Finite Group Databasebeta

← #22563052

Subgroups — #22563052

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
2C22×(C2×D67)⋊C4C211yesmaximal, characteristic
3C2×C22⋊C4×D67C211yesmaximal, characteristic
4#16863287C211yesmaximal, characteristic, complemented
5#16863287C211yesmaximal, characteristic, complemented
6#16863285C211yesmaximal, characteristic, complemented
7#16863285C211yesmaximal, characteristic, complemented
8#16863269C211yesmaximal, characteristic, complemented
9#16863268C211yesmaximal, characteristic, complemented
10#16863258C211yesmaximal, characteristic, complemented
11#16863255C211yesmaximal, characteristic, complemented
12#16863251C211yesmaximal, characteristic
13C22⋊C4×Dic67C211yesmaximal, characteristic, complemented
14C4×(C2×D67)⋊C4C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C67C211yesmaximal, characteristic, complemented
16C24×D67C2211yescharacteristic
17C23×Dic67C2211yescharacteristic
18C22×C4×D67C2211yescharacteristic
19C22×C4×D67C2211yescharacteristic
20#14524000—24no—
21C2×Dic67⋊C4C2211yescharacteristic, complemented
22C2×Dic67⋊C4C2211yescharacteristic, complemented
23C2×Dic67⋊C4C2211yescharacteristic, complemented
24C2×Dic67⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic67C2211yescharacteristic
26C2×C22⋊Dic67C2211yescharacteristic
27C2×(C2×D67)⋊C4—24no—
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
36C2×(C2×D67)⋊C4C2211yescharacteristic
37C2×(C2×D67)⋊C4C2211yescharacteristic
38C22⋊C4×D67C418yescomplemented
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
47C23×C4×C67C2211yescharacteristic
48C2×C22⋊C4×C67C2211yescharacteristic
49C2×C22⋊C4×C67C2211yescharacteristic
50C2×C42×C67C2211yescharacteristic, complemented
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×Dic67—22no—
64C22×Dic67—22no—
65C22×Dic67—41no—
66C22×Dic67C2311yescharacteristic
67C22×Dic67C2311yescharacteristic
68C22×Dic67C2311yescharacteristic
69C22×Dic67C2311yescharacteristic
70C22×Dic67C2311yescharacteristic
71C22×Dic67C2311yescharacteristic
72C2×C4×D67—22no—
73C2×C4×D67—22no—
74C2×C4×D67—41no—
75C2×C4×D67—41no—
76C2×C4×D67C2×C414yescomplemented
77C2×C4×D67C2×C414yescomplemented
78Dic67⋊C4—22no—
79Dic67⋊C4—22no—
80Dic67⋊C4—41no—
81Dic67⋊C4—41no—
82C22⋊Dic67—22no—
83C22⋊Dic67C2×C414yes—
84(C2×D67)⋊C4—22no—
85(C2×D67)⋊C4—22no—
86(C2×D67)⋊C4—22no—
87(C2×D67)⋊C4—22no—
88(C2×D67)⋊C4—22no—
89(C2×D67)⋊C4—41no—
90(C2×D67)⋊C4—41no—
91(C2×D67)⋊C4C2×C414yescomplemented
92(C2×D67)⋊C4C2×C414yescomplemented
93(C2×D67)⋊C4D414yes—
94C4×Dic67—22no—
95C4×Dic67—41no—
96C24×C67C2311yescharacteristic
97C22×C4×C67—22no—
98C22×C4×C67—22no—
99C22×C4×C67—41no—
100C22×C4×C67C2311yescharacteristic
101C22×C4×C67C2311yescharacteristic
102C22×C4×C67C2311yescharacteristic
103C22×C4×C67C2311yescharacteristic
104C22×C4×C67C2311yescharacteristic
105C22×C4×C67C2311yescharacteristic
106C22⋊C4×C67—22no—
107C22⋊C4×C67C2×C414yes—
108C42×C67—22no—
109C42×C67—41no—
110C23×C67—21no—
111C23×C67—21no—
112C23×C67—21no—
113C23×C67—21no—
114C23×C67C22×C412yes—
115C23×C67#4112yes—
116C23×C67#4112yes—
117C23×C67C2411yescharacteristic
118C22×D67—21no—
119C22×D67—21no—
120C22×D67—21no—
121C22×D67—21no—
122C22×D67—28no—
123C22×D67—21no—
124C22×D67—21no—
125C22×D67—21no—
126C22×D67—21no—
127C22×D67—21no—
128C22×D67—21no—
129C22×D67—28no—
130C22×D67—44no—
131C22×D67—44no—
132C22×D67—44no—
133C22×D67—44no—
134C22×D67—44no—
135C22×D67C22×C412yes—
136C22×D67C22×C412yes—
137C22×D67C2×D412yes—
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×C67—44no—
155C2×C4×C67C22×C412yes—
156C2×C4×C67C22×C412yes—
157C2×C4×C67C2×D412yes—
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×Dic67—21no—
173C2×Dic67—44no—
174C2×Dic67C22×C412yes—
175C2×Dic67C22×C412yes—
176C2×Dic67C2×D412yes—
177C4×D67—28no—
178C4×D67—44no—
179C22×C67—22no—
180C22×C67—22no—
181C22×C67—22no—
182C22×C67—22no—
183C22×C67—22no—
184C22×C67—22no—
185C22×C67#11714yescomplemented
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—416no—
203C2×D67—41no—
