Finite Group Databasebeta

← #22570387

Subgroups — #22570387

Order 8576.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863870C211yesmaximal, characteristic, complemented
2#16863734C211yesmaximal, characteristic, complemented
3#16863717C211yesmaximal, characteristic, complemented
4#16863683C211yesmaximal, characteristic, complemented
5#16863673C211yesmaximal, characteristic, complemented
6#16863608C211yesmaximal, characteristic, complemented
7#16863608C211yesmaximal, characteristic, complemented
8#16863546C211yesmaximal, characteristic, complemented
9#16863529C211yesmaximal, characteristic
10#16863506C211yesmaximal, characteristic, complemented
11#16863494C211yesmaximal, characteristic, complemented
12C4×#11761615C211yesmaximal, characteristic, complemented
13C4×#11761613C211yesmaximal, characteristic, complemented
14#16863457C211yesmaximal, characteristic, complemented
15#542×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C2×#11761615C2211yescharacteristic
18C2×#11761615C2211yescharacteristic
19C2×#11761613C2211yescharacteristic
20C2×#11761613C2211yescharacteristic
21C4×D268C2211yescharacteristic
22C4×D268C2211yescharacteristic
23C4×D268C2211yescharacteristic
24C4×D268C2211yescharacteristic
25#14524005—21no—
26#14524005—21no—
27#14524002C2211yescharacteristic
28C22⋊C4×D67—21no—
29C22⋊C4×D67—21no—
30#14523976C2211yescharacteristic, complemented
31#14523976C2211yescharacteristic, complemented
32#14523974C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35C4⋊(C67⋊C8)C2211yescharacteristic, complemented
36#14523967C2211yescharacteristic, complemented
37#14523967C2211yescharacteristic, complemented
38#14523964C2211yescharacteristic, complemented
39#14523964C2211yescharacteristic, complemented
40#14523963C2211yescharacteristic
41#14523963C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44C22⋊(C67⋊C8)C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46#14523954C2211yescharacteristic, complemented
47C4×C67⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C67C2211yescharacteristic, complemented
49#126×C67—21no—
50#124×C67—21no—
51#124×C67C2211yescharacteristic, complemented
52#123×C67—21no—
53C22⋊Q8×C67—21no—
54C22⋊Q8×C67C2211yescharacteristic
55#121×C67C2211yescharacteristic
56C4×Q8×C67C2211yescharacteristic, complemented
57C4×D4×C67C2211yescharacteristic, complemented
58#117×C67C2211yescharacteristic
59C23×D67—21no—
60C2×D268—21no—
61C2×D268—22no—
62C2×D268—22no—
63C2×D268C2311yescharacteristic
64C2×D268C2311yescharacteristic
65C2×C4×D67—21no—
66C2×C4×D67—21no—
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69#11761615—22no—
70#11761615—22no—
71#11761613—22no—
72#11761613—22no—
73C2×C67⋊C8C2311yescharacteristic
74C2×C67⋊C8C2311yescharacteristic
75C2×C67⋊C8C2311yescharacteristic
76C2×C67⋊C8C2311yescharacteristic
77C4⋊Dic67C2311yescharacteristic
78C4⋊Dic67C2311yescharacteristic
79C22⋊Dic67—21no—
80(C2×D67)⋊C4—21no—
81(C2×D67)⋊C4—21no—
82(C2×D67)⋊C4—21no—
83(C2×D67)⋊C4—21no—
84C4×Dic67—21no—
85C2×Q8×C67—21no—
86C2×Q8×C67C2311yescharacteristic
87C2×D4×C67—21no—
88C2×D4×C67C2311yescharacteristic
89C22×C4×C67—21no—
90C22×C4×C67C2311yescharacteristic
91C4⋊C4×C67—21no—
92C4⋊C4×C67—21no—
93C4⋊C4×C67—21no—
94C4⋊C4×C67D411yescharacteristic
95C4⋊C4×C67D411yescharacteristic
96C4⋊C4×C67C2311yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C22⋊C4×C67—21no—
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67D411yescharacteristic
104C22⋊C4×C67D411yescharacteristic
105C42×C67—21no—
106C42×C67C2311yescharacteristic
107C42×C67C2311yescharacteristic
108C23×C67—21no—
109C23×C67C2×D411yescharacteristic
110C22×D67—21no—
111C22×D67—21no—
112C22×D67—24no—
113C22×D67—81no—
114C22×D67—81no—
115Q8×C67—21no—
116Q8×C67—21no—
117Q8×C67—81no—
118D4×C67—21no—
119D4×C67—21no—
120D4×C67—81no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—81no—
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2×D411yescharacteristic
130C2×C4×C67C2×D411yescharacteristic
131C2×C4×C67C2×D411yescharacteristic
132C2×C4×C67C2411yescharacteristic
133C2×Dic67—21no—
134C2×Dic67—21no—
135D268—24no—
136D268—21no—
137D268—21no—
138D268#4112yes—
139D268#4112yes—
140C4×D67—81no—
141C4×D67—81no—
142C4×D67—81no—
143C4×D67—81no—
144C67⋊C8—21no—
145C67⋊C8—21no—
146C67⋊C8—21no—
147C67⋊C8—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67C22×D411yescharacteristic
155C2×D67—22no—
156C2×D67—22no—
157C2×D67—41no—
158C2×D67—41no—
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—84no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—21no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67—41no—
