Finite Group Databasebeta

← #16855434

Subgroups — #16855434

Order 3968.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521848C211yesmaximal, characteristic, complemented
2C2×#11758491C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D31C211yesmaximal, characteristic, complemented
4#14521814C211yesmaximal, characteristic, complemented
5#14521320C211yesmaximal, characteristic, complemented
6#14521319C211yesmaximal, characteristic, complemented
7#14521318C211yesmaximal, characteristic, complemented
8#14521317C211yesmaximal, characteristic, complemented
9#14521315C211yesmaximal, characteristic, complemented
10#14521314C211yesmaximal, characteristic, complemented
11#14521309C211yesmaximal, characteristic, complemented
12#14521096C211yesmaximal, characteristic, complemented
13#14521032C211yesmaximal, characteristic, complemented
14#100×D31C211yesmaximal, characteristic, complemented
15C31×#471C211yesmaximal, characteristic, complemented
16#41×D31C2211yescharacteristic
17C2×#92499C2211yescharacteristic
18C2×D4×D31C2211yescharacteristic
19C31×#137C2211yescharacteristic, complemented
20C31×AGL(1,ℤ/8)C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic, complemented
22C31×#131C2211yescharacteristic, complemented
23#11758526C2211yescharacteristic
24#11758526C2211yescharacteristic, complemented
25#11758525C2211yescharacteristic, complemented
26#11758519C2211yescharacteristic, complemented
27C2×#92481C2211yescharacteristic, complemented
28#11758509C2211yescharacteristic
29C2×#92479C2211yescharacteristic, complemented
30#11758496—21no—
31#11758495—21no—
32#11758493—21no—
33#11758491—21no—
34#11758491—21no—
35#11758491—21no—
36#11758491—21no—
37#11758491—21no—
38#11758491—21no—
39SD16×D31—21no—
40#11758488—21no—
41D8×D31—21no—
42#11758485C2211yescharacteristic
43#11758485C2211yescharacteristic, complemented
44#11758484—21no—
45#11758484C2211yescharacteristic, complemented
46M16×D31—21no—
47M16×D31C2211yescharacteristic
48M16×D31C2211yescharacteristic
49M16×D31C2211yescharacteristic
50C2×D248C2211yescharacteristic, complemented
51#11758478—21no—
52#11758478C2211yescharacteristic, complemented
53C2×#92469—21no—
54C2×#92469C2211yescharacteristic
55C31×#108C2211yescharacteristic, complemented
56C31×#101C2211yescharacteristic, complemented
57C31×#100C2211yescharacteristic, complemented
58#11758424C2211yescharacteristic, complemented
59#11758421C2211yescharacteristic, complemented
60#11758411C2211yescharacteristic, complemented
61#11758411C2211yescharacteristic, complemented
62#11758411C2211yescharacteristic, complemented
63#11758411C2211yescharacteristic, complemented
64#11758410C2211yescharacteristic, complemented
65#11758410C2211yescharacteristic, complemented
66#11758407C2211yescharacteristic, complemented
67C23×D31—21no—
68#41×C31C2311yescharacteristic
69C2×Q8×C31C2311yescharacteristic
70C2×D4×C31C2311yescharacteristic
71C2×C22⋊D31—21no—
72#92499—21no—
73#92499—21no—
74#92499—21no—
75#92499—21no—
