Finite Group Databasebeta

← #16855081

Subgroups — #16855081

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#14521886C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D31C212yesmaximal, complemented
3#14521617C211yesmaximal, characteristic, complemented
4#14521224C211yesmaximal, characteristic, complemented
5#14521222C211yesmaximal, characteristic, complemented
6#14521221C211yesmaximal, characteristic, complemented
7#14521218C212yesmaximal, complemented
8#14521215C212yesmaximal, complemented
9#14521064C212yesmaximal, complemented
10#14521032C211yesmaximal, characteristic, complemented
11C31×#454C211yesmaximal, characteristic, complemented
12C31×2−1+4C2211yescharacteristic, complemented
13#11758566—22no—
14#11758566—22no—
15#11758566C2212yescomplemented
16#41×D31—22no—
17#41×D31—22no—
18#41×D31C2212yes—
19#11758564—22no—
20#11758564C2211yescharacteristic, complemented
21C2×#92499—22no—
22C2×#92499C2211yescharacteristic
23C2×D4×D31C2211yescharacteristic, complemented
24C31×AGL(1,ℤ/8)C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758526C2212yescomplemented
27#11758524C2211yescharacteristic, complemented
28#11758517C2211yescharacteristic, complemented
29#11758509C2212yescomplemented
30#11758491—22no—
31SD16×D31—22no—
32#11758488—22no—
33D8×D31—22no—
34#11758485C2212yescomplemented
35M16×D31C2212yescomplemented
36#11758461—22no—
37#11758459—22no—
38#11758451C2211yescharacteristic, complemented
39#11758449C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#101C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758424C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758411C2212yescomplemented
46#11758393C2212yescomplemented
47C23×D31—21no—
48#41×C31—22no—
49#41×C31—22no—
50#41×C31C2312yes—
51C2×Q8×C31—22no—
52C2×Q8×C31C2311yescharacteristic
53C2×D4×C31C2311yescharacteristic
54C2×C22⋊D31—22no—
55C2×C22⋊D31—21no—
56#92499—22no—
57#92499—24no—
58#92499—22no—
59#92499—22no—
60#92499—22no—
61#92499—22no—
62#92499D412yescomplemented
63Q8×D31—22no—
64Q8×D31—22no—
65Q8×D31D412yescomplemented
66#92497—22no—
67#92497—22no—
68#92497D412yescomplemented
69D4×D31—24no—
70D4×D31—22no—
71D4×D31—24no—
72D4×D31—22no—
73D4×D31—22no—
74D4×D31—22no—
75D4×D31—22no—
76D4×D31D412yescomplemented
77#92495—24no—
78#92495—24no—
79#92495—22no—
80#92495—22no—
81#92495—22no—
82#92495C2312yes—
83C2×D124—24no—
84C2×D124—22no—
85C2×D124—22no—
86C2×D124D412yes—
87C2×D124C2311yescharacteristic
88C2×D124C2311yescharacteristic
89C2×C4×D31—24no—
90C2×C4×D31—22no—
91C2×C4×D31D412yes—
92C2×C4×D31C2311yescharacteristic
93SD16×C31—22no—
94D8×C31—22no—
95M16×C31C2312yes—
96C22⋊C4×C31—22no—
97C42×C31C2311yescharacteristic
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—21no—
112C22×D31—24no—
113C22×D31—22no—
114C22×D31—21no—
115C22×D31—42no—
116C22×D31—82no—
117C22×D31C2×D412yes—
118C22×D31C2×D411yescharacteristic
119Q8×C31—22no—
120Q8×C31—22no—
121Q8×C31C2×D412yes—
122D4×C31—22no—
123D4×C31—22no—
124D4×C31—22no—
125D4×C31C2×D412yes—
