Finite Group Databasebeta

← #21457433

Subgroups — #21457433

Order 7808.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14519804C211yesmaximal, characteristic, complemented
2#16840431C211yesmaximal, characteristic, complemented
3D4×C22⋊D61C211yesmaximal, characteristic, complemented
4#16840137C211yesmaximal, characteristic, complemented
5#16840015C211yesmaximal, characteristic, complemented
6#16840012C211yesmaximal, characteristic, complemented
7#16839960C211yesmaximal, characteristic, complemented
8#16839955C211yesmaximal, characteristic, complemented
9#16839907C211yesmaximal, characteristic, complemented
10#16839906C211yesmaximal, characteristic, complemented
11#16839896C211yesmaximal, characteristic, complemented
12#16839894C211yesmaximal, characteristic, complemented
13SD16×Dic61C211yesmaximal, characteristic, complemented
14#16839838C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic, complemented
16C2×#11758226C2211yescharacteristic, complemented
17C22×C22⋊D61—21no—
18C2×D4×D61C2211yescharacteristic
19#14519804—24no—
20C2×#11758201C2211yescharacteristic, complemented
21C2×D488C2211yescharacteristic, complemented
22C2×#11758199C2211yescharacteristic
23C2×#11758199C2211yescharacteristic, complemented
24C2×#11758199C2211yescharacteristic, complemented
25C2×#11758189C2211yescharacteristic, complemented
26#14519773—21no—
27#14519773C2211yescharacteristic
28#14519773C2211yescharacteristic, complemented
29#14519751—21no—
30#14519751C2211yescharacteristic
31#14519751C2211yescharacteristic, complemented
32#14519749—21no—
33#14519748—21no—
34#14519748C2211yescharacteristic
35#14519748C2211yescharacteristic, complemented
36#14519746—21no—
37#14519745—21no—
38Q8×Dic61C2211yescharacteristic, complemented
39D4×Dic61C2211yescharacteristic
40C4×C22⋊D61—21no—
41C4×C22⋊D61C2211yescharacteristic
42#14519709C2211yescharacteristic, complemented
43#14519706C2211yescharacteristic, complemented
44#14519701C2211yescharacteristic, complemented
45#14519699C2211yescharacteristic, complemented
46#14519699C2211yescharacteristic, complemented
47#14519698C2211yescharacteristic
48#14519698C2211yescharacteristic, complemented
49#14519698C2211yescharacteristic, complemented
50C22⋊#92418C2211yescharacteristic, complemented
51#14519689C2211yescharacteristic, complemented
52C8×Dic61C2211yescharacteristic, complemented
53C2×#41×C61C2211yescharacteristic, complemented
54C22×D4×C61C2211yescharacteristic
55AGL(1,ℤ/8)×C61—24no—
56C2×SD16×C61C2211yescharacteristic
57C2×SD16×C61C2211yescharacteristic, complemented
58C2×D8×C61C2211yescharacteristic
59C2×D8×C61C2211yescharacteristic, complemented
60C2×M16×C61C2211yescharacteristic, complemented
61C24×C61—21no—
62C23×D61—21no—
63#41×C61—22no—
64#41×C61—41no—
65C2×Q8×C61C2311yescharacteristic
66C2×D4×C61—22no—
67C2×D4×C61—21no—
68C2×D4×C61—21no—
69C2×D4×C61—41no—
70C2×D4×C61C2311yescharacteristic
71C2×D4×C61C2311yescharacteristic
72C2×D4×C61C2311yescharacteristic
73C22×C4×C61—21no—
74C22×C4×C61C2311yescharacteristic
75C2×C22⋊D61—21no—
76C2×C22⋊D61—21no—
77C2×C22⋊D61—21no—
78C2×C22⋊D61—21no—
79C2×C22⋊D61—21no—
80C2×C22⋊D61—21no—
81C2×C22⋊D61—21no—
82C2×C22⋊D61—21no—
83C2×C22⋊D61—21no—
84C2×C22⋊D61—44no—
85C2×C22⋊D61D411yescharacteristic
86C22×Dic61—21no—
87#11758226—22no—
88#11758226—41no—
89D4×D61—22no—
90D4×D61—41no—
91C2×D244—21no—
92C2×D244C2311yescharacteristic
93C2×D244C2311yescharacteristic
94C2×C4×D61—21no—
95C2×C4×D61C2311yescharacteristic
96SD16×C61—22no—
97SD16×C61D414yescomplemented
98D8×C61—22no—
99D8×C61—22no—
100M16×C61—22no—
101C2×C8×C61C2311yescharacteristic
102C2×C8×C61C2311yescharacteristic
103C22⋊Dic61—21no—
104C22⋊Dic61—21no—
105C22⋊Dic61—21no—
106C22⋊Dic61D411yescharacteristic
