Finite Group Databasebeta

← #16855069

Subgroups — #16855069

Order 3968.

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

normal subgroups only

This group has 762 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
1H(5,𝔽2)×D31C211yesmaximal, characteristic, complemented
2#14521848C212yesmaximal, complemented
3#14521617C211yesmaximal, characteristic, complemented
4#14521226C211yesmaximal, characteristic, complemented
5#14521225C212yesmaximal, complemented
6#14521219C211yesmaximal, characteristic, complemented
7#14521218C212yesmaximal, complemented
8#14521217C211yesmaximal, characteristic, complemented
9#14521064C212yesmaximal, complemented
10#100×D31C211yesmaximal, characteristic, complemented
11C31×#455C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2211yescharacteristic
13#11758566—22no—
14#11758566C2212yes—
15#41×D31—22no—
16#41×D31C2212yes—
17C2×#92499C2211yescharacteristic, complemented
18#11758561—22no—
19#11758561—22no—
20#11758561C2211yescharacteristic
21C2×D4×D31—22no—
22C2×D4×D31—22no—
23C2×D4×D31C2211yescharacteristic
24C31×#137C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758526C2212yescomplemented
27#11758524C2211yescharacteristic, 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#11758459—22no—
38#11758451C2211yescharacteristic, complemented
39#11758449C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#100C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758421C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758411C2212yescomplemented
46#11758393C2212yescomplemented
47C23×D31—22no—
48C23×D31D412yes—
49#41×C31—22no—
50#41×C31C2312yes—
51C2×Q8×C31C2311yescharacteristic, complemented
52C2×D4×C31—22no—
53C2×D4×C31—22no—
54C2×D4×C31C2311yescharacteristic
55C2×C22⋊D31—22no—
56C2×C22⋊D31—24no—
57C2×C22⋊D31—22no—
58C2×C22⋊D31—22no—
59C2×C22⋊D31D412yes—
60#92499—22no—
61#92499—22no—
62#92499—22no—
63#92499D412yescomplemented
64Q8×D31D412yescomplemented
65#92497—24no—
66#92497—22no—
67#92497—22no—
68#92497D412yescomplemented
69D4×D31—24no—
70D4×D31—24no—
71D4×D31—24no—
72D4×D31—22no—
73D4×D31—24no—
74D4×D31—22no—
75D4×D31—22no—
76D4×D31—22no—
77D4×D31—22no—
78D4×D31D412yescomplemented
79#92495—24no—
80#92495—22no—
81#92495—22no—
82#92495C2312yes—
83C2×D124—22no—
84C2×D124—22no—
85C2×D124—21no—
86C2×D124—22no—
87C2×D124C2311yescharacteristic
88C2×D124C2311yescharacteristic, complemented
89C2×C4×D31—22no—
90C2×C4×D31—22no—
91C2×C4×D31—21no—
92C2×C4×D31C2311yescharacteristic
93Q16×C31—22no—
94SD16×C31—22no—
95M16×C31C2312yescomplemented
96C22⋊C4×C31—22no—
97C42×C31C2311yescharacteristic, complemented
98#92482—22no—
99#92481—22no—
100#92481—22no—
101#92479—22no—
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—22no—
104Dic31⋊C4—21no—
105C4×Dic31C2311yescharacteristic, complemented
106#92474C2312yescomplemented
107D248—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—22no—
112C23×C31C2×D412yes—
113C22×D31—24no—
114C22×D31—22no—
115C22×D31—22no—
116C22×D31—22no—
117C22×D31—22no—
118C22×D31—21no—
119C22×D31—82no—
120C22×D31—84no—
121C22×D31C2×D412yes—
122C22×D31C2×D411yescharacteristic
123Q8×C31—22no—
