Finite Group Databasebeta

← #16855057

Subgroups — #16855057

Order 3968.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521885C211yesmaximal, characteristic, complemented
2#14521847C211yesmaximal, characteristic, complemented
3#14521595C211yesmaximal, characteristic, complemented
4#14521230C212yesmaximal, complemented
5#14521225C212yesmaximal, complemented
6#14521220C212yesmaximal, complemented
7#14521215C212yesmaximal, complemented
8C2×#11758425C211yesmaximal, characteristic, complemented
9C2×#11758413C211yesmaximal, characteristic, complemented
10#14521031C211yesmaximal, characteristic, complemented
11C31×#575C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#41×C31C2211yescharacteristic, complemented
14#11758567—21no—
15#11758567C2212yes—
16#11758566—21no—
17#11758566C2212yes—
18#41×D31—22no—
19#41×D31—22no—
20#11758564—21no—
21#11758561—21no—
22#11758561—22no—
23C2×#92495—22no—
24C2×#92495C2211yescharacteristic
25C2×#92495C2211yescharacteristic
26C31×#137—21no—
27C31×#137C2212yescomplemented
28C31×AGL(1,ℤ/8)—21no—
29C31×AGL(1,ℤ/8)C2212yescomplemented
30C31×#135—22no—
31C2×M16×C31C2211yescharacteristic, complemented
32#11758528—21no—
33#11758527—22no—
34#11758526—21no—
35#11758524C2211yescharacteristic, complemented
36#11758521C2211yescharacteristic, complemented
37#11758519C2212yescomplemented
38#11758517C2211yescharacteristic, complemented
39#11758516—22no—
40#11758513C2211yescharacteristic, complemented
41#11758509C2212yescomplemented
42C4×C22⋊D31—22no—
43C2×C4×Dic31C2211yescharacteristic, complemented
44C2×#92474C2211yescharacteristic, complemented
45#11758425C2212yescomplemented
46#11758425C2212yescomplemented
47#11758413C2212yescomplemented
48#11758413C2212yescomplemented
49#11758412C2212yescomplemented
50#11758411C2212yescomplemented
51#41×C31—21no—
52#41×C31—21no—
53#41×C31—21no—
54#41×C31—21no—
55#41×C31—21no—
56#41×C31—21no—
57#41×C31D412yescomplemented
58#41×C31C2312yes—
59C2×Q8×C31D411yescharacteristic, complemented
60C2×Q8×C31C2311yescharacteristic
61C2×D4×C31—21no—
62C2×D4×C31—21no—
63C2×D4×C31D411yescharacteristic, complemented
64C2×D4×C31C2311yescharacteristic
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31C2311yescharacteristic
68C2×C22⋊D31—24no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31—22no—
71C2×C22⋊D31D412yes—
72C22×Dic31—21no—
73#92499—22no—
74#92499—22no—
75Q8×D31—22no—
76Q8×D31—22no—
77#92497—24no—
78#92497—22no—
79#92497—22no—
80#92497—24no—
81D4×D31—24no—
82D4×D31—22no—
83D4×D31—24no—
84D4×D31—22no—
85#92495—24no—
86#92495—24no—
87#92495—21no—
88#92495—21no—
89#92495—21no—
90#92495—21no—
91#92495—21no—
92#92495—21no—
93#92495—22no—
94#92495—22no—
95#92495D412yescomplemented
96#92495C2312yes—
97C2×D124—22no—
98C2×D124D411yescharacteristic, complemented
99C2×D124C2311yescharacteristic
100C2×C4×D31—24no—
101C2×C4×D31—21no—
102C2×C4×D31D412yes—
103C2×Dic62—22no—
104C2×Dic62D411yescharacteristic, complemented
105C2×Dic62C2311yescharacteristic
106Q16×C31—22no—
107SD16×C31—22no—
108SD16×C31—22no—
109D8×C31—22no—
110M16×C31—21no—
111M16×C31C2312yes—
112C2×C8×C31—21no—
113C22⋊Dic31—22no—
114#92482—22no—
115#92481—22no—
116#92480—22no—
117#92479—22no—
118(C2×D31)⋊C4—22no—
119Dic31⋊C4—22no—
120C4×Dic31—21no—
121C4×Dic31C2311yescharacteristic
122C4×Dic31C2311yescharacteristic
123#92474—21no—
124#92474C2312yes—
125C2×C31⋊C8—21no—
126C23×C31—21no—
127C23×C31—21no—
128C23×C31C2×D411yescharacteristic
129C22×D31—24no—
130C22×D31—21no—
