Finite Group Databasebeta

← #16855061

Subgroups — #16855061

Order 3968.

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

normal subgroups only

This group has 724 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
2#14521848C212yesmaximal, complemented
3#14521598C211yesmaximal, characteristic, complemented
4#14521230C212yesmaximal, complemented
5#14521226C211yesmaximal, characteristic, complemented
6#14521223C212yesmaximal, complemented
7#14521219C211yesmaximal, characteristic, complemented
8#14521217C211yesmaximal, characteristic, complemented
9#104×D31C212yesmaximal, complemented
10#14521030C211yesmaximal, characteristic, complemented
11C31×#455C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2211yescharacteristic
13#11758567—22no—
14#11758567C2212yes—
15#41×D31—22no—
16#41×D31C2212yes—
17C2×#92499C2211yescharacteristic, complemented
18#11758561—22no—
19#11758561—22no—
20#11758561C2211yescharacteristic
21C2×#92497—22no—
22C2×#92497—22no—
23C2×#92497C2211yescharacteristic
24C31×#137C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758526C2212yescomplemented
27#11758521C2211yescharacteristic, complemented
28#11758519C2212yescomplemented
29#11758517C2211yescharacteristic, complemented
30#11758496—22no—
31#11758495—22no—
32#11758493—22no—
33#11758491—22no—
34#11758485C2212yescomplemented
35M16×D31C2212yescomplemented
36#11758461—22no—
37(C22⋊D31)⋊C4—22no—
38#11758452C2211yescharacteristic, complemented
39C42×D31C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#100C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758421C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758412C2212yescomplemented
46#11758393C2212yescomplemented
47#41×C31—22no—
48#41×C31C2312yes—
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—22no—
51C2×D4×C31—22no—
52C2×D4×C31C2311yescharacteristic
53C2×C22⋊D31—24no—
54C2×C22⋊D31—22no—
55C2×C22⋊D31—22no—
56C2×C22⋊D31—22no—
57C2×C22⋊D31D412yes—
58C22×Dic31—22no—
59C22×Dic31D412yes—
60#92499—22no—
61#92499—22no—
62#92499D412yescomplemented
63Q8×D31—22no—
64Q8×D31D412yescomplemented
65#92497—24no—
66#92497—24no—
67#92497—24no—
68#92497—22no—
69#92497—22no—
70#92497—24no—
71#92497—22no—
72#92497—22no—
73#92497—22no—
74#92497D412yescomplemented
75D4×D31—24no—
76D4×D31—22no—
77D4×D31—22no—
78D4×D31D412yescomplemented
79#92495—24no—
80#92495—22no—
81#92495—22no—
82#92495C2312yes—
83C2×D124—21no—
84C2×D124C2311yescharacteristic
85C2×C4×D31—22no—
86C2×C4×D31—22no—
87C2×C4×D31—21no—
88C2×C4×D31—21no—
89C2×C4×D31C2311yescharacteristic
90C2×Dic62—22no—
91C2×Dic62—22no—
92C2×Dic62C2311yescharacteristic
93Q16×C31—22no—
94SD16×C31—22no—
95M16×C31C2312yes—
96C22⋊C4×C31—22no—
97C42×C31C2311yescharacteristic
98#92482—22no—
99#92481—22no—
100#92481—22no—
101#92479—22no—
102(C2×D31)⋊C4—22no—
103Dic31⋊C4—22no—
104C4×Dic31—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107D248—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—22no—
112C23×C31C2×D412yes—
113C22×D31—22no—