124Q8×C31C2×D412yes—
125D4×C31—24no—
126D4×C31—22no—
127D4×C31—22no—
128D4×C31C2×D412yes—
129C2×C4×C31—22no—
130C2×C4×C31—22no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31C2411yescharacteristic
134C22⋊D31—24no—
135C22⋊D31—24no—
136C22⋊D31—22no—
137C22⋊D31—22no—
138C22⋊D31—24no—
139C22⋊D31—22no—
140C22⋊D31—82no—
141C2×Dic31—22no—
142C2×Dic31—22no—
143C2×Dic31—21no—
144C2×Dic31C2×D411yescharacteristic
145D124—24no—
146D124—22no—
147D124—22no—
148D124—22no—
149D124—41no—
150D124—41no—
151D124—82no—
152D124C2×D412yes—
153C4×D31—24no—
154C4×D31—22no—
155C4×D31—22no—
156C4×D31—22no—
157C4×D31—42no—
158C4×D31C2×D412yes—
159C4×D31C2×D412yes—
160Dic62—22no—
161Dic62C2×D412yes—
162C8×C31—22no—
163C31⋊C8—22no—
164#3019—311nomaximal, Sylow
165C22×C31—22no—
166C22×C31—22no—
167C22×C31—42no—
168C22×C31—82no—
169C22×C31C22×D411yescharacteristic
170C2×D31—22no—
171C2×D31—22no—
172C2×D31—41no—
173C2×D31—44no—
174C2×D31—44no—
175C2×D31—42no—
176C2×D31—42no—
177C2×D31—41no—
178C2×D31—82no—
179C2×D31—84no—
180C2×D31—82no—
181C2×D31#12011yescharacteristic
182C2×D31#12011yescharacteristic
183C4×C31—22no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31C22×D412yes—
187Dic31—22no—
188Dic31—41no—
189Dic31#12012yes—
190C2×H(5,𝔽2)—311no—
191#575—312no—
192#535—311no—
193#455—312no—
194#455—311no—
195#455D3111yescharacteristic, complemented
196#453—311no—
197#453—312no—
198#453—311no—
199#421—312no—
200C2×#100—311no—
201C2×C31—21no—
202C2×C31—42no—
203C2×C31—42no—
204C2×C31C2×#12011yescharacteristic
205D31—21no—
206D31—21no—
207D31—42no—
208D31—42no—
209D31—81no—
210H(5,𝔽2)—312no—
211H(5,𝔽2)—311no—
212H(5,𝔽2)—622no—
213H(5,𝔽2)—622no—
214H(5,𝔽2)—622no—
215H(5,𝔽2)C2×D3111yescharacteristic
216C2×#41—312no—
217C2×#41—311no—
218C2×#41—622no—
219C22×D4—311no—
220C22×D4—622no—
221C22×D4—622no—
222#137—312no—
223#137—622no—
224#137C2×D3112yescomplemented
225AGL(1,ℤ/8)—312no—
226AGL(1,ℤ/8)—312no—
227AGL(1,ℤ/8)—622no—
228#135—622no—
229#135—622no—
230C2×M16—312no—
231#127—311no—
232#127—311no—
233#124—311no—
234#124C2×D3111yescharacteristic, complemented
235#121—622no—
236#120—622no—
237#117—311no—
238#104—312no—
239#104—312no—
240#104—312no—
241#104C2×D3112yescomplemented
242#100—311no—
243#100—312no—
244#100C2×D3111yescharacteristic, complemented
245C31#301911yescharacteristic, complemented, Sylow
246C24—312no—
247C24—622no—
248#41—22no—
249#41—312no—
250#41—312no—
251#41—312no—
252#41—622no—
253#41—622no—
254#41—624no—
255#41—622no—
256#41—622no—
257#41—622no—
258#41—622no—
259#41—624no—
260#41—622no—
261#41C22×D3112yes—
262C2×Q8—312no—
263C2×Q8C22×D3111yescharacteristic, complemented
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—312no—
267C2×D4—312no—
268C2×D4—311no—
269C2×D4—311no—
270C2×D4—622no—
271C2×D4—622no—
272C2×D4—621no—
273C2×D4—622no—
274C2×D4—622no—
275C2×D4—622no—
276C2×D4—622no—
277C2×D4—622no—
278C2×D4—624no—
279C2×D4—624no—
280C2×D4—624no—
281C2×D4—624no—
282C2×D4—624no—
283C2×D4—622no—
284C2×D4—622no—
285C2×D4—622no—
286C2×D4—622no—
287C2×D4C22×D3111yescharacteristic
288C22×C4—311no—
289C22×C4—622no—
290C22×C4—621no—
291C22×C4—622no—
292Q16—22no—
293Q16—622no—