76#92499D411yescharacteristic
77Q8×D31D411yescharacteristic
78#92497—21no—
79#92497D411yescharacteristic
80D4×D31—21no—
81D4×D31—21no—
82D4×D31—21no—
83D4×D31—21no—
84D4×D31—21no—
85D4×D31D411yescharacteristic
86#92495—21no—
87#92495C2311yescharacteristic
88C2×D124—21no—
89C2×D124C2311yescharacteristic
90C2×D124C2311yescharacteristic
91C2×C4×D31—21no—
92C2×C4×D31—21no—
93C2×C4×D31C2311yescharacteristic
94Q16×C31—21no—
95SD16×C31—21no—
96SD16×C31—21no—
97SD16×C31—21no—
98SD16×C31—21no—
99D8×C31—21no—
100M16×C31—21no—
101M16×C31C2311yescharacteristic
102M16×C31C2311yescharacteristic
103M16×C31C2311yescharacteristic
104C2×C8×C31—21no—
105C2×C8×C31C2311yescharacteristic
106#92482—21no—
107#92481—21no—
108#92481—21no—
109#92481—21no—
110#92481—21no—
111#92481—21no—
112#92480—21no—
113#92479—21no—
114#92479—21no—
115#92479—21no—
116#92479—21no—
117#92479—21no—
118#92474—21no—
119#92474C2311yescharacteristic
120#92474C2311yescharacteristic
121#92474C2311yescharacteristic
122C2×C31⋊C8—21no—
123C2×C31⋊C8C2311yescharacteristic
124D248—21no—
125D248—21no—
126D248—21no—
127D248—21no—
128#92470—21no—
129#92470—21no—
130#92469—21no—
131#92469—21no—
132#92469—21no—
133#92469—21no—
134#92469—21no—
135#92469D411yescharacteristic
136#92469D411yescharacteristic
137#92469D411yescharacteristic
138#92469D411yescharacteristic
139C8×D31—21no—
140C8×D31—21no—
141C8×D31—21no—
142C8×D31—21no—
143C8×D31—21no—
144C23×C31—21no—
145C22×D31—21no—
146C22×D31—21no—
147C22×D31—21no—
148C22×D31—41no—
149C22×D31—41no—
150C22×D31—82no—
151C22×D31#4111yescharacteristic
152Q8×C31—21no—
153Q8×C31—21no—
154Q8×C31C2×D411yescharacteristic
155D4×C31—21no—
156D4×C31—21no—
157D4×C31—21no—
158D4×C31C2×D411yescharacteristic
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31C2411yescharacteristic
162C22⋊D31—21no—
163C22⋊D31—21no—
164C22⋊D31—21no—
165C22⋊D31—41no—
166C22⋊D31—41no—
167C2×Dic31—21no—
168C2×Dic31#4111yescharacteristic
169D124—21no—
170D124—21no—
171D124—21no—
172D124—21no—
173D124—21no—
174D124—41no—
175D124—41no—
176D124C2×D411yescharacteristic
177C4×D31—21no—
178C4×D31—21no—
179C4×D31—21no—
180C4×D31—41no—
181C4×D31—41no—
182C4×D31C2×D411yescharacteristic
183C4×D31C2×D411yescharacteristic
184C4×D31C2×D411yescharacteristic
185C4×D31C2×D411yescharacteristic
186Dic62—21no—
187Dic62C2×D411yescharacteristic
188C8×C31—21no—
189C8×C31—21no—
190C8×C31—21no—
191C8×C31C2×D411yescharacteristic
192C8×C31C2×D411yescharacteristic
193C31⋊C8—21no—
194C31⋊C8—21no—
195C31⋊C8—21no—
196C31⋊C8C2×D411yescharacteristic
197C31⋊C8C2×D411yescharacteristic
198#3068—311nomaximal, Sylow
199C22×C31—21no—
200C22×C31—41no—
201C22×C31—81no—
202C22×C31C2×#4111yescharacteristic
203C2×D31—21no—
204C2×D31—41no—
205C2×D31—41no—
206C2×D31—41no—
207C2×D31—81no—
208C2×D31—82no—
209C2×D31—82no—
210C2×D31—81no—
211C2×D31—81no—
212C2×D31#12111yescharacteristic
213C2×D31#12111yescharacteristic
214C4×C31—21no—
215C4×C31—41no—
216C4×C31C22×D411yescharacteristic
217C4×C31C22×D411yescharacteristic