114C22×D31—22no—
115C22×D31—21no—
116C22×D31C2×D411yescharacteristic
117Q8×C31—22no—
118Q8×C31C2×D412yes—
119D4×C31—24no—
120D4×C31—22no—
121D4×C31—22no—
122D4×C31C2×D412yes—
123C2×C4×C31—22no—
124C2×C4×C31—22no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31C2411yescharacteristic
128C22⋊D31—24no—
129C22⋊D31—24no—
130C22⋊D31—22no—
131C22⋊D31—24no—
132C22⋊D31—22no—
133C22⋊D31—22no—
134C22⋊D31—44no—
135C2×Dic31—24no—
136C2×Dic31—22no—
137C2×Dic31—21no—
138C2×Dic31—22no—
139C2×Dic31—22no—
140C2×Dic31—22no—
141C2×Dic31C2×D412yes—
142C2×Dic31C2×D411yescharacteristic
143D124—22no—
144D124—22no—
145D124—42no—
146D124C2×D412yes—
147C4×D31—24no—
148C4×D31—22no—
149C4×D31—22no—
150C4×D31—22no—
151C4×D31—42no—
152C4×D31—41no—
153C4×D31—41no—
154C4×D31C2×D412yes—
155C4×D31C2×D412yes—
156Dic62—24no—
157Dic62—22no—
158Dic62—22no—
159Dic62C2×D412yes—
160C8×C31—22no—
161C31⋊C8—22no—
162#3018—311nomaximal, Sylow
163C22×C31—22no—
164C22×C31—22no—
165C22×C31—42no—
166C22×C31—82no—
167C22×C31C22×D411yescharacteristic
168C2×D31—22no—
169C2×D31—41no—
170C2×D31—42no—
171C2×D31—41no—
172C2×D31—82no—
173C2×D31—82no—
174C2×D31#12012yes—
175C4×C31—22no—
176C4×C31—41no—
177C4×C31—41no—
178C4×C31C22×D412yes—
179Dic31—22no—
180Dic31—22no—
181Dic31—22no—
182Dic31#12011yescharacteristic
183Dic31#12011yescharacteristic
184#585—311no—
185#575—312no—
186#532—311no—
187#456—312no—
188#455—311no—
189#455D3111yescharacteristic, complemented
190#454—312no—
191#453—311no—
192#453—311no—
193C2×#104—312no—
194#413—311no—
195C2×C31—21no—
196C2×C31—42no—
197C2×C31—42no—
198C2×C31C2×#12011yescharacteristic
199D31—22no—
200D31—42no—
201D31—81no—
2022−1+4—312no—
2032−1+4—622no—
204H(5,𝔽2)—311no—
205H(5,𝔽2)—622no—
206H(5,𝔽2)—622no—
207H(5,𝔽2)C2×D3111yescharacteristic
208C2×#41—311no—
209C2×#41—312no—
210C2×#41—311no—
211C2×#41—622no—
212C2×#41—622no—
213C2×#41—622no—
214#137—312no—
215#137—622no—
216#137C2×D3112yescomplemented
217AGL(1,ℤ/8)—312no—
218AGL(1,ℤ/8)—312no—
219AGL(1,ℤ/8)—622no—
220#135—622no—
221#135—622no—
222C2×M16—312no—
223C4⋊Q8—311no—
224#127—311no—
225#124—311no—
226#124C2×D3111yescharacteristic, complemented
227#121—622no—
228C4×D4—622no—
229C2×C42—311no—
230#104—312no—
231#104—312no—
232#104—312no—
233#104C2×D3112yescomplemented
234#101—312no—
235#100—311no—
236#100C2×D3111yescharacteristic, complemented
237C31#301811yescharacteristic, complemented, Sylow
238#41—22no—
239#41—312no—
240#41—312no—
241#41—312no—
242#41—622no—
243#41—622no—
244#41—624no—
245#41—622no—
246#41—622no—
247#41—622no—
248#41—624no—
249#41—622no—
250#41—622no—
251#41—622no—
252#41—624no—
253#41—624no—
254#41—624no—
255#41—622no—
256#41C22×D3112yes—
257C2×Q8—311no—
258C2×Q8—312no—
259C2×Q8—622no—
260C2×Q8—622no—
261C2×Q8—622no—
262C2×Q8C22×D3111yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—312no—
266C2×D4—312no—
267C2×D4—311no—
268C2×D4—621no—
269C2×D4—622no—
270C2×D4—622no—
271C2×D4—624no—
272C2×D4—622no—
273C2×D4—624no—
274C2×D4—622no—
275C2×D4—622no—
