Finite Group Databasebeta

← #16855070

Subgroups — #16855070

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#14521849C211yesmaximal, characteristic, complemented
3#14521847C211yesmaximal, characteristic, complemented
4#14521618C211yesmaximal, characteristic, complemented
5#14521228C214yesmaximal, complemented
6#14521218C214yesmaximal, complemented
7#14521067C211yesmaximal, characteristic, complemented
8C2×#11758411C211yesmaximal, characteristic, complemented
9C31×#421C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758566—22no—
12#11758566—22no—
13#11758566—22no—
14#11758566C2214yes—
15C2×#92499—21no—
16C2×D4×D31—22no—
17C2×D4×D31—21no—
18C2×#92495—21no—
19C2×#92495C2211yescharacteristic
20C2×#92495C2211yescharacteristic, complemented
21C22×D124—21no—
22C22×D124C2211yescharacteristic
23C2×M16×C31C2211yescharacteristic, complemented
24C31×#117C2211yescharacteristic, complemented
25#11758528C2212yescomplemented
26#11758527—22no—
27#11758526C2212yescomplemented
28#11758519—21no—
29#11758509—21no—
30C2×#92474C2211yescharacteristic, complemented
31#11758486—21no—
32#11758486C2212yescomplemented
33#11758485—21no—
34#11758485C2212yescomplemented
35#11758481—22no—
36#11758472—22no—
37#11758459—22no—
38#11758452C2212yescomplemented
39#11758451C2212yescomplemented
40C31×#104C2214yescomplemented
41#11758411C2214yescomplemented
42#11758393C2214yescomplemented
43C23×D31—22no—
44C23×D31D412yes—
45#41×C31—21no—
46#41×C31—21no—
47#41×C31D412yescomplemented
48#41×C31C2312yes—
49C2×Q8×C31D411yescharacteristic, complemented
50C2×D4×C31—21no—
51C2×D4×C31D411yescharacteristic, complemented
52C22×C4×C31—21no—
53C22×C4×C31C2311yescharacteristic
54C2×C22⋊D31—22no—
55C2×C22⋊D31—21no—
56C2×C22⋊D31—21no—
57#92499—24no—
58#92499—24no—
59D4×D31—28no—
60D4×D31—24no—
61D4×D31—28no—
62D4×D31—24no—
63#92495—24no—
64#92495—21no—
65#92495—21no—
66#92495—21no—
67#92495—21no—
68#92495—21no—
69#92495—21no—
70#92495—24no—
71#92495D412yescomplemented
72#92495C2312yes—
73C2×D124—22no—
74C2×D124—24no—
75C2×D124—21no—
76C2×D124—21no—
77C2×D124—21no—
78C2×D124—21no—
79C2×D124—24no—
80C2×D124—24no—
81C2×D124D412yes—
82C2×D124D411yescharacteristic, complemented
83C2×D124C2312yes—
84C2×D124C2311yescharacteristic, complemented
85C2×C4×D31—22no—
86C2×C4×D31—21no—
87C2×C4×D31—21no—
88C2×Dic62D411yescharacteristic, complemented
89C2×Dic62C2311yescharacteristic, complemented
90M16×C31—21no—
91M16×C31C2312yescomplemented
92C2×C8×C31—21no—
93C4⋊C4×C31—21no—
94C22⋊C4×C31—21no—
95C42×C31C2312yescomplemented
96#92482—22no—
97#92481—22no—
98#92480—22no—
99#92479—22no—
100(C2×D31)⋊C4—24no—
101#92474—21no—
102#92474C2312yescomplemented
103C2×C31⋊C8—21no—
104Dic124—22no—
105D248—22no—
106#92470—22no—
107#92470—22no—
108C23×C31—21no—
109C23×C31C2×D411yescharacteristic
110C22×D31—22no—
111C22×D31—21no—
112C22×D31—28no—
113C22×D31—22no—
114C22×D31—21no—
115C22×D31—82no—
116C22×D31—84no—
117C22×D31C2×D414yes—
118Q8×C31—21no—
119Q8×C31C2×D412yes—
120D4×C31—22no—
121D4×C31—22no—
122D4×C31—21no—