126C2×C4×C31—24no—
127C2×C4×C31—22no—
128C2×C4×C31—21no—
129C2×C4×C31C2×D412yes—
130C2×C4×C31C2411yescharacteristic
131C22⋊D31—24no—
132C22⋊D31—22no—
133C22⋊D31—22no—
134C22⋊D31—22no—
135C22⋊D31—41no—
136C22⋊D31—41no—
137C22⋊D31—82no—
138C2×Dic31—22no—
139C2×Dic31—21no—
140C2×Dic31C2×D411yescharacteristic
141D124—24no—
142D124—22no—
143D124—22no—
144D124—24no—
145D124—22no—
146D124—22no—
147D124—82no—
148D124C2×D412yes—
149C4×D31—24no—
150C4×D31—24no—
151C4×D31—22no—
152C4×D31—22no—
153C4×D31—22no—
154C4×D31—22no—
155C4×D31—22no—
156C4×D31C2×D412yes—
157C4×D31C2×D412yes—
158Dic62—22no—
159Dic62—22no—
160Dic62C2×D412yes—
161C8×C31—22no—
162C31⋊C8—22no—
163#3021—311nomaximal, Sylow
164C22×C31—22no—
165C22×C31—42no—
166C22×C31—41no—
167C22×C31C22×D411yescharacteristic
168C2×D31—22no—
169C2×D31—22no—
170C2×D31—41no—
171C2×D31—42no—
172C2×D31—42no—
173C2×D31—41no—
174C2×D31—82no—
175C2×D31—84no—
176C2×D31—84no—
177C2×D31#12012yes—
178C4×C31—22no—
179C4×C31—22no—
180C4×C31—41no—
181C4×C31C22×D412yes—
182Dic31—22no—
183Dic31—41no—
184Dic31#12011yescharacteristic
185Dic31#12011yescharacteristic
186#585—311no—
187C2×AGL(1,ℤ/8)—312no—
188#535—311no—
189#454—311no—
190#454—311no—
191#454—311no—
192#454D3111yescharacteristic, complemented
193#453—312no—
194#453—312no—
195#421—312no—
196#413—311no—
197C2×C31—21no—
198C2×C31—42no—
199C2×C31—81no—
200C2×C31C2×#12011yescharacteristic
201D31—22no—
202D31—42no—
203D31—42no—
204D31—81no—
2052−1+4—311no—
2062−1+4—622no—
2072−1+4C2×D3111yescharacteristic, complemented
208H(5,𝔽2)—312no—
209H(5,𝔽2)—622no—
210H(5,𝔽2)—622no—
211C2×#41—311no—
212C2×#41—312no—
213C2×#41—622no—
214C2×#41—622no—
215C2×#41—622no—
216C22×D4—311no—
217AGL(1,ℤ/8)—312no—
218AGL(1,ℤ/8)—312no—
219AGL(1,ℤ/8)—312no—
220AGL(1,ℤ/8)—622no—
221AGL(1,ℤ/8)—622no—
222AGL(1,ℤ/8)C2×D3112yescomplemented
223C2×SD16—622no—
224C2×D8—622no—
225C2×M16—312no—
226#127—311no—
227#127—311no—
228#124—311no—
229#124C2×D3111yescharacteristic, complemented
230#121—622no—
231#120—622no—
232#117—311no—
233#104—312no—
234#104—312no—
235#104—312no—
236#104C2×D3112yescomplemented
237#101—311no—
238#101C2×D3111yescharacteristic, complemented
239#100—312no—
240C31#302111yescharacteristic, complemented, Sylow
241C24—621no—
242#41—22no—
243#41—22no—
244#41—312no—
245#41—312no—
246#41—312no—
247#41—622no—
248#41—622no—
249#41—624no—
250#41—624no—
251#41—622no—
252#41—622no—
253#41—622no—
254#41—622no—
255#41—622no—
256#41—622no—
257#41—624no—
258#41—622no—
259#41—622no—
260#41C22×D3112yes—
261C2×Q8—22no—
262C2×Q8—312no—
263C2×Q8—622no—
264C2×Q8—622no—
265C2×Q8C22×D3111yescharacteristic
266C2×D4—312no—
267C2×D4—312no—
268C2×D4—311no—
269C2×D4—311no—
270C2×D4—622no—
271C2×D4—624no—
272C2×D4—622no—
273C2×D4—622no—
274C2×D4—622no—
275C2×D4—621no—
276C2×D4—622no—
277C2×D4—624no—
278C2×D4—622no—
279C2×D4—624no—
280C2×D4—622no—
281C2×D4—622no—
282C2×D4C22×D3111yescharacteristic
283C22×C4—311no—
284C22×C4—312no—
285C22×C4—622no—
286C22×C4—624no—
287SD16—22no—