218Dic31—21no—
219Dic31#12111yescharacteristic
220Dic31#12111yescharacteristic
221#575—311no—
222C2×AGL(1,ℤ/8)—311no—
223C2×AGL(1,ℤ/8)—311no—
224#568—311no—
225#472—311no—
226#472—311no—
227#472—311no—
228#472—311no—
229#471—311no—
230#471—311no—
231#471—311no—
232#471D3111yescharacteristic, complemented
233#430—311no—
234#413—311no—
235C2×#100—311no—
236C2×C31—21no—
237C2×C31—41no—
238C2×C31—81no—
239C2×C31C2×#12111yescharacteristic
240D31—21no—
241D31—21no—
242D31—41no—
243D31—81no—
244D31—81no—
245C2×#41—311no—
246C2×#41—311no—
247C22×D4—311no—
248#137—311no—
249#137—621no—
250#137C2×D3111yescharacteristic, complemented
251AGL(1,ℤ/8)—311no—
252AGL(1,ℤ/8)—311no—
253AGL(1,ℤ/8)—311no—
254AGL(1,ℤ/8)—311no—
255AGL(1,ℤ/8)—311no—
256AGL(1,ℤ/8)—621no—
257AGL(1,ℤ/8)—621no—
258AGL(1,ℤ/8)—621no—
259AGL(1,ℤ/8)—621no—
260AGL(1,ℤ/8)—621no—
261AGL(1,ℤ/8)—621no—
262AGL(1,ℤ/8)—621no—
263AGL(1,ℤ/8)C2×D3111yescharacteristic
264#135—621no—
265#135—621no—
266C2×SD16—311no—
267C2×SD16—621no—
268C2×SD16C2×D3111yescharacteristic, complemented
269C2×D8—311no—
270C2×D8—311no—
271C2×D8—621no—
272#131—311no—
273#131—311no—
274#131—311no—
275#131—621no—
276#131—621no—
277#131C2×D3111yescharacteristic, complemented
278C2×M16—311no—
279C2×M16—311no—
280C2×M16—311no—
281C2×M16—311no—
282C2×M16—621no—
283C2×M16—621no—
284#108—311no—
285#108—311no—
286#108—311no—
287#108C2×D3111yescharacteristic, complemented
288#101—311no—
289#101C2×D3111yescharacteristic, complemented
290#100—311no—
291#100—311no—
292#100—311no—
293#100—311no—
294#100—311no—
295#100C2×D3111yescharacteristic, complemented
296C31#306811yescharacteristic, complemented, Sylow
297C24—621no—
298#41—311no—
299#41—311no—
300#41—311no—
301#41—621no—
302#41—621no—
303#41—621no—
304#41—621no—
305#41—621no—
306#41—621no—
307#41C22×D3111yescharacteristic
308C2×Q8—311no—
309C2×Q8C22×D3111yescharacteristic
310C2×D4—311no—
311C2×D4—311no—
312C2×D4—311no—
313C2×D4—621no—
314C2×D4—621no—
315C2×D4—621no—
316C2×D4—621no—
317C2×D4—621no—
318C2×D4—621no—
319C2×D4—621no—
320C2×D4C22×D3111yescharacteristic
321C22×C4—311no—
322C22×C4—621no—
323C22×C4—621no—
324Q16—21no—
325Q16—621no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—621no—
331SD16—621no—
332SD16—621no—
333SD16—621no—
334SD16—621no—
335SD16—621no—
336SD16—621no—
337SD16—621no—
338D8—21no—
339D8—621no—
340D8—621no—
341D8—621no—
342D8—621no—
343D8—621no—
344D8—621no—
345D8—621no—
346D8—621no—
347D8—621no—
348M16—21no—
349M16—311no—
350M16—311no—
351M16—311no—
352M16—311no—
353M16—311no—
354M16—311no—
355M16—311no—
356M16—621no—
357M16—621no—
358M16—621no—
359M16—621no—
360M16—621no—
361M16—621no—
362M16C22×D3111yescharacteristic
363M16C22×D3111yescharacteristic
364M16C22×D3111yescharacteristic
365C2×C8—21no—
366C2×C8—311no—
367C2×C8—621no—
368C2×C8—621no—
369C2×C8—621no—
370C2×C8—621no—
371C2×C8—621no—
372C2×C8—621no—
373C2×C8C22×D3111yescharacteristic
374C23—21no—
375C23—311no—
376C23—621no—
377C23—621no—
378C23—621no—