276C2×D4C22×D3111yescharacteristic
277C22×C4—311no—
278C22×C4—312no—
279C22×C4—622no—
280C22×C4—621no—
281C22×C4—622no—
282C22×C4—621no—
283C22×C4—622no—
284Q16—22no—
285Q16—622no—
286SD16—22no—
287SD16—622no—
288SD16—622no—
289SD16—622no—
290D8—622no—
291D8—622no—
292M16—312no—
293M16—622no—
294M16C22×D3112yes—
295C2×C8—622no—
296C4⋊C4—622no—
297C22⋊C4—22no—
298C22⋊C4—622no—
299C42—311no—
300C42—621no—
301C42C22×D3111yescharacteristic
302C23—22no—
303C23—311no—
304C23—622no—
305C23—622no—
306C23—621no—
307C23D4×D3112yes—
308Q8—22no—
309Q8—312no—
310Q8—622no—
311Q8—624no—
312Q8—622no—
313Q8D4×D3112yes—
314D4—24no—
315D4—22no—
316D4—22no—
317D4—312no—
318D4—622no—
319D4—622no—
320D4—624no—
321D4—622no—
322D4—624no—
323D4—624no—
324D4—622no—
325D4—622no—
326D4—1242no—
327D4—1244no—
328D4D4×D3112yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—311no—
334C2×C4—312no—
335C2×C4—312no—
336C2×C4—312no—
337C2×C4—622no—
338C2×C4—622no—
339C2×C4—624no—
340C2×C4—622no—
341C2×C4—622no—
342C2×C4—622no—
343C2×C4—624no—
344C2×C4—622no—
345C2×C4—621no—
346C2×C4—622no—
347C2×C4—1241no—
348C2×C4—1241no—
349C2×C4—1242no—
350C2×C4C23×D3111yescharacteristic
351C8—22no—
352C8—622no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—82no—
357C22—312no—
358C22—622no—
359C22—1241no—
360C22—1242no—
361C22—1241no—
362C22—2482no—
363C22—2482no—
364C22C2×D4×D3111yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—311no—
369C4—311no—
370C4—622no—
371C4—622no—
372C4—622no—
373C4C2×D4×D3112yes—
374C2—21no—
375C2—42no—
376C2—42no—
377C2—622no—
378C2—1242no—
379C2—2481no—
380C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#455C2
socleC2×C31C2×#120
hypercenter#455D31
hypocenterC31#3018
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #14521884C2semidirect
2. #14521848C2semidirect
3. #14521598C2semidirect
4. #14521230C2semidirect
5. #14521226C2semidirect
6. #14521223C2semidirect
7. #14521219C2semidirect
8. #14521217C2semidirect
9. #104×D31C2semidirect
10. #14521030C2semidirect
11. C31×#455C2semidirect
17. C2×#92499C22semidirect
24. C31×#137C22semidirect
25. C31×#124C22semidirect
26. #11758526C22semidirect
27. #11758521C22semidirect
28. #11758519C22semidirect
29. #11758517C22semidirect
34. #11758485C22semidirect
35. M16×D31C22semidirect
38. #11758452C22semidirect
39. C42×D31C22semidirect
40. C31×#104C22semidirect
41. C31×#100C22semidirect
42. #11758425C22semidirect
43. #11758421C22semidirect
44. #11758413C22semidirect
45. #11758412C22semidirect
46. #11758393C22semidirect
62. #92499D4semidirect
64. Q8×D31D4semidirect
74. #92497D4semidirect
78. D4×D31D4semidirect
189. #455D31semidirect
216. #137C2×D31semidirect
226. #124C2×D31semidirect
233. #104C2×D31semidirect
236. #100C2×D31semidirect
237. C31#3018semidirect

Maximal subgroups

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

subgroupindex
#145218842
#145218482
#145215982
#145212302
#145212262
#145212232
#145212192
#145212172
#104×D312
#145210302
C31×#4552
#301831