294SD16—22no—
295SD16—622no—
296SD16—622no—
297SD16—622no—
298D8—622no—
299D8—622no—
300M16—312no—
301M16—622no—
302M16C22×D3112yescomplemented
303C2×C8—622no—
304C4⋊C4—621no—
305C22⋊C4—22no—
306C22⋊C4—621no—
307C22⋊C4—622no—
308C42—311no—
309C42C22×D3111yescharacteristic, complemented
310C23—22no—
311C23—311no—
312C23—312no—
313C23—622no—
314C23—622no—
315C23—622no—
316C23—622no—
317C23—624no—
318C23—621no—
319C23—2482no—
320C23—2484no—
321C23D4×D3112yes—
322Q8—22no—
323Q8—312no—
324Q8—622no—
325Q8D4×D3112yes—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—312no—
330D4—622no—
331D4—624no—
332D4—624no—
333D4—622no—
334D4—622no—
335D4—624no—
336D4—624no—
337D4—622no—
338D4—622no—
339D4—622no—
340D4—1241no—
341D4—1241no—
342D4—2482no—
343D4—2482no—
344D4D4×D3112yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—311no—
350C2×C4—312no—
351C2×C4—312no—
352C2×C4—622no—
353C2×C4—622no—
354C2×C4—624no—
355C2×C4—622no—
356C2×C4—622no—
357C2×C4—622no—
358C2×C4—621no—
359C2×C4—1242no—
360C2×C4C23×D3111yescharacteristic
361C8—22no—
362C8—622no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—82no—
367C22—311no—
368C22—311no—
369C22—622no—
370C22—622no—
371C22—1241no—
372C22—1242no—
373C22—1242no—
374C22—1244no—
375C22—1244no—
376C22—1241no—
377C22—2482no—
378C22—2482no—
379C22—2484no—
380C22C2×D4×D3111yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—312no—
385C4—622no—
386C4—1241no—
387C4C2×D4×D3112yes—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—621no—
392C2—621no—
393C2—1242no—
394C2—1242no—
395C2—2481no—
396C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#455C2
socleC2×C31C2×#120
hypercenter#455D31
hypocenterC31#3019
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. H(5,𝔽2)×D31C2semidirect
2. #14521848C2semidirect
3. #14521617C2semidirect
4. #14521226C2semidirect
5. #14521225C2semidirect
6. #14521219C2semidirect
7. #14521218C2semidirect
8. #14521217C2semidirect
9. #14521064C2semidirect
10. #100×D31C2semidirect
11. C31×#455C2semidirect
17. C2×#92499C22semidirect
24. C31×#137C22semidirect
25. C31×#124C22semidirect
26. #11758526C22semidirect
27. #11758524C22semidirect
28. #11758519C22semidirect
29. #11758517C22semidirect
34. #11758485C22semidirect
35. M16×D31C22semidirect
38. #11758451C22semidirect
39. #11758449C22semidirect
40. C31×#104C22semidirect
41. C31×#100C22semidirect
42. #11758425C22semidirect
43. #11758421C22semidirect
44. #11758413C22semidirect
45. #11758411C22semidirect
46. #11758393C22semidirect
51. C2×Q8×C31C23semidirect
63. #92499D4semidirect
64. Q8×D31D4semidirect
68. #92497D4semidirect
78. D4×D31D4semidirect
88. C2×D124C23semidirect
95. M16×C31C23semidirect
97. C42×C31C23semidirect
105. C4×Dic31C23semidirect
106. #92474C23semidirect
195. #455D31semidirect
224. #137C2×D31semidirect
234. #124C2×D31semidirect
241. #104C2×D31semidirect
244. #100C2×D31semidirect
245. C31#3019semidirect
263. C2×Q8C22×D31semidirect
302. M16C22×D31semidirect
309. C42C22×D31semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D312
#145218482
#145216172
#145212262
#145212252
#145212192
#145212182
#145212172
#145210642
#100×D312
C31×#4552
#301931