Finite Group Databasebeta

← #16855066

Subgroups — #16855066

Order 3968.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#11758566C211yesmaximal, characteristic, complemented
2#14521847C211yesmaximal, characteristic, complemented
3#14521613C211yesmaximal, characteristic, complemented
4#14521225C214yesmaximal, complemented
5#14521215C214yesmaximal, complemented
6#14521066C211yesmaximal, characteristic, complemented
7#14521065C211yesmaximal, characteristic, complemented
8C2×#11758411C211yesmaximal, characteristic, complemented
9C31×#575C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C2×#41×C31C2211yescharacteristic, complemented
12#11758566—22no—
13#11758566—22no—
14#11758566—22no—
15#11758566C2214yes—
16C2×#92499—21no—
17C2×D4×D31—22no—
18C2×D4×D31—21no—
19C2×#92495—21no—
20C2×#92495C2211yescharacteristic
21C22×D124—21no—
22C22×D124C2211yescharacteristic
23C31×#137—21no—
24C31×#137C2212yescomplemented
25C31×AGL(1,ℤ/8)—21no—
26C31×AGL(1,ℤ/8)C2212yescomplemented
27C31×#135—22no—
28C2×M16×C31C2211yescharacteristic, complemented
29#11758528—21no—
30#11758527—22no—
31#11758526—21no—
32#11758524C2212yescomplemented
33#11758519C2212yescomplemented
34#11758517C2212yescomplemented
35#11758514—22no—
36#11758509C2212yescomplemented
37#11758507—22no—
38#11758503C2211yescharacteristic, complemented
39C2×#92474C2211yescharacteristic, complemented
40#11758425C2214yescomplemented
41#11758413C2214yescomplemented
42#11758411C2214yescomplemented
43C23×D31—22no—
44C23×D31D412yes—
45#41×C31—21no—
46#41×C31—21no—
47#41×C31—21no—
48#41×C31—21no—
49#41×C31—21no—
50#41×C31—21no—
51#41×C31D412yescomplemented
52#41×C31C2312yes—
53C2×Q8×C31D411yescharacteristic, complemented
54C2×Q8×C31C2311yescharacteristic, complemented
55C2×D4×C31—21no—
56C2×D4×C31—21no—
57C2×D4×C31D411yescharacteristic, complemented
58C2×D4×C31C2311yescharacteristic, complemented
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31C2311yescharacteristic
62C2×C22⋊D31—22no—
63C2×C22⋊D31—21no—
64C2×C22⋊D31—24no—
65#92499—24no—
66#92499—24no—
67D4×D31—28no—
68D4×D31—24no—
69D4×D31—28no—
70D4×D31—24no—
71#92495—24no—
72#92495—21no—
73#92495—21no—
74#92495—24no—
75#92495D412yescomplemented
76#92495C2312yes—
77C2×D124—22no—
78C2×D124—24no—
79C2×D124—21no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124—21no—
83C2×D124—24no—
84C2×D124D412yes—
85C2×D124D411yescharacteristic, complemented
86C2×D124C2312yes—
87C2×C4×D31—22no—
88C2×C4×D31—21no—
89C2×Dic62D411yescharacteristic, complemented
90Q16×C31—22no—
91SD16×C31—22no—
92SD16×C31—22no—
93D8×C31—22no—
94M16×C31—21no—
95M16×C31C2312yescomplemented
96C2×C8×C31—21no—
97C22⋊Dic31—21no—
98#92482—22no—
99#92481—22no—
100#92480—22no—
101#92479—22no—
102(C2×D31)⋊C4—24no—
103C4⋊Dic31—21no—
104C4×Dic31C2312yescomplemented
105#92474—21no—
106#92474C2312yescomplemented
107C2×C31⋊C8—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31C2×D411yescharacteristic
111C22×D31—22no—
112C22×D31—21no—
113C22×D31—28no—
114C22×D31—22no—
115C22×D31—82no—
116C22×D31—84no—
117C22×D31C2×D414yes—
118Q8×C31—21no—
119Q8×C31—21no—
120Q8×C31—21no—
121Q8×C31C2×D412yes—