107#11758201—22no—
108#11758199—22no—
109#11758199—41no—
110#11758199—41no—
111#11758198—21no—
112#11758198—21no—
113#11758198D411yescharacteristic
114C4⋊Dic61—21no—
115C4⋊Dic61C2311yescharacteristic
116C4⋊Dic61C2311yescharacteristic
117#11758196D411yescharacteristic
118C4×Dic61—21no—
119C4×Dic61C2311yescharacteristic
120C2×#92418C2311yescharacteristic
121C2×#92418C2311yescharacteristic
122D488—22no—
123#11758189—22no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—81no—
130C23×C61C2×D411yescharacteristic
131C22×D61—21no—
132C22×D61—21no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—81no—
136C22×D61C2×D411yescharacteristic
137Q8×C61—21no—
138Q8×C61C2×D412yescomplemented
139D4×C61—21no—
140D4×C61—21no—
141D4×C61—21no—
142D4×C61—21no—
143D4×C61—21no—
144D4×C61—42no—
145D4×C61—42no—
146D4×C61C2×D412yes—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—42no—
151C2×C4×C61C2×D411yescharacteristic
152C2×C4×C61C2411yescharacteristic
153C22⋊D61—22no—
154C22⋊D61—42no—
155C22⋊D61—42no—
156C22⋊D61—42no—
157C22⋊D61—42no—
158C22⋊D61—42no—
159C22⋊D61—81no—
160C22⋊D61—81no—
161C22⋊D61—81no—
162C22⋊D61—81no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—42no—
166C2×Dic61C2×D411yescharacteristic
167C2×Dic61C2×D411yescharacteristic
168C2×Dic61C2×D411yescharacteristic
169D244—21no—
170D244—21no—
171D244—21no—
172D244—21no—
173D244—42no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—42no—
177C8×C61—21no—
178C8×C61C2×D412yescomplemented
179#92418—21no—
180#92418—21no—
181C22×C61—22no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—84no—
194C22×C61C22×D411yescharacteristic
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—41no—
202C2×D61—41no—
203C2×D61—41no—
204C2×D61—84no—
205C4×C61—22no—
206C4×C61—21no—
207C4×C61C22×D411yescharacteristic
208C4×C61H(5,𝔽2)11yescharacteristic
209Dic61—21no—
210Dic61—21no—
211Dic61—41no—
212Dic61—41no—
213Dic61AGL(1,ℤ/8)12yescomplemented
214#3274—611nomaximal, Sylow
215C2×C61—42no—
216C2×C61—41no—
217C2×C61—81no—
218C2×C61—81no—
219C2×C61D4211yescharacteristic
220C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C61#57611yescharacteristic
222D61—41no—
223D61—81no—
224D61—81no—
225C2×AGL(1,ℤ/8)—611no—
226C2×AGL(1,ℤ/8)D6111yescharacteristic, complemented
227#550—611no—
228D42—611no—
229#493—611no—
230#469—611no—
231#469—611no—
232#460—611no—
233#459—611no—
234#449—611no—
235#449—611no—
236#447—611no—
237#447—611no—
238C4×SD16—611no—
239#436—611no—
240C61#327411yescharacteristic, complemented, Sylow
241C2×#41—611no—
242C2×#41C2×D6111yescharacteristic, complemented
243C22×D4—611no—
244C22×D4—1221no—
245C22×D4C2×D6111yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—1224no—
248C2×SD16—611no—
249C2×SD16C2×D6111yescharacteristic
250C2×SD16C2×D6111yescharacteristic, complemented
251C2×D8—611no—
252C2×D8—611no—
253C2×D8—611no—
254C2×D8—611no—
255C2×D8C2×D6111yescharacteristic
256C2×D8C2×D6111yescharacteristic, complemented
257C2×M16—611no—
258C2×M16C2×D6111yescharacteristic, complemented
259#127—611no—
260#127—611no—
261#127—1221no—
262#121—611no—
263#121—611no—
264#121—611no—
265#121—611no—
266#121—1221no—
267#121—1221no—
268#121—1221no—
269#120—1221no—
270#120—1221no—
271C4×Q8—611no—
272C4×D4—611no—
273C4×D4—611no—
274C4×D4—1221no—
275#106—611no—
276C4⋊C8—611no—
277#103—611no—
278#102—611no—
279#102—611no—
280#102—611no—
281#102—611no—
282#102—611no—
283C22⋊C8—611no—
284C22⋊C8—611no—
285C4×C8—611no—
286C24—21no—
287C24—1221no—
288#41—22no—
289#41—41no—
290#41—1222no—
291#41—2441no—
292C2×Q8C22×D6111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—611no—
298C2×D4—611no—
299C2×D4—611no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—1221no—
306C2×D4—1222no—
307C2×D4—1221no—