204C2×D67—41no—
205C2×D67—41no—
206C2×D67—41no—
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—24no—
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—42no—
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#388—671no—
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⋊C4D6711yescharacteristic, complemented
247C25—671no—
248C23×C4—671no—
249C23×C4—671no—
250C23×C4—671no—
251C23×C4C2×D6711yescharacteristic
252#117—1344no—
253C2×C4⋊C4—671no—
254C2×C4⋊C4—671no—
255C2×C4⋊C4—671no—
256C2×C4⋊C4—671no—
257C2×C22⋊C4—671no—
258C2×C22⋊C4—671no—
259C2×C22⋊C4—671no—
260C2×C22⋊C4—671no—
261C2×C22⋊C4—671no—
262C2×C22⋊C4—671no—
263C2×C22⋊C4—678no—
264C2×C22⋊C4—671no—
265C2×C22⋊C4—671no—
266C2×C22⋊C4—671no—
267C2×C22⋊C4—671no—
268C2×C22⋊C4—671no—
269C2×C22⋊C4—671no—
270C2×C22⋊C4—1344no—
271C2×C22⋊C4C2×D6711yescharacteristic
272C2×C22⋊C4C2×D6711yescharacteristic
273C2×C42—671no—
274C2×C42—671no—
275C2×C42C2×D6711yescharacteristic, complemented
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—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—2681no—
299C22×C4—2681no—
300C22×C4—2681no—
301C22×C4—674no—
302C22×C4—671no—
303C22×C4—671no—
304C22×C4—671no—
305C22×C4—674no—
306C22×C4—671no—
307C22×C4—671no—
308C22×C4—671no—
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—2681no—
320C4⋊C4—2681no—
321C4⋊C4—1342no—
322C4⋊C4—1342no—
323C22⋊C4—22no—
324C22⋊C4—2681no—
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×D6714yes—
337C42—22no—
338C42—41no—
339C42—2681no—
340C42—1342no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—2684no—
346C23—2684no—
347C23—2684no—
348C23—2684no—
349C23—2684no—
350C23—672no—
351C23—672no—
352C23—672no—
353C23—672no—
354C23—1348no—
355C23—1348no—
356C23—1341no—
357C23—1341no—
358C23—1341no—
359C23—1341no—
360C23—1341no—
361C23—1341no—
362C23—1341no—
363C23—1341no—
364C23—1341no—
365C23—1341no—
366C23C2×C4×D6712yes—
367C23#1176162812yes—
368C23#1176162812yes—
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—44no—
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—22no—
414C22—22no—
415C22—22no—
416C22—26816no—
417C22—2681no—
418C22—2681no—
419C22—2681no—
420C22—2681no—
421C22—674no—
422C22—1342no—
423C22—1342no—
424C22—1342no—
425C22—1342no—
426C22—1342no—
427C22—1342no—
428C22—1342no—
429C22—1342no—
430C22—1342no—
431C22#1452400014yescomplemented
432C22C22×C4×D6711yescharacteristic
433C22C2×D4×D6711yescharacteristic
434C22#41×D6711yescharacteristic
435C22C2×#1176162811yescharacteristic
436C22C2×#1176162811yescharacteristic
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—24no—
452C2—2682no—
453C2—1344no—
454C2C2×#1452400011yescharacteristic
455C2C4×D4×D6711yescharacteristic
456C2#1686386811yescharacteristic
457C2#1686403611yescharacteristic
458C2#1686404211yescharacteristic
459C2#1686407011yescharacteristic
460C2#1686407511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
FittingC4×C22⋊C4×C67C2
socleC23×C67C24
hypercenterC4×C22⋊C4D67
hypocenterC67#2342
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14524000C2semidirect
4. #16863287C2semidirect
5. #16863287C2semidirect
6. #16863285C2semidirect
7. #16863285C2semidirect
8. #16863269C2semidirect
9. #16863268C2semidirect
10. #16863258C2semidirect
11. #16863255C2semidirect
13. C22⋊C4×Dic67C2semidirect
14. C4×(C2×D67)⋊C4C2semidirect
15. C4×C22⋊C4×C67C2semidirect
21. C2×Dic67⋊C4C22semidirect
22. C2×Dic67⋊C4C22semidirect
23. C2×Dic67⋊C4C22semidirect
24. C2×Dic67⋊C4C22semidirect
38. C22⋊C4×D67C4semidirect
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
50. C2×C42×C67C22semidirect
51. C2×C42×C67C22semidirect
52. #95×C67C22semidirect
53. #95×C67C22semidirect
76. C2×C4×D67C2×C4semidirect
77. C2×C4×D67C2×C4semidirect
91. (C2×D67)⋊C4C2×C4semidirect
92. (C2×D67)⋊C4C2×C4semidirect
185. C22×C67#117semidirect
231. C67#2342semidirect
246. C4×C22⋊C4D67semidirect
275. C2×C42C2×D67semidirect
276. C2×C42C2×D67semidirect
283. #95C2×D67semidirect
284. #95C2×D67semidirect
431. C22#14524000semidirect