Finite Group Databasebeta

← #16857247

Subgroups — #16857247

Order 3968.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758509C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3#14521720C211yesmaximal, characteristic, complemented
4#14521439C211yesmaximal, characteristic, complemented
5#14521297C211yesmaximal, characteristic, complemented
6#14521283C211yesmaximal, characteristic, complemented
7#14521268C211yesmaximal, characteristic, complemented
8#14521244C211yesmaximal, characteristic, complemented
9#14521209C211yesmaximal, characteristic, complemented
10#14521194C211yesmaximal, characteristic, complemented
11#14521192C211yesmaximal, characteristic, complemented
12#14521183C211yesmaximal, characteristic, complemented
13#14521149C211yesmaximal, characteristic, complemented
14#14521121C211yesmaximal, characteristic, complemented
15C31×#546C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×#92495C2211yescharacteristic
19C2×#92495C2211yescharacteristic, complemented
20C31×#124C2211yescharacteristic
21C31×#123—21no—
22C31×C22⋊Q8C2211yescharacteristic
23C31×#121—21no—
24C31×#121C2211yescharacteristic
25C31×#120—21no—
26C4×D4×C31C2211yescharacteristic
27C4×D4×C31C2211yescharacteristic
28C2×C22⋊C4×C31—21no—
29#11758527—24no—
30C2×#92482C2211yescharacteristic, complemented
31C2×#92481C2211yescharacteristic, complemented
32C2×#92481C2211yescharacteristic, complemented
33#11758517—21no—
34C2×#92480C2211yescharacteristic
35C2×#92480C2211yescharacteristic
36#11758509—24no—
37C2×#92479C2211yescharacteristic
38C2×#92479C2211yescharacteristic
39C2×#92479C2211yescharacteristic
40C2×#92479C2211yescharacteristic
41#11758507C2211yescharacteristic
42#11758507C2211yescharacteristic, complemented
43C2×#92474C2211yescharacteristic
44C22×C31⋊C8C2211yescharacteristic, complemented
45#11758470—21no—
46#11758461—21no—
47#11758461—21no—
48#11758451C2211yescharacteristic, complemented
49C4×Dic62C2211yescharacteristic, complemented
50#11758423C2211yescharacteristic, complemented
51#11758420C2211yescharacteristic
52#11758420C2211yescharacteristic
53C22⋊(C31⋊C8)C2211yescharacteristic
54C22⋊(C31⋊C8)C2211yescharacteristic, complemented
55#11758398C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic, complemented
57#11758397C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59C4⋊(C31⋊C8)C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C24×C31—21no—
62#41×C31—22no—
63#41×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31—22no—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—41no—
70C2×D4×C31D411yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C2×D4×C31C2311yescharacteristic
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31C2311yescharacteristic
76C22×C4×C31C2311yescharacteristic
77C2×C22⋊D31—21no—
78C2×C22⋊D31—21no—
79C2×C22⋊D31—21no—
80C2×C22⋊D31—21no—
81#92495—22no—
82#92495—22no—
83#92495—41no—
84#92495—41no—
85C2×D124—21no—
86C2×D124—21no—
87C2×D124C2311yescharacteristic
88C2×D124C2311yescharacteristic
89C2×C4×D31—21no—
90C2×C4×D31—21no—
91C2×Dic62C2311yescharacteristic
92C4⋊C4×C31—21no—
93C4⋊C4×C31D411yescharacteristic
94C4⋊C4×C31C2311yescharacteristic
95C22⋊C4×C31—21no—
96C22⋊C4×C31—21no—
97C22⋊C4×C31—21no—
98C22⋊C4×C31—21no—
99C22⋊C4×C31—21no—
100C22⋊C4×C31D411yescharacteristic
101C22⋊C4×C31D411yescharacteristic
102C42×C31C2311yescharacteristic
103#92482—22no—
