Finite Group Databasebeta

← #21457308

Subgroups — #21457308

Order 7808.

546 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
1C2×SD16×D61C211yesmaximal, characteristic, complemented
2#16840419C211yesmaximal, characteristic, complemented
3D4×C22⋊D61C211yesmaximal, characteristic, complemented
4#16840150C211yesmaximal, characteristic, complemented
5#16840008C211yesmaximal, characteristic, complemented
6#16839995C211yesmaximal, characteristic, complemented
7#16839975C211yesmaximal, characteristic, complemented
8#16839960C211yesmaximal, characteristic, complemented
9#16839913C211yesmaximal, characteristic, complemented
10#16839907C211yesmaximal, characteristic, complemented
11#16839899C211yesmaximal, characteristic, complemented
12#16839894C211yesmaximal, characteristic, complemented
13#16839863C211yesmaximal, characteristic, complemented
14#16839831C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic, complemented
16C2×Q8×D61C2211yescharacteristic, complemented
17C22×C22⋊D61—21no—
18C2×D4×D61C2211yescharacteristic
19C2×#11758190C2211yescharacteristic, complemented
20C2×#11758200C2211yescharacteristic
21C2×#11758200C2211yescharacteristic, complemented
22C2×#11758201C2211yescharacteristic, complemented
23SD16×D61—24no—
24C2×#11758199C2211yescharacteristic, complemented
25C2×C8×D61C2211yescharacteristic, complemented
26#14519773C2211yescharacteristic
27#14519764—21no—
28#14519764C2211yescharacteristic, complemented
29#14519756—21no—
30#14519756C2211yescharacteristic, complemented
31#14519751C2211yescharacteristic
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#14519711C2211yescharacteristic, complemented
43#14519706C2211yescharacteristic, complemented
44#14519702C2211yescharacteristic, complemented
45#14519701C2211yescharacteristic, complemented
46#14519699C2211yescharacteristic, complemented
47#14519698C2211yescharacteristic
48#14519698C2211yescharacteristic, complemented
49#14519698C2211yescharacteristic, complemented
50C22⋊#92418C2211yescharacteristic, complemented
51#14519689C2211yescharacteristic, complemented
52#14519687C2211yescharacteristic, 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—44no—
82C2×C22⋊D61D411yescharacteristic
83C22×Dic61—21no—
84Q8×D61—22no—
85Q8×D61—41no—
86D4×D61—22no—
87D4×D61—41no—
88C2×D244C2311yescharacteristic
89C2×C4×D61—21no—
90C2×C4×D61C2311yescharacteristic
91C2×Dic122—21no—
92C2×Dic122C2311yescharacteristic
93SD16×C61—22no—
94SD16×C61D414yescomplemented
95D8×C61—22no—
96D8×C61—22no—
97M16×C61—22no—
98C2×C8×C61C2311yescharacteristic
99C2×C8×C61C2311yescharacteristic
100C22⋊Dic61—21no—
101C22⋊Dic61—21no—
102C22⋊Dic61—21no—
103C22⋊Dic61—21no—
104C22⋊Dic61—21no—
105C22⋊Dic61—21no—
106C22⋊Dic61D411yescharacteristic
107#11758201—22no—
108#11758200—22no—
109#11758200—41no—
110#11758199—41no—
111#11758198—21no—
112#11758198D411yescharacteristic
113C4⋊Dic61—21no—
114C4⋊Dic61C2311yescharacteristic
115C4⋊Dic61C2311yescharacteristic
116#11758196—21no—
117#11758196D411yescharacteristic
118C4×Dic61—21no—
119C4×Dic61C2311yescharacteristic
120C2×#92418C2311yescharacteristic
121C2×#92418C2311yescharacteristic
122#11758190—22no—
123C8×D61—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—42no—
133C22×D61—42no—
134C22×D61—81no—
135C22×D61C2×D411yescharacteristic
136Q8×C61—21no—
