Finite Group Databasebeta

← #16857230

Subgroups — #16857230

Order 3968.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758489C211yesmaximal, characteristic, complemented
2#14521698C211yesmaximal, characteristic, complemented
3#14521693C211yesmaximal, characteristic, complemented
4#14521440C211yesmaximal, characteristic, complemented
5#14521307C211yesmaximal, characteristic, complemented
6#14521275C211yesmaximal, characteristic, complemented
7#14521262C211yesmaximal, characteristic, complemented
8#14521252C211yesmaximal, characteristic, complemented
9#14521211C211yesmaximal, characteristic, complemented
10#14521208C211yesmaximal, characteristic, complemented
11#14521200C211yesmaximal, characteristic, complemented
12#14521191C211yesmaximal, characteristic, complemented
13#14521161C211yesmaximal, characteristic, complemented
14#14521118C211yesmaximal, characteristic, complemented
15C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×#92497C2211yescharacteristic
19C2×#92497C2211yescharacteristic, complemented
20C31×AGL(1,ℤ/8)—24no—
21C2×SD16×C31C2211yescharacteristic
22C2×SD16×C31C2211yescharacteristic
23C2×D8×C31C2211yescharacteristic
24C2×D8×C31C2211yescharacteristic
25C2×M16×C31C2211yescharacteristic
26#11758530—21no—
27C2×C22⋊Dic31—21no—
28#11758523C2211yescharacteristic, complemented
29#11758521C2211yescharacteristic, complemented
30C2×#92482C2211yescharacteristic, complemented
31#11758516—21no—
32#11758516—21no—
33#11758515C2211yescharacteristic
34#11758513C2211yescharacteristic
35#11758512—21no—
36#11758512—21no—
37D4×Dic31C2211yescharacteristic
38D4×Dic31C2211yescharacteristic, complemented
39C2×#92480C2211yescharacteristic
40C2×#92480C2211yescharacteristic
41C2×#92480C2211yescharacteristic, complemented
42#11758506—21no—
43C4×C22⋊D31C2211yescharacteristic
44#11758501C2211yescharacteristic
45#11758501C2211yescharacteristic
46C2×Dic31⋊C4—21no—
47#11758489—24no—
48C2×Dic124C2211yescharacteristic, complemented
49C2×C8×D31C2211yescharacteristic, complemented
50#11758423C2211yescharacteristic, complemented
51#11758420C2211yescharacteristic
52#11758420C2211yescharacteristic
53#11758420C2211yescharacteristic, complemented
54C22⋊(C31⋊C8)C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic, complemented
56#11758405C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59#11758403C2211yescharacteristic, complemented
60Dic31⋊C8C2211yescharacteristic, 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—41no—
69C2×D4×C31C2311yescharacteristic
70C2×D4×C31C2311yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31—21no—
77C2×C22⋊D31D411yescharacteristic
78C22×Dic31—21no—
79C22×Dic31—21no—
80C22×Dic31—21no—
81C22×Dic31—21no—
82#92497—22no—
83#92497—22no—
84#92497—41no—
85#92497—41no—
86C2×C4×D31C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88C2×Dic62C2311yescharacteristic
89SD16×C31—22no—
90SD16×C31—22no—
91D8×C31—22no—
92D8×C31D414yescomplemented
93M16×C31—22no—
94C2×C8×C31C2311yescharacteristic
95C2×C8×C31C2311yescharacteristic
96C22⋊Dic31—21no—
97C22⋊Dic31—21no—
98C22⋊Dic31—21no—
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103C22⋊Dic31D411yescharacteristic
104#92482—41no—
105#92480—22no—
106#92480—22no—