131C22×D31C2×D412yes—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135Q8×C31C2×D412yes—
136D4×C31—22no—
137D4×C31—22no—
138D4×C31—21no—
139D4×C31—21no—
140D4×C31—21no—
141D4×C31—41no—
142D4×C31—41no—
143D4×C31C2×D412yes—
144C2×C4×C31—21no—
145C2×C4×C31—22no—
146C2×C4×C31—22no—
147C2×C4×C31—21no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2×D411yescharacteristic
152C2×C4×C31C2×D411yescharacteristic
153C2×C4×C31C2×D411yescharacteristic
154C2×C4×C31C2×D411yescharacteristic
155C2×C4×C31C2411yescharacteristic
156C22⋊D31—24no—
157C22⋊D31—24no—
158C22⋊D31—22no—
159C22⋊D31—22no—
160C22⋊D31—22no—
161C22⋊D31—22no—
162C22⋊D31—44no—
163C2×Dic31—24no—
164C2×Dic31—21no—
165C2×Dic31—21no—
166C2×Dic31—21no—
167C2×Dic31—42no—
168C2×Dic31C2×D412yes—
169D124—24no—
170D124—21no—
171D124—21no—
172D124—21no—
173D124C2×D412yes—
174C4×D31—24no—
175C4×D31—24no—
176C4×D31—22no—
177C4×D31—22no—
178C4×D31—22no—
179C4×D31—22no—
180Dic62—24no—
181Dic62—21no—
182Dic62—21no—
183Dic62—21no—
184Dic62C2×D412yes—
185C8×C31—22no—
186C31⋊C8—22no—
187#3018—311nomaximal, Sylow
188C22×C31—22no—
189C22×C31—41no—
190C22×C31—42no—
191C22×C31—41no—
192C22×C31—81no—
193C22×C31#12011yescharacteristic
194C22×C31#12011yescharacteristic
195C22×C31C22×D411yescharacteristic
196C2×D31—22no—
197C2×D31—22no—
198C2×D31—42no—
199C2×D31—81no—
200C2×D31—84no—
201C4×C31—22no—
202C4×C31—41no—
203C4×C31#12011yescharacteristic
204C4×C31#12011yescharacteristic
205C4×C31C22×D411yescharacteristic
206C4×C31C22×D411yescharacteristic
207Dic31—22no—
208Dic31—22no—
209Dic31—22no—
210Dic31—41no—
211#585—311no—
212#575—311no—
213#575D3111yescharacteristic, complemented
214#532—311no—
215#456—312no—
216#455—312no—
217#454—312no—
218#453—312no—
219C2×#104—311no—
220C2×#104—311no—
221#413—311no—
222C2×C31—21no—
223C2×C31—21no—
224C2×C31—21no—
225C2×C31—42no—
226C2×C31—81no—
227C2×C31C2×#12011yescharacteristic
228D31—42no—
229D31—42no—
2302−1+4—312no—
2312−1+4—621no—
2322−1+4—621no—
233H(5,𝔽2)—312no—
234H(5,𝔽2)—622no—
235H(5,𝔽2)—621no—
236H(5,𝔽2)—621no—
237C2×#41—311no—
238C2×#41—311no—
239C2×#41—622no—
240C2×#41—622no—
241C2×#41—622no—
242C2×#41C2×D3111yescharacteristic
243C2×#41C2×D3111yescharacteristic, complemented
244#137—21no—
245#137—312no—
246#137—621no—
247#137C2×D3112yescomplemented
248AGL(1,ℤ/8)—21no—
249AGL(1,ℤ/8)—312no—
250AGL(1,ℤ/8)—621no—
251AGL(1,ℤ/8)C2×D3112yescomplemented
252#135—22no—
253#135—622no—
254C2×M16—311no—
255C2×M16C2×D3111yescharacteristic, complemented
256C4⋊Q8—311no—
257#127—311no—
258#124—311no—
259#124—311no—
260#121—622no—
261C4×D4—622no—
262C2×C42—311no—
263#104—312no—
264#104—312no—
265#104—312no—
266#104—312no—
267#101—312no—
268#100—312no—
269C31#301811yescharacteristic, complemented, Sylow
270#41—21no—
271#41—21no—
272#41—21no—
273#41—21no—
274#41—21no—
275#41—21no—
276#41—312no—
277#41—312no—
278#41—622no—
279#41—622no—
280#41—624no—
281#41—624no—
282#41—624no—
283#41—624no—
284#41—622no—
285#41—622no—
286#41—622no—
287#41—622no—
288#41—621no—
289#41—621no—
290#41—621no—
291#41—621no—
292#41—621no—
293#41—621no—
294#41C22⋊D3112yescomplemented
295#41C22×D3112yes—
296C2×Q8—311no—
297C2×Q8—311no—
298C2×Q8—622no—
299C2×Q8—622no—
300C2×Q8—622no—
301C2×Q8C22⋊D3111yescharacteristic, complemented
302C2×Q8C22×D3111yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—312no—
306C2×D4—311no—
307C2×D4—311no—
308C2×D4—622no—
309C2×D4—624no—
310C2×D4—624no—
311C2×D4—622no—
312C2×D4—624no—
313C2×D4—621no—
314C2×D4—622no—
315C2×D4—622no—
