Finite Group Databasebeta

← #16855058

Subgroups — #16855058

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#14521886C211yesmaximal, characteristic, complemented
2#14521845C212yesmaximal, complemented
3#14521598C211yesmaximal, characteristic, complemented
4#14521234C211yesmaximal, characteristic, complemented
5#14521232C211yesmaximal, characteristic, complemented
6#14521228C212yesmaximal, complemented
7#14521221C211yesmaximal, characteristic, complemented
8#14521215C212yesmaximal, complemented
9#104×D31C212yesmaximal, complemented
10#14521032C211yesmaximal, characteristic, complemented
11C31×#454C211yesmaximal, characteristic, complemented
12C31×2−1+4C2211yescharacteristic
13#11758566—22no—
14#11758566—22no—
15#11758566C2212yes—
16#41×D31—22no—
17#41×D31—22no—
18#41×D31C2212yes—
19#11758564—22no—
20#11758564C2211yescharacteristic
21C2×#92499—22no—
22C2×#92499C2211yescharacteristic
23C2×#92497C2211yescharacteristic, complemented
24C31×AGL(1,ℤ/8)C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758528C2212yescomplemented
27#11758521C2211yescharacteristic, complemented
28#11758517C2211yescharacteristic, complemented
29#11758509C2212yescomplemented
30#11758493—22no—
31#11758492—22no—
32#11758489—22no—
33#11758488—22no—
34#11758486C2212yescomplemented
35M16×D31C2212yescomplemented
36#11758461—22no—
37(C22⋊D31)⋊C4—22no—
38#11758452C2211yescharacteristic, complemented
39C42×D31C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#101C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758424C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758411C2212yescomplemented
46#11758393C2212yescomplemented
47#41×C31—22no—
48#41×C31—22no—
49#41×C31C2312yes—
50C2×Q8×C31—22no—
51C2×Q8×C31C2311yescharacteristic
52C2×D4×C31C2311yescharacteristic
53C2×C22⋊D31—22no—
54C2×C22⋊D31—21no—
55C22×Dic31—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#92497—22no—
69#92497—22no—
70#92497D412yescomplemented
71D4×D31—24no—
72D4×D31—22no—
73D4×D31—24no—
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×D124D412yes—
86C2×D124C2311yescharacteristic
87C2×C4×D31—24no—
88C2×C4×D31—22no—
89C2×C4×D31—21no—
90C2×C4×D31D412yes—
91C2×C4×D31C2311yescharacteristic
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94D8×C31—22no—
95M16×C31C2312yes—
96C22⋊C4×C31—22no—
97C42×C31C2311yescharacteristic
98#92482—22no—
99#92480—22no—
100#92480—22no—
101#92479—22no—
102(C2×D31)⋊C4—22no—
103Dic31⋊C4—22no—
104C4×Dic31—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107Dic124—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—21no—
112C22×D31—24no—
113C22×D31—22no—
114C22×D31C2×D412yes—
115C22×D31C2×D411yescharacteristic
116Q8×C31—22no—
117Q8×C31—22no—
118Q8×C31C2×D412yes—
119D4×C31—22no—
120D4×C31—22no—
121D4×C31—22no—
122D4×C31C2×D412yes—
123C2×C4×C31—24no—
124C2×C4×C31—22no—
125C2×C4×C31—21no—
126C2×C4×C31C2×D412yes—
127C2×C4×C31C2411yescharacteristic
128C22⋊D31—24no—
129C22⋊D31—22no—
130C22⋊D31—22no—
131C22⋊D31—22no—
132C22⋊D31—44no—
133C22⋊D31—42no—
134C2×Dic31—21no—
135C2×Dic31—22no—
136C2×Dic31—21no—
137C2×Dic31—41no—
138C2×Dic31—41no—
139C2×Dic31C2×D411yescharacteristic
140D124—24no—
141D124—22no—
142D124—22no—
143D124—24no—
144D124—22no—
145D124C2×D412yes—
146C4×D31—24no—
147C4×D31—24no—
148C4×D31—22no—
149C4×D31—22no—
150C4×D31—22no—
151C4×D31—22no—
152C4×D31—22no—
153C4×D31—42no—
154C4×D31C2×D412yes—
155C4×D31C2×D412yes—
156Dic62—22no—
157Dic62—22no—
158Dic62—22no—
159Dic62C2×D412yes—
160C8×C31—22no—
161C31⋊C8—22no—
162#3018—311nomaximal, Sylow
163C22×C31—22no—
164C22×C31—42no—
165C22×C31—41no—
166C22×C31C22×D411yescharacteristic
167C2×D31—22no—
168C2×D31—22no—
169C2×D31—41no—
170C2×D31—42no—
171C2×D31—82no—
172C2×D31—84no—
173C2×D31#12012yes—
174C4×C31—22no—
175C4×C31—22no—
176C4×C31—41no—
177C4×C31C22×D412yes—
178Dic31—22no—
179Dic31—22no—
180Dic31—41no—
181Dic31#12011yescharacteristic
182Dic31#12011yescharacteristic
183#585—311no—
184#575—312no—
185#532—311no—
186#456—311no—
187#456—311no—
188#455—312no—
189#454—311no—
190#454D3111yescharacteristic, complemented
191#453—312no—
192C2×#104—312no—
193#413—311no—
194C2×C31—21no—
195C2×C31—42no—
196C2×C31—81no—
197C2×C31C2×#12011yescharacteristic
198D31—22no—
199D31—42no—
200D31—42no—
2012−1+4—311no—
2022−1+4—622no—