122D4×C31—22no—
123D4×C31—22no—
124D4×C31—21no—
125D4×C31—21no—
126D4×C31—21no—
127D4×C31—41no—
128D4×C31—41no—
129D4×C31C2×D412yes—
130C2×C4×C31—21no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—21no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2×D411yescharacteristic
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2×D411yescharacteristic
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C22⋊D31—28no—
143C22⋊D31—22no—
144C22⋊D31—22no—
145C22⋊D31—84no—
146C2×Dic31—21no—
147C2×Dic31—22no—
148D124—28no—
149D124—22no—
150D124—21no—
151D124—22no—
152D124—22no—
153D124—22no—
154D124C2×D412yes—
155C4×D31—28no—
156C4×D31—22no—
157C4×D31—22no—
158Dic62—21no—
159Dic62C2×D412yes—
160C8×C31—22no—
161C31⋊C8—22no—
162#3019—311nomaximal, Sylow
163C22×C31—22no—
164C22×C31—41no—
165C22×C31—42no—
166C22×C31—42no—
167C22×C31—41no—
168C22×C31#12011yescharacteristic
169C22×C31#12011yescharacteristic
170C22×C31C22×D411yescharacteristic
171C2×D31—24no—
172C2×D31—22no—
173C2×D31—44no—
174C2×D31—42no—
175C2×D31—44no—
176C2×D31—82no—
177C2×D31—88no—
178C4×C31—22no—
179C4×C31—41no—
180C4×C31#12011yescharacteristic
181C4×C31#12011yescharacteristic
182C4×C31C22×D411yescharacteristic
183C4×C31C22×D411yescharacteristic
184Dic31—22no—
185Dic31—42no—
186C2×H(5,𝔽2)—311no—
187#575—311no—
188#575D3111yescharacteristic, complemented
189#535—311no—
190#455—314no—
191#453—314no—
192#421—311no—
193#421—311no—
194C2×#100—311no—
195C2×C31—21no—
196C2×C31—21no—
197C2×C31—21no—
198C2×C31—42no—
199C2×C31—81no—
200C2×C31C2×#12011yescharacteristic
201D31—44no—
202D31—42no—
203H(5,𝔽2)—314no—
204H(5,𝔽2)—622no—
205H(5,𝔽2)—622no—
206H(5,𝔽2)—622no—
207C2×#41—311no—
208C2×#41—621no—
209C2×#41—621no—
210C2×#41C2×D3111yescharacteristic
211C2×#41C2×D3111yescharacteristic, complemented
212C22×D4—311no—
213C22×D4—621no—
214C22×D4—622no—
215C22×D4—621no—
216#137—21no—
217#137—312no—
218#137—621no—
219#137C2×D3112yescomplemented
220AGL(1,ℤ/8)—21no—
221AGL(1,ℤ/8)—312no—
222AGL(1,ℤ/8)—621no—
223AGL(1,ℤ/8)C2×D3112yescomplemented
224#135—22no—
225#135—622no—
226C2×M16—311no—
227C2×M16C2×D3111yescharacteristic, complemented
228#127—312no—
229#124—312no—
230#121—622no—
231#120—622no—
232#117—311no—
233#104—314no—
234#104—314no—
235#100—314no—
236C31#301911yescharacteristic, complemented, Sylow
237C24—312no—
238C24—622no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—312no—
246#41—312no—
247#41—624no—
248#41—624no—
249#41—624no—
250#41—624no—
251#41—621no—
252#41—621no—
253#41C22⋊D3112yescomplemented
254#41C22×D3112yes—
255C2×Q8—311no—
256C2×Q8C22⋊D3111yescharacteristic, complemented
257C2×Q8C22×D3111yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—311no—
261C2×D4—312no—
262C2×D4—312no—
263C2×D4—624no—
264C2×D4—622no—
265C2×D4—624no—
266C2×D4—628no—
267C2×D4—628no—
268C2×D4—624no—
269C2×D4—622no—
270C2×D4—621no—
271C2×D4—624no—
272C2×D4—624no—
273C2×D4—621no—
274C2×D4—621no—
275C2×D4—621no—
276C2×D4—621no—
277C2×D4C22⋊D3111yescharacteristic, complemented
278C2×D4C22×D3111yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—622no—
282C22×C4—621no—
