Finite Group Databasebeta

← #22563053

Subgroups — #22563053

Order 8576.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524000C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D67C211yesmaximal, characteristic
3C2×C22⋊C4×D67C211yesmaximal, characteristic
4#16863290C211yesmaximal, characteristic, complemented
5#16863289C211yesmaximal, characteristic, complemented
6#16863284C211yesmaximal, characteristic, complemented
7#16863270C211yesmaximal, characteristic, complemented
8#16863267C211yesmaximal, characteristic, complemented
9#16863258C211yesmaximal, characteristic, complemented
10#16863253C211yesmaximal, characteristic, complemented
11#16863251C211yesmaximal, characteristic
12C4×(C2×D67)⋊C4C211yesmaximal, characteristic, complemented
13#16863243C211yesmaximal, characteristic, complemented
14#16863243C211yesmaximal, characteristic, complemented
15#388×C67C211yesmaximal, characteristic, complemented
16C24×D67C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C22×C4×D67C2211yescharacteristic
19C22×C4×D67C2211yescharacteristic
20C22×C4×D67C2211yescharacteristic
21#14524000—24no—
22C2×C4⋊Dic67C2211yescharacteristic, complemented
23C2×Dic67⋊C4C2211yescharacteristic, complemented
24C2×Dic67⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic67C2211yescharacteristic
26C2×C22⋊Dic67C2211yescharacteristic
27C2×C22⋊Dic67C2211yescharacteristic
28C2×(C2×D67)⋊C4C2211yescharacteristic
29C2×(C2×D67)⋊C4C2211yescharacteristic
30C2×(C2×D67)⋊C4C2211yescharacteristic
31C2×(C2×D67)⋊C4C2211yescharacteristic
32C2×(C2×D67)⋊C4C2211yescharacteristic
33C2×(C2×D67)⋊C4C2211yescharacteristic
34C2×(C2×D67)⋊C4C2211yescharacteristic
35C2×(C2×D67)⋊C4C2211yescharacteristic
36C22⋊C4×D67—24no—
37C22⋊C4×D67C418yescomplemented
38C2×C4×Dic67C2211yescharacteristic, complemented
39C2×C4×Dic67C2211yescharacteristic, complemented
40C2×C4×Dic67C2211yescharacteristic, complemented
41C67⋊#95C2211yescharacteristic, complemented
42C67⋊#95C2211yescharacteristic, complemented
43C67⋊#95C2211yescharacteristic, complemented
44C67⋊#95C2211yescharacteristic, complemented
45C67⋊#95C2211yescharacteristic, complemented
46C67⋊#95C2211yescharacteristic, complemented
47C2×C4⋊C4×C67C2211yescharacteristic, complemented
48C2×C22⋊C4×C67C2211yescharacteristic
49C2×C22⋊C4×C67C2211yescharacteristic
50C2×C22⋊C4×C67C2211yescharacteristic
51C2×C42×C67C2211yescharacteristic, complemented
52#95×C67C2211yescharacteristic, complemented
53#95×C67C2211yescharacteristic, complemented
54C23×D67—22no—
55C23×D67—22no—
56C23×D67—22no—
57C23×D67—22no—
58C23×D67—41no—
59C23×D67—41no—
60C23×D67C2×C414yes—
61C23×D67C2311yescharacteristic
62C23×D67C2311yescharacteristic
63C22×Dic67C2311yescharacteristic
64C22×Dic67C2311yescharacteristic
65C22×Dic67C2311yescharacteristic
66C22×Dic67C2311yescharacteristic
67C22×Dic67C2311yescharacteristic
68C22×Dic67C2311yescharacteristic
69C2×C4×D67—22no—
70C2×C4×D67—22no—