137Q8×C61C2×D412yescomplemented
138D4×C61—21no—
139D4×C61—21no—
140D4×C61—21no—
141D4×C61—21no—
142D4×C61—21no—
143D4×C61—42no—
144D4×C61—42no—
145D4×C61C2×D412yes—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—42no—
150C2×C4×C61C2×D411yescharacteristic
151C2×C4×C61C2411yescharacteristic
152C22⋊D61—22no—
153C22⋊D61—42no—
154C22⋊D61—42no—
155C22⋊D61—42no—
156C22⋊D61—42no—
157C22⋊D61—42no—
158C22⋊D61—81no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—42no—
163C2×Dic61C2×D411yescharacteristic
164C2×Dic61C2×D411yescharacteristic
165C2×Dic61C2×D411yescharacteristic
166D244—21no—
167D244—21no—
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—42no—
171Dic122—21no—
172Dic122—21no—
173Dic122—42no—
174C8×C61—21no—
175C8×C61C2×D412yescomplemented
176#92418—21no—
177#92418—21no—
178C22×C61—22no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—84no—
191C22×C61C22×D411yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—41no—
195C2×D61—41no—
196C2×D61—41no—
197C2×D61—84no—
198C2×D61AGL(1,ℤ/8)12yescomplemented
199C4×C61—22no—
200C4×C61—21no—
201C4×C61C22×D411yescharacteristic
202C4×C61H(5,𝔽2)11yescharacteristic
203Dic61—21no—
204Dic61—21no—
205Dic61—21no—
206Dic61—41no—
207Dic61—41no—
208Dic61—41no—
209#3265—611nomaximal, Sylow
210C2×C61—42no—
211C2×C61—41no—
212C2×C61—81no—
213C2×C61—81no—
214C2×C61D4211yescharacteristic
215C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C61#57711yescharacteristic
217D61—22no—
218D61—81no—
219C2×AGL(1,ℤ/8)D6111yescharacteristic, complemented
220C22×SD16—611no—
221#548—611no—
222D42—611no—
223#496—611no—
224#468—611no—
225#466—611no—
226#463—611no—
227#460—611no—
228#450—611no—
229#449—611no—
230#448—611no—
231#447—611no—
232#440—611no—
233#435—611no—
234C61#326511yescharacteristic, complemented, Sylow
235C2×#41C2×D6111yescharacteristic, complemented
236C22×Q8—611no—
237C22×D4—611no—
238C22×D4—1221no—
239C22×D4C2×D6111yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—611no—
242C2×SD16—611no—
243C2×SD16—611no—
244C2×SD16—611no—
245C2×SD16—1224no—
246C2×SD16C2×D6111yescharacteristic
247C2×SD16C2×D6111yescharacteristic, complemented
248C2×D8—611no—
249C2×D8C2×D6111yescharacteristic
250C2×D8C2×D6111yescharacteristic, complemented
251C2×M16C2×D6111yescharacteristic, complemented
252C22×C8—611no—
253#127—611no—
254#124—611no—
255#124—1221no—
256C22⋊Q8—611no—
257C22⋊Q8—1221no—
258#121—611no—
259#121—611no—
260#121—611no—
261#121—1221no—
262#121—1221no—
263#120—1221no—
264#120—1221no—
265C4×Q8—611no—
266C4×D4—611no—
267C4×D4—611no—
268C4×D4—1221no—
269#107—611no—
270C4⋊C8—611no—
271#103—611no—
272#103—611no—
273#102—611no—
274#102—611no—
275#102—611no—
276#102—611no—
277C22⋊C8—611no—
278C22⋊C8—611no—
279#97—611no—
280C24—21no—
281C24—1221no—
282#41—22no—
283#41—41no—
284C2×Q8—611no—
285C2×Q8—1221no—
286C2×Q8—1222no—
287C2×Q8—2441no—
288C2×Q8C22×D6111yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—611no—
294C2×D4—611no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—1222no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—2441no—
303C2×D4—2444no—
304C2×D4C22×D6111yescharacteristic
305C2×D4C22×D6111yescharacteristic
306C2×D4C22×D6111yescharacteristic
307C22×C4—21no—
308C22×C4—611no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4C22×D6111yescharacteristic
312SD16—22no—