283C22×C4C22×D3111yescharacteristic
284Q16—22no—
285Q16—622no—
286SD16—22no—
287SD16—22no—
288SD16—622no—
289SD16—622no—
290D8—22no—
291D8—622no—
292M16—21no—
293M16—312no—
294M16—621no—
295M16C22×D3112yescomplemented
296C2×C8—21no—
297C2×C8—621no—
298C4⋊C4—621no—
299C22⋊C4—624no—
300C22⋊C4—621no—
301C42—312no—
302C23—21no—
303C23—21no—
304C23—314no—
305C23—628no—
306C23—621no—
307C23—622no—
308C23—622no—
309C23—2482no—
310C23—2484no—
311C23D4×D3111yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—312no—
316Q8—621no—
317Q8C2×C22⋊D3112yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—312no—
326D4—628no—
327D4—628no—
328D4—622no—
329D4—622no—
330D4—621no—
331D4—622no—
332D4—622no—
333D4—622no—
334D4—622no—
335D4—2484no—
336D4C2×C22⋊D3112yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—628no—
344C2×C4—621no—
345C2×C4—622no—
346C2×C4—622no—
347C2×C4—622no—
348C2×C4D4×D3111yescharacteristic
349C2×C4D4×D3111yescharacteristic
350C2×C4D4×D3111yescharacteristic
351C2×C4C2×C22⋊D3111yescharacteristic
352C2×C4C2×C22⋊D3111yescharacteristic
353C2×C4C23×D3111yescharacteristic
354C8—22no—
355C8—622no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—622no—
362C22—624no—
363C22—1244no—
364C22—1244no—
365C22—1242no—
366C22—2482no—
367C22—2488no—
368C22#1175851411yescharacteristic
369C22#1175851411yescharacteristic
370C22C2×D4×D3111yescharacteristic
371C4—22no—
372C4—41no—
373C4—622no—
374C4—1242no—
375C4#1175851411yescharacteristic
376C4#1175851411yescharacteristic
377C4C2×D4×D3111yescharacteristic
378C4C22×C22⋊D3111yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—1244no—
385C2—1242no—
386C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#575C2
socleC2×C31C2×#120
hypercenter#575D31
hypocenterC31#3019
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. #14521847C2semidirect
3. #14521613C2semidirect
4. #14521225C2semidirect
5. #14521215C2semidirect
6. #14521066C2semidirect
7. #14521065C2semidirect
8. C2×#11758411C2semidirect
9. C31×#575C2semidirect
11. C2×#41×C31C22semidirect
24. C31×#137C22semidirect
26. C31×AGL(1,ℤ/8)C22semidirect
28. C2×M16×C31C22semidirect
32. #11758524C22semidirect
33. #11758519C22semidirect
34. #11758517C22semidirect
36. #11758509C22semidirect
38. #11758503C22semidirect
39. C2×#92474C22semidirect
40. #11758425C22semidirect
41. #11758413C22semidirect
42. #11758411C22semidirect
51. #41×C31D4semidirect
53. C2×Q8×C31D4semidirect
54. C2×Q8×C31C23semidirect
57. C2×D4×C31D4semidirect
58. C2×D4×C31C23semidirect
75. #92495D4semidirect
85. C2×D124D4semidirect
89. C2×Dic62D4semidirect
95. M16×C31C23semidirect
104. C4×Dic31C23semidirect
106. #92474C23semidirect
188. #575D31semidirect
211. C2×#41C2×D31semidirect
219. #137C2×D31semidirect
223. AGL(1,ℤ/8)C2×D31semidirect
227. C2×M16C2×D31semidirect
236. C31#3019semidirect
253. #41C22⋊D31semidirect
256. C2×Q8C22⋊D31semidirect
257. C2×Q8C22×D31semidirect
277. C2×D4C22⋊D31semidirect
278. C2×D4C22×D31semidirect
295. M16C22×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
#145218472
#145216132
#145212252
#145212152
#145210662
#145210652
C2×#117584112
C31×#5752
#301931