379C23—1241no—
380C23—1241no—
381C23—2482no—
382Q8—21no—
383Q8—21no—
384Q8—311no—
385Q8—621no—
386Q8D4×D3111yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—311no—
391D4—621no—
392D4—621no—
393D4—621no—
394D4—621no—
395D4—621no—
396D4—621no—
397D4—621no—
398D4—621no—
399D4—1241no—
400D4—1241no—
401D4—1241no—
402D4—1241no—
403D4D4×D3111yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—311no—
407C2×C4—311no—
408C2×C4—311no—
409C2×C4—311no—
410C2×C4—311no—
411C2×C4—621no—
412C2×C4—621no—
413C2×C4—621no—
414C2×C4—621no—
415C2×C4—1241no—
416C2×C4—1241no—
417C2×C4C23×D3111yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—311no—
422C8—311no—
423C8—621no—
424C8—621no—
425C8—621no—
426C8D4×D3111yescharacteristic
427C8D4×D3111yescharacteristic
428C22—21no—
429C22—41no—
430C22—81no—
431C22—311no—
432C22—311no—
433C22—621no—
434C22—1241no—
435C22—1241no—
436C22—1241no—
437C22—2481no—
438C22—2481no—
439C22—2481no—
440C22—2482no—
441C22—2482no—
442C22#41×D3111yescharacteristic
443C4—21no—
444C4—41no—
445C4—311no—
446C4—311no—
447C4—621no—
448C4C2×D4×D3111yescharacteristic
449C4C2×D4×D3111yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—621no—
454C2—621no—
455C2—1241no—
456C2—2481no—
457C2—2481no—
458C2#121×D3111yescharacteristic
subgroupG/H
centerC2#121×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#471C2
socleC2×C31C2×#121
hypercenter#471D31
hypocenterC31#3068
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #14521848C2semidirect
2. C2×#11758491C2semidirect
3. AGL(1,ℤ/8)×D31C2semidirect
4. #14521814C2semidirect
5. #14521320C2semidirect
6. #14521319C2semidirect
7. #14521318C2semidirect
8. #14521317C2semidirect
9. #14521315C2semidirect
10. #14521314C2semidirect
11. #14521309C2semidirect
12. #14521096C2semidirect
13. #14521032C2semidirect
14. #100×D31C2semidirect
15. C31×#471C2semidirect
19. C31×#137C22semidirect
21. C2×SD16×C31C22semidirect
22. C31×#131C22semidirect
24. #11758526C22semidirect
25. #11758525C22semidirect
26. #11758519C22semidirect
27. C2×#92481C22semidirect
29. C2×#92479C22semidirect
43. #11758485C22semidirect
45. #11758484C22semidirect
50. C2×D248C22semidirect
52. #11758478C22semidirect
55. C31×#108C22semidirect
56. C31×#101C22semidirect
57. C31×#100C22semidirect
58. #11758424C22semidirect
59. #11758421C22semidirect
60. #11758411C22semidirect
61. #11758411C22semidirect
62. #11758411C22semidirect
63. #11758411C22semidirect
64. #11758410C22semidirect
65. #11758410C22semidirect
66. #11758407C22semidirect
232. #471D31semidirect
250. #137C2×D31semidirect
268. C2×SD16C2×D31semidirect
277. #131C2×D31semidirect
287. #108C2×D31semidirect
289. #101C2×D31semidirect
295. #100C2×D31semidirect
296. C31#3068semidirect

Maximal subgroups

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

subgroupindex
#145218482
C2×#117584912
AGL(1,ℤ/8)×D312
#145218142
#145213202
#145213192
#145213182
#145213172
#145213152
#145213142
#145213092
#145210962
#145210322
#100×D312
C31×#4712
#306831