71C2×C4×D67—22no—
72C2×C4×D67—22no—
73C2×C4×D67—22no—
74C2×C4×D67—22no—
75C2×C4×D67—41no—
76C2×C4×D67—41no—
77C2×C4×D67—41no—
78C2×C4×D67—41no—
79C2×C4×D67C2×C414yescomplemented
80C2×C4×D67C2×C414yescomplemented
81C4⋊Dic67—41no—
82Dic67⋊C4—22no—
83Dic67⋊C4—22no—
84C22⋊Dic67—22no—
85C22⋊Dic67—41no—
86C22⋊Dic67C2×C414yescomplemented
87(C2×D67)⋊C4—22no—
88(C2×D67)⋊C4—22no—
89(C2×D67)⋊C4—22no—
90(C2×D67)⋊C4—22no—
91(C2×D67)⋊C4—22no—
92(C2×D67)⋊C4C2×C414yes—
93(C2×D67)⋊C4C2×C414yes—
94(C2×D67)⋊C4D414yes—
95C4×Dic67—22no—
96C4×Dic67—41no—
97C4×Dic67—41no—
98C24×C67C2311yescharacteristic
99C22×C4×C67C2311yescharacteristic
100C22×C4×C67C2311yescharacteristic
101C22×C4×C67C2311yescharacteristic
102C22×C4×C67C2311yescharacteristic
103C22×C4×C67C2311yescharacteristic
104C22×C4×C67C2311yescharacteristic
105C4⋊C4×C67—41no—
106C22⋊C4×C67—22no—
107C22⋊C4×C67—41no—
108C22⋊C4×C67C2×C414yescomplemented
109C42×C67—22no—
110C23×C67—21no—
111C23×C67—21no—
112C23×C67—21no—
113C23×C67—21no—
114C23×C67—21no—
115C23×C67—21no—
116C23×C67C22×C412yes—
117C23×C67C2411yescharacteristic
118C22×D67—21no—
119C22×D67—21no—
120C22×D67—21no—
121C22×D67—21no—
122C22×D67—21no—
123C22×D67—21no—
124C22×D67—21no—
125C22×D67—21no—
126C22×D67—28no—
127C22×D67—28no—
128C22×D67—44no—
129C22×D67—44no—
130C22×D67—44no—
131C22×D67—44no—
132C22×D67—44no—
133C22×D67C22×C412yes—
134C22×D67C22×C412yes—
135C22×D67C2×D412yes—
136C22×D67#4112yes—
137C22×D67#4112yes—
138C22×D67#4112yes—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—21no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—21no—
147C2×C4×C67—21no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—21no—
151C2×C4×C67—21no—
152C2×C4×C67—21no—
153C2×C4×C67—21no—
154C2×C4×C67C22×C412yes—
155C2×C4×C67C22×C412yes—
156C2×C4×C67C2×D412yes—
157C2×Dic67—21no—
158C2×Dic67—21no—
159C2×Dic67—21no—
160C2×Dic67—21no—
161C2×Dic67—21no—
162C2×Dic67—21no—
163C2×Dic67—21no—
164C2×Dic67—21no—
165C2×Dic67—21no—
166C2×Dic67—21no—
167C2×Dic67—21no—
168C2×Dic67—21no—
169C2×Dic67—21no—
170C2×Dic67—21no—
171C2×Dic67—21no—
172C2×Dic67C22×C412yes—
173C2×Dic67C22×C412yes—
174C2×Dic67C2×D412yes—
175C4×D67—28no—
176C4×D67—44no—
177C4×D67—44no—
178C4×D67—44no—
179C22×C67—22no—
180C22×C67—22no—
181C22×C67—22no—
182C22×C67—41no—
183C22×C67—41no—
184C22×C67—41no—
185C22×C67—41no—
186C22×C67C23×C411yescharacteristic
187C22×C67C22×D411yescharacteristic
188C22×C67C2×#4111yescharacteristic
189C22×C67C2×#4111yescharacteristic
190C22×C67C2×#4111yescharacteristic
191C22×C67H(5,𝔽2)11yescharacteristic
192C22×C67H(5,𝔽2)11yescharacteristic
193C2×D67—22no—
194C2×D67—22no—
195C2×D67—22no—
196C2×D67—22no—
197C2×D67—22no—
198C2×D67—22no—
199C2×D67—22no—
200C2×D67—22no—
201C2×D67—22no—
202C2×D67—22no—
203C2×D67—22no—
204C2×D67—22no—
205C2×D67—416no—
206C2×D67#11714yescomplemented
207C2×D67C4×D414yes—
208C4×C67—22no—
209C4×C67—22no—
210C4×C67—22no—