104#92481—22no—
105#92481—41no—
106#92480—22no—
107#92480D414yescomplemented
108#92479—22no—
109#92479—22no—
110#92479—22no—
111#92479—41no—
112(C2×D31)⋊C4—21no—
113(C2×D31)⋊C4—21no—
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116C4×Dic31—21no—
117#92474—22no—
118C2×C31⋊C8—22no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121C2×C31⋊C8C2311yescharacteristic
122C2×C31⋊C8C2311yescharacteristic
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—81no—
128C23×C31C2×D411yescharacteristic
129C23×C31C2×D411yescharacteristic
130C22×D31—21no—
131C22×D31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134D4×C31—21no—
135D4×C31—21no—
136D4×C31—21no—
137D4×C31—41no—
138D4×C31—42no—
139D4×C31—42no—
140D4×C31—81no—
141D4×C31C2×D412yes—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—42no—
149C2×C4×C31—81no—
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2×D411yescharacteristic
152C2×C4×C31C2×D411yescharacteristic
153C2×C4×C31C2×D411yescharacteristic
154C2×C4×C31C2411yescharacteristic
155C22⋊D31—42no—
156C22⋊D31—42no—
157C22⋊D31—81no—
158C22⋊D31—81no—
159C2×Dic31—21no—
160C2×Dic31—21no—
161D124—21no—
162D124—21no—
163D124—21no—
164D124—21no—
165D124—42no—
166D124—81no—
167C4×D31—42no—
168C4×D31—42no—
169Dic62—21no—
170Dic62C2×D412yes—
171C31⋊C8—21no—
172C31⋊C8—21no—
173C31⋊C8—21no—
174C31⋊C8C2×D412yes—
175#3268—311nomaximal, Sylow
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—22no—
179C22×C31—21no—
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31—84no—
189C22×C31C22×D411yescharacteristic
190C2×D31—41no—
191C2×D31—41no—
192C2×D31—41no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—41no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—41no—
201C4×C31—41no—
202C4×C31C22×D411yescharacteristic
203C4×C31H(5,𝔽2)11yescharacteristic
204Dic31—22no—
205Dic31—41no—
206C2×AGL(1,ℤ/8)—311no—
207C2×#135—311no—
208#550—311no—
209#546D3111yescharacteristic, complemented
210#496—311no—
211#468—311no—
212#466—311no—
213#464—311no—
214#459—311no—
215#452—311no—
216#449—311no—
217#449—311no—
218#447—311no—
219#440—311no—
220#435—311no—
221C2×C31—41no—
222C2×C31—41no—
223C2×C31—42no—
224C2×C31—81no—
225C2×C31D4211yescharacteristic
226C2×C31#57511yescharacteristic
227C2×C31#57611yescharacteristic
228D31—81no—
229D31—81no—
230C2×#41—311no—
231C2×#41—311no—
232C2×#41C2×D3111yescharacteristic
233C22×D4C2×D3111yescharacteristic
234AGL(1,ℤ/8)—624no—
235#135—624no—
236C2×Q16—311no—
237C2×SD16—311no—
238C2×SD16—311no—
239C2×SD16—311no—
240C2×SD16—311no—
241C2×D8—311no—
242C2×D8—311no—
243C2×D8—311no—
244C2×D8—311no—
245C2×M16—311no—
246C22×C8—311no—
247#127—311no—
248#127—621no—
249#124C2×D3111yescharacteristic
250#123—21no—
251C22⋊Q8C2×D3111yescharacteristic
252#121—21no—
253#121—311no—
254#121—311no—
255#121—621no—
256#121—621no—
257#121C2×D3111yescharacteristic
258#120—21no—
259C4×Q8—311no—
260C4×D4—621no—
261C4×D4C2×D3111yescharacteristic
262C4×D4C2×D3111yescharacteristic
263C2×C22⋊C4—21no—
264#107—311no—
265C4⋊C8—311no—
266#103—311no—
267#103—311no—
268#102—311no—
269#102—311no—
270#102—311no—
271#102—311no—
272C22⋊C8—311no—
273C22⋊C8—311no—
274#97—311no—
275C31#326811yescharacteristic, complemented, Sylow
276C24—21no—
277#41—22no—
278#41—41no—
279#41—622no—
280#41—622no—
281#41—1241no—
282#41—1241no—
283C2×Q8—311no—
284C2×Q8C22×D3111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—311no—