107#92480—41no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4D411yescharacteristic
110C4⋊Dic31C2311yescharacteristic
111C4⋊Dic31C2311yescharacteristic
112Dic31⋊C4—21no—
113Dic31⋊C4—21no—
114Dic31⋊C4—21no—
115Dic31⋊C4—21no—
116Dic31⋊C4—21no—
117Dic31⋊C4D411yescharacteristic
118C4×Dic31C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121Dic124—22no—
122C8×D31—22no—
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—81no—
129C23×C31C2×D411yescharacteristic
130C22×D31C2×D411yescharacteristic
131Q8×C31—21no—
132Q8×C31—21no—
133D4×C31—21no—
134D4×C31—21no—
135D4×C31—21no—
136D4×C31—21no—
137D4×C31—42no—
138D4×C31—42no—
139D4×C31C2×D412yes—
140D4×C31C2×D412yes—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—42no—
145C2×C4×C31C2×D411yescharacteristic
146C2×C4×C31C2411yescharacteristic
147C22⋊D31—41no—
148C22⋊D31—42no—
149C22⋊D31—42no—
150C22⋊D31—81no—
151C2×Dic31—21no—
152C2×Dic31—21no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31—42no—
156C2×Dic31—42no—
157C2×Dic31—42no—
158C2×Dic31—81no—
159C2×Dic31C2×D411yescharacteristic
160C2×Dic31C2×D411yescharacteristic
161C2×Dic31C2×D411yescharacteristic
162C4×D31—21no—
163C4×D31—21no—
164Dic62—21no—
165Dic62—21no—
166Dic62—21no—
167Dic62—21no—
168C8×C31—21no—
169C8×C31C2×D412yes—
170C31⋊C8—21no—
171C31⋊C8—21no—
172#3267—311nomaximal, Sylow
173C22×C31—22no—
174C22×C31—22no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—84no—
186C22×C31C22×D411yescharacteristic
187C2×D31—21no—
188C2×D31—21no—
189C2×D31—21no—
190C4×C31—21no—
191C4×C31—41no—
192C4×C31C22×D411yescharacteristic
193C4×C31H(5,𝔽2)11yescharacteristic
194Dic31—21no—
195Dic31—21no—
196Dic31—21no—
197Dic31—41no—
198Dic31—41no—
199Dic31—41no—
200Dic31—41no—
201C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
202C2×#135—311no—
203#547—311no—
204#546—311no—
205#497—311no—
206#470—311no—
207#465—311no—
208#463—311no—
209#461—311no—
210#452—311no—
211#452—311no—
212#450—311no—
213#449—311no—
214#442—311no—
215#435—311no—
216C2×C31—41no—
217C2×C31—42no—
218C2×C31—42no—
219C2×C31—81no—
220C2×C31D4211yescharacteristic
221C2×C31#57511yescharacteristic
222C2×C31#57711yescharacteristic
223D31—41no—
224C2×#41—311no—
225C2×#41—311no—
226C2×#41C2×D3111yescharacteristic
227C22×D4C2×D3111yescharacteristic
228AGL(1,ℤ/8)—24no—
229#135—624no—
230C2×Q16—311no—
231C2×Q16—311no—
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—311no—
235C2×SD16C2×D3111yescharacteristic
236C2×SD16C2×D3111yescharacteristic
237C2×D8C2×D3111yescharacteristic
238C2×D8C2×D3111yescharacteristic
239C2×M16C2×D3111yescharacteristic
240C22×C8—311no—
241C4⋊Q8—311no—
242#124—311no—
243#123—621no—
244#123—621no—
245#123—621no—
246C22⋊Q8—311no—
247C22⋊Q8—311no—
248C22⋊Q8—311no—
249#121—311no—
250#121—621no—
251#121—621no—
252#120—621no—
253C4×D4—311no—
254C4×D4—311no—
255C4×D4—311no—
256C2×C4⋊C4—621no—
257C2×C22⋊C4—621no—
258#106—311no—
259C4⋊C8—311no—
260#103—311no—
261#103—311no—
262#103—311no—
263#102—311no—
264#102—311no—
265#102—311no—
266C22⋊C8—311no—
267C22⋊C8—311no—
268#97—311no—
269C31#326711yescharacteristic, complemented, Sylow
270C24—21no—
271#41—22no—
272#41—41no—
273#41—622no—
274#41—622no—
275#41—1241no—
276#41—1241no—
277C2×Q8—311no—
278C2×Q8—311no—