211C4×C67—41no—
212C4×C67—41no—
213C4×C67—41no—
214Dic67—22no—
215Dic67—22no—
216Dic67—22no—
217Dic67—41no—
218Dic67—41no—
219Dic67—41no—
220C2×C67—42no—
221C2×C67C2×#11711yescharacteristic
222C2×C67C2×C4×D411yescharacteristic
223C2×C67#51811yescharacteristic
224C2×C67#54611yescharacteristic
225C2×C67#54611yescharacteristic
226C2×C67#55111yescharacteristic
227C2×C67#55111yescharacteristic
228D67—24no—
229D67—24no—
230#2342—671nomaximal, Sylow
231C67#234211yescharacteristic, complemented, Sylow
232C2×#117—671no—
233C22×C22⋊C4—671no—
234C22×C22⋊C4—671no—
235#388—671no—
236#388—671no—
237#388—671no—
238#388D6711yescharacteristic, complemented
239#385—671no—
240#385—671no—
241#381—671no—
242#380—671no—
243#379—671no—
244C4×C22⋊C4—671no—
245C4×C22⋊C4—671no—
246C4×C22⋊C4—671no—
247C25—671no—
248C23×C4—671no—
249C23×C4—671no—
250C23×C4—671no—
251C23×C4—671no—
252#117—1344no—
253C2×C4⋊C4—671no—
254C2×C4⋊C4—671no—
255C2×C4⋊C4—671no—
256C2×C4⋊C4C2×D6711yescharacteristic, complemented
257C2×C22⋊C4—671no—
258C2×C22⋊C4—671no—
259C2×C22⋊C4—671no—
260C2×C22⋊C4—671no—
261C2×C22⋊C4—671no—
262C2×C22⋊C4—678no—
263C2×C22⋊C4—671no—
264C2×C22⋊C4—671no—
265C2×C22⋊C4—671no—
266C2×C22⋊C4—671no—
267C2×C22⋊C4—671no—
268C2×C22⋊C4—671no—
269C2×C22⋊C4—1344no—
270C2×C22⋊C4C2×D6711yescharacteristic
271C2×C22⋊C4C2×D6711yescharacteristic
272C2×C22⋊C4C2×D6711yescharacteristic
273C2×C42—671no—
274C2×C42—671no—
275C2×C42—671no—
276C2×C42C2×D6711yescharacteristic, complemented
277#95—671no—
278#95—671no—
279#95—671no—
280#95—671no—
281#95—671no—
282#95—671no—
283#95C2×D6711yescharacteristic, complemented
284#95C2×D6711yescharacteristic, complemented
285C24—2681no—
286C24—2681no—
287C24—674no—
288C24—671no—
289C24—671no—
290C24—1342no—
291C24—1342no—
292C24—1342no—
293C24—1342no—
294C24C22×D6711yescharacteristic
295C22×C4—2681no—
296C22×C4—2681no—
297C22×C4—2681no—
298C22×C4—2681no—
299C22×C4—674no—
300C22×C4—671no—
301C22×C4—671no—
302C22×C4—671no—
303C22×C4—674no—
304C22×C4—671no—
305C22×C4—671no—
306C22×C4—671no—
307C22×C4—1342no—
308C22×C4—1342no—
309C22×C4—1342no—
310C22×C4—1342no—
311C22×C4—1342no—
312C22×C4—1342no—
313C22×C4C22×D6711yescharacteristic
314C22×C4C22×D6711yescharacteristic
315C22×C4C22×D6711yescharacteristic
316C22×C4C22×D6711yescharacteristic
317C22×C4C22×D6711yescharacteristic
318C22×C4C22×D6711yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—2681no—
321C4⋊C4—1342no—
322C4⋊C4—1342no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—2681no—
326C22⋊C4—674no—
327C22⋊C4—674no—
328C22⋊C4—674no—
329C22⋊C4—674no—
330C22⋊C4—1342no—
331C22⋊C4—1342no—
332C22⋊C4—1342no—
333C22⋊C4—1342no—
334C22⋊C4—1342no—
335C22⋊C4—1342no—
336C22⋊C4C4×D6714yescomplemented
337C42—22no—
338C42—2681no—
339C42—2681no—
340C42—1342no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2684no—
348C23—2684no—
349C23—2684no—
350C23—2684no—
351C23—2684no—
352C23—672no—