313SD16—1222no—
314SD16—1222no—
315SD16—1222no—
316SD16—2441no—
317SD16C22⋊D6114yescomplemented
318D8—22no—
319D8—22no—
320D8—2441no—
321M16—22no—
322C2×C8—611no—
323C2×C8—611no—
324C2×C8—1222no—
325C2×C8C22×D6111yescharacteristic
326C2×C8C22×D6111yescharacteristic
327C4⋊C4—611no—
328C4⋊C4—611no—
329C4⋊C4—611no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C22⋊C4—611no—
333C22⋊C4—611no—
334C22⋊C4—1221no—
335C22⋊C4—1221no—
336C22⋊C4—1221no—
337C22⋊C4—1221no—
338C22⋊C4—1221no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C42—611no—
342C42—1221no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—611no—
350C23—1221no—
351C23—4881no—
352C23—2442no—
353C23—2442no—
354C23D4×D6111yescharacteristic
355Q8—21no—
356Q8—1221no—
357Q8—1221no—
358Q8—2442no—
359Q8C2×C22⋊D6112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—1222no—
368D4—1221no—
369D4—1221no—
370D4—4881no—
371D4—2442no—
372D4—2442no—
373D4—2442no—
374D4—2442no—
375D4—2442no—
376D4C2×C22⋊D6112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—611no—
382C2×C4—611no—
383C2×C4—611no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4D4×D6111yescharacteristic
392C2×C4C23×D6111yescharacteristic
393C8—21no—
394C8—1221no—
395C8—1221no—
396C8C2×C22⋊D6112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—612no—
411C22—1221no—
412C22—1221no—
413C22—4884no—
414C22—2441no—
415C22—2441no—
416C22—2441no—
417C22C2×D4×D6111yescharacteristic
418C4—21no—
419C4—22no—
420C4—1221no—
421C4—1221no—
422C4—1221no—
423C4—2441no—
424C4—2441no—
425C4—2441no—
426C4C22×C22⋊D6111yescharacteristic
427C4#1451982211yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—1222no—
433C2—4881no—
434C2D4×C22⋊D6111yescharacteristic
435C2AGL(1,ℤ/8)×D6111yescharacteristic
436C2#1684057211yescharacteristic
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#3265
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×SD16×D61C2semidirect
2. #16840419C2semidirect
3. D4×C22⋊D61C2semidirect
4. #16840150C2semidirect
5. #16840008C2semidirect
6. #16839995C2semidirect
7. #16839975C2semidirect
8. #16839960C2semidirect
9. #16839913C2semidirect
10. #16839907C2semidirect
11. #16839899C2semidirect
12. #16839894C2semidirect
13. #16839863C2semidirect
14. #16839831C2semidirect
15. C2×AGL(1,ℤ/8)×C61C2semidirect
16. C2×Q8×D61C22semidirect
19. C2×#11758190C22semidirect
21. C2×#11758200C22semidirect
22. C2×#11758201C22semidirect
24. C2×#11758199C22semidirect
25. C2×C8×D61C22semidirect
28. #14519764C22semidirect
30. #14519756C22semidirect
35. #14519748C22semidirect
38. Q8×Dic61C22semidirect
42. #14519711C22semidirect
43. #14519706C22semidirect
44. #14519702C22semidirect
45. #14519701C22semidirect
46. #14519699C22semidirect
48. #14519698C22semidirect
49. #14519698C22semidirect
50. C22⋊#92418C22semidirect
51. #14519689C22semidirect
52. #14519687C22semidirect
53. C2×#41×C61C22semidirect
57. C2×SD16×C61C22semidirect
59. C2×D8×C61C22semidirect
60. C2×M16×C61C22semidirect
94. SD16×C61D4semidirect
137. Q8×C61C2×D4semidirect
175. C8×C61C2×D4semidirect
198. C2×D61AGL(1,ℤ/8)semidirect
219. C2×AGL(1,ℤ/8)D61semidirect
234. C61#3265semidirect
235. C2×#41C2×D61semidirect
247. C2×SD16C2×D61semidirect
250. C2×D8C2×D61semidirect
251. C2×M16C2×D61semidirect
317. SD16C22⋊D61semidirect
359. Q8C2×C22⋊D61semidirect
396. C8C2×C22⋊D61semidirect