279C2×Q8C22×D3111yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—311no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4C22×D3111yescharacteristic
289C2×D4C22×D3111yescharacteristic
290C2×D4C22×D3111yescharacteristic
291C22×C4—21no—
292C22×C4—311no—
293C22×C4—621no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4C22×D3111yescharacteristic
298Q16—622no—
299Q16—1241no—
300SD16—22no—
301SD16—22no—
302SD16—622no—
303SD16—622no—
304SD16—1241no—
305D8—22no—
306D8C22⋊D3114yescomplemented
307M16—22no—
308C2×C8—311no—
309C2×C8—311no—
310C2×C8—622no—
311C2×C8C22×D3111yescharacteristic
312C2×C8C22×D3111yescharacteristic
313C4⋊C4—311no—
314C4⋊C4—311no—
315C4⋊C4—311no—
316C4⋊C4—621no—
317C4⋊C4—621no—
318C4⋊C4—621no—
319C4⋊C4—621no—
320C4⋊C4—621no—
321C22⋊C4—311no—
322C22⋊C4—311no—
323C22⋊C4—621no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C22⋊C4—621no—
330C22⋊C4—621no—
331C42—311no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—311no—
339C23D4×D3111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—621no—
343Q8—621no—
344Q8—621no—
345Q8—621no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—1241no—
353D4—1242no—
354D4—1242no—
355D4—2481no—
356D4C2×C22⋊D3112yes—
357D4C2×C22⋊D3112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—311no—
363C2×C4—311no—
364C2×C4—311no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—621no—
368C2×C4—621no—
369C2×C4—621no—
370C2×C4—621no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—1242no—
374C2×C4—2481no—
375C2×C4D4×D3111yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—21no—
378C8—621no—
379C8—621no—
380C8C2×C22⋊D3112yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—621no—
395C22—621no—
396C22—621no—
397C22C2×D4×D3111yescharacteristic
398C4—21no—
399C4—41no—
400C4—621no—
401C4—621no—
402C4—621no—
403C4—1241no—
404C4—1241no—
405C4—1241no—
406C4—1241no—
407C4#1175856111yescharacteristic
408C4C22×C22⋊D3111yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—1241no—
414C2D4×C22⋊D3111yescharacteristic
415C2#1452184511yescharacteristic
416C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3267
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11758489C2semidirect
2. #14521698C2semidirect
3. #14521693C2semidirect
4. #14521440C2semidirect
5. #14521307C2semidirect
6. #14521275C2semidirect
7. #14521262C2semidirect
8. #14521252C2semidirect
9. #14521211C2semidirect
10. #14521208C2semidirect
11. #14521200C2semidirect
12. #14521191C2semidirect
13. #14521161C2semidirect
14. #14521118C2semidirect
15. C2×C31×AGL(1,ℤ/8)C2semidirect
19. C2×#92497C22semidirect
28. #11758523C22semidirect
29. #11758521C22semidirect
30. C2×#92482C22semidirect
38. D4×Dic31C22semidirect
41. C2×#92480C22semidirect
48. C2×Dic124C22semidirect
49. C2×C8×D31C22semidirect
50. #11758423C22semidirect
53. #11758420C22semidirect
55. (C2×D31)⋊C8C22semidirect
56. #11758405C22semidirect
57. #11758404C22semidirect
58. #11758404C22semidirect
59. #11758403C22semidirect
60. Dic31⋊C8C22semidirect
92. D8×C31D4semidirect
201. C2×AGL(1,ℤ/8)D31semidirect
269. C31#3267semidirect
306. D8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584892
#145216982
#145216932
#145214402
#145213072
#145212752
#145212622
#145212522
#145212112
#145212082
#145212002
#145211912
#145211612
#145211182
C2×C31×AGL(1,ℤ/8)2
#326731