291C2×D4—311no—
292C2×D4—621no—
293C2×D4—621no—
294C2×D4—621no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4C22⋊D3111yescharacteristic
299C2×D4C22×D3111yescharacteristic
300C2×D4C22×D3111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—621no—
304C22×C4—621no—
305C22×C4C22×D3111yescharacteristic
306C22×C4C22×D3111yescharacteristic
307Q16—622no—
308SD16—314no—
309SD16—622no—
310SD16—622no—
311SD16—1241no—
312D8—622no—
313D8—622no—
314D8—622no—
315D8—1241no—
316M16—622no—
317C2×C8—311no—
318C2×C8—311no—
319C2×C8—311no—
320C2×C8—311no—
321C2×C8—622no—
322C4⋊C4—21no—
323C4⋊C4—311no—
324C4⋊C4—621no—
325C4⋊C4C22⋊D3111yescharacteristic
326C4⋊C4C22×D3111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—621no—
333C22⋊C4—621no—
334C22⋊C4C22⋊D3111yescharacteristic
335C22⋊C4C22⋊D3111yescharacteristic
336C42—621no—
337C42C22×D3111yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—621no—
344C23—621no—
345C23C2×C22⋊D3111yescharacteristic
346C23C2×C22⋊D3111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—312no—
350Q8—621no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—621no—
359D4—621no—
360D4—621no—
361D4—621no—
362D4—1242no—
363D4—1242no—
364D4—1242no—
365D4—2481no—
366D4—2481no—
367D4—2481no—
368D4D4×D3112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—621no—
378C2×C4—621no—
379C2×C4—1242no—
380C2×C4—1242no—
381C2×C4C2×C22⋊D3111yescharacteristic
382C2×C4C2×C22⋊D3111yescharacteristic
383C2×C4C2×C22⋊D3111yescharacteristic
384C2×C4C2×C22⋊D3111yescharacteristic
385C2×C4C23×D3111yescharacteristic
386C8—312no—
387C8—621no—
388C8—621no—
389C8—621no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—1241no—
404C22—1241no—
405C22—1241no—
406C22—1241no—
407C22—1241no—
408C22—1241no—
409C22C22×C22⋊D3111yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—622no—
417C4—1241no—
418C4C2×D4×D3111yescharacteristic
419C4#1175856111yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—2481no—
425C2—2481no—
426C2D4×C22⋊D3111yescharacteristic
427C2#1452184711yescharacteristic
428C2#1452185211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#546C2
socleC22×C31C22×D4
hypercenter#546D31
hypocenterC31#3268
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#11758509C2semidirect
2. C2×#11758527C2semidirect
3. #14521720C2semidirect
4. #14521439C2semidirect
5. #14521297C2semidirect
6. #14521283C2semidirect
7. #14521268C2semidirect
8. #14521244C2semidirect
9. #14521209C2semidirect
10. #14521194C2semidirect
11. #14521192C2semidirect
12. #14521183C2semidirect
13. #14521149C2semidirect
14. #14521121C2semidirect
15. C31×#546C2semidirect
19. C2×#92495C22semidirect
30. C2×#92482C22semidirect
31. C2×#92481C22semidirect
32. C2×#92481C22semidirect
42. #11758507C22semidirect
44. C22×C31⋊C8C22semidirect
48. #11758451C22semidirect
49. C4×Dic62C22semidirect
50. #11758423C22semidirect
54. C22⋊(C31⋊C8)C22semidirect
55. #11758398C22semidirect
56. #11758397C22semidirect
57. #11758397C22semidirect
58. #11758395C22semidirect
59. C4⋊(C31⋊C8)C22semidirect
60. #11758391C22semidirect
107. #92480D4semidirect
209. #546D31semidirect
275. C31#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#117585092
C2×#117585272
#145217202
#145214392
#145212972
#145212832
#145212682
#145212442
#145212092
#145211942
#145211922
#145211832
#145211492
#145211212
C31×#5462
#326831