×C4×D31—22no—
88C2×C4×D31—22no—
89C2×C4×D31C2311yescharacteristic
90C2×Dic62—22no—
91C2×Dic62—22no—
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94D8×C31—22no—
95M16×C31C2312yes—
96C42×C31C2311yescharacteristic
97C22⋊Dic31—22no—
98#92481—22no—
99#92480—22no—
100#92479—22no—
101#92479—22no—
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—22no—
104Dic31⋊C4—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107D248—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—22no—
112C23×C31—21no—
113C23×C31C2×D412yes—
114C22×D31—22no—
115C22×D31—22no—
116C22×D31—21no—
117C22×D31—42no—
118C22×D31—82no—
119C22×D31C2×D411yescharacteristic
120Q8×C31C2×D412yes—
121D4×C31—24no—
122D4×C31—22no—
123D4×C31—22no—
124D4×C31—22no—
125D4×C31—82no—
126D4×C31C2×D412yes—
127C2×C4×C31—22no—
128C2×C4×C31—22no—
129C2×C4×C31—21no—
130C2×C4×C31C2411yescharacteristic
131C22⋊D31—24no—
132C22⋊D31—24no—
133C22⋊D31—22no—
134C22⋊D31—24no—
135C22⋊D31—22no—
136C22⋊D31—22no—
137C22⋊D31—41no—
138C22⋊D31—41no—
139C22⋊D31—82no—
140C2×Dic31—24no—
141C2×Dic31—22no—
142C2×Dic31—22no—
143C2×Dic31—22no—
144C2×Dic31—22no—
145C2×Dic31—21no—
146C2×Dic31C2×D412yes—
147C2×Dic31C2×D411yescharacteristic
148D124—22no—
149D124—22no—
150D124C2×D412yes—
151C4×D31—24no—
152C4×D31—22no—
153C4×D31—22no—
154C4×D31—22no—
155C4×D31C2×D412yes—
156C4×D31C2×D412yes—
157Dic62—24no—
158Dic62—22no—
159Dic62—22no—
160Dic62C2×D412yes—
161C8×C31—22no—
162C31⋊C8—22no—
163#3021—311nomaximal, Sylow
164C22×C31—22no—
165C22×C31—22no—
166C22×C31—42no—
167C22×C31—42no—
168C22×C31—41no—
169C22×C31—82no—
170C22×C31C22×D411yescharacteristic
171C2×D31—22no—
172C2×D31—41no—
173C2×D31—42no—
174C2×D31—41no—
175C2×D31—82no—
176C2×D31—84no—
177C2×D31—82no—
178C2×D31#12012yes—
179C4×C31—22no—
180C4×C31—41no—
181C4×C31C22×D412yes—
182Dic31—22no—
183Dic31—22no—
184Dic31—41no—
185Dic31#12011yescharacteristic
186Dic31#12011yescharacteristic
187#585—311no—
188C2×AGL(1,ℤ/8)—312no—
189#535—311no—
190#454—312no—
191#454—312no—
192#453—311no—
193#453—311no—
194#453—311no—
195#453D3111yescharacteristic, complemented
196#421—312no—
197#413—311no—
198C2×C31—21no—
199C2×C31—42no—
200C2×C31—42no—
201C2×C31—81no—
202C2×C31C2×#12011yescharacteristic
203D31—22no—
204D31—42no—
205D31—81no—
2062−1+4—312no—
2072−1+4—622no—
208H(5,𝔽2)—311no—
209H(5,𝔽2)—622no—
210H(5,𝔽2)—622no—
211H(5,𝔽2)C2×D3111yescharacteristic, complemented
212C2×#41—311no—
213C2×#41—312no—
214C2×#41—622no—
215C2×#41—622no—
216C2×#41—622no—
217C22×D4—311no—
218AGL(1,ℤ/8)—312no—
219AGL(1,ℤ/8)—312no—
220AGL(1,ℤ/8)—312no—
221AGL(1,ℤ/8)—622no—
222AGL(1,ℤ/8)—622no—
223AGL(1,ℤ/8)C2×D3112yescomplemented
224C2×SD16—622no—
225C2×D8—622no—
226C2×M16—312no—
227#127—311no—
228#127C2×D3111yescharacteristic, complemented
229#124—311no—
230#124—311no—
231#121—622no—
232#120—622no—
233#117—311no—
234#104—312no—
235#104—312no—
236#104—312no—
237#104C2×D3112yescomplemented
238#101—312no—
239#100—311no—
240#100C2×D3111yescharacteristic, complemented
241C31#302111yescharacteristic, complemented, Sylow
242C24—621no—
243#41—22no—
244#41—312no—
245#41—312no—
246#41—312no—
247#41—622no—
248#41—622no—
249#41—624no—
250#41—622no—
251#41—622no—
252#41—622no—
253#41—624no—
254#41—622no—
255#41—624no—
256#41—624no—
257#41—624no—
258#41—622no—
259#41C22×D3112yes—
260C2×Q8—311no—
261C2×Q8—312no—
262C2×Q8—622no—
263C2×Q8—622no—