2032−1+4C2×D3111yescharacteristic
204H(5,𝔽2)—312no—
205H(5,𝔽2)—622no—
206H(5,𝔽2)—622no—
207C2×#41—311no—
208C2×#41—312no—
209C2×#41—311no—
210C2×#41—622no—
211C2×#41—622no—
212C2×#41—622no—
213#137—312no—
214#137—312no—
215#137—622no—
216AGL(1,ℤ/8)—312no—
217AGL(1,ℤ/8)—622no—
218AGL(1,ℤ/8)C2×D3112yescomplemented
219#135—622no—
220#135—622no—
221C2×M16—312no—
222C4⋊Q8—311no—
223#127—311no—
224#124—311no—
225#124C2×D3111yescharacteristic, complemented
226#121—622no—
227C4×D4—622no—
228C2×C42—311no—
229#104—312no—
230#104—312no—
231#104—312no—
232#104C2×D3112yescomplemented
233#101—311no—
234#101C2×D3111yescharacteristic, complemented
235#100—312no—
236C31#301811yescharacteristic, complemented, Sylow
237#41—22no—
238#41—22no—
239#41—312no—
240#41—312no—
241#41—312no—
242#41—622no—
243#41—622no—
244#41—624no—
245#41—624no—
246#41—622no—
247#41—622no—
248#41—622no—
249#41—622no—
250#41—622no—
251#41—622no—
252#41—622no—
253#41—622no—
254#41—624no—
255#41—622no—
256#41—622no—
257#41C22×D3112yes—
258C2×Q8—22no—
259C2×Q8—312no—
260C2×Q8—311no—
261C2×Q8—622no—
262C2×Q8—622no—
263C2×Q8C22×D3111yescharacteristic
264C2×D4—312no—
265C2×D4—312no—
266C2×D4—311no—
267C2×D4—622no—
268C2×D4—624no—
269C2×D4—622no—
270C2×D4—624no—
271C2×D4—622no—
272C2×D4—624no—
273C2×D4—622no—
274C2×D4—622no—
275C2×D4—621no—
276C2×D4C22×D3111yescharacteristic
277C22×C4—311no—
278C22×C4—312no—
279C22×C4—622no—
280C22×C4—624no—
281C22×C4—621no—
282C22×C4—621no—
283Q16—622no—
284Q16—622no—
285SD16—22no—
286SD16—622no—
287SD16—622no—
288SD16—622no—
289D8—22no—
290D8—622no—
291M16—312no—
292M16—622no—
293M16C22×D3112yes—
294C2×C8—622no—
295C4⋊C4—622no—
296C22⋊C4—22no—
297C22⋊C4—622no—
298C42—311no—
299C42—621no—
300C42C22×D3111yescharacteristic
301C23—21no—
302C23—311no—
303C23—312no—
304C23—622no—
305C23—624no—
306Q8—22no—
307Q8—22no—
308Q8—312no—
309Q8—622no—
310Q8—622no—
311Q8—622no—
312Q8D4×D3112yes—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—312no—
317D4—624no—
318D4—622no—
319D4—624no—
320D4—622no—
321D4—622no—
322D4—622no—
323D4—624no—
324D4—622no—
325D4—622no—
326D4—1244no—
327D4—1242no—
328D4D4×D3112yes—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—311no—
333C2×C4—312no—
334C2×C4—312no—
335C2×C4—622no—
336C2×C4—622no—
337C2×C4—622no—
338C2×C4—622no—
339C2×C4—624no—
340C2×C4—624no—
341C2×C4—622no—
342C2×C4—621no—
343C2×C4—622no—
344C2×C4—621no—
345C2×C4—1242no—
346C2×C4—1241no—
347C2×C4—1241no—
348C2×C4D4×D3112yes—
349C2×C4C23×D3111yescharacteristic
350C8—22no—
351C8—622no—
352C22—22no—
353C22—42no—
354C22—41no—
355C22—312no—
356C22—622no—
357C22—622no—
358C22—1241no—
359C22—1242no—
360C22—2482no—
361C22—2484no—
362C22C2×D4×D3111yescharacteristic
363C4—22no—
364C4—22no—
365C4—41no—
366C4—311no—
367C4—311no—
368C4—622no—
369C4—622no—
370C4—1241no—
371C4C2×D4×D3112yes—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—622no—
376C2—1242no—
377C2—1242no—
378C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#454C2
socleC2×C31C2×#120
hypercenter#454D31
hypocenterC31#3018
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. #14521845C2semidirect
3. #14521598C2semidirect
4. #14521234C2semidirect
5. #14521232C2semidirect
6. #14521228C2semidirect
7. #14521221C2semidirect
8. #14521215C2semidirect
9. #104×D31C2semidirect
10. #14521032C2semidirect
11. C31×#454C2semidirect
23. C2×#92497C22semidirect
24. C31×AGL(1,ℤ/8)C22semidirect
25. C31×#124C22semidirect
26. #11758528C22semidirect
27. #11758521C22semidirect
28. #11758517C22semidirect
29. #11758509C22semidirect
34. #11758486C22semidirect
35. M16×D31C22semidirect
38. #11758452C22semidirect
39. C42×D31C22semidirect
40. C31×#104C22semidirect
41. C31×#101C22semidirect
42. #11758425C22semidirect
43. #11758424C22semidirect
44. #11758413C22semidirect
45. #11758411C22semidirect
46. #11758393C22semidirect
62. #92499D4semidirect
65. Q8×D31D4semidirect
70. #92497D4semidirect
76. D4×D31D4semidirect
190. #454D31semidirect
218. AGL(1,ℤ/8)C2×D31semidirect
225. #124C2×D31semidirect
232. #104C2×D31semidirect
234. #101C2×D31semidirect
236. C31#3018semidirect

Maximal subgroups

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

subgroupindex
#145218862
#145218452
#145215982
#145212342
#145212322
#145212282
#145212212
#145212152
#104×D312
#145210322
C31×#4542
#301831