170C4×C67C2×#4111yescharacteristic
171C4×C67C2×#4111yescharacteristic
172Dic67—41no—
173Dic67—41no—
174C2×C67—41no—
175C2×C67—81no—
176C2×C67#52511yescharacteristic
177C2×C67#57611yescharacteristic
178C2×C67#57711yescharacteristic
179D67—42no—
180D67—42no—
181#3118—671nomaximal, Sylow
182C67#311811yescharacteristic, complemented, Sylow
183#542D6711yescharacteristic, complemented
184#518—671no—
185#485—671no—
186#482—671no—
187#476—671no—
188#474—671no—
189#463—671no—
190#463—671no—
191#450—671no—
192#447—671no—
193#442—671no—
194#440—671no—
195C4×SD16—671no—
196C4×D8—671no—
197#433—671no—
198C22×D4—671no—
199C2×SD16—671no—
200C2×SD16—671no—
201C2×D8—671no—
202C2×D8—671no—
203C4⋊Q8C2×D6711yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D6711yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D6711yescharacteristic
210#121C2×D6711yescharacteristic
211C4×Q8C2×D6711yescharacteristic, complemented
212C4×D4—671no—
213C4×D4—671no—
214C4×D4—671no—
215C4×D4—671no—
216C4×D4—1341no—
217C4×D4—1341no—
218C4×D4C2×D6711yescharacteristic, complemented
219#117—671no—
220#117C2×D6711yescharacteristic
221C2×C22⋊C4—1341no—
222C2×C22⋊C4—1341no—
223#107—671no—
224#107—671no—
225#106—671no—
226#106—671no—
227C4⋊C8—671no—
228C4⋊C8—671no—
229#103—671no—
230#103—671no—
231#102—671no—
232#102—671no—
233#102—671no—
234#102—671no—
235#102—671no—
236#102—671no—
237C22⋊C8—671no—
238C22⋊C8—671no—
239#97—671no—
240C4×C8—671no—
241C24—1341no—
242C2×Q8—21no—
243C2×Q8C22×D6711yescharacteristic
244C2×D4—21no—
245C2×D4—671no—
246C2×D4—671no—
247C2×D4—1342no—
248C2×D4—1342no—
249C2×D4—1341no—
250C2×D4C22×D6711yescharacteristic
251C22×C4—21no—
252C22×C4—1341no—
253C22×C4—1341no—
254C22×C4—1341no—
255C22×C4—1341no—
256C22×C4C22×D6711yescharacteristic
257SD16—1342no—
258SD16—1342no—
259D8—1342no—
260D8—1342no—
261C2×C8—671no—
262C2×C8—671no—
263C2×C8—671no—
264C2×C8—671no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—671no—
269C4⋊C4—671no—
270C4⋊C4C22⋊D6711yescharacteristic
271C4⋊C4C22⋊D6711yescharacteristic
272C4⋊C4C22×D6711yescharacteristic
273C4⋊C4C22×D6711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C22⋊C4C22⋊D6711yescharacteristic
285C22⋊C4C22⋊D6711yescharacteristic
286C42—21no—
287C42—1341no—
288C42C22×D6711yescharacteristic
289C42C22×D6711yescharacteristic
290C23—21no—
291C23—5361no—
292C23—5361no—
293C23—1344no—
294C23—1341no—
295C23—1341no—
296C23C2×C22⋊D6711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—672no—
304D4—672no—
305D4—1344no—
306D4—1341no—
307D4—1341no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—5361no—
315C2×C4—5361no—
316C2×C4—5361no—
317C2×C4—5361no—
318C2×C4—1341no—
319C2×C4—1341no—
320C2×C4C2×C22⋊D6711yescharacteristic
321C2×C4C2×C22⋊D6711yescharacteristic
322C2×C4C2×C22⋊D6711yescharacteristic
323C2×C4C2×C22⋊D6711yescharacteristic
324C2×C4C2×C22⋊D6711yescharacteristic
325C2×C4C23×D6711yescharacteristic
326C8—1341no—
327C8—1341no—
328C8—1341no—
329C8—1341no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—2681no—
337C22—2681no—
338C22—2681no—
339C22—2681no—
340C22—5364no—
341C22—1342no—
342C22—1342no—
343C22C22×C22⋊D6711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—2681no—
353C4—2681no—
354C4#41×D6711yescharacteristic
355C4#41×D6711yescharacteristic
356C2—41no—
357C2—81no—
358C2—2682no—
359C2—2682no—
360C2#1686390311yescharacteristic
361C2#1686419911yescharacteristic
362C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#542×C67C2
socleC22×C67C22×D4
hypercenter#542D67
hypocenterC67#3118
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #16863870C2semidirect
2. #16863734C2semidirect
3. #16863717C2semidirect
4. #16863683C2semidirect
5. #16863673C2semidirect
6. #16863608C2semidirect
7. #16863608C2semidirect
8. #16863546C2semidirect
10. #16863506C2semidirect
11. #16863494C2semidirect
12. C4×#11761615C2semidirect
13. C4×#11761613C2semidirect
14. #16863457C2semidirect
15. #542×C67C2semidirect
30. #14523976C22semidirect
31. #14523976C22semidirect
32. #14523974C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. C4⋊(C67⋊C8)C22semidirect
36. #14523967C22semidirect
37. #14523967C22semidirect
38. #14523964C22semidirect
39. #14523964C22semidirect
46. #14523954C22semidirect
47. C4×C67⋊C8C22semidirect
48. C4⋊Q8×C67C22semidirect
51. #124×C67C22semidirect
56. C4×Q8×C67C22semidirect
57. C4×D4×C67C22semidirect
182. C67#3118semidirect
183. #542D67semidirect
203. C4⋊Q8C2×D67semidirect
206. #124C2×D67semidirect
211. C4×Q8C2×D67semidirect
218. C4×D4C2×D67semidirect