264C2×Q8—622no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—312no—
269C2×D4—312no—
270C2×D4—311no—
271C2×D4—622no—
272C2×D4—622no—
273C2×D4—621no—
274C2×D4—622no—
275C2×D4—622no—
276C2×D4—624no—
277C2×D4—622no—
278C2×D4—624no—
279C2×D4—622no—
280C2×D4—622no—
281C2×D4C22×D3111yescharacteristic
282C2×D4C22×D3111yescharacteristic
283C22×C4—311no—
284C22×C4—312no—
285C22×C4—622no—
286C22×C4—622no—
287C22×C4—622no—
288SD16—22no—
289SD16—622no—
290SD16—622no—
291SD16—622no—
292D8—22no—
293D8—622no—
294D8—622no—
295D8—622no—
296M16—312no—
297M16—622no—
298M16C22×D3112yes—
299C2×C8—622no—
300C4⋊C4—621no—
301C22⋊C4—622no—
302C22⋊C4—621no—
303C22⋊C4—622no—
304C42—311no—
305C42C22×D3111yescharacteristic
306C23—22no—
307C23—21no—
308C23—311no—
309C23—622no—
310C23—622no—
311C23—621no—
312C23—1242no—
313C23—2482no—
314C23D4×D3112yes—
315Q8—312no—
316Q8—622no—
317Q8—624no—
318Q8—622no—
319Q8D4×D3112yes—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—82no—
325D4—312no—
326D4—622no—
327D4—622no—
328D4—624no—
329D4—622no—
330D4—624no—
331D4—624no—
332D4—622no—
333D4—622no—
334D4—1241no—
335D4—1241no—
336D4—2482no—
337D4D4×D3112yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—311no—
342C2×C4—312no—
343C2×C4—312no—
344C2×C4—312no—
345C2×C4—622no—
346C2×C4—622no—
347C2×C4—624no—
348C2×C4—622no—
349C2×C4—622no—
350C2×C4—622no—
351C2×C4—624no—
352C2×C4—622no—
353C2×C4—622no—
354C2×C4—621no—
355C2×C4C23×D3111yescharacteristic
356C8—22no—
357C8—622no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—82no—
364C22—312no—
365C22—622no—
366C22—1241no—
367C22—1242no—
368C22—1241no—
369C22—2482no—
370C22—2482no—
371C22—2484no—
372C22C2×D4×D3111yescharacteristic
373C4—22no—
374C4—41no—
375C4—311no—
376C4—311no—
377C4—622no—
378C4—622no—
379C4—1241no—
380C4C2×D4×D3112yes—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—81no—
385C2—622no—
386C2—1242no—
387C2—2481no—
388C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#453C2
socleC2×C31C2×#120
hypercenter#453D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #14521884C2semidirect
2. AGL(1,ℤ/8)×D31C2semidirect
3. #14521612C2semidirect
4. #14521223C2semidirect
5. #14521220C2semidirect
6. #14521219C2semidirect
7. #14521217C2semidirect
8. #14521216C2semidirect
9. #14521064C2semidirect
10. #14521030C2semidirect
11. C31×#453C2semidirect
12. C31×H(5,𝔽2)C22semidirect
14. #11758567C22semidirect
19. #11758561C22semidirect
23. C2×D4×D31C22semidirect
24. C31×AGL(1,ℤ/8)C22semidirect
25. C31×#127C22semidirect
26. #11758526C22semidirect
27. #11758517C22semidirect
30. #11758513C22semidirect
31. #11758509C22semidirect
36. #11758485C22semidirect
37. M16×D31C22semidirect
38. #11758452C22semidirect
39. #11758449C22semidirect
40. C31×#104C22semidirect
41. C31×#100C22semidirect
42. #11758425C22semidirect
43. #11758421C22semidirect
44. #11758413C22semidirect
45. #11758412C22semidirect
46. #11758393C22semidirect
63. #92499D4semidirect
65. Q8×D31D4semidirect
75. #92497D4semidirect
81. D4×D31D4semidirect
195. #453D31semidirect
211. H(5,𝔽2)C2×D31semidirect
223. AGL(1,ℤ/8)C2×D31semidirect
228. #127C2×D31semidirect
237. #104C2×D31semidirect
240. #100C2×D31semidirect
241. C31#3021semidirect

Maximal subgroups

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

subgroupindex
#145218842
AGL(1,ℤ/8)×D312
#145216122
#145212232
#145212202
#145212192
#145212172
#145212162
#145210642
#145210302
C31×#4532
#302131