123D4×C31C2×D412yes—
124C2×C4×C31—21no—
125C2×C4×C31—22no—
126C2×C4×C31—22no—
127C2×C4×C31—22no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2411yescharacteristic
134C22⋊D31—28no—
135C22⋊D31—22no—
136C22⋊D31—22no—
137C22⋊D31—41no—
138C22⋊D31—41no—
139C2×Dic31—21no—
140C2×Dic31—21no—
141D124—28no—
142D124—22no—
143D124—21no—
144D124—22no—
145D124—22no—
146D124—22no—
147D124—21no—
148D124—21no—
149D124—84no—
150D124C2×D412yes—
151C4×D31—28no—
152C4×D31—22no—
153C4×D31—22no—
154C4×D31—41no—
155C4×D31—41no—
156Dic62—21no—
157Dic62—21no—
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#12011yescharacteristic
167C22×C31#12011yescharacteristic
168C22×C31C22×D411yescharacteristic
169C2×D31—24no—
170C2×D31—22no—
171C2×D31—44no—
172C2×D31—42no—
173C2×D31—44no—
174C2×D31—42no—
175C2×D31—41no—
176C2×D31—82no—
177C2×D31—88no—
178C4×C31—22no—
179C4×C31—42no—
180C4×C31#12011yescharacteristic
181C4×C31#12011yescharacteristic
182C4×C31C22×D411yescharacteristic
183C4×C31C22×D411yescharacteristic
184Dic31—22no—
185Dic31—41no—
186C2×H(5,𝔽2)—311no—
187#575—311no—
188#575—311no—
189#535—311no—
190#455—314no—
191#453—314no—
192#421—311no—
193#421D3111yescharacteristic, complemented
194C2×#100—311no—
195C2×C31—21no—
196C2×C31—21no—
197C2×C31—21no—
198C2×C31—42no—
199C2×C31C2×#12011yescharacteristic
200D31—44no—
201D31—42no—
202D31—81no—
203H(5,𝔽2)—314no—
204H(5,𝔽2)—622no—
205H(5,𝔽2)—622no—
206H(5,𝔽2)—622no—
207C2×#41—311no—
208C2×#41—311no—
209C2×#41—621no—
210C2×#41—621no—
211C2×#41C2×D3111yescharacteristic
212C22×D4—311no—
213C22×D4—621no—
214C22×D4—622no—
215C22×D4—621no—
216#137—312no—
217#137—312no—
218#137—621no—
219#137—621no—
220AGL(1,ℤ/8)—312no—
221AGL(1,ℤ/8)—312no—
222AGL(1,ℤ/8)—621no—
223AGL(1,ℤ/8)—621no—
224#135—622no—
225#135—622no—
226C2×M16—311no—
227C2×M16C2×D3111yescharacteristic, complemented
228#127—312no—
229#124—312no—
230#121—622no—
231#120—622no—
232#117C2×D3111yescharacteristic, complemented
233#104—314no—
234#104C2×D3114yescomplemented
235#100—314no—
236C31#301911yescharacteristic, complemented, Sylow
237C24—312no—
238C24—622no—
239#41—21no—
240#41—21no—
241#41—312no—
242#41—312no—
243#41—624no—
244#41—624no—
245#41—624no—
246#41—624no—
247#41—621no—
248#41—621no—
249#41—621no—
250#41—621no—
251#41—621no—
252#41—621no—
253#41D12412yescomplemented
254#41C22×D3112yes—
255C2×Q8—311no—
256C2×Q8—311no—
257C2×Q8D12411yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—311no—
260C2×D4—312no—
261C2×D4—312no—
262C2×D4—311no—
263C2×D4—624no—
264C2×D4—622no—
265C2×D4—624no—
266C2×D4—624no—
267C2×D4—628no—
268C2×D4—628no—
269C2×D4—624no—
270C2×D4—622no—
271C2×D4—621no—
272C2×D4—624no—
273C2×D4—621no—
274C2×D4—621no—
275C2×D4—621no—
276C2×D4—621no—
277C2×D4—621no—
278C2×D4D12411yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—622no—
281C22×C4—621no—
282C22×C4—621no—
283C22×C4C22×D3111yescharacteristic
284Q16—622no—
285Q16—622no—
286SD16—622no—
287SD16—622no—
288SD16—622no—
289SD16—622no—
290D8—622no—
291D8—622no—
292M16—21no—