288SD16—622no—
289SD16—622no—
290SD16—622no—
291D8—22no—
292D8—622no—
293D8—622no—
294D8—622no—
295M16—312no—
296M16—622no—
297M16C22×D3112yes—
298C2×C8—622no—
299C4⋊C4—621no—
300C22⋊C4—22no—
301C22⋊C4—621no—
302C22⋊C4—622no—
303C42—311no—
304C42C22×D3111yescharacteristic
305C23—21no—
306C23—311no—
307C23—312no—
308C23—622no—
309C23—624no—
310C23—621no—
311C23—1242no—
312C23—2482no—
313Q8—22no—
314Q8—22no—
315Q8—312no—
316Q8—622no—
317Q8—622no—
318Q8D4×D3112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—312no—
323D4—624no—
324D4—622no—
325D4—624no—
326D4—622no—
327D4—622no—
328D4—622no—
329D4—624no—
330D4—622no—
331D4—622no—
332D4—622no—
333D4—1241no—
334D4—1241no—
335D4—2482no—
336D4—2482no—
337D4D4×D3112yes—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—311no—
342C2×C4—312no—
343C2×C4—312no—
344C2×C4—622no—
345C2×C4—622no—
346C2×C4—622no—
347C2×C4—622no—
348C2×C4—624no—
349C2×C4—624no—
350C2×C4—622no—
351C2×C4—622no—
352C2×C4—621no—
353C2×C4D4×D3112yes—
354C2×C4C23×D3111yescharacteristic
355C8—22no—
356C8—622no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—312no—
361C22—622no—
362C22—622no—
363C22—1241no—
364C22—1242no—
365C22—1242no—
366C22—1241no—
367C22—2482no—
368C22—2484no—
369C22—2484no—
370C22C2×D4×D3111yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—311no—
375C4—311no—
376C4—622no—
377C4—1241no—
378C4C2×D4×D3112yes—
379C2—21no—
380C2—42no—
381C2—81no—
382C2—622no—
383C2—1242no—
384C2—1242no—
385C2—2481no—
386C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#454C2
socleC2×C31C2×#120
hypercenter#454D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #14521886C2semidirect
2. AGL(1,ℤ/8)×D31C2semidirect
3. #14521617C2semidirect
4. #14521224C2semidirect
5. #14521222C2semidirect
6. #14521221C2semidirect
7. #14521218C2semidirect
8. #14521215C2semidirect
9. #14521064C2semidirect
10. #14521032C2semidirect
11. C31×#454C2semidirect
12. C31×2−1+4C22semidirect
15. #11758566C22semidirect
20. #11758564C22semidirect
23. C2×D4×D31C22semidirect
24. C31×AGL(1,ℤ/8)C22semidirect
25. C31×#124C22semidirect
26. #11758526C22semidirect
27. #11758524C22semidirect
28. #11758517C22semidirect
29. #11758509C22semidirect
34. #11758485C22semidirect
35. M16×D31C22semidirect
38. #11758451C22semidirect
39. #11758449C22semidirect
40. C31×#104C22semidirect
41. C31×#101C22semidirect
42. #11758425C22semidirect
43. #11758424C22semidirect
44. #11758413C22semidirect
45. #11758411C22semidirect
46. #11758393C22semidirect
62. #92499D4semidirect
65. Q8×D31D4semidirect
68. #92497D4semidirect
76. D4×D31D4semidirect
192. #454D31semidirect
207. 2−1+4C2×D31semidirect
222. AGL(1,ℤ/8)C2×D31semidirect
229. #124C2×D31semidirect
236. #104C2×D31semidirect
238. #101C2×D31semidirect
240. C31#3021semidirect

Maximal subgroups

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

subgroupindex
#145218862
AGL(1,ℤ/8)×D312
#145216172
#145212242
#145212222
#145212212
#145212182
#145212152
#145210642
#145210322
C31×#4542
#302131