353C23—672no—
354C23—672no—
355C23—672no—
356C23—672no—
357C23—672no—
358C23—1348no—
359C23—1348no—
360C23—1341no—
361C23—1341no—
362C23—1341no—
363C23—1341no—
364C23—1341no—
365C23—1341no—
366C23—1341no—
367C23—1341no—
368C23C2×C4×D6712yes—
369C23C23×D6711yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—2684no—
386C2×C4—2684no—
387C2×C4—2684no—
388C2×C4—672no—
389C2×C4—672no—
390C2×C4—672no—
391C2×C4—1348no—
392C2×C4—1341no—
393C2×C4—1341no—
394C2×C4—1341no—
395C2×C4—1341no—
396C2×C4—1341no—
397C2×C4—1341no—
398C2×C4—1341no—
399C2×C4—1341no—
400C2×C4—1341no—
401C2×C4—1341no—
402C2×C4—1341no—
403C2×C4—1341no—
404C2×C4—1341no—
405C2×C4—1341no—
406C2×C4—1341no—
407C2×C4C2×C4×D6712yes—
408C2×C4C2×C4×D6712yes—
409C2×C4D4×D6712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—26816no—
418C22—674no—
419C22—674no—
420C22—1342no—
421C22—1342no—
422C22—1342no—
423C22—1342no—
424C22—1342no—
425C22—1342no—
426C22—1342no—
427C22—1342no—
428C22—1342no—
429C22—1342no—
430C22—1342no—
431C22—1342no—
432C22C22×C4×D6711yescharacteristic
433C22C2×D4×D6711yescharacteristic
434C22#41×D6711yescharacteristic
435C22#41×D6711yescharacteristic
436C22#41×D6711yescharacteristic
437C22#1452408811yescharacteristic
438C22#1452408911yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—2681no—
446C4—2681no—
447C4—2681no—
448C4—1342no—
449C4—1342no—
450C4—1342no—
451C2—42no—
452C2—1344no—
453C2—1344no—
454C2#117×D6711yescharacteristic
455C2C4×D4×D6711yescharacteristic
456C2#1686386811yescharacteristic
457C2#1686403811yescharacteristic
458C2#1686404311yescharacteristic
459C2#1686407111yescharacteristic
460C2#1686407611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#388×C67C2
socleC23×C67C24
hypercenter#388D67
hypocenterC67#2342
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#14524000C2semidirect
4. #16863290C2semidirect
5. #16863289C2semidirect
6. #16863284C2semidirect
7. #16863270C2semidirect
8. #16863267C2semidirect
9. #16863258C2semidirect
10. #16863253C2semidirect
12. C4×(C2×D67)⋊C4C2semidirect
13. #16863243C2semidirect
14. #16863243C2semidirect
15. #388×C67C2semidirect
22. C2×C4⋊Dic67C22semidirect
23. C2×Dic67⋊C4C22semidirect
24. C2×Dic67⋊C4C22semidirect
37. C22⋊C4×D67C4semidirect
38. C2×C4×Dic67C22semidirect
39. C2×C4×Dic67C22semidirect
40. C2×C4×Dic67C22semidirect
41. C67⋊#95C22semidirect
42. C67⋊#95C22semidirect
43. C67⋊#95C22semidirect
44. C67⋊#95C22semidirect
45. C67⋊#95C22semidirect
46. C67⋊#95C22semidirect
47. C2×C4⋊C4×C67C22semidirect
51. C2×C42×C67C22semidirect
52. #95×C67C22semidirect
53. #95×C67C22semidirect
79. C2×C4×D67C2×C4semidirect
80. C2×C4×D67C2×C4semidirect
86. C22⋊Dic67C2×C4semidirect
108. C22⋊C4×C67C2×C4semidirect
206. C2×D67#117semidirect
231. C67#2342semidirect
238. #388D67semidirect
256. C2×C4⋊C4C2×D67semidirect
276. C2×C42C2×D67semidirect
283. #95C2×D67semidirect
284. #95C2×D67semidirect
336. C22⋊C4C4×D67semidirect