308C2×D4—1221no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—2441no—
312C2×D4—2444no—
313C2×D4C22×D6111yescharacteristic
314C2×D4C22×D6111yescharacteristic
315C2×D4C22×D6111yescharacteristic
316C22×C4—21no—
317C22×C4—611no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4C22×D6111yescharacteristic
321SD16—22no—
322SD16—1222no—
323SD16C22⋊D6114yescomplemented
324D8—22no—
325D8—22no—
326D8—1222no—
327D8—1222no—
328D8—2441no—
329D8—2441no—
330M16—22no—
331M16—1222no—
332C2×C8—611no—
333C2×C8—611no—
334C2×C8C22×D6111yescharacteristic
335C2×C8C22×D6111yescharacteristic
336C4⋊C4—611no—
337C4⋊C4—611no—
338C4⋊C4—611no—
339C4⋊C4—1221no—
340C22⋊C4—611no—
341C22⋊C4—611no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C42—611no—
348C42—1221no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—611no—
356C23—1221no—
357C23—1221no—
358C23—4881no—
359C23—2442no—
360C23—2442no—
361C23D4×D6111yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D6112yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—1222no—
372D4—1221no—
373D4—1221no—
374D4—1221no—
375D4—1221no—
376D4—4881no—
377D4—4881no—
378D4—4881no—
379D4—4881no—
380D4—2442no—
381D4—2442no—
382D4—2442no—
383D4—2442no—
384D4—2442no—
385D4—2442no—
386D4C2×C22⋊D6112yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—611no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4D4×D6111yescharacteristic
401C2×C4C23×D6111yescharacteristic
402C8—21no—
403C8—1221no—
404C8—1221no—
405C8C2×C22⋊D6112yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—1221no—
420C22—1221no—
421C22—1221no—
422C22—4884no—
423C22—2441no—
424C22—2441no—
425C22—2441no—
426C22—2441no—
427C22—2441no—
428C22—2441no—
429C22C2×D4×D6111yescharacteristic
430C4—21no—
431C4—22no—
432C4—612no—
433C4—1221no—
434C4—1221no—
435C4—2441no—
436C4—2441no—
437C4C22×C22⋊D6111yescharacteristic
438C4#1451982211yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—4881no—
444C2—4881no—
445C2—2441no—
446C2D4×C22⋊D6111yescharacteristic
447C2AGL(1,ℤ/8)×D6111yescharacteristic
448C2#1684056611yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC2×AGL(1,ℤ/8)×C61C2
socleC22×C61C22×D4
hypercenterC2×AGL(1,ℤ/8)D61
hypocenterC61#3274
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
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×#14519804C2semidirect
2. #16840431C2semidirect
3. D4×C22⋊D61C2semidirect
4. #16840137C2semidirect
5. #16840015C2semidirect
6. #16840012C2semidirect
7. #16839960C2semidirect
8. #16839955C2semidirect
9. #16839907C2semidirect
10. #16839906C2semidirect
11. #16839896C2semidirect
12. #16839894C2semidirect
13. SD16×Dic61C2semidirect
14. #16839838C2semidirect
15. C2×AGL(1,ℤ/8)×C61C2semidirect
16. C2×#11758226C22semidirect
20. C2×#11758201C22semidirect
21. C2×D488C22semidirect
23. C2×#11758199C22semidirect
24. C2×#11758199C22semidirect
25. C2×#11758189C22semidirect
28. #14519773C22semidirect
31. #14519751C22semidirect
35. #14519748C22semidirect
38. Q8×Dic61C22semidirect
42. #14519709C22semidirect
43. #14519706C22semidirect
44. #14519701C22semidirect
45. #14519699C22semidirect
46. #14519699C22semidirect
48. #14519698C22semidirect
49. #14519698C22semidirect
50. C22⋊#92418C22semidirect
51. #14519689C22semidirect
52. C8×Dic61C22semidirect
53. C2×#41×C61C22semidirect
57. C2×SD16×C61C22semidirect
59. C2×D8×C61C22semidirect
60. C2×M16×C61C22semidirect
97. SD16×C61D4semidirect
138. Q8×C61C2×D4semidirect
178. C8×C61C2×D4semidirect
213. Dic61AGL(1,ℤ/8)semidirect
226. C2×AGL(1,ℤ/8)D61semidirect
240. C61#3274semidirect
242. C2×#41C2×D61semidirect
250. C2×SD16C2×D61semidirect
256. C2×D8C2×D61semidirect
258. C2×M16C2×D61semidirect
323. SD16C22⋊D61semidirect
363. Q8C2×C22⋊D61semidirect
405. C8C2×C22⋊D61semidirect