293M16—312no—
294M16—621no—
295M16C22×D3112yescomplemented
296C2×C8—21no—
297C2×C8—621no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—624no—
301C42C22×D3112yescomplemented
302C23—21no—
303C23—314no—
304C23—628no—
305C23—621no—
306C23—622no—
307C23—622no—
308C23—621no—
309C23—2482no—
310C23—2484no—
311C23D4×D3111yescharacteristic
312Q8—21no—
313Q8—312no—
314Q8—621no—
315Q8—621no—
316Q8—621no—
317Q8C2×D12412yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—312no—
322D4—628no—
323D4—628no—
324D4—622no—
325D4—622no—
326D4—621no—
327D4—622no—
328D4—622no—
329D4—622no—
330D4—621no—
331D4—622no—
332D4—621no—
333D4—1241no—
334D4—1241no—
335D4—2484no—
336D4C2×D12412yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—628no—
342C2×C4—621no—
343C2×C4—622no—
344C2×C4—622no—
345C2×C4—621no—
346C2×C4—1241no—
347C2×C4—1241no—
348C2×C4C2×D12411yescharacteristic
349C2×C4C2×D12411yescharacteristic
350C2×C4D4×D3111yescharacteristic
351C2×C4D4×D3111yescharacteristic
352C2×C4D4×D3111yescharacteristic
353C2×C4C23×D3111yescharacteristic
354C8—22no—
355C8—622no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—622no—
360C22—624no—
361C22—1244no—
362C22—1244no—
363C22—1242no—
364C22—1242no—
365C22—1241no—
366C22—2482no—
367C22—2488no—
368C22#1175845911yescharacteristic
369C22#1175845911yescharacteristic
370C22C2×D4×D3111yescharacteristic
371C4—22no—
372C4—42no—
373C4—622no—
374C4—1241no—
375C4#1175845911yescharacteristic
376C4#1175845911yescharacteristic
377C4C22×D12411yescharacteristic
378C4C2×D4×D3111yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—1244no—
384C2—1242no—
385C2—2481no—
386C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#421C2
socleC2×C31C2×#120
hypercenter#421D31
hypocenterC31#3019
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. #14521849C2semidirect
3. #14521847C2semidirect
4. #14521618C2semidirect
5. #14521228C2semidirect
6. #14521218C2semidirect
7. #14521067C2semidirect
8. C2×#11758411C2semidirect
9. C31×#421C2semidirect
20. C2×#92495C22semidirect
23. C2×M16×C31C22semidirect
24. C31×#117C22semidirect
25. #11758528C22semidirect
27. #11758526C22semidirect
30. C2×#92474C22semidirect
32. #11758486C22semidirect
34. #11758485C22semidirect
38. #11758452C22semidirect
39. #11758451C22semidirect
40. C31×#104C22semidirect
41. #11758411C22semidirect
42. #11758393C22semidirect
47. #41×C31D4semidirect
49. C2×Q8×C31D4semidirect
51. C2×D4×C31D4semidirect
71. #92495D4semidirect
82. C2×D124D4semidirect
84. C2×D124C23semidirect
88. C2×Dic62D4semidirect
89. C2×Dic62C23semidirect
91. M16×C31C23semidirect
95. C42×C31C23semidirect
102. #92474C23semidirect
193. #421D31semidirect
227. C2×M16C2×D31semidirect
232. #117C2×D31semidirect
234. #104C2×D31semidirect
236. C31#3019semidirect
253. #41D124semidirect
257. C2×Q8D124semidirect
278. C2×D4D124semidirect
295. M16C22×D31semidirect
301. C42C22×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
#145218492
#145218472
#145216182
#145212282
#145212182
#145210672
C2×#117584112
C31×#4212
#301931