316C2×D4C22⋊D3111yescharacteristic, complemented
317C2×D4C22×D3111yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—312no—
321C22×C4—624no—
322C22×C4—621no—
323C22×C4—621no—
324C22×C4C22×D3111yescharacteristic
325Q16—22no—
326Q16—622no—
327SD16—22no—
328SD16—22no—
329SD16—622no—
330SD16—622no—
331D8—22no—
332D8—622no—
333M16—21no—
334M16—312no—
335M16—621no—
336M16C22×D3112yes—
337C2×C8—21no—
338C2×C8—621no—
339C4⋊C4—622no—
340C22⋊C4—622no—
341C22⋊C4—622no—
342C42—311no—
343C42—311no—
344C42—621no—
345C23—21no—
346C23—21no—
347C23—312no—
348C23—624no—
349C23—621no—
350C23D4×D3111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—312no—
355Q8—624no—
356Q8—621no—
357Q8—621no—
358Q8—621no—
359Q8C2×C22⋊D3112yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—312no—
368D4—624no—
369D4—624no—
370D4—624no—
371D4—622no—
372D4—622no—
373D4—622no—
374D4—622no—
375D4—621no—
376D4—621no—
377D4—621no—
378D4—1244no—
379D4C2×C22⋊D3112yes—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—312no—
387C2×C4—624no—
388C2×C4—624no—
389C2×C4—624no—
390C2×C4—621no—
391C2×C4—621no—
392C2×C4—622no—
393C2×C4—622no—
394C2×C4—622no—
395C2×C4—622no—
396C2×C4—621no—
397C2×C4—1242no—
398C2×C4D4×D3111yescharacteristic
399C2×C4D4×D3111yescharacteristic
400C2×C4D4×D3111yescharacteristic
401C2×C4C2×C22⋊D3111yescharacteristic
402C2×C4C2×C22⋊D3111yescharacteristic
403C2×C4C23×D3111yescharacteristic
404C8—22no—
405C8—622no—
406C22—22no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—81no—
411C22—622no—
412C22—622no—
413C22—1242no—
414C22—2481no—
415C22—2484no—
416C22#1175851411yescharacteristic
417C22#1175851411yescharacteristic
418C22C2×D4×D3111yescharacteristic
419C4—22no—
420C4—41no—
421C4—622no—
422C4—622no—
423C4—622no—
424C4—1241no—
425C4#1175851411yescharacteristic
426C4#1175851411yescharacteristic
427C4C2×D4×D3111yescharacteristic
428C4C22×C22⋊D3111yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—81no—
434C2—1242no—
435C2—1242no—
436C2C2×#1175851411yescharacteristic
subgroupG/H
centerC4#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#575C2
socleC2×C31C2×#120
hypercenter#575D31
hypocenterC31#3018
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521847C2semidirect
3. #14521595C2semidirect
4. #14521230C2semidirect
5. #14521225C2semidirect
6. #14521220C2semidirect
7. #14521215C2semidirect
8. C2×#11758425C2semidirect
9. C2×#11758413C2semidirect
10. #14521031C2semidirect
11. C31×#575C2semidirect
13. C2×#41×C31C22semidirect
27. C31×#137C22semidirect
29. C31×AGL(1,ℤ/8)C22semidirect
31. C2×M16×C31C22semidirect
35. #11758524C22semidirect
36. #11758521C22semidirect
37. #11758519C22semidirect
38. #11758517C22semidirect
40. #11758513C22semidirect
41. #11758509C22semidirect
43. C2×C4×Dic31C22semidirect
44. C2×#92474C22semidirect
45. #11758425C22semidirect
46. #11758425C22semidirect
47. #11758413C22semidirect
48. #11758413C22semidirect
49. #11758412C22semidirect
50. #11758411C22semidirect
57. #41×C31D4semidirect
59. C2×Q8×C31D4semidirect
63. C2×D4×C31D4semidirect
95. #92495D4semidirect
98. C2×D124D4semidirect
104. C2×Dic62D4semidirect
213. #575D31semidirect
243. C2×#41C2×D31semidirect
247. #137C2×D31semidirect
251. AGL(1,ℤ/8)C2×D31semidirect
255. C2×M16C2×D31semidirect
269. C31#3018semidirect
294. #41C22⋊D31semidirect
301. C2×Q8C22⋊D31semidirect
316. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145218472
#145215952
#145212302
#145212252
#145212202
#145212152
C2×#117584252
C2×#117584132
#145210